diff --git a/.gitignore b/.gitignore
index 8015250fd006f864a33ddfca9ef31cf0b80940f9..d92b2408600b3dabede036b511093012be8cbcbf 100644
--- a/.gitignore
+++ b/.gitignore
@@ -1,2 +1,3 @@
/info4/kapitel-*/.ipynb_checkpoints
/info4/kapitel-*/exports
+/info4/kapitel-*/*/__pycache__/
diff --git a/info4/kapitel-0/Logik.ipynb b/info4/kapitel-0/Logik.ipynb
index 7ac1ca5af7d43b983235bd1c4788fc28aba3b593..4b4978fd637f5d1bad67fa3eff5bbe514179a232 100644
--- a/info4/kapitel-0/Logik.ipynb
+++ b/info4/kapitel-0/Logik.ipynb
@@ -12,7 +12,7 @@
"* Heinrich-Heine Universität Düsseldorf\n",
"\n",
"Grundlagen der Logik und Mengentheorie sind nicht im Skript.\n",
- "Hier definieren wir einige Grundlagen und Notationen die im Skript verwendet werden.\n",
+ "Hier definieren wir einige Grundlagen und Notationen, die im Skript verwendet werden.\n",
"Ein gutes Verständnis dieser Grundlagen und Notationen ist für das Verständnis des Skripts, aber auch anderer Teile der Informatik unumgänglich.\n",
"\n",
"Die Folien für diese Grundlagen sind als Jupyter Notebooks erstellt worden und im [Gitlab der Informatik](https://gitlab.cs.uni-duesseldorf.de/general/stups/prob-teaching-notebooks) erhältlich.\n"
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- "<text text-anchor=\"middle\" x=\"338\" y=\"-14.8\" font-family=\"Times,serif\" font-size=\"14.00\" fill=\"#000000\">1</text>\n",
+ "<polygon fill=\"white\" stroke=\"black\" points=\"334,-0.5 334,-36.5 388,-36.5 388,-0.5 334,-0.5\"/>\n",
+ "<text text-anchor=\"middle\" x=\"361\" y=\"-14.8\" font-family=\"Times,serif\" font-size=\"14.00\">1</text>\n",
"</g>\n",
"<!-- Node11->Node9 -->\n",
"<g id=\"edge11\" class=\"edge\">\n",
"<title>Node11->Node9</title>\n",
- "<path fill=\"none\" stroke=\"#000000\" d=\"M310.997,-35.0018C302.5314,-40.1752 293.0484,-45.9704 284.0611,-51.4627\"/>\n",
- "<polygon fill=\"#000000\" stroke=\"#000000\" points=\"282.0027,-48.6187 275.295,-56.8197 285.6529,-54.5917 282.0027,-48.6187\"/>\n",
+ "<path fill=\"none\" stroke=\"black\" d=\"M333.78,-34.7C325.39,-39.89 315.96,-45.71 307.01,-51.25\"/>\n",
+ "<polygon fill=\"black\" stroke=\"black\" points=\"304.94,-48.41 298.27,-56.65 308.62,-54.37 304.94,-48.41\"/>\n",
"</g>\n",
"</g>\n",
"</svg>"
@@ -800,7 +800,7 @@
"\n",
"Man nennt so eine Interpretation ein <b>Modell</b> für die Formel.\n",
"\n",
- "* eine Interpretation $i$ so dass $i(\\phi)={\\color{olive} TRUE}$ ist ein <b>Modell</b> für $\\phi$\n",
+ "* eine Interpretation $i$ so dass $i(\\phi)={\\color{olive}{TRUE}}$ ist ein <b>Modell</b> für $\\phi$\n",
"* eine Formel $\\phi$ die mindestens ein Modell hat heißt <b>erfüllbar</b>\n",
"* eine Formel $\\phi$ wo alle Interpretationen auch Modelle sind wird eine <b>Tautologie</b> genannt\n",
"* eine Formel ohne Modell heißt <b>unerfüllbar</b> oder ein Widerspruch\n",
diff --git a/info4/kapitel-0/Mengentheorie.ipynb b/info4/kapitel-0/Mengentheorie.ipynb
index cd73a13428dc225e0be1c84c57601b035653c93b..327f7097067fc1ec4b644dddf357d7555396426a 100644
--- a/info4/kapitel-0/Mengentheorie.ipynb
+++ b/info4/kapitel-0/Mengentheorie.ipynb
@@ -5,15 +5,17 @@
"metadata": {},
"source": [
"# Theoretische Informatik - Vorlesung 0 - Teil 2 Mengentheorie\n",
- "April 2020\n",
- "Michael Leuschel\n",
- "Heinrich-Heine Universität Düsseldorf\n",
+ "\n",
+ "* April 2020\n",
+ "* Michael Leuschel\n",
+ "* Lehrstuhl Softwaretechnik und Programmiersprachen\n",
+ "* Heinrich-Heine Universität Düsseldorf\n",
"\n",
"Grundlagen der Logik und Mengentheorie sind nicht im Skript.\n",
"Hier definieren wir einige Grundlagen und Notationen die im Skript verwendet werden.\n",
"Ein gutes Verständnis dieser Grundlagen und Notationen ist für das Verständnis des Skripts, aber auch anderer Teile der Informatik unumgänglich.\n",
"\n",
- "Auswertung der Formeln erfolgt mit dem [Jupyter kernel](https://gitlab.cs.uni-duesseldorf.de/general/stups/prob2-jupyter-kernel) für [ProB](https://www3.hhu.de/stups/prob)\n",
+ "Auswertung der Formeln erfolgt mit dem [Jupyter Kernel](https://gitlab.cs.uni-duesseldorf.de/general/stups/prob2-jupyter-kernel) für [ProB](https://www3.hhu.de/stups/prob)\n",
"\n",
"\n",
"Um dieses Notebook zu starten kann man entweder selber Jupyter und den [ProB Kernel](https://gitlab.cs.uni-duesseldorf.de/general/stups/prob2-jupyter-kernel) installieren.\n",
@@ -62,13 +64,13 @@
"Sie hat keine Elemente:\n",
" * $z = \\emptyset \\Leftrightarrow \\forall(a).(a\\not\\in z)$\n",
"\n",
- "Zwei Mengen $x$ und $y$ sind gleich gdw wenn sie die gleichen Elemente haben:\n",
+ "Zwei Mengen $x$ und $y$ sind gleich gdw sie die gleichen Elemente haben:\n",
" * $\\forall(x,y).(x=y \\Leftrightarrow \\forall(a).(a\\in x \\Leftrightarrow a \\in y))$"
]
},
{
"cell_type": "code",
- "execution_count": 17,
+ "execution_count": 1,
"metadata": {},
"outputs": [
{
@@ -80,7 +82,7 @@
"FALSE"
]
},
- "execution_count": 17,
+ "execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
@@ -91,7 +93,7 @@
},
{
"cell_type": "code",
- "execution_count": 18,
+ "execution_count": 2,
"metadata": {},
"outputs": [
{
@@ -103,7 +105,7 @@
"FALSE"
]
},
- "execution_count": 18,
+ "execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
@@ -114,7 +116,7 @@
},
{
"cell_type": "code",
- "execution_count": 19,
+ "execution_count": 3,
"metadata": {},
"outputs": [
{
@@ -126,7 +128,7 @@
"TRUE"
]
},
- "execution_count": 19,
+ "execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
@@ -147,7 +149,7 @@
},
{
"cell_type": "code",
- "execution_count": 20,
+ "execution_count": 4,
"metadata": {},
"outputs": [
{
@@ -159,7 +161,7 @@
"TRUE"
]
},
- "execution_count": 20,
+ "execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
@@ -177,7 +179,7 @@
},
{
"cell_type": "code",
- "execution_count": 22,
+ "execution_count": 5,
"metadata": {},
"outputs": [
{
@@ -189,7 +191,7 @@
"FALSE"
]
},
- "execution_count": 22,
+ "execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
@@ -200,7 +202,7 @@
},
{
"cell_type": "code",
- "execution_count": 23,
+ "execution_count": 6,
"metadata": {},
"outputs": [
{
@@ -212,7 +214,7 @@
"FALSE"
]
},
- "execution_count": 23,
+ "execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
@@ -230,7 +232,7 @@
},
{
"cell_type": "code",
- "execution_count": 24,
+ "execution_count": 7,
"metadata": {},
"outputs": [
{
@@ -242,7 +244,7 @@
"TRUE"
]
},
- "execution_count": 24,
+ "execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
@@ -271,7 +273,7 @@
},
{
"cell_type": "code",
- "execution_count": 25,
+ "execution_count": 8,
"metadata": {},
"outputs": [
{
@@ -283,7 +285,7 @@
"{2,3,5,7}"
]
},
- "execution_count": 25,
+ "execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
@@ -294,7 +296,7 @@
},
{
"cell_type": "code",
- "execution_count": 26,
+ "execution_count": 9,
"metadata": {},
"outputs": [
{
@@ -306,7 +308,7 @@
"{5}"
]
},
- "execution_count": 26,
+ "execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
@@ -317,7 +319,7 @@
},
{
"cell_type": "code",
- "execution_count": 27,
+ "execution_count": 10,
"metadata": {},
"outputs": [
{
@@ -329,7 +331,7 @@
"{2,3}"
]
},
- "execution_count": 27,
+ "execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
@@ -351,7 +353,7 @@
},
{
"cell_type": "code",
- "execution_count": 28,
+ "execution_count": 11,
"metadata": {},
"outputs": [
{
@@ -363,7 +365,7 @@
"{2,3,5}"
]
},
- "execution_count": 28,
+ "execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
@@ -381,7 +383,7 @@
},
{
"cell_type": "code",
- "execution_count": 29,
+ "execution_count": 12,
"metadata": {},
"outputs": [
{
@@ -393,7 +395,7 @@
"{a∣a > 10}"
]
},
- "execution_count": 29,
+ "execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
@@ -404,7 +406,7 @@
},
{
"cell_type": "code",
- "execution_count": 30,
+ "execution_count": 13,
"metadata": {},
"outputs": [
{
@@ -416,7 +418,7 @@
"{2}"
]
},
- "execution_count": 30,
+ "execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
@@ -438,7 +440,7 @@
},
{
"cell_type": "code",
- "execution_count": 31,
+ "execution_count": 14,
"metadata": {},
"outputs": [
{
@@ -460,7 +462,7 @@
"\ty = {2,5}"
]
},
- "execution_count": 31,
+ "execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
@@ -471,7 +473,7 @@
},
{
"cell_type": "code",
- "execution_count": 32,
+ "execution_count": 15,
"metadata": {},
"outputs": [
{
@@ -493,7 +495,7 @@
"\ty = {2,5}"
]
},
- "execution_count": 32,
+ "execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
@@ -504,7 +506,7 @@
},
{
"cell_type": "code",
- "execution_count": 33,
+ "execution_count": 16,
"metadata": {},
"outputs": [
{
@@ -526,7 +528,7 @@
"\ty = {2,5}"
]
},
- "execution_count": 33,
+ "execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
@@ -544,7 +546,7 @@
},
{
"cell_type": "code",
- "execution_count": 34,
+ "execution_count": 17,
"metadata": {},
"outputs": [
{
@@ -556,7 +558,7 @@
"{1,2,3,4,5,6,7,8,9,10}"
]
},
- "execution_count": 34,
+ "execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
@@ -579,7 +581,7 @@
},
{
"cell_type": "code",
- "execution_count": 35,
+ "execution_count": 18,
"metadata": {},
"outputs": [
{
@@ -591,7 +593,7 @@
"{2,3,44,55}"
]
},
- "execution_count": 35,
+ "execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
@@ -602,7 +604,7 @@
},
{
"cell_type": "code",
- "execution_count": 36,
+ "execution_count": 19,
"metadata": {},
"outputs": [
{
@@ -614,7 +616,7 @@
"{2,3,44,55}"
]
},
- "execution_count": 36,
+ "execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
@@ -668,7 +670,7 @@
},
{
"cell_type": "code",
- "execution_count": 37,
+ "execution_count": 20,
"metadata": {},
"outputs": [
{
@@ -680,7 +682,7 @@
"{1,3,7,9}"
]
},
- "execution_count": 37,
+ "execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
@@ -691,7 +693,7 @@
},
{
"cell_type": "code",
- "execution_count": 38,
+ "execution_count": 21,
"metadata": {},
"outputs": [
{
@@ -703,7 +705,7 @@
"{1,3,5,7,9,10}"
]
},
- "execution_count": 38,
+ "execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
@@ -714,7 +716,7 @@
},
{
"cell_type": "code",
- "execution_count": 39,
+ "execution_count": 22,
"metadata": {},
"outputs": [
{
@@ -726,7 +728,7 @@
"{1,2,3,4,6,7,8,9}"
]
},
- "execution_count": 39,
+ "execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
@@ -737,7 +739,7 @@
},
{
"cell_type": "code",
- "execution_count": 40,
+ "execution_count": 23,
"metadata": {},
"outputs": [
{
@@ -749,7 +751,7 @@
"{1,3,7,9}"
]
},
- "execution_count": 40,
+ "execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
@@ -779,9 +781,23 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 24,
"metadata": {},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\emptyset$"
+ ],
+ "text/plain": [
+ "∅"
+ ]
+ },
+ "execution_count": 24,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"{2,4} ∩ (ℤ \\ {2,4})"
]
@@ -797,7 +813,7 @@
"* $\\forall a. a\\in x \\Rightarrow a \\in y$\n",
"\n",
"Wir benutzen diese Schreibweise $x \\subseteq y$.\n",
- "In diesem Fall ist $y$ auch eine Obermenge von $y$, geschrieben als $y \\supseteq x$.\n",
+ "In diesem Fall ist $y$ auch eine Obermenge von $x$, geschrieben als $y \\supseteq x$.\n",
"\n",
"Für die echte Teilmenge benutzen wir folgende Schreibweise und Definition:\n",
"* $x \\subset y$ $\\Leftrightarrow$ $(x \\subseteq y \\wedge x\\neq y)$."
@@ -805,7 +821,7 @@
},
{
"cell_type": "code",
- "execution_count": 41,
+ "execution_count": 25,
"metadata": {},
"outputs": [
{
@@ -825,7 +841,7 @@
"\ty = (1 ‥ 5)"
]
},
- "execution_count": 41,
+ "execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
@@ -836,7 +852,7 @@
},
{
"cell_type": "code",
- "execution_count": 42,
+ "execution_count": 26,
"metadata": {},
"outputs": [
{
@@ -848,7 +864,7 @@
"FALSE"
]
},
- "execution_count": 42,
+ "execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
@@ -859,7 +875,7 @@
},
{
"cell_type": "code",
- "execution_count": 43,
+ "execution_count": 27,
"metadata": {},
"outputs": [
{
@@ -871,7 +887,7 @@
"TRUE"
]
},
- "execution_count": 43,
+ "execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
@@ -892,7 +908,7 @@
},
{
"cell_type": "code",
- "execution_count": 44,
+ "execution_count": 28,
"metadata": {},
"outputs": [
{
@@ -904,7 +920,7 @@
"3"
]
},
- "execution_count": 44,
+ "execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
@@ -915,7 +931,7 @@
},
{
"cell_type": "code",
- "execution_count": 45,
+ "execution_count": 29,
"metadata": {},
"outputs": [
{
@@ -927,7 +943,7 @@
"3"
]
},
- "execution_count": 45,
+ "execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
@@ -938,7 +954,7 @@
},
{
"cell_type": "code",
- "execution_count": 46,
+ "execution_count": 30,
"metadata": {},
"outputs": [
{
@@ -950,7 +966,7 @@
"0"
]
},
- "execution_count": 46,
+ "execution_count": 30,
"metadata": {},
"output_type": "execute_result"
}
@@ -968,7 +984,7 @@
},
{
"cell_type": "code",
- "execution_count": 47,
+ "execution_count": 31,
"metadata": {},
"outputs": [
{
@@ -1015,7 +1031,7 @@
},
{
"cell_type": "code",
- "execution_count": 48,
+ "execution_count": 32,
"metadata": {},
"outputs": [
{
@@ -1047,7 +1063,7 @@
"\tO = 0"
]
},
- "execution_count": 48,
+ "execution_count": 32,
"metadata": {},
"output_type": "execute_result"
}
@@ -1065,7 +1081,7 @@
},
{
"cell_type": "code",
- "execution_count": 49,
+ "execution_count": 33,
"metadata": {},
"outputs": [
{
@@ -1097,7 +1113,7 @@
"\tO = 3"
]
},
- "execution_count": 49,
+ "execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
@@ -1116,7 +1132,7 @@
},
{
"cell_type": "code",
- "execution_count": 50,
+ "execution_count": 34,
"metadata": {},
"outputs": [
{
@@ -1148,7 +1164,7 @@
"\tO = 4"
]
},
- "execution_count": 50,
+ "execution_count": 34,
"metadata": {},
"output_type": "execute_result"
}
@@ -1167,7 +1183,7 @@
},
{
"cell_type": "code",
- "execution_count": 51,
+ "execution_count": 35,
"metadata": {},
"outputs": [
{
@@ -1199,7 +1215,7 @@
"\tO = 0"
]
},
- "execution_count": 51,
+ "execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
@@ -1220,7 +1236,7 @@
},
{
"cell_type": "code",
- "execution_count": 52,
+ "execution_count": 36,
"metadata": {},
"outputs": [
{
@@ -1232,7 +1248,7 @@
"{(9↦5↦6↦7↦1↦0↦8↦2)}"
]
},
- "execution_count": 52,
+ "execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
@@ -1248,7 +1264,7 @@
},
{
"cell_type": "code",
- "execution_count": 53,
+ "execution_count": 37,
"metadata": {},
"outputs": [
{
@@ -1263,7 +1279,7 @@
"9\t5\t6\t7\t1\t0\t8\t2\n"
]
},
- "execution_count": 53,
+ "execution_count": 37,
"metadata": {},
"output_type": "execute_result"
}
@@ -1300,7 +1316,7 @@
},
{
"cell_type": "code",
- "execution_count": 54,
+ "execution_count": 38,
"metadata": {},
"outputs": [
{
@@ -1315,7 +1331,7 @@
"2\t0\t3\t4\t1\t8\t9\n"
]
},
- "execution_count": 54,
+ "execution_count": 38,
"metadata": {},
"output_type": "execute_result"
}
@@ -1353,7 +1369,7 @@
},
{
"cell_type": "code",
- "execution_count": 55,
+ "execution_count": 39,
"metadata": {},
"outputs": [
{
@@ -1365,7 +1381,7 @@
"3"
]
},
- "execution_count": 55,
+ "execution_count": 39,
"metadata": {},
"output_type": "execute_result"
}
@@ -1376,7 +1392,7 @@
},
{
"cell_type": "code",
- "execution_count": 56,
+ "execution_count": 40,
"metadata": {},
"outputs": [
{
@@ -1388,7 +1404,7 @@
"1"
]
},
- "execution_count": 56,
+ "execution_count": 40,
"metadata": {},
"output_type": "execute_result"
}
@@ -1399,7 +1415,7 @@
},
{
"cell_type": "code",
- "execution_count": 57,
+ "execution_count": 41,
"metadata": {},
"outputs": [
{
@@ -1411,7 +1427,7 @@
"2"
]
},
- "execution_count": 57,
+ "execution_count": 41,
"metadata": {},
"output_type": "execute_result"
}
@@ -1422,7 +1438,7 @@
},
{
"cell_type": "code",
- "execution_count": 58,
+ "execution_count": 42,
"metadata": {},
"outputs": [
{
@@ -1434,7 +1450,7 @@
"TRUE"
]
},
- "execution_count": 58,
+ "execution_count": 42,
"metadata": {},
"output_type": "execute_result"
}
@@ -1445,7 +1461,7 @@
},
{
"cell_type": "code",
- "execution_count": 59,
+ "execution_count": 43,
"metadata": {},
"outputs": [
{
@@ -1457,7 +1473,7 @@
"FALSE"
]
},
- "execution_count": 59,
+ "execution_count": 43,
"metadata": {},
"output_type": "execute_result"
}
@@ -1478,7 +1494,7 @@
},
{
"cell_type": "code",
- "execution_count": 60,
+ "execution_count": 44,
"metadata": {},
"outputs": [
{
@@ -1490,7 +1506,7 @@
"FALSE"
]
},
- "execution_count": 60,
+ "execution_count": 44,
"metadata": {},
"output_type": "execute_result"
}
@@ -1501,7 +1517,7 @@
},
{
"cell_type": "code",
- "execution_count": 61,
+ "execution_count": 45,
"metadata": {},
"outputs": [
{
@@ -1513,7 +1529,7 @@
"0"
]
},
- "execution_count": 61,
+ "execution_count": 45,
"metadata": {},
"output_type": "execute_result"
}
@@ -1524,7 +1540,7 @@
},
{
"cell_type": "code",
- "execution_count": 62,
+ "execution_count": 46,
"metadata": {},
"outputs": [
{
@@ -1536,7 +1552,7 @@
"1"
]
},
- "execution_count": 62,
+ "execution_count": 46,
"metadata": {},
"output_type": "execute_result"
}
@@ -1547,7 +1563,7 @@
},
{
"cell_type": "code",
- "execution_count": 63,
+ "execution_count": 47,
"metadata": {},
"outputs": [
{
@@ -1559,7 +1575,7 @@
"TRUE"
]
},
- "execution_count": 63,
+ "execution_count": 47,
"metadata": {},
"output_type": "execute_result"
}
@@ -1570,7 +1586,7 @@
},
{
"cell_type": "code",
- "execution_count": 64,
+ "execution_count": 48,
"metadata": {},
"outputs": [
{
@@ -1582,7 +1598,7 @@
"FALSE"
]
},
- "execution_count": 64,
+ "execution_count": 48,
"metadata": {},
"output_type": "execute_result"
}
@@ -1593,7 +1609,7 @@
},
{
"cell_type": "code",
- "execution_count": 65,
+ "execution_count": 49,
"metadata": {},
"outputs": [
{
@@ -1605,7 +1621,7 @@
"TRUE"
]
},
- "execution_count": 65,
+ "execution_count": 49,
"metadata": {},
"output_type": "execute_result"
}
@@ -1624,7 +1640,7 @@
},
{
"cell_type": "code",
- "execution_count": 66,
+ "execution_count": 50,
"metadata": {},
"outputs": [
{
@@ -1636,7 +1652,7 @@
"{∅,{1},{1,2},{2}}"
]
},
- "execution_count": 66,
+ "execution_count": 50,
"metadata": {},
"output_type": "execute_result"
}
@@ -1647,7 +1663,7 @@
},
{
"cell_type": "code",
- "execution_count": 67,
+ "execution_count": 51,
"metadata": {},
"outputs": [
{
@@ -1659,7 +1675,7 @@
"{∅}"
]
},
- "execution_count": 67,
+ "execution_count": 51,
"metadata": {},
"output_type": "execute_result"
}
@@ -1670,7 +1686,7 @@
},
{
"cell_type": "code",
- "execution_count": 68,
+ "execution_count": 52,
"metadata": {},
"outputs": [
{
@@ -1682,7 +1698,7 @@
"{∅,{1},{1,2},{1,3},{2},{1,2,3},{2,3},{3}}"
]
},
- "execution_count": 68,
+ "execution_count": 52,
"metadata": {},
"output_type": "execute_result"
}
@@ -1693,7 +1709,7 @@
},
{
"cell_type": "code",
- "execution_count": 69,
+ "execution_count": 53,
"metadata": {},
"outputs": [
{
@@ -1722,7 +1738,7 @@
"{3}\n"
]
},
- "execution_count": 69,
+ "execution_count": 53,
"metadata": {},
"output_type": "execute_result"
}
@@ -1733,7 +1749,7 @@
},
{
"cell_type": "code",
- "execution_count": 70,
+ "execution_count": 54,
"metadata": {},
"outputs": [
{
@@ -1754,7 +1770,7 @@
"{{1}}\n"
]
},
- "execution_count": 70,
+ "execution_count": 54,
"metadata": {},
"output_type": "execute_result"
}
@@ -1765,7 +1781,7 @@
},
{
"cell_type": "code",
- "execution_count": 71,
+ "execution_count": 55,
"metadata": {},
"outputs": [
{
@@ -1777,7 +1793,7 @@
"8"
]
},
- "execution_count": 71,
+ "execution_count": 55,
"metadata": {},
"output_type": "execute_result"
}
@@ -1788,7 +1804,7 @@
},
{
"cell_type": "code",
- "execution_count": 72,
+ "execution_count": 56,
"metadata": {},
"outputs": [
{
@@ -1800,7 +1816,7 @@
"1024"
]
},
- "execution_count": 72,
+ "execution_count": 56,
"metadata": {},
"output_type": "execute_result"
}
@@ -1811,7 +1827,7 @@
},
{
"cell_type": "code",
- "execution_count": 73,
+ "execution_count": 57,
"metadata": {},
"outputs": [
{
@@ -1823,7 +1839,7 @@
"1267650600228229401496703205376"
]
},
- "execution_count": 73,
+ "execution_count": 57,
"metadata": {},
"output_type": "execute_result"
}
@@ -1834,7 +1850,7 @@
},
{
"cell_type": "code",
- "execution_count": 74,
+ "execution_count": 58,
"metadata": {},
"outputs": [
{
@@ -1846,7 +1862,7 @@
"340282366920938463463374607431768211456"
]
},
- "execution_count": 74,
+ "execution_count": 58,
"metadata": {},
"output_type": "execute_result"
}
@@ -1857,7 +1873,7 @@
},
{
"cell_type": "code",
- "execution_count": 75,
+ "execution_count": 59,
"metadata": {},
"outputs": [
{
@@ -1869,7 +1885,7 @@
"115792089237316195423570985008687907853269984665640564039457584007913129639936"
]
},
- "execution_count": 75,
+ "execution_count": 59,
"metadata": {},
"output_type": "execute_result"
}
@@ -1915,7 +1931,7 @@
"## Relationen vs Prädikate\n",
"Eine Relation ist die explizite Darstellung eines Prädikats als Menge:\n",
" \n",
- "* Prädikate können mit logischen Junktoren und Quantoren verarabeitet werden:\n",
+ "* Prädikate können mit logischen Junktoren und Quantoren verarbeitet werden:\n",
" * $\\exists x. is\\_rich(x)$, $\\forall x.(is\\_poor(x) \\Rightarrow \\neg is\\_rich(x))$\n",
" \n",
"* Relationen können mit mengentheoretischen Operationen verarbeitet werden:\n",
@@ -1938,7 +1954,7 @@
},
{
"cell_type": "code",
- "execution_count": 78,
+ "execution_count": 60,
"metadata": {},
"outputs": [
{
@@ -1950,7 +1966,7 @@
"{(1↦4),(1↦5),(2↦4),(2↦5)}"
]
},
- "execution_count": 78,
+ "execution_count": 60,
"metadata": {},
"output_type": "execute_result"
}
@@ -1961,7 +1977,7 @@
},
{
"cell_type": "code",
- "execution_count": 79,
+ "execution_count": 61,
"metadata": {},
"outputs": [
{
@@ -1973,7 +1989,7 @@
"{(1↦4),(2↦4)}"
]
},
- "execution_count": 79,
+ "execution_count": 61,
"metadata": {},
"output_type": "execute_result"
}
@@ -1984,7 +2000,7 @@
},
{
"cell_type": "code",
- "execution_count": 80,
+ "execution_count": 62,
"metadata": {},
"outputs": [
{
@@ -1996,7 +2012,7 @@
"{(1↦FALSE),(1↦TRUE),(2↦FALSE),(2↦TRUE)}"
]
},
- "execution_count": 80,
+ "execution_count": 62,
"metadata": {},
"output_type": "execute_result"
}
@@ -2007,7 +2023,7 @@
},
{
"cell_type": "code",
- "execution_count": 81,
+ "execution_count": 63,
"metadata": {},
"outputs": [
{
@@ -2019,7 +2035,7 @@
"{(1↦1),(1↦2),(1↦3),(2↦1),(2↦2),(2↦3),(3↦1),(3↦2),(3↦3)}"
]
},
- "execution_count": 81,
+ "execution_count": 63,
"metadata": {},
"output_type": "execute_result"
}
@@ -2030,7 +2046,7 @@
},
{
"cell_type": "code",
- "execution_count": 82,
+ "execution_count": 64,
"metadata": {},
"outputs": [
{
@@ -2042,7 +2058,7 @@
"100"
]
},
- "execution_count": 82,
+ "execution_count": 64,
"metadata": {},
"output_type": "execute_result"
}
@@ -2060,7 +2076,7 @@
},
{
"cell_type": "code",
- "execution_count": 83,
+ "execution_count": 65,
"metadata": {},
"outputs": [
{
@@ -2072,7 +2088,7 @@
"{(2↦3)}"
]
},
- "execution_count": 83,
+ "execution_count": 65,
"metadata": {},
"output_type": "execute_result"
}
@@ -2083,7 +2099,7 @@
},
{
"cell_type": "code",
- "execution_count": 86,
+ "execution_count": 66,
"metadata": {},
"outputs": [
{
@@ -2095,7 +2111,7 @@
"{(3↦2)}"
]
},
- "execution_count": 86,
+ "execution_count": 66,
"metadata": {},
"output_type": "execute_result"
}
@@ -2126,7 +2142,7 @@
},
{
"cell_type": "code",
- "execution_count": 87,
+ "execution_count": 67,
"metadata": {},
"outputs": [
{
@@ -2138,7 +2154,7 @@
"{(0↦1),(0↦2),(0↦3),(0↦4),(0↦5),(0↦6),(0↦7),(0↦8),(0↦9),(1↦2),(1↦3),(1↦4),(1↦5),(1↦6),(1↦7),(1↦8),(1↦9),(2↦3),(2↦4),(2↦5),(2↦6),(2↦7),(2↦8),(2↦9),(3↦4),(3↦5),(3↦6),(3↦7),(3↦8),(3↦9),(4↦5),(4↦6),(4↦7),(4↦8),(4↦9),(5↦6),(5↦7),(5↦8),(5↦9),(6↦7),(6↦8),(6↦9),(7↦8),(7↦9),(8↦9)}"
]
},
- "execution_count": 87,
+ "execution_count": 67,
"metadata": {},
"output_type": "execute_result"
}
@@ -2157,7 +2173,7 @@
},
{
"cell_type": "code",
- "execution_count": 88,
+ "execution_count": 68,
"metadata": {},
"outputs": [
{
@@ -2169,7 +2185,7 @@
"{(2↦1),(4↦2),(6↦3),(8↦4),(10↦5)}"
]
},
- "execution_count": 88,
+ "execution_count": 68,
"metadata": {},
"output_type": "execute_result"
}
@@ -2187,7 +2203,7 @@
},
{
"cell_type": "code",
- "execution_count": 89,
+ "execution_count": 69,
"metadata": {},
"outputs": [
{
@@ -2196,83 +2212,96 @@
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@@ -2281,7 +2310,7 @@
"<Dot visualization: expr_as_graph [(\"halb\",{a,b|a:1..10 & b:1..10 & b*2=a})]>"
]
},
- "execution_count": 89,
+ "execution_count": 69,
"metadata": {},
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}
@@ -2300,7 +2329,7 @@
},
{
"cell_type": "code",
- "execution_count": 90,
+ "execution_count": 70,
"metadata": {},
"outputs": [
{
@@ -2312,7 +2341,7 @@
"{(1↦\"nein\"),(2↦\"nein\"),(3↦\"ja\"),(4↦\"nein\"),(5↦\"nein\"),(6↦\"ja\"),(7↦\"nein\")}"
]
},
- "execution_count": 90,
+ "execution_count": 70,
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@@ -2323,7 +2352,7 @@
},
{
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- "execution_count": 91,
+ "execution_count": 71,
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@@ -2434,7 +2479,7 @@
"<Dot visualization: expr_as_graph [(\"teilbar\",{x,y|x:1..7 & (x mod 3=0 => y=\"ja\") & (x mod 3>0 => y=\"nein\")})]>"
]
},
- "execution_count": 91,
+ "execution_count": 71,
"metadata": {},
"output_type": "execute_result"
}
@@ -2447,12 +2492,12 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "Das Kartesische Produkt stellt die Relation dar die immer wahr ist (für die angegebenen Basismengen). Als Graph ist dies der [vollständige Graph](https://de.wikipedia.org/wiki/Vollständiger_Graph) über die Basismengen:"
+ "Das Kartesische Produkt stellt die Relation dar, die immer wahr ist (für die angegebenen Basismengen). Als Graph ist dies der [vollständige Graph](https://de.wikipedia.org/wiki/Vollständiger_Graph) über die Basismengen:"
]
},
{
"cell_type": "code",
- "execution_count": 92,
+ "execution_count": 72,
"metadata": {},
"outputs": [
{
@@ -2461,7 +2506,7 @@
"Preference changed: DOT_ENGINE = circo\n"
]
},
- "execution_count": 92,
+ "execution_count": 72,
"metadata": {},
"output_type": "execute_result"
}
@@ -2472,7 +2517,7 @@
},
{
"cell_type": "code",
- "execution_count": 93,
+ "execution_count": 73,
"metadata": {},
"outputs": [
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@@ -2671,7 +2746,7 @@
"<Dot visualization: expr_as_graph [(\"K5\",(1..5)*(1..5))]>"
]
},
- "execution_count": 93,
+ "execution_count": 73,
"metadata": {},
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}
@@ -2682,7 +2757,7 @@
},
{
"cell_type": "code",
- "execution_count": 94,
+ "execution_count": 74,
"metadata": {},
"outputs": [
{
@@ -2691,7 +2766,7 @@
"Preference changed: DOT_ENGINE = circo\n"
]
},
- "execution_count": 94,
+ "execution_count": 74,
"metadata": {},
"output_type": "execute_result"
}
@@ -2702,7 +2777,7 @@
},
{
"cell_type": "code",
- "execution_count": 95,
+ "execution_count": 75,
"metadata": {},
"outputs": [
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+ "<title>9->10</title>\n",
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+ "<title>9->8</title>\n",
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+ "<title>9->6</title>\n",
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+ "<title>9->5</title>\n",
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+ "<title>9->4</title>\n",
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+ "<g id=\"edge18\" class=\"edge\">\n",
+ "<title>9->3</title>\n",
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+ "<g id=\"edge19\" class=\"edge\">\n",
+ "<title>9->2</title>\n",
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+ "<title>9->1</title>\n",
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+ "<title>8->10</title>\n",
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+ "<g id=\"edge22\" class=\"edge\">\n",
+ "<title>8->9</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M183.91,-39.05C227.12,-74.87 335.63,-154.05 392.35,-191.5\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"390.7,-194.61 400.99,-197.14 394.53,-188.74 390.7,-194.61\"/>\n",
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+ "<g id=\"edge23\" class=\"edge\">\n",
+ "<title>8->8</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M192.81,-27.72C202.59,-27.86 210.57,-25.62 210.57,-20.98 210.57,-18.15 207.6,-16.21 203.08,-15.16\"/>\n",
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+ "<g id=\"edge24\" class=\"edge\">\n",
+ "<title>8->7</title>\n",
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+ "<g id=\"edge25\" class=\"edge\">\n",
+ "<title>8->6</title>\n",
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+ "<title>8->2</title>\n",
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+ "<title>8->1</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M192.77,-26.67C210.2,-27.86 233.07,-28.04 252.19,-27.23\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"252.57,-30.71 262.37,-26.67 252.19,-23.72 252.57,-30.71\"/>\n",
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+ "<g id=\"edge31\" class=\"edge\">\n",
+ "<title>7->10</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M92.41,-97.77C150.26,-101.31 285.95,-101.5 352.13,-98.34\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"352.72,-101.81 362.51,-97.78 352.34,-94.82 352.72,-101.81\"/>\n",
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+ "<title>7->9</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M92.58,-106.73C156.98,-131.6 318.48,-184.2 391.37,-204.2\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"390.48,-207.58 401.05,-206.8 392.3,-200.82 390.48,-207.58\"/>\n",
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+ "<g id=\"edge33\" class=\"edge\">\n",
+ "<title>7->8</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M92.49,-81.28C108.01,-71.41 127.23,-57.54 142.09,-45.53\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"144.39,-48.17 149.85,-39.09 139.92,-42.79 144.39,-48.17\"/>\n",
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- "<g id=\"edge68\" class=\"edge\"><title>7->7</title>\n",
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+ "<g id=\"edge34\" class=\"edge\">\n",
+ "<title>7->7</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M92.54,-100.57C102.32,-100.71 110.3,-98.46 110.3,-93.82 110.3,-91 107.34,-89.06 102.81,-88.01\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"102.81,-84.49 92.54,-87.08 102.19,-91.46 102.81,-84.49\"/>\n",
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- "<text text-anchor=\"middle\" x=\"56.0286\" y=\"-144.591\" font-family=\"Times,serif\" font-size=\"14.00\">K10</text>\n",
+ "<g id=\"edge35\" class=\"edge\">\n",
+ "<title>7->6</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M53.95,-111.95C45.77,-131.06 35.73,-161.37 30.11,-183.58\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"26.67,-182.91 27.79,-193.45 33.48,-184.51 26.67,-182.91\"/>\n",
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+ "<g id=\"edge36\" class=\"edge\">\n",
+ "<title>7->5</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M61.49,-111.93C57.85,-152.73 57.59,-252.09 60.72,-301.17\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"57.25,-301.69 61.49,-311.4 64.23,-301.16 57.25,-301.69\"/>\n",
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- "<text text-anchor=\"middle\" x=\"123.935\" y=\"-232.078\" font-family=\"Times,serif\" font-size=\"14.00\">K10</text>\n",
+ "<g id=\"edge37\" class=\"edge\">\n",
+ "<title>7->4</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M67.69,-111.87C79.98,-163.59 128.05,-312.46 152.18,-374.63\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"149.03,-376.18 155.97,-384.18 155.53,-373.59 149.03,-376.18\"/>\n",
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- "<text text-anchor=\"middle\" x=\"182.889\" y=\"-227.856\" font-family=\"Times,serif\" font-size=\"14.00\">K10</text>\n",
+ "<g id=\"edge38\" class=\"edge\">\n",
+ "<title>7->3</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M74.43,-111.88C106.94,-164.08 216.19,-314.95 265.35,-376.19\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"262.85,-378.66 271.87,-384.21 268.28,-374.25 262.85,-378.66\"/>\n",
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- "<g id=\"edge78\" class=\"edge\"><title>7->2</title>\n",
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- "<text text-anchor=\"middle\" x=\"228.2\" y=\"-191.321\" font-family=\"Times,serif\" font-size=\"14.00\">K10</text>\n",
+ "<g id=\"edge39\" class=\"edge\">\n",
+ "<title>7->2</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M84.5,-112.17C137.34,-155.5 285.2,-263.23 353.9,-308.82\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"352.38,-312.01 362.66,-314.58 356.23,-306.16 352.38,-312.01\"/>\n",
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+ "<g id=\"edge40\" class=\"edge\">\n",
+ "<title>7->1</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M92.41,-89.74C132.58,-79.57 207.97,-55.33 252.98,-38.33\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"254.29,-41.58 262.37,-34.72 251.78,-35.05 254.29,-41.58\"/>\n",
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+ "<g id=\"edge41\" class=\"edge\">\n",
+ "<title>6->10</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M54.23,-206.74C118.57,-189.87 279.96,-137.66 352.94,-110.36\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"354.52,-113.5 362.63,-106.68 352.04,-106.95 354.52,-113.5\"/>\n",
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+ "<g id=\"edge42\" class=\"edge\">\n",
+ "<title>6->9</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M54.25,-215.31C124.36,-219.32 311.05,-219.48 390.95,-215.81\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"391.24,-219.3 401.05,-215.3 390.88,-212.31 391.24,-219.3\"/>\n",
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+ "<g id=\"edge43\" class=\"edge\">\n",
+ "<title>6->8</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M45.65,-193.51C73.96,-159.72 126.62,-87.39 151.77,-47.92\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"154.95,-49.44 157.26,-39.1 149.01,-45.74 154.95,-49.44\"/>\n",
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+ "<g id=\"edge44\" class=\"edge\">\n",
+ "<title>6->7</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M38.35,-193.57C46.53,-174.46 56.57,-144.15 62.19,-121.94\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"65.63,-122.61 64.51,-112.07 58.82,-121.01 65.63,-122.61\"/>\n",
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+ "<g id=\"edge45\" class=\"edge\">\n",
+ "<title>6->6</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M54.24,-218.44C64.02,-218.58 72,-216.34 72,-211.7 72,-208.87 69.04,-206.93 64.51,-205.88\"/>\n",
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+ "<g id=\"edge46\" class=\"edge\">\n",
+ "<title>6->5</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M27.77,-229.84C31.86,-249.17 41.49,-279.95 50.19,-302.21\"/>\n",
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+ "<title>6->4</title>\n",
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+ "<title>6->3</title>\n",
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+ "<title>6->2</title>\n",
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+ "<title>3->6</title>\n",
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+ "<title>3->5</title>\n",
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+ "<title>3->4</title>\n",
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+ "<title>3->1</title>\n",
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},
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+ "execution_count": 75,
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},
{
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- "execution_count": 96,
+ "execution_count": 76,
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+ "<title>20->12</title>\n",
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+ "<title>20->3</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M803.34,-313.46C789.5,-372.15 760.22,-555.72 753.16,-626.55\"/>\n",
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+ "<g id=\"node19\" class=\"node\">\n",
+ "<title>2</title>\n",
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- "<g id=\"edge38\" class=\"edge\"><title>20->2</title>\n",
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- "<text text-anchor=\"middle\" x=\"818.398\" y=\"-409.822\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge19\" class=\"edge\">\n",
+ "<title>20->2</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M805.77,-313.36C802.02,-356.16 801.78,-463.32 805.02,-514.68\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"801.54,-515.04 805.77,-524.75 808.52,-514.52 801.54,-515.04\"/>\n",
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- "<text text-anchor=\"middle\" x=\"752.543\" y=\"-179.127\" font-family=\"Times,serif\" font-size=\"14.00\">1</text>\n",
+ "<g id=\"node20\" class=\"node\">\n",
+ "<title>1</title>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"779.54,-201.33 725.54,-201.33 725.54,-165.33 779.54,-165.33 779.54,-201.33\"/>\n",
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"</g>\n",
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- "<g id=\"edge40\" class=\"edge\"><title>20->1</title>\n",
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+ "<g id=\"edge20\" class=\"edge\">\n",
+ "<title>20->1</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M805.72,-276.89C798.71,-258.73 784.87,-230.73 772.94,-210.07\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"775.92,-208.24 767.8,-201.44 769.91,-211.82 775.92,-208.24\"/>\n",
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"</g>\n",
"<!-- 19->20 -->\n",
- "<g id=\"edge42\" class=\"edge\"><title>19->20</title>\n",
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+ "<g id=\"edge21\" class=\"edge\">\n",
+ "<title>19->20</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M831.18,-400.81C830.11,-380.37 825.14,-347 819.85,-323.23\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"823.22,-322.29 817.49,-313.38 816.42,-323.92 823.22,-322.29\"/>\n",
+ "<text text-anchor=\"middle\" x=\"840.02\" y=\"-365.82\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge44\" class=\"edge\"><title>19->19</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M856.382,-425.814C866.165,-425.959 874.141,-423.711 874.141,-419.07 874.141,-416.243 871.179,-414.303 866.651,-413.252\"/>\n",
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+ "<g id=\"edge22\" class=\"edge\">\n",
+ "<title>19->19</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M856.38,-425.81C866.16,-425.96 874.14,-423.71 874.14,-419.07 874.14,-416.24 871.18,-414.3 866.65,-413.25\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"866.66,-409.74 856.38,-412.33 866.03,-416.71 866.66,-409.74\"/>\n",
+ "<text text-anchor=\"middle\" x=\"888.64\" y=\"-415.37\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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- "<g id=\"edge46\" class=\"edge\"><title>19->18</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"684.965,-119.534 676.885,-112.68 678.927,-123.076 684.965,-119.534\"/>\n",
- "<text text-anchor=\"middle\" x=\"766.683\" y=\"-249.814\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge23\" class=\"edge\">\n",
+ "<title>19->18</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M823.36,-400.62C800.5,-346.04 719,-185.48 682.01,-121.41\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"684.96,-119.53 676.89,-112.68 678.93,-123.08 684.96,-119.53\"/>\n",
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- "<g id=\"edge48\" class=\"edge\"><title>19->17</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"578.081,-61.2207 569.074,-55.6412 572.636,-65.6195 578.081,-61.2207\"/>\n",
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+ "<g id=\"edge24\" class=\"edge\">\n",
+ "<title>19->17</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M819.62,-400.94C780.58,-339.08 633.76,-136.54 575.7,-63.84\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"578.08,-61.22 569.07,-55.64 572.64,-65.62 578.08,-61.22\"/>\n",
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- "<g id=\"edge50\" class=\"edge\"><title>19->16</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"460.429,-40.2923 450.691,-36.1203 455.702,-45.4549 460.429,-40.2923\"/>\n",
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+ "<g id=\"edge25\" class=\"edge\">\n",
+ "<title>19->16</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M814.95,-400.82C757.14,-336.08 539.45,-118.03 458.53,-43.3\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"460.43,-40.29 450.69,-36.12 455.7,-45.45 460.43,-40.29\"/>\n",
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- "<g id=\"edge52\" class=\"edge\"><title>19->15</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"341.574,-56.1665 331.312,-53.5308 337.69,-61.9905 341.574,-56.1665\"/>\n",
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+ "<g id=\"edge26\" class=\"edge\">\n",
+ "<title>19->15</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M809.28,-401.07C730.71,-337.89 441.39,-127.44 339.71,-59.13\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"341.57,-56.17 331.31,-53.53 337.69,-61.99 341.57,-56.17\"/>\n",
+ "<text text-anchor=\"middle\" x=\"560\" y=\"-233.9\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge54\" class=\"edge\"><title>19->14</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"230.22,-106.286 219.672,-105.289 227.296,-112.646 230.22,-106.286\"/>\n",
- "<text text-anchor=\"middle\" x=\"509.373\" y=\"-263.458\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge27\" class=\"edge\">\n",
+ "<title>19->14</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M801.91,-401.81C700.43,-344.62 344.41,-163.12 228.84,-109.5\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"230.22,-106.29 219.67,-105.29 227.3,-112.65 230.22,-106.29\"/>\n",
+ "<text text-anchor=\"middle\" x=\"500.87\" y=\"-244.46\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge56\" class=\"edge\"><title>19->13</title>\n",
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+ "<g id=\"edge28\" class=\"edge\">\n",
+ "<title>19->13</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M802.01,-407.3C690.64,-365.71 268.48,-228.42 140.55,-192.02\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"141.39,-188.62 130.82,-189.29 139.5,-195.36 141.39,-188.62\"/>\n",
+ "<text text-anchor=\"middle\" x=\"456.78\" y=\"-288.46\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge58\" class=\"edge\"><title>19->12</title>\n",
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+ "<g id=\"edge29\" class=\"edge\">\n",
+ "<title>19->12</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M801.72,-411.98C683.78,-387.98 219.67,-314.35 84.04,-297.94\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"84.09,-294.42 73.76,-296.74 83.28,-301.38 84.09,-294.42\"/>\n",
+ "<text text-anchor=\"middle\" x=\"457.38\" y=\"-343.76\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<g id=\"edge30\" class=\"edge\">\n",
+ "<title>19->11</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M801.99,-416.42C682.37,-411.09 202.14,-410.96 64.21,-416.01\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"63.99,-412.52 54.14,-416.42 64.27,-419.51 63.99,-412.52\"/>\n",
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- "<g id=\"edge62\" class=\"edge\"><title>19->10</title>\n",
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+ "<g id=\"edge31\" class=\"edge\">\n",
+ "<title>19->10</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M801.79,-420.7C683.88,-434.02 218.99,-507.5 83.73,-533.99\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"82.96,-530.57 73.84,-535.96 84.33,-537.44 82.96,-530.57\"/>\n",
+ "<text text-anchor=\"middle\" x=\"428.26\" y=\"-481.14\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge64\" class=\"edge\"><title>19->9</title>\n",
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+ "<g id=\"edge32\" class=\"edge\">\n",
+ "<title>19->9</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M802.05,-425C690.74,-455.71 268.64,-592.69 140.62,-639.39\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"139.05,-636.24 130.88,-642.98 141.47,-642.81 139.05,-636.24\"/>\n",
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- "<g id=\"edge66\" class=\"edge\"><title>19->8</title>\n",
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+ "<g id=\"edge33\" class=\"edge\">\n",
+ "<title>19->8</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M802.08,-429.63C700.7,-475.69 343.48,-657.51 228.35,-721.37\"/>\n",
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+ "<g id=\"edge34\" class=\"edge\">\n",
+ "<title>19->7</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M801.99,-434.95C711.89,-494.52 423.64,-703.84 332.2,-775.97\"/>\n",
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+ "<g id=\"edge35\" class=\"edge\">\n",
+ "<title>19->6</title>\n",
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+ "<g id=\"edge36\" class=\"edge\">\n",
+ "<title>19->5</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M812.02,-437.15C761.64,-498.69 615.33,-699.87 567.33,-773.37\"/>\n",
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+ "<g id=\"edge37\" class=\"edge\">\n",
+ "<title>19->4</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M815.97,-437.31C783.64,-491.41 702.18,-650.97 673.62,-715.76\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"670.24,-714.75 669.5,-725.32 676.67,-717.52 670.24,-714.75\"/>\n",
+ "<text text-anchor=\"middle\" x=\"730.29\" y=\"-580.34\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge76\" class=\"edge\"><title>19->3</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"753.507,-626.143 754.537,-636.688 760.307,-627.803 753.507,-626.143\"/>\n",
- "<text text-anchor=\"middle\" x=\"802.054\" y=\"-531.762\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge38\" class=\"edge\">\n",
+ "<title>19->3</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M819.15,-437.15C802.09,-478.05 769.49,-577.79 756.96,-626.77\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"753.51,-626.14 754.54,-636.69 760.31,-627.8 753.51,-626.14\"/>\n",
+ "<text text-anchor=\"middle\" x=\"773.55\" y=\"-520.76\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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- "<g id=\"edge78\" class=\"edge\"><title>19->2</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M821.212,-437.11C815.909,-457.46 810.388,-490.808 808.214,-514.648\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"804.707,-514.621 807.464,-524.851 811.688,-515.135 804.707,-514.621\"/>\n",
- "<text text-anchor=\"middle\" x=\"800.713\" y=\"-469.679\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge39\" class=\"edge\">\n",
+ "<title>19->2</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M821.21,-437.11C815.91,-457.46 810.39,-490.81 808.21,-514.65\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"804.71,-514.62 807.46,-524.85 811.69,-515.13 804.71,-514.62\"/>\n",
+ "<text text-anchor=\"middle\" x=\"800.21\" y=\"-479.68\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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"<!-- 19->1 -->\n",
- "<g id=\"edge80\" class=\"edge\"><title>19->1</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"769.564,-209.485 762.578,-201.52 763.072,-212.103 769.564,-209.485\"/>\n",
- "<text text-anchor=\"middle\" x=\"782.743\" y=\"-305.701\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge40\" class=\"edge\">\n",
+ "<title>19->1</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M827.15,-400.97C817.74,-359.92 785.46,-259.6 766.33,-210.83\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"769.56,-209.48 762.58,-201.52 763.07,-212.1 769.56,-209.48\"/>\n",
+ "<text text-anchor=\"middle\" x=\"782.24\" y=\"-309.7\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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- "<g id=\"edge82\" class=\"edge\"><title>18->20</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"782.042,-271.593 791.096,-277.095 787.45,-267.147 782.042,-271.593\"/>\n",
- "<text text-anchor=\"middle\" x=\"715.457\" y=\"-195.847\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge41\" class=\"edge\">\n",
+ "<title>18->20</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M672.35,-112.95C694.14,-148.8 751.21,-227.94 784.56,-269.14\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"782.04,-271.59 791.1,-277.09 787.45,-267.15 782.04,-271.59\"/>\n",
+ "<text text-anchor=\"middle\" x=\"713.96\" y=\"-179.85\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 18->19 -->\n",
- "<g id=\"edge84\" class=\"edge\"><title>18->19</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M669.596,-113.05C692.453,-167.625 773.959,-328.186 810.948,-392.259\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"807.99,-394.134 816.07,-400.988 814.027,-390.592 807.99,-394.134\"/>\n",
- "<text text-anchor=\"middle\" x=\"754.272\" y=\"-241.454\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge42\" class=\"edge\">\n",
+ "<title>18->19</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M669.6,-113.05C692.45,-167.62 773.96,-328.19 810.95,-392.26\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"807.99,-394.13 816.07,-400.99 814.03,-390.59 807.99,-394.13\"/>\n",
+ "<text text-anchor=\"middle\" x=\"725.77\" y=\"-241.45\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 18->18 -->\n",
- "<g id=\"edge86\" class=\"edge\"><title>18->18</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M691.055,-101.341C700.838,-101.486 708.814,-99.2383 708.814,-94.5976 708.814,-91.7698 705.852,-89.8304 701.324,-88.7796\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"701.328,-85.2659 691.055,-87.8542 700.7,-92.2377 701.328,-85.2659\"/>\n",
- "<text text-anchor=\"middle\" x=\"720.867\" y=\"-90.3976\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge43\" class=\"edge\">\n",
+ "<title>18->18</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M691.05,-101.34C700.84,-101.49 708.81,-99.24 708.81,-94.6 708.81,-91.77 705.85,-89.83 701.32,-88.78\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"701.33,-85.27 691.05,-87.85 700.7,-92.24 701.33,-85.27\"/>\n",
+ "<text text-anchor=\"middle\" x=\"723.31\" y=\"-90.9\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 18->17 -->\n",
- "<g id=\"edge88\" class=\"edge\"><title>18->17</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M641.059,-76.5378C625.916,-67.5533 605.724,-57.0839 588.385,-49.1557\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"589.656,-45.8909 579.096,-45.0382 586.82,-52.2904 589.656,-45.8909\"/>\n",
- "<text text-anchor=\"middle\" x=\"608.722\" y=\"-70.6468\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge44\" class=\"edge\">\n",
+ "<title>18->17</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M641.06,-76.54C625.92,-67.55 605.72,-57.08 588.39,-49.16\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"589.66,-45.89 579.1,-45.04 586.82,-52.29 589.66,-45.89\"/>\n",
+ "<text text-anchor=\"middle\" x=\"600.22\" y=\"-51.65\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 18->16 -->\n",
- "<g id=\"edge90\" class=\"edge\"><title>18->16</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M636.794,-80.9981C594.517,-64.3738 512.683,-37.6433 465,-24.755\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"465.646,-21.3059 455.085,-22.151 463.868,-28.0764 465.646,-21.3059\"/>\n",
- "<text text-anchor=\"middle\" x=\"546.897\" y=\"-60.6765\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge45\" class=\"edge\">\n",
+ "<title>18->16</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M636.79,-81C594.52,-64.37 512.68,-37.64 465,-24.76\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"465.65,-21.31 455.08,-22.15 463.87,-28.08 465.65,-21.31\"/>\n",
+ "<text text-anchor=\"middle\" x=\"536.4\" y=\"-56.68\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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"<!-- 18->15 -->\n",
- "<g id=\"edge92\" class=\"edge\"><title>18->15</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M636.783,-86.4606C573.244,-72.5952 414.299,-47.2587 341.547,-39.2135\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"341.771,-35.7178 331.46,-38.1554 341.04,-42.6796 341.771,-35.7178\"/>\n",
- "<text text-anchor=\"middle\" x=\"490.165\" y=\"-47.6371\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge46\" class=\"edge\">\n",
+ "<title>18->15</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M636.78,-86.46C573.24,-72.6 414.3,-47.26 341.55,-39.21\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"341.77,-35.72 331.46,-38.16 341.04,-42.68 341.77,-35.72\"/>\n",
+ "<text text-anchor=\"middle\" x=\"474.66\" y=\"-51.64\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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"<!-- 18->14 -->\n",
- "<g id=\"edge94\" class=\"edge\"><title>18->14</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M636.771,-91.2406C556.871,-86.8922 321.729,-86.7235 229.784,-90.7345\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"229.373,-87.2502 219.554,-91.2306 229.712,-94.242 229.373,-87.2502\"/>\n",
- "<text text-anchor=\"middle\" x=\"433.277\" y=\"-77.7875\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge47\" class=\"edge\">\n",
+ "<title>18->14</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M636.77,-91.24C556.87,-86.89 321.73,-86.72 229.78,-90.73\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"229.37,-87.25 219.55,-91.23 229.71,-94.24 229.37,-87.25\"/>\n",
+ "<text text-anchor=\"middle\" x=\"418.78\" y=\"-79.79\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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"<!-- 18->13 -->\n",
- "<g id=\"edge96\" class=\"edge\"><title>18->13</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M636.687,-95.7653C544.936,-105.535 246.111,-152.692 140.766,-173.795\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"139.996,-170.38 130.901,-175.814 141.4,-177.238 139.996,-170.38\"/>\n",
- "<text text-anchor=\"middle\" x=\"387.727\" y=\"-119.58\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge48\" class=\"edge\">\n",
+ "<title>18->13</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M636.69,-95.77C544.94,-105.54 246.11,-152.69 140.77,-173.8\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"140,-170.38 130.9,-175.81 141.4,-177.24 140,-170.38\"/>\n",
+ "<text text-anchor=\"middle\" x=\"374.23\" y=\"-123.58\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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"<!-- 18->12 -->\n",
- "<g id=\"edge98\" class=\"edge\"><title>18->12</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M636.782,-100.297C538.051,-127.218 196.892,-237.881 83.5516,-279.496\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"82.0809,-276.308 73.9249,-283.071 84.5178,-282.87 82.0809,-276.308\"/>\n",
- "<text text-anchor=\"middle\" x=\"357.167\" y=\"-173.696\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge49\" class=\"edge\">\n",
+ "<title>18->12</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M636.78,-100.3C538.05,-127.22 196.89,-237.88 83.55,-279.5\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"82.08,-276.31 73.92,-283.07 84.52,-282.87 82.08,-276.31\"/>\n",
+ "<text text-anchor=\"middle\" x=\"345.67\" y=\"-178.7\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge100\" class=\"edge\"><title>18->11</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M636.757,-105.159C535.374,-151.218 178.15,-333.037 63.0253,-396.9\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"61.2365,-393.89 54.2147,-401.824 64.6515,-400.001 61.2365,-393.89\"/>\n",
- "<text text-anchor=\"middle\" x=\"343.891\" y=\"-234.83\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge50\" class=\"edge\">\n",
+ "<title>18->11</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M636.76,-105.16C535.37,-151.22 178.15,-333.04 63.03,-396.9\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"61.24,-393.89 54.21,-401.82 64.65,-400 61.24,-393.89\"/>\n",
+ "<text text-anchor=\"middle\" x=\"335.39\" y=\"-239.83\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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"<!-- 18->10 -->\n",
- "<g id=\"edge102\" class=\"edge\"><title>18->10</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M636.768,-110.671C535.393,-178.069 178.824,-437.033 75.0989,-518.449\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"72.6329,-515.937 66.9618,-524.886 76.9759,-521.427 72.6329,-515.937\"/>\n",
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+ "<g id=\"edge51\" class=\"edge\">\n",
+ "<title>18->10</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M636.77,-110.67C535.39,-178.07 178.82,-437.03 75.1,-518.45\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"72.63,-515.94 66.96,-524.89 76.98,-521.43 72.63,-515.94\"/>\n",
+ "<text text-anchor=\"middle\" x=\"341.43\" y=\"-303.36\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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"<!-- 18->9 -->\n",
- "<g id=\"edge104\" class=\"edge\"><title>18->9</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"122.371,-626.972 118.198,-636.71 127.533,-631.7 122.371,-626.972\"/>\n",
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+ "<g id=\"edge52\" class=\"edge\">\n",
+ "<title>18->9</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M641.96,-112.62C552.32,-194.74 214.08,-532.88 125.1,-629.17\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"122.37,-626.97 118.2,-636.71 127.53,-631.7 122.37,-626.97\"/>\n",
+ "<text text-anchor=\"middle\" x=\"369.03\" y=\"-374.7\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge106\" class=\"edge\"><title>18->8</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"205.254,-715.026 202.678,-725.303 211.1,-718.875 205.254,-715.026\"/>\n",
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+ "<g id=\"edge53\" class=\"edge\">\n",
+ "<title>18->8</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M647.4,-112.89C574.32,-204.06 280.22,-608.69 208.21,-716.89\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"205.25,-715.03 202.68,-725.3 211.1,-718.88 205.25,-715.03\"/>\n",
+ "<text text-anchor=\"middle\" x=\"413.31\" y=\"-403.69\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge108\" class=\"edge\"><title>18->7</title>\n",
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+ "<g id=\"edge54\" class=\"edge\">\n",
+ "<title>18->7</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M651.87,-112.76C596.54,-208.84 367.77,-657.56 314.93,-773.26\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"311.68,-771.95 310.76,-782.51 318.06,-774.82 311.68,-771.95\"/>\n",
+ "<text text-anchor=\"middle\" x=\"468.9\" y=\"-431.81\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<g id=\"edge55\" class=\"edge\">\n",
+ "<title>18->6</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M655.39,-112.9C617.55,-210.86 467.25,-672.98 434.27,-792.09\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"430.84,-791.35 431.61,-801.92 437.6,-793.18 430.84,-791.35\"/>\n",
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+ "<g id=\"edge56\" class=\"edge\">\n",
+ "<title>18->5</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M658.55,-112.75C637.61,-208.58 566.64,-655.66 553.58,-772.51\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"550.1,-772.16 552.53,-782.47 557.06,-772.89 550.1,-772.16\"/>\n",
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+ "<g id=\"edge57\" class=\"edge\">\n",
+ "<title>18->4</title>\n",
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+ "<g id=\"edge58\" class=\"edge\">\n",
+ "<title>18->3</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M664.08,-112.69C671.56,-194.19 724.14,-527.51 744.88,-626.79\"/>\n",
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+ "<g id=\"edge59\" class=\"edge\">\n",
+ "<title>18->2</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M666.76,-112.77C684.11,-182.37 765.01,-432.09 796.91,-515.6\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"793.66,-516.89 800.56,-524.94 800.18,-514.35 793.66,-516.89\"/>\n",
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+ "<g id=\"edge60\" class=\"edge\">\n",
+ "<title>18->1</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M674.81,-112.9C685.81,-126.1 703.08,-143.94 718.58,-158.41\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"716.27,-161.04 726.01,-165.2 720.99,-155.87 716.27,-161.04\"/>\n",
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- "<text text-anchor=\"middle\" x=\"660.677\" y=\"-169.663\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge61\" class=\"edge\">\n",
+ "<title>17->20</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M565.29,-55.79C604.77,-100.99 721.43,-218.11 778.07,-269.94\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"775.88,-272.68 785.65,-276.8 780.58,-267.49 775.88,-272.68\"/>\n",
+ "<text text-anchor=\"middle\" x=\"657.18\" y=\"-151.66\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"803.068,-395.48 812.075,-401.059 808.513,-391.081 803.068,-395.48\"/>\n",
- "<text text-anchor=\"middle\" x=\"696.487\" y=\"-211.106\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge62\" class=\"edge\">\n",
+ "<title>17->19</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M561.53,-55.76C600.57,-117.62 747.39,-320.16 805.45,-392.86\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"803.07,-395.48 812.07,-401.06 808.51,-391.08 803.07,-395.48\"/>\n",
+ "<text text-anchor=\"middle\" x=\"668.99\" y=\"-213.11\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"626.165,-86.3365 636.726,-87.1892 629.002,-79.9371 626.165,-86.3365\"/>\n",
- "<text text-anchor=\"middle\" x=\"607.1\" y=\"-53.1806\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge63\" class=\"edge\">\n",
+ "<title>17->18</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M574.76,-55.69C589.91,-64.67 610.1,-75.14 627.44,-83.07\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"626.17,-86.34 636.73,-87.19 629,-79.94 626.17,-86.34\"/>\n",
+ "<text text-anchor=\"middle\" x=\"586.6\" y=\"-73.18\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge128\" class=\"edge\"><title>17->17</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"589.523,-28.2981 579.249,-30.8864 588.894,-35.2698 589.523,-28.2981\"/>\n",
- "<text text-anchor=\"middle\" x=\"609.061\" y=\"-33.4298\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge64\" class=\"edge\">\n",
+ "<title>17->17</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M579.25,-44.37C589.03,-44.52 597.01,-42.27 597.01,-37.63 597.01,-34.8 594.05,-32.86 589.52,-31.81\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"589.52,-28.3 579.25,-30.89 588.89,-35.27 589.52,-28.3\"/>\n",
+ "<text text-anchor=\"middle\" x=\"611.51\" y=\"-33.93\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge130\" class=\"edge\"><title>17->16</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"465.572,-14.126 455.27,-16.5993 464.866,-21.0903 465.572,-14.126\"/>\n",
- "<text text-anchor=\"middle\" x=\"494.114\" y=\"-29.3581\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge65\" class=\"edge\">\n",
+ "<title>17->16</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M524.79,-27.49C507.37,-23.57 484.54,-19.8 465.44,-17.63\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"465.57,-14.13 455.27,-16.6 464.87,-21.09 465.57,-14.13\"/>\n",
+ "<text text-anchor=\"middle\" x=\"509.61\" y=\"-26.36\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge132\" class=\"edge\"><title>17->15</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"341.136,-29.1491 331.352,-33.2138 341.536,-36.1377 341.136,-29.1491\"/>\n",
- "<text text-anchor=\"middle\" x=\"433.001\" y=\"-19.7166\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge66\" class=\"edge\">\n",
+ "<title>17->15</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M524.53,-33.2C479.93,-30.31 391.72,-30.12 341.47,-32.64\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"341.14,-29.15 331.35,-33.21 341.54,-36.14 341.14,-29.15\"/>\n",
+ "<text text-anchor=\"middle\" x=\"447.5\" y=\"-36.72\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge134\" class=\"edge\"><title>17->14</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"228.755,-80.8969 219.693,-86.3875 230.201,-87.7458 228.755,-80.8969\"/>\n",
- "<text text-anchor=\"middle\" x=\"378.379\" y=\"-67.9917\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge67\" class=\"edge\">\n",
+ "<title>17->14</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M524.98,-38.13C461.45,-44.34 302.55,-69.3 229.78,-84.26\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"228.75,-80.9 219.69,-86.39 230.2,-87.75 228.75,-80.9\"/>\n",
+ "<text text-anchor=\"middle\" x=\"362.88\" y=\"-49.99\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge136\" class=\"edge\"><title>17->13</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M524.918,-42.8823C447.909,-63.3881 227.728,-134.724 140.153,-167.286\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"138.758,-164.071 130.635,-170.872 141.227,-170.621 138.758,-164.071\"/>\n",
- "<text text-anchor=\"middle\" x=\"335.536\" y=\"-112.884\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge68\" class=\"edge\">\n",
+ "<title>17->13</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M524.92,-42.88C447.91,-63.39 227.73,-134.72 140.15,-167.29\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"138.76,-164.07 130.63,-170.87 141.23,-170.62 138.76,-164.07\"/>\n",
+ "<text text-anchor=\"middle\" x=\"318.04\" y=\"-93.88\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge138\" class=\"edge\"><title>17->12</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M524.805,-47.914C439.729,-86.1556 178.846,-218.866 82.6871,-272.536\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"80.9051,-269.523 73.907,-277.478 84.3383,-275.623 80.9051,-269.523\"/>\n",
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+ "<g id=\"edge69\" class=\"edge\">\n",
+ "<title>17->12</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M524.8,-47.91C439.73,-86.16 178.85,-218.87 82.69,-272.54\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"80.91,-269.52 73.91,-277.48 84.34,-275.62 80.91,-269.52\"/>\n",
+ "<text text-anchor=\"middle\" x=\"289.25\" y=\"-149.03\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge140\" class=\"edge\"><title>17->11</title>\n",
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+ "<g id=\"edge70\" class=\"edge\">\n",
+ "<title>17->11</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M524.86,-53.51C434.76,-113.08 146.51,-322.4 55.07,-394.53\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"52.74,-391.91 47.1,-400.88 57.1,-397.39 52.74,-391.91\"/>\n",
+ "<text text-anchor=\"middle\" x=\"275.46\" y=\"-212.82\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge142\" class=\"edge\"><title>17->10</title>\n",
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+ "<g id=\"edge71\" class=\"edge\">\n",
+ "<title>17->10</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M529.87,-55.74C446.14,-132.21 150.1,-428.16 67.89,-517.43\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"65.19,-515.19 61.04,-524.94 70.36,-519.91 65.19,-515.19\"/>\n",
+ "<text text-anchor=\"middle\" x=\"284.38\" y=\"-290.38\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge144\" class=\"edge\"><title>17->9</title>\n",
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+ "<g id=\"edge72\" class=\"edge\">\n",
+ "<title>17->9</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M535.5,-55.94C464.74,-144.06 188.71,-523.83 119.38,-628.21\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"116.42,-626.34 113.85,-636.62 122.27,-630.18 116.42,-626.34\"/>\n",
+ "<text text-anchor=\"middle\" x=\"312.94\" y=\"-345.88\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge146\" class=\"edge\"><title>17->8</title>\n",
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+ "<g id=\"edge73\" class=\"edge\">\n",
+ "<title>17->8</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M540.06,-55.79C484.73,-151.87 255.97,-600.59 203.12,-716.29\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"199.87,-714.98 198.96,-725.54 206.25,-717.86 199.87,-714.98\"/>\n",
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+ "<g id=\"edge74\" class=\"edge\">\n",
+ "<title>17->7</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M543.7,-55.78C504.79,-156.77 344.65,-649.16 310.39,-772.48\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"306.96,-771.73 307.72,-782.3 313.72,-773.57 306.96,-771.73\"/>\n",
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+ "<g id=\"edge75\" class=\"edge\">\n",
+ "<title>17->6</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M546.84,-55.85C524.63,-158.72 444.02,-666.7 429.77,-792.11\"/>\n",
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+ "<title>17->5</title>\n",
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+ "<g id=\"edge78\" class=\"edge\">\n",
+ "<title>17->3</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M555.36,-55.86C578.02,-143.48 700.53,-521.21 740.33,-627.12\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"737.18,-628.69 744.03,-636.78 743.72,-626.19 737.18,-628.69\"/>\n",
+ "<text text-anchor=\"middle\" x=\"633.35\" y=\"-345.29\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge158\" class=\"edge\"><title>17->2</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"789.023,-517.826 797.046,-524.746 795.089,-514.334 789.023,-517.826\"/>\n",
- "<text text-anchor=\"middle\" x=\"689.077\" y=\"-274.543\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge79\" class=\"edge\">\n",
+ "<title>17->2</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M558.23,-55.64C590.98,-131.47 740.13,-424.71 791.92,-515.85\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"789.02,-517.83 797.05,-524.75 795.09,-514.33 789.02,-517.83\"/>\n",
+ "<text text-anchor=\"middle\" x=\"660.58\" y=\"-289.54\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge160\" class=\"edge\"><title>17->1</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M569.827,-55.8608C602.822,-83.5181 673.744,-135.401 716.912,-163.79\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"715.123,-166.801 725.417,-169.307 718.933,-160.929 715.123,-166.801\"/>\n",
- "<text text-anchor=\"middle\" x=\"651.369\" y=\"-93.6253\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge80\" class=\"edge\">\n",
+ "<title>17->1</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M569.83,-55.86C602.82,-83.52 673.74,-135.4 716.91,-163.79\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"715.12,-166.8 725.42,-169.31 718.93,-160.93 715.12,-166.8\"/>\n",
+ "<text text-anchor=\"middle\" x=\"628.87\" y=\"-113.63\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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- "<g id=\"edge162\" class=\"edge\"><title>16->20</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M447.445,-36.1793C508.126,-85.6124 694.764,-221.504 773.71,-274.098\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"772.07,-277.21 782.344,-279.798 775.927,-271.368 772.07,-277.21\"/>\n",
- "<text text-anchor=\"middle\" x=\"602.577\" y=\"-162.939\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge81\" class=\"edge\">\n",
+ "<title>16->20</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M447.44,-36.18C508.13,-85.61 694.76,-221.5 773.71,-274.1\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"772.07,-277.21 782.34,-279.8 775.93,-271.37 772.07,-277.21\"/>\n",
+ "<text text-anchor=\"middle\" x=\"596.08\" y=\"-143.94\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->19 -->\n",
- "<g id=\"edge164\" class=\"edge\"><title>16->19</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M442.258,-36.2511C500.075,-100.991 717.763,-319.045 798.68,-393.77\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"796.782,-396.778 806.521,-400.95 801.509,-391.616 796.782,-396.778\"/>\n",
- "<text text-anchor=\"middle\" x=\"609.469\" y=\"-221.811\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge82\" class=\"edge\">\n",
+ "<title>16->19</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M442.26,-36.25C500.07,-100.99 717.76,-319.05 798.68,-393.77\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"796.78,-396.78 806.52,-400.95 801.51,-391.62 796.78,-396.78\"/>\n",
+ "<text text-anchor=\"middle\" x=\"605.97\" y=\"-203.81\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->18 -->\n",
- "<g id=\"edge166\" class=\"edge\"><title>16->18</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M455.09,-31.5996C497.368,-48.2238 579.201,-74.9544 626.884,-87.8426\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"626.238,-91.2917 636.8,-90.4466 628.017,-84.5213 626.238,-91.2917\"/>\n",
- "<text text-anchor=\"middle\" x=\"536.987\" y=\"-67.5211\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge83\" class=\"edge\">\n",
+ "<title>16->18</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M455.09,-31.6C497.37,-48.22 579.2,-74.95 626.88,-87.84\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"626.24,-91.29 636.8,-90.45 628.02,-84.52 626.24,-91.29\"/>\n",
+ "<text text-anchor=\"middle\" x=\"555.49\" y=\"-63.52\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->17 -->\n",
- "<g id=\"edge168\" class=\"edge\"><title>16->17</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M455.293,-28.1442C472.709,-32.0633 495.534,-35.8302 514.637,-37.9991\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"514.507,-41.5038 524.809,-39.0305 515.213,-34.5395 514.507,-41.5038\"/>\n",
- "<text text-anchor=\"middle\" x=\"485.965\" y=\"-17.8717\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge84\" class=\"edge\">\n",
+ "<title>16->17</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M455.29,-28.14C472.71,-32.06 495.53,-35.83 514.64,-38\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"514.51,-41.5 524.81,-39.03 515.21,-34.54 514.51,-41.5\"/>\n",
+ "<text text-anchor=\"middle\" x=\"499.46\" y=\"-36.87\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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- "<g id=\"edge170\" class=\"edge\"><title>16->16</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M455.311,-24.7434C465.094,-24.8884 473.07,-22.6406 473.07,-18 473.07,-15.1721 470.109,-13.2328 465.581,-12.182\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"465.585,-8.66827 455.311,-11.2566 464.957,-15.64 465.585,-8.66827\"/>\n",
- "<text text-anchor=\"middle\" x=\"485.124\" y=\"-13.8\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge85\" class=\"edge\">\n",
+ "<title>16->16</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M455.31,-24.74C465.09,-24.89 473.07,-22.64 473.07,-18 473.07,-15.17 470.11,-13.23 465.58,-12.18\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"465.59,-8.67 455.31,-11.26 464.96,-15.64 465.59,-8.67\"/>\n",
+ "<text text-anchor=\"middle\" x=\"487.57\" y=\"-14.3\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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- "<g id=\"edge172\" class=\"edge\"><title>16->15</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M400.905,-16.5964C383.366,-18.1331 360.327,-21.5653 341.128,-25.4135\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"340.257,-22.0202 331.202,-27.5201 341.711,-28.8677 340.257,-22.0202\"/>\n",
- "<text text-anchor=\"middle\" x=\"370.016\" y=\"-6.80495\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge86\" class=\"edge\">\n",
+ "<title>16->15</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M400.9,-16.6C383.37,-18.13 360.33,-21.57 341.13,-25.41\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"340.26,-22.02 331.2,-27.52 341.71,-28.87 340.26,-22.02\"/>\n",
+ "<text text-anchor=\"middle\" x=\"356.52\" y=\"-9.8\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge174\" class=\"edge\"><title>16->14</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M401.033,-22.157C358.704,-32.9008 276.755,-59.2693 229.101,-77.2458\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"227.557,-74.0897 219.48,-80.9458 230.07,-80.6232 227.557,-74.0897\"/>\n",
- "<text text-anchor=\"middle\" x=\"318.067\" y=\"-57.5014\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge87\" class=\"edge\">\n",
+ "<title>16->14</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M401.03,-22.16C358.7,-32.9 276.75,-59.27 229.1,-77.25\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"227.56,-74.09 219.48,-80.95 230.07,-80.62 227.56,-74.09\"/>\n",
+ "<text text-anchor=\"middle\" x=\"329.57\" y=\"-53.5\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->13 -->\n",
- "<g id=\"edge176\" class=\"edge\"><title>16->13</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M400.865,-27.5488C342.28,-53.3134 204.353,-123.343 139.04,-160.21\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"137.273,-157.189 130.322,-165.184 140.742,-163.269 137.273,-157.189\"/>\n",
- "<text text-anchor=\"middle\" x=\"275.952\" y=\"-101.68\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge88\" class=\"edge\">\n",
+ "<title>16->13</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M400.86,-27.55C342.28,-53.31 204.35,-123.34 139.04,-160.21\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"137.27,-157.19 130.32,-165.18 140.74,-163.27 137.27,-157.19\"/>\n",
+ "<text text-anchor=\"middle\" x=\"284.45\" y=\"-97.68\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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"<!-- 16->12 -->\n",
- "<g id=\"edge178\" class=\"edge\"><title>16->12</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M400.951,-33.3037C330.814,-79.1137 144.29,-214.523 73.9419,-270.564\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"71.5496,-267.997 65.9544,-276.994 75.939,-273.45 71.5496,-267.997\"/>\n",
- "<text text-anchor=\"middle\" x=\"245.446\" y=\"-159.734\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge89\" class=\"edge\">\n",
+ "<title>16->12</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M400.95,-33.3C330.81,-79.11 144.29,-214.52 73.94,-270.56\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"71.55,-268 65.95,-276.99 75.94,-273.45 71.55,-268\"/>\n",
+ "<text text-anchor=\"middle\" x=\"222.95\" y=\"-140.73\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->11 -->\n",
- "<g id=\"edge180\" class=\"edge\"><title>16->11</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M405.377,-36.1872C334.299,-100.645 117.176,-317.67 48.197,-393.073\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"45.3761,-390.974 41.2611,-400.737 50.5663,-395.671 45.3761,-390.974\"/>\n",
- "<text text-anchor=\"middle\" x=\"237.787\" y=\"-221.43\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge90\" class=\"edge\">\n",
+ "<title>16->11</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M405.38,-36.19C334.3,-100.65 117.18,-317.67 48.2,-393.07\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"45.38,-390.97 41.26,-400.74 50.57,-395.67 45.38,-390.97\"/>\n",
+ "<text text-anchor=\"middle\" x=\"212.29\" y=\"-218.43\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge182\" class=\"edge\"><title>16->10</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M411.324,-36.298C347.873,-114.827 123.86,-422.986 62.2885,-516.196\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"59.2234,-514.488 56.6899,-524.776 65.0858,-518.314 59.2234,-514.488\"/>\n",
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+ "<g id=\"edge91\" class=\"edge\">\n",
+ "<title>16->10</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M411.32,-36.3C347.87,-114.83 123.86,-422.99 62.29,-516.2\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"59.22,-514.49 56.69,-524.78 65.09,-518.31 59.22,-514.49\"/>\n",
+ "<text text-anchor=\"middle\" x=\"222.31\" y=\"-280.05\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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"<!-- 16->9 -->\n",
- "<g id=\"edge184\" class=\"edge\"><title>16->9</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M415.931,-36.2593C364.06,-125.934 163.457,-519.37 114.348,-627.34\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"111.094,-626.043 110.201,-636.6 117.483,-628.904 111.094,-626.043\"/>\n",
- "<text text-anchor=\"middle\" x=\"279.14\" y=\"-334.6\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge92\" class=\"edge\">\n",
+ "<title>16->9</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M415.93,-36.26C364.06,-125.93 163.46,-519.37 114.35,-627.34\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"111.09,-626.04 110.2,-636.6 117.48,-628.9 111.09,-626.04\"/>\n",
+ "<text text-anchor=\"middle\" x=\"250.64\" y=\"-335.6\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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"<!-- 16->8 -->\n",
- "<g id=\"edge186\" class=\"edge\"><title>16->8</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"195.096,-714.754 195.862,-725.321 201.853,-716.582 195.096,-714.754\"/>\n",
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+ "<g id=\"edge93\" class=\"edge\">\n",
+ "<title>16->8</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M419.64,-36.3C381.81,-134.27 231.51,-596.39 198.52,-715.49\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"195.1,-714.75 195.86,-725.32 201.85,-716.58 195.1,-714.75\"/>\n",
+ "<text text-anchor=\"middle\" x=\"294.58\" y=\"-379.7\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge188\" class=\"edge\"><title>16->7</title>\n",
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+ "<g id=\"edge94\" class=\"edge\">\n",
+ "<title>16->7</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M422.91,-36.22C400.69,-139.09 320.09,-647.07 305.84,-772.48\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"302.35,-772.14 304.76,-782.45 309.31,-772.89 302.35,-772.14\"/>\n",
+ "<text text-anchor=\"middle\" x=\"349.87\" y=\"-408.15\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge190\" class=\"edge\"><title>16->6</title>\n",
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+ "<g id=\"edge95\" class=\"edge\">\n",
+ "<title>16->6</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M425.86,-36.26C419.98,-140.84 419.81,-663.54 425.36,-791.73\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"421.88,-792.09 425.86,-801.91 428.87,-791.75 421.88,-792.09\"/>\n",
+ "<text text-anchor=\"middle\" x=\"411.11\" y=\"-417.79\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<g id=\"edge96\" class=\"edge\">\n",
+ "<title>16->5</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M428.71,-36.16C439.03,-138.87 519.31,-646.83 544.78,-772.4\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"541.39,-773.31 546.86,-782.39 548.25,-771.88 541.39,-773.31\"/>\n",
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+ "<g id=\"edge97\" class=\"edge\">\n",
+ "<title>16->4</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M431.62,-36.28C457.39,-134.27 607.6,-597.18 651.81,-715.8\"/>\n",
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+ "<g id=\"edge98\" class=\"edge\">\n",
+ "<title>16->3</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M434.63,-36.11C473.95,-125.64 674.89,-520.47 735.42,-627.81\"/>\n",
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+ "<g id=\"edge101\" class=\"edge\">\n",
+ "<title>15->20</title>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"771.96,-283.98 782.51,-284.94 774.85,-277.61 771.96,-283.98\"/>\n",
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+ "<title>15->19</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M323.99,-55.63C402.56,-118.81 691.88,-329.27 793.56,-397.57\"/>\n",
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+ "<g id=\"edge103\" class=\"edge\">\n",
+ "<title>15->18</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M331.16,-45.77C394.7,-59.63 553.65,-84.97 626.4,-93.01\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"626.18,-96.51 636.49,-94.07 626.91,-89.55 626.18,-96.51\"/>\n",
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+ "<g id=\"edge104\" class=\"edge\">\n",
+ "<title>15->17</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M331.61,-42.06C376.21,-44.95 464.42,-45.14 514.67,-42.62\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"515,-46.11 524.79,-42.05 514.61,-39.12 515,-46.11\"/>\n",
+ "<text text-anchor=\"middle\" x=\"408.64\" y=\"-46.14\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<g id=\"edge105\" class=\"edge\">\n",
+ "<title>15->16</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M331.3,-39.03C348.84,-37.5 371.88,-34.06 391.08,-30.22\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"391.95,-33.61 401,-28.11 390.49,-26.76 391.95,-33.61\"/>\n",
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+ "<g id=\"edge106\" class=\"edge\">\n",
+ "<title>15->15</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M331.37,-44.37C341.16,-44.52 349.13,-42.27 349.13,-37.63 349.13,-34.8 346.17,-32.86 341.64,-31.81\"/>\n",
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+ "<g id=\"edge107\" class=\"edge\">\n",
+ "<title>15->14</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M277.01,-45.05C261.08,-51.87 240.89,-61.93 224.2,-71.28\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"222.27,-68.36 215.35,-76.38 225.76,-74.43 222.27,-68.36\"/>\n",
+ "<text text-anchor=\"middle\" x=\"236.11\" y=\"-61.97\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"126.836,-156.104 121.339,-165.161 131.285,-161.509 126.836,-156.104\"/>\n",
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+ "<g id=\"edge108\" class=\"edge\">\n",
+ "<title>15->13</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M276.83,-51.76C237.97,-76.59 167.38,-127.77 129.38,-158.55\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"126.84,-156.1 121.34,-165.16 131.28,-161.51 126.84,-156.1\"/>\n",
+ "<text text-anchor=\"middle\" x=\"217.6\" y=\"-108.95\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"63.8093,-267.315 59.786,-277.116 69.0433,-271.963 63.8093,-267.315\"/>\n",
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+ "<g id=\"edge109\" class=\"edge\">\n",
+ "<title>15->12</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M280.29,-55.94C229.88,-101.07 113.63,-217.25 66.7,-269.33\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"63.81,-267.31 59.79,-277.12 69.04,-271.96 63.81,-267.31\"/>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"39.1828,-390.333 36.7136,-400.636 45.0689,-394.122 39.1828,-390.333\"/>\n",
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+ "<g id=\"edge110\" class=\"edge\">\n",
+ "<title>15->11</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M287.01,-55.71C236.63,-117.25 90.33,-318.43 42.32,-391.93\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"39.18,-390.33 36.71,-400.64 45.07,-394.12 39.18,-390.33\"/>\n",
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+ "<g id=\"edge111\" class=\"edge\">\n",
+ "<title>15->10</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M291.69,-55.85C247.17,-131.95 98.09,-424.26 57.01,-515.6\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"53.69,-514.44 52.85,-525 60.1,-517.27 53.69,-514.44\"/>\n",
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+ "<g id=\"edge112\" class=\"edge\">\n",
+ "<title>15->9</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M295.62,-55.66C261.54,-142.82 138.64,-520.58 109.54,-626.89\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"106.15,-626.02 106.95,-636.59 112.91,-627.83 106.15,-626.02\"/>\n",
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+ "<g id=\"edge113\" class=\"edge\">\n",
+ "<title>15->8</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M298.87,-55.78C277.93,-151.61 206.96,-598.7 193.9,-715.54\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"190.42,-715.19 192.85,-725.5 197.38,-715.92 190.42,-715.19\"/>\n",
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+ "<g id=\"edge114\" class=\"edge\">\n",
+ "<title>15->7</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M301.88,-55.74C296.06,-156.49 295.89,-647.97 301.36,-772.03\"/>\n",
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+ "<title>15->6</title>\n",
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+ "<title>15->5</title>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"777.59,-520.65 787.31,-524.85 782.33,-515.5 777.59,-520.65\"/>\n",
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+ "<g id=\"edge120\" class=\"edge\">\n",
+ "<title>15->1</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M331.44,-50.19C408.57,-79.67 628.05,-151.11 715.72,-175.43\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"714.93,-178.84 725.5,-178.09 716.77,-172.08 714.93,-178.84\"/>\n",
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- "<text text-anchor=\"middle\" x=\"491.989\" y=\"-204.435\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge121\" class=\"edge\">\n",
+ "<title>14->20</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M219.51,-106.62C318.41,-143.84 659.06,-254.65 772.47,-286.65\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"771.84,-290.11 782.41,-289.41 773.71,-283.36 771.84,-290.11\"/>\n",
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- "<text text-anchor=\"middle\" x=\"512.095\" y=\"-241.81\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge122\" class=\"edge\">\n",
+ "<title>14->19</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M219.56,-111.85C321.03,-169.04 677.06,-350.55 792.63,-404.17\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"791.25,-407.38 801.8,-408.38 794.17,-401.02 791.25,-407.38\"/>\n",
+ "<text text-anchor=\"middle\" x=\"491.59\" y=\"-261.81\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"626.768,-101.945 636.587,-97.9647 626.429,-94.9533 626.768,-101.945\"/>\n",
- "<text text-anchor=\"middle\" x=\"422.864\" y=\"-103.008\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge123\" class=\"edge\">\n",
+ "<title>14->18</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M219.37,-97.95C299.27,-102.3 534.41,-102.47 626.36,-98.46\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"626.77,-101.95 636.59,-97.96 626.43,-94.95 626.77,-101.95\"/>\n",
+ "<text text-anchor=\"middle\" x=\"408.36\" y=\"-102.01\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<text text-anchor=\"middle\" x=\"365.956\" y=\"-55.8357\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge124\" class=\"edge\">\n",
+ "<title>14->17</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M219.36,-94.1C282.89,-87.89 441.78,-62.93 514.55,-47.97\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"515.58,-51.33 524.64,-45.84 514.13,-44.48 515.58,-51.33\"/>\n",
+ "<text text-anchor=\"middle\" x=\"352.46\" y=\"-74.84\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"392.841,-38.508 400.918,-31.6519 390.328,-31.9744 392.841,-38.508\"/>\n",
- "<text text-anchor=\"middle\" x=\"308.331\" y=\"-70.6963\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge125\" class=\"edge\">\n",
+ "<title>14->16</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M219.36,-90.44C261.69,-79.7 343.64,-53.33 391.3,-35.35\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"392.84,-38.51 400.92,-31.65 390.33,-31.97 392.84,-38.51\"/>\n",
+ "<text text-anchor=\"middle\" x=\"290.83\" y=\"-66.7\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"274.191,-63.8673 281.113,-55.8464 270.7,-57.8 274.191,-63.8673\"/>\n",
- "<text text-anchor=\"middle\" x=\"239.853\" y=\"-57.8586\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge126\" class=\"edge\">\n",
+ "<title>14->15</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M219.45,-87.17C235.38,-80.36 255.57,-70.29 272.26,-60.94\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"274.19,-63.87 281.11,-55.85 270.7,-57.8 274.19,-63.87\"/>\n",
+ "<text text-anchor=\"middle\" x=\"260.35\" y=\"-77.86\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<g id=\"edge127\" class=\"edge\">\n",
+ "<title>14->14</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M219.57,-101.34C229.35,-101.49 237.33,-99.24 237.33,-94.6 237.33,-91.77 234.37,-89.83 229.84,-88.78\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"229.84,-85.27 219.57,-87.85 229.21,-92.24 229.84,-85.27\"/>\n",
+ "<text text-anchor=\"middle\" x=\"251.83\" y=\"-90.9\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge256\" class=\"edge\"><title>14->13</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"118.424,-155.233 114.54,-165.09 123.723,-159.807 118.424,-155.233\"/>\n",
- "<text text-anchor=\"middle\" x=\"154.416\" y=\"-141.932\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge128\" class=\"edge\">\n",
+ "<title>14->13</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M165.73,-112.78C151.27,-125.7 133.87,-143.14 121.1,-157.48\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"118.42,-155.23 114.54,-165.09 123.72,-159.81 118.42,-155.23\"/>\n",
+ "<text text-anchor=\"middle\" x=\"128.92\" y=\"-138.93\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge258\" class=\"edge\"><title>14->12</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"57.2913,-266.77 54.9718,-277.108 63.2318,-270.473 57.2913,-266.77\"/>\n",
- "<text text-anchor=\"middle\" x=\"105.11\" y=\"-177.368\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge129\" class=\"edge\">\n",
+ "<title>14->12</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M173.8,-112.77C144.07,-148.31 86.87,-226.88 60.42,-268.37\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"57.29,-266.77 54.97,-277.11 63.23,-270.47 57.29,-266.77\"/>\n",
+ "<text text-anchor=\"middle\" x=\"131.61\" y=\"-194.37\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge260\" class=\"edge\"><title>14->11</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"33.4279,-390.276 32.6876,-400.845 39.8572,-393.045 33.4279,-390.276\"/>\n",
- "<text text-anchor=\"middle\" x=\"93.9809\" y=\"-240.865\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge130\" class=\"edge\">\n",
+ "<title>14->11</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M179.16,-112.84C146.82,-166.94 65.36,-326.5 36.8,-391.29\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"33.43,-390.28 32.69,-400.84 39.86,-393.04 33.43,-390.28\"/>\n",
+ "<text text-anchor=\"middle\" x=\"93.48\" y=\"-255.87\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<g id=\"edge131\" class=\"edge\">\n",
+ "<title>14->10</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M183.37,-112.67C155.68,-181.97 74.64,-430.87 52.13,-515\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"48.69,-514.31 49.57,-524.87 55.47,-516.07 48.69,-514.31\"/>\n",
+ "<text text-anchor=\"middle\" x=\"103.25\" y=\"-317.64\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<g id=\"edge132\" class=\"edge\">\n",
+ "<title>14->9</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M186.78,-112.83C168.47,-194.43 115.69,-526.55 104.91,-626.33\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"101.4,-626.25 103.88,-636.55 108.37,-626.95 101.4,-626.25\"/>\n",
+ "<text text-anchor=\"middle\" x=\"131.34\" y=\"-373.38\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge266\" class=\"edge\"><title>14->8</title>\n",
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+ "<g id=\"edge133\" class=\"edge\">\n",
+ "<title>14->8</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M189.91,-112.63C184.32,-202.77 184.14,-604.52 189.35,-715.22\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"185.86,-715.49 189.89,-725.29 192.85,-715.12 185.86,-715.49\"/>\n",
+ "<text text-anchor=\"middle\" x=\"175.13\" y=\"-417.73\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<g id=\"edge134\" class=\"edge\">\n",
+ "<title>14->7</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M192.86,-112.76C202.25,-208.63 272.87,-655.7 296.8,-772.52\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"293.41,-773.42 298.9,-782.48 300.26,-771.97 293.41,-773.42\"/>\n",
+ "<text text-anchor=\"middle\" x=\"230.33\" y=\"-446.44\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<g id=\"edge135\" class=\"edge\">\n",
+ "<title>14->6</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M195.88,-112.88C221.64,-210.86 371.85,-673.77 416.07,-792.4\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"412.84,-793.77 419.66,-801.88 419.39,-791.29 412.84,-793.77\"/>\n",
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+ "<g id=\"edge136\" class=\"edge\">\n",
+ "<title>14->5</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M199.02,-112.73C241.51,-208.84 470.31,-658.3 535.06,-773.55\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"532.14,-775.49 540.12,-782.46 538.23,-772.04 532.14,-775.49\"/>\n",
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+ "<g id=\"edge137\" class=\"edge\">\n",
+ "<title>14->4</title>\n",
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+ "<title>14->3</title>\n",
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+ "<title>14->2</title>\n",
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+ "<g id=\"edge140\" class=\"edge\">\n",
+ "<title>14->1</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M219.62,-102.11C311.52,-121.38 609.89,-168.78 715.27,-181.03\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"715.1,-184.53 725.43,-182.16 715.88,-177.57 715.1,-184.53\"/>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"772.015,-295.984 782.351,-293.655 772.819,-289.03 772.015,-295.984\"/>\n",
- "<text text-anchor=\"middle\" x=\"450.45\" y=\"-248.288\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge141\" class=\"edge\">\n",
+ "<title>13->20</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M130.83,-190.51C240.01,-212.95 646.19,-277.42 772.07,-292.47\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"772.02,-295.98 782.35,-293.66 772.82,-289.03 772.02,-295.98\"/>\n",
+ "<text text-anchor=\"middle\" x=\"436.95\" y=\"-245.29\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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- "<g id=\"edge284\" class=\"edge\"><title>13->19</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"791.347,-413.773 801.921,-413.105 793.239,-407.034 791.347,-413.773\"/>\n",
- "<text text-anchor=\"middle\" x=\"465.457\" y=\"-286.537\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge142\" class=\"edge\">\n",
+ "<title>13->19</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M130.73,-195.1C242.1,-236.69 664.26,-373.97 792.19,-410.37\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"791.35,-413.77 801.92,-413.11 793.24,-407.03 791.35,-413.77\"/>\n",
+ "<text text-anchor=\"middle\" x=\"446.96\" y=\"-306.54\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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- "<g id=\"edge286\" class=\"edge\"><title>13->18</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"627.415,-107.544 636.511,-102.11 626.012,-100.687 627.415,-107.544\"/>\n",
- "<text text-anchor=\"middle\" x=\"377.685\" y=\"-127.944\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge143\" class=\"edge\">\n",
+ "<title>13->18</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M130.72,-182.16C222.48,-172.39 521.3,-125.23 626.65,-104.13\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"627.42,-107.54 636.51,-102.11 626.01,-100.69 627.42,-107.54\"/>\n",
+ "<text text-anchor=\"middle\" x=\"364.18\" y=\"-146.94\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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- "<g id=\"edge288\" class=\"edge\"><title>13->17</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"516.847,-56.8859 524.971,-50.0845 514.379,-50.3355 516.847,-56.8859\"/>\n",
- "<text text-anchor=\"middle\" x=\"320.07\" y=\"-99.6729\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge144\" class=\"edge\">\n",
+ "<title>13->17</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M130.69,-178.07C207.7,-157.57 427.88,-86.23 515.45,-53.67\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"516.85,-56.89 524.97,-50.08 514.38,-50.34 516.85,-56.89\"/>\n",
+ "<text text-anchor=\"middle\" x=\"308.57\" y=\"-119.67\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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- "<g id=\"edge290\" class=\"edge\"><title>13->16</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"394.395,-44.1383 401.347,-36.1427 390.926,-38.0583 394.395,-44.1383\"/>\n",
- "<text text-anchor=\"middle\" x=\"255.716\" y=\"-91.2476\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge145\" class=\"edge\">\n",
+ "<title>13->16</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M130.8,-173.78C189.39,-148.01 327.32,-77.98 392.63,-41.12\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"394.4,-44.14 401.35,-36.14 390.93,-38.06 394.4,-44.14\"/>\n",
+ "<text text-anchor=\"middle\" x=\"247.22\" y=\"-111.25\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<text text-anchor=\"middle\" x=\"212.627\" y=\"-123.602\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge146\" class=\"edge\">\n",
+ "<title>13->15</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M130.9,-169.19C169.76,-144.36 240.35,-93.19 278.36,-62.41\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"280.89,-64.85 286.39,-55.8 276.45,-59.45 280.89,-64.85\"/>\n",
+ "<text text-anchor=\"middle\" x=\"190.13\" y=\"-119.6\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge294\" class=\"edge\"><title>13->14</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"177.501,-122.692 181.385,-112.835 172.201,-118.118 177.501,-122.692\"/>\n",
- "<text text-anchor=\"middle\" x=\"163.509\" y=\"-149.593\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge147\" class=\"edge\">\n",
+ "<title>13->14</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M130.2,-165.15C144.65,-152.22 162.06,-134.79 174.82,-120.44\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"177.5,-122.69 181.38,-112.83 172.2,-118.12 177.5,-122.69\"/>\n",
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- "<g id=\"edge296\" class=\"edge\"><title>13->13</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"141.112,-173.995 130.838,-176.584 140.484,-180.967 141.112,-173.995\"/>\n",
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+ "<g id=\"edge148\" class=\"edge\">\n",
+ "<title>13->13</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M130.84,-190.07C140.62,-190.22 148.6,-187.97 148.6,-183.33 148.6,-180.5 145.64,-178.56 141.11,-177.51\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"141.11,-174 130.84,-176.58 140.48,-180.97 141.11,-174\"/>\n",
+ "<text text-anchor=\"middle\" x=\"163.1\" y=\"-179.63\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge298\" class=\"edge\"><title>13->12</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"50.9865,-266.293 50.4315,-276.873 57.4633,-268.948 50.9865,-266.293\"/>\n",
- "<text text-anchor=\"middle\" x=\"57.3475\" y=\"-223.187\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge149\" class=\"edge\">\n",
+ "<title>13->12</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M88.37,-201.4C77.35,-219.26 63.2,-246.75 54.33,-267.37\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"50.99,-266.29 50.43,-276.87 57.46,-268.95 50.99,-266.29\"/>\n",
+ "<text text-anchor=\"middle\" x=\"56.85\" y=\"-238.19\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge300\" class=\"edge\"><title>13->11</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"27.9637,-390.4 28.9933,-400.945 34.7641,-392.059 27.9637,-390.4\"/>\n",
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+ "<g id=\"edge150\" class=\"edge\">\n",
+ "<title>13->11</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M93.61,-201.41C76.55,-242.31 43.94,-342.05 31.41,-391.03\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"27.96,-390.4 28.99,-400.94 34.76,-392.06 27.96,-390.4\"/>\n",
+ "<text text-anchor=\"middle\" x=\"77.01\" y=\"-300.02\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"43.7605,-514.45 46.3416,-524.726 50.7319,-515.083 43.7605,-514.45\"/>\n",
- "<text text-anchor=\"middle\" x=\"86.3391\" y=\"-355.998\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge151\" class=\"edge\">\n",
+ "<title>13->10</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M97.43,-201.65C83.58,-260.35 54.31,-443.91 47.25,-514.74\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"43.76,-514.45 46.34,-524.73 50.73,-515.08 43.76,-514.45\"/>\n",
+ "<text text-anchor=\"middle\" x=\"57.84\" y=\"-362\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"96.689,-626.904 100.788,-636.674 103.676,-626.48 96.689,-626.904\"/>\n",
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+ "<g id=\"edge152\" class=\"edge\">\n",
+ "<title>13->9</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M100.78,-201.52C95.74,-273.57 95.54,-539.22 100.18,-626.69\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"96.69,-626.9 100.79,-636.67 103.68,-626.48 96.69,-626.9\"/>\n",
+ "<text text-anchor=\"middle\" x=\"114.98\" y=\"-402.9\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<g id=\"edge153\" class=\"edge\">\n",
+ "<title>13->8</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M103.86,-201.42C111.34,-282.92 163.93,-616.24 184.66,-715.52\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"181.27,-716.38 186.8,-725.41 188.11,-714.9 181.27,-716.38\"/>\n",
+ "<text text-anchor=\"middle\" x=\"129.76\" y=\"-462.27\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<g id=\"edge154\" class=\"edge\">\n",
+ "<title>13->7</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M106.95,-201.56C129.61,-289.17 252.12,-666.91 291.92,-772.82\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"288.77,-774.38 295.61,-782.47 295.31,-771.88 288.77,-774.38\"/>\n",
+ "<text text-anchor=\"middle\" x=\"184.94\" y=\"-490.99\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<g id=\"edge155\" class=\"edge\">\n",
+ "<title>13->6</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M110.16,-201.44C149.47,-290.96 350.42,-685.8 410.95,-793.14\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"408.04,-795.11 416.03,-802.06 414.12,-791.64 408.04,-795.11\"/>\n",
+ "<text text-anchor=\"middle\" x=\"246.05\" y=\"-501.09\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<g id=\"edge156\" class=\"edge\">\n",
+ "<title>13->5</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M113.88,-201.57C171.01,-289.69 447.63,-670.79 529.33,-774.44\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"526.79,-776.86 535.75,-782.51 532.27,-772.5 526.79,-776.86\"/>\n",
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+ "<g id=\"edge157\" class=\"edge\">\n",
+ "<title>13->4</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M118.3,-201.55C193.18,-284.15 531.45,-622.72 634.19,-718.34\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"632.01,-721.09 641.73,-725.31 636.76,-715.95 632.01,-721.09\"/>\n",
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+ "<g id=\"edge158\" class=\"edge\">\n",
+ "<title>13->3</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M124,-201.43C216.7,-275.29 598.23,-552.72 717.07,-633\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"715.23,-635.99 725.49,-638.66 719.13,-630.18 715.23,-635.99\"/>\n",
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+ "<g id=\"edge159\" class=\"edge\">\n",
+ "<title>13->2</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M130.84,-200.43C240.4,-261.96 648.74,-470.12 773.25,-528.06\"/>\n",
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+ "<g id=\"edge160\" class=\"edge\">\n",
+ "<title>13->1</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M130.66,-186.24C233.11,-191.2 597.39,-191.35 715.37,-186.68\"/>\n",
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+ "<g id=\"edge161\" class=\"edge\">\n",
+ "<title>12->20</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M73.78,-297.84C189.17,-303.09 639.02,-303.23 772.18,-298.26\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"772.44,-301.75 782.29,-297.85 772.15,-294.76 772.44,-301.75\"/>\n",
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- "<text text-anchor=\"middle\" x=\"431.894\" y=\"-366.043\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge162\" class=\"edge\">\n",
+ "<title>12->19</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M74.06,-302.22C191.99,-326.23 656.1,-399.86 791.73,-416.26\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"791.68,-419.78 802.02,-417.47 792.49,-412.83 791.68,-419.78\"/>\n",
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- "<text text-anchor=\"middle\" x=\"347.277\" y=\"-183.634\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge163\" class=\"edge\">\n",
+ "<title>12->18</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M73.66,-289.43C172.39,-262.51 513.55,-151.85 626.89,-110.23\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"628.36,-113.42 636.52,-106.66 625.93,-106.86 628.36,-113.42\"/>\n",
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- "<text text-anchor=\"middle\" x=\"288.892\" y=\"-156.338\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge164\" class=\"edge\">\n",
+ "<title>12->17</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M73.83,-284.85C158.91,-246.61 419.79,-113.9 515.95,-60.23\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"517.73,-63.24 524.73,-55.29 514.3,-57.14 517.73,-63.24\"/>\n",
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- "<text text-anchor=\"middle\" x=\"245.254\" y=\"-168.999\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge165\" class=\"edge\">\n",
+ "<title>12->16</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M73.75,-279.83C143.89,-234.02 330.41,-98.61 400.76,-42.57\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"403.15,-45.14 408.75,-36.14 398.76,-39.68 403.15,-45.14\"/>\n",
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- "<text text-anchor=\"middle\" x=\"166.265\" y=\"-154.93\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge166\" class=\"edge\">\n",
+ "<title>12->15</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M70.47,-276.83C120.88,-231.7 237.14,-115.51 284.06,-63.43\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"286.95,-65.45 290.98,-55.65 281.72,-60.8 286.95,-65.45\"/>\n",
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- "<text text-anchor=\"middle\" x=\"133.847\" y=\"-203.962\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge167\" class=\"edge\">\n",
+ "<title>12->14</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M65.16,-276.96C94.89,-241.42 152.08,-162.85 178.54,-121.37\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"181.67,-122.96 183.99,-112.62 175.73,-119.26 181.67,-122.96\"/>\n",
+ "<text text-anchor=\"middle\" x=\"107.35\" y=\"-202.96\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge336\" class=\"edge\"><title>12->13</title>\n",
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- "<text text-anchor=\"middle\" x=\"64.8799\" y=\"-232.873\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge168\" class=\"edge\">\n",
+ "<title>12->13</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M61.86,-277.06C72.88,-259.2 87.03,-231.71 95.9,-211.09\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"99.24,-212.17 99.8,-201.59 92.76,-209.51 99.24,-212.17\"/>\n",
+ "<text text-anchor=\"middle\" x=\"93.38\" y=\"-232.87\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge338\" class=\"edge\"><title>12->12</title>\n",
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- "<text text-anchor=\"middle\" x=\"103.683\" y=\"-290.933\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge169\" class=\"edge\">\n",
+ "<title>12->12</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M73.87,-301.88C83.65,-302.02 91.63,-299.77 91.63,-295.13 91.63,-292.3 88.67,-290.37 84.14,-289.31\"/>\n",
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+ "<text text-anchor=\"middle\" x=\"106.13\" y=\"-291.43\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge340\" class=\"edge\"><title>12->11</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"22.1958,-390.684 24.9527,-400.914 29.1769,-391.197 22.1958,-390.684\"/>\n",
- "<text text-anchor=\"middle\" x=\"18.202\" y=\"-345.741\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge170\" class=\"edge\">\n",
+ "<title>12->11</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M38.7,-313.17C33.4,-333.52 27.88,-366.87 25.7,-390.71\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"22.2,-390.68 24.95,-400.91 29.18,-391.2 22.2,-390.68\"/>\n",
+ "<text text-anchor=\"middle\" x=\"17.7\" y=\"-355.74\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge342\" class=\"edge\"><title>12->10</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"38.6595,-515.041 42.8892,-524.754 45.6404,-514.523 38.6595,-515.041\"/>\n",
- "<text text-anchor=\"middle\" x=\"29.5171\" y=\"-409.822\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge171\" class=\"edge\">\n",
+ "<title>12->10</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M42.89,-313.36C39.14,-356.16 38.89,-463.32 42.14,-514.68\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"38.66,-515.04 42.89,-524.75 45.64,-514.52 38.66,-515.04\"/>\n",
+ "<text text-anchor=\"middle\" x=\"57.02\" y=\"-402.82\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge344\" class=\"edge\"><title>12->9</title>\n",
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+ "<g id=\"edge172\" class=\"edge\">\n",
+ "<title>12->9</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M46.33,-313.21C50.96,-371.61 79.82,-555.48 95.17,-626.46\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"91.82,-627.48 97.43,-636.47 98.65,-625.95 91.82,-627.48\"/>\n",
+ "<text text-anchor=\"middle\" x=\"56.25\" y=\"-473.64\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<g id=\"edge173\" class=\"edge\">\n",
+ "<title>12->8</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M49.58,-313.31C66.93,-382.9 147.82,-632.63 179.73,-716.14\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"176.48,-717.43 183.37,-725.47 183,-714.88 176.48,-717.43\"/>\n",
+ "<text text-anchor=\"middle\" x=\"100.15\" y=\"-518.52\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge348\" class=\"edge\"><title>12->7</title>\n",
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+ "<g id=\"edge174\" class=\"edge\">\n",
+ "<title>12->7</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M52.85,-313.14C85.6,-388.97 234.75,-682.21 286.54,-773.35\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"283.64,-775.33 291.67,-782.25 289.71,-771.84 283.64,-775.33\"/>\n",
+ "<text text-anchor=\"middle\" x=\"155.2\" y=\"-547.05\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<g id=\"edge175\" class=\"edge\">\n",
+ "<title>12->6</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M56.6,-313.22C106.93,-391.57 331.68,-701.3 404.98,-793.9\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"402.41,-796.3 411.39,-801.92 407.88,-791.93 402.41,-796.3\"/>\n",
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+ "<g id=\"edge176\" class=\"edge\">\n",
+ "<title>12->5</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M61.04,-313.21C130.09,-389.76 426.62,-686.62 522.23,-775.37\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"520.08,-778.15 529.81,-782.35 524.83,-773 520.08,-778.15\"/>\n",
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+ "<g id=\"edge177\" class=\"edge\">\n",
+ "<title>12->4</title>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"626.5,-724.8 636.76,-727.46 630.4,-718.99 626.5,-724.8\"/>\n",
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+ "<g id=\"edge178\" class=\"edge\">\n",
+ "<title>12->3</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M73.87,-312.24C183.44,-373.76 591.77,-581.92 716.29,-639.87\"/>\n",
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+ "<g id=\"edge179\" class=\"edge\">\n",
+ "<title>12->2</title>\n",
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+ "<g id=\"edge180\" class=\"edge\">\n",
+ "<title>12->1</title>\n",
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+ "<g id=\"edge181\" class=\"edge\">\n",
+ "<title>11->20</title>\n",
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+ "<g id=\"edge182\" class=\"edge\">\n",
+ "<title>11->19</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M54.15,-421.72C173.77,-427.05 654.01,-427.18 791.93,-422.13\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"792.15,-425.62 802,-421.72 791.87,-418.63 792.15,-425.62\"/>\n",
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+ "<g id=\"edge183\" class=\"edge\">\n",
+ "<title>11->18</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M54.06,-408.51C155.44,-362.45 512.66,-180.63 627.79,-116.77\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"629.58,-119.78 636.6,-111.84 626.16,-113.67 629.58,-119.78\"/>\n",
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- "<text text-anchor=\"middle\" x=\"281.043\" y=\"-216.479\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge184\" class=\"edge\">\n",
+ "<title>11->17</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M54.15,-403.19C144.25,-343.62 432.5,-134.3 523.94,-62.17\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"526.27,-64.79 531.91,-55.82 521.91,-59.31 526.27,-64.79\"/>\n",
+ "<text text-anchor=\"middle\" x=\"274.54\" y=\"-236.48\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<text text-anchor=\"middle\" x=\"239.284\" y=\"-229.241\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge185\" class=\"edge\">\n",
+ "<title>11->16</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M49.69,-400.88C120.77,-336.43 337.89,-119.4 406.87,-44\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"409.69,-46.1 413.81,-36.33 404.5,-41.4 409.69,-46.1\"/>\n",
+ "<text text-anchor=\"middle\" x=\"242.78\" y=\"-226.24\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge372\" class=\"edge\"><title>11->15</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"291.95,-66.367 294.419,-56.0639 286.064,-62.5784 291.95,-66.367\"/>\n",
- "<text text-anchor=\"middle\" x=\"153.466\" y=\"-219.684\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge186\" class=\"edge\">\n",
+ "<title>11->15</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M44.12,-401C94.5,-339.45 240.81,-138.27 288.81,-64.77\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"291.95,-66.37 294.42,-56.06 286.06,-62.58 291.95,-66.37\"/>\n",
+ "<text text-anchor=\"middle\" x=\"180.97\" y=\"-236.68\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"185.899,-123.392 186.64,-112.823 179.47,-120.624 185.899,-123.392\"/>\n",
- "<text text-anchor=\"middle\" x=\"97.3463\" y=\"-250.403\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge187\" class=\"edge\">\n",
+ "<title>11->14</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M40.17,-400.83C72.5,-346.73 153.96,-187.17 182.53,-122.38\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"185.9,-123.39 186.64,-112.82 179.47,-120.62 185.9,-123.39\"/>\n",
+ "<text text-anchor=\"middle\" x=\"125.85\" y=\"-265.4\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"102.634,-211.998 101.604,-201.453 95.8335,-210.338 102.634,-211.998\"/>\n",
- "<text text-anchor=\"middle\" x=\"82.0865\" y=\"-305.979\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge188\" class=\"edge\">\n",
+ "<title>11->13</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M36.99,-400.99C54.05,-360.09 86.65,-260.35 99.18,-211.37\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"102.63,-212 101.6,-201.45 95.83,-210.34 102.63,-212\"/>\n",
+ "<text text-anchor=\"middle\" x=\"53.59\" y=\"-309.98\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"51.434,-323.52 48.6771,-313.29 44.4529,-323.006 51.434,-323.52\"/>\n",
- "<text text-anchor=\"middle\" x=\"55.4278\" y=\"-360.062\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge189\" class=\"edge\">\n",
+ "<title>11->12</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M34.93,-401.03C40.23,-380.68 45.75,-347.33 47.93,-323.49\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"51.43,-323.52 48.68,-313.29 44.45,-323.01 51.43,-323.52\"/>\n",
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- "<g id=\"edge380\" class=\"edge\"><title>11->11</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"64.5146,-409.739 54.2408,-412.327 63.8863,-416.71 64.5146,-409.739\"/>\n",
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+ "<g id=\"edge190\" class=\"edge\">\n",
+ "<title>11->11</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M54.24,-425.81C64.02,-425.96 72,-423.71 72,-419.07 72,-416.24 69.04,-414.3 64.51,-413.25\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"64.51,-409.74 54.24,-412.33 63.89,-416.71 64.51,-409.74\"/>\n",
+ "<text text-anchor=\"middle\" x=\"86.5\" y=\"-415.37\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<g id=\"edge382\" class=\"edge\"><title>11->10</title>\n",
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+ "<g id=\"edge191\" class=\"edge\">\n",
+ "<title>11->10</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M24.96,-437.33C26.04,-457.77 31,-491.14 36.29,-514.91\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"32.92,-515.85 38.65,-524.76 39.72,-514.22 32.92,-515.85\"/>\n",
+ "<text text-anchor=\"middle\" x=\"16.12\" y=\"-479.92\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<g id=\"edge192\" class=\"edge\">\n",
+ "<title>11->9</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M28.99,-437.17C38.4,-478.22 70.68,-578.54 89.81,-627.31\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"86.58,-628.66 93.56,-636.62 93.07,-626.04 86.58,-628.66\"/>\n",
+ "<text text-anchor=\"middle\" x=\"66.65\" y=\"-521.04\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<g id=\"edge193\" class=\"edge\">\n",
+ "<title>11->8</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M32.78,-437.52C55.64,-492.1 137.14,-652.66 174.13,-716.73\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"171.18,-718.61 179.26,-725.46 177.21,-715.06 171.18,-718.61\"/>\n",
+ "<text text-anchor=\"middle\" x=\"88.96\" y=\"-580.93\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<g id=\"edge194\" class=\"edge\">\n",
+ "<title>11->7</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M36.52,-437.2C75.56,-499.06 222.38,-701.6 280.44,-774.3\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"278.06,-776.92 287.07,-782.5 283.51,-772.52 278.06,-776.92\"/>\n",
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+ "<g id=\"edge195\" class=\"edge\">\n",
+ "<title>11->6</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M41.19,-437.32C99,-502.06 316.69,-720.12 397.61,-794.84\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"395.71,-797.85 405.45,-802.02 400.44,-792.69 395.71,-797.85\"/>\n",
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- "<g id=\"edge392\" class=\"edge\"><title>11->5</title>\n",
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+ "<g id=\"edge196\" class=\"edge\">\n",
+ "<title>11->5</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M46.86,-437.08C125.43,-500.25 414.75,-710.71 516.43,-779.01\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"514.57,-781.97 524.83,-784.61 518.45,-776.15 514.57,-781.97\"/>\n",
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+ "<g id=\"edge197\" class=\"edge\">\n",
+ "<title>11->4</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M54.23,-436.33C155.71,-493.52 511.73,-675.02 627.3,-728.64\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"625.92,-731.86 636.47,-732.85 628.84,-725.5 625.92,-731.86\"/>\n",
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+ "<g id=\"edge198\" class=\"edge\">\n",
+ "<title>11->3</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M54.13,-430.84C165.5,-472.43 587.67,-609.72 715.59,-646.12\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"714.75,-649.52 725.32,-648.85 716.64,-642.78 714.75,-649.52\"/>\n",
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+ "<g id=\"edge199\" class=\"edge\">\n",
+ "<title>11->2</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M54.43,-426.16C172.36,-450.16 636.47,-523.8 772.1,-540.2\"/>\n",
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+ "<title>11->1</title>\n",
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+ "<title>10->20</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M73.72,-537.02C189.1,-505.07 639.71,-358.83 772.5,-310.48\"/>\n",
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+ "<g id=\"edge203\" class=\"edge\">\n",
+ "<title>10->18</title>\n",
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+ "<g id=\"edge204\" class=\"edge\">\n",
+ "<title>10->17</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M68.77,-524.9C152.5,-448.43 448.54,-152.48 530.75,-63.2\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"533.45,-65.45 537.6,-55.7 528.27,-60.73 533.45,-65.45\"/>\n",
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+ "<g id=\"edge205\" class=\"edge\">\n",
+ "<title>10->16</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M63.38,-524.71C126.83,-446.18 350.84,-138.02 412.41,-44.81\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"415.48,-46.52 418.01,-36.23 409.61,-42.69 415.48,-46.52\"/>\n",
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+ "<g id=\"edge206\" class=\"edge\">\n",
+ "<title>10->15</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M59.07,-524.78C103.59,-448.69 252.67,-156.38 293.76,-65.04\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"297.07,-66.2 297.91,-55.64 290.67,-63.37 297.07,-66.2\"/>\n",
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+ "<g id=\"edge207\" class=\"edge\">\n",
+ "<title>10->14</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M55.59,-524.93C83.28,-455.64 164.32,-206.74 186.83,-122.6\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"190.27,-123.3 189.39,-112.74 183.49,-121.54 190.27,-123.3\"/>\n",
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+ "<g id=\"edge208\" class=\"edge\">\n",
+ "<title>10->13</title>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"106.47,-211.89 103.89,-201.61 99.5,-211.25 106.47,-211.89\"/>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"54.6,-323.1 50.3704,-313.386 47.6192,-323.618 54.6,-323.1\"/>\n",
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+ "<g id=\"edge209\" class=\"edge\">\n",
+ "<title>10->12</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M50.37,-524.78C54.12,-481.99 54.37,-374.83 51.12,-323.46\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"54.6,-323.1 50.37,-313.39 47.62,-323.62 54.6,-323.1\"/>\n",
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+ "<g id=\"edge210\" class=\"edge\">\n",
+ "<title>10->11</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M48.67,-524.75C47.59,-504.31 42.63,-470.94 37.34,-447.17\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"40.71,-446.23 34.98,-437.31 33.91,-447.85 40.71,-446.23\"/>\n",
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+ "<g id=\"edge211\" class=\"edge\">\n",
+ "<title>10->10</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M73.87,-549.75C83.65,-549.9 91.63,-547.65 91.63,-543.01 91.63,-540.18 88.67,-538.24 84.14,-537.19\"/>\n",
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+ "<g id=\"edge212\" class=\"edge\">\n",
+ "<title>10->9</title>\n",
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+ "<g id=\"edge213\" class=\"edge\">\n",
+ "<title>10->8</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M55.17,-561.36C76.95,-597.21 134.03,-676.35 167.38,-717.56\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"164.86,-720 173.91,-725.51 170.27,-715.56 164.86,-720\"/>\n",
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+ "<g id=\"edge214\" class=\"edge\">\n",
+ "<title>10->7</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M59.91,-561.17C99.39,-606.37 216.05,-723.49 272.69,-775.32\"/>\n",
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+ "<g id=\"edge215\" class=\"edge\">\n",
+ "<title>10->6</title>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"390.63,-802.22 400.9,-804.81 394.49,-796.38 390.63,-802.22\"/>\n",
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+ "<g id=\"edge216\" class=\"edge\">\n",
+ "<title>10->5</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M73.73,-560.56C158.62,-608.81 419.24,-741.71 515.66,-786.07\"/>\n",
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+ "<title>10->4</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M73.82,-555.03C172.71,-592.25 513.36,-703.06 626.77,-735.06\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"626.14,-738.52 636.71,-737.82 628.01,-731.78 626.14,-738.52\"/>\n",
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+ "<g id=\"edge218\" class=\"edge\">\n",
+ "<title>10->3</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M73.86,-550.19C183.04,-572.63 589.22,-637.1 715.1,-652.15\"/>\n",
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+ "<title>10->2</title>\n",
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+ "<title>9->17</title>\n",
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+ "<title>9->16</title>\n",
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+ "<title>9->15</title>\n",
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+ "<title>9->14</title>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"194.52,-123.16 192.05,-112.86 187.56,-122.46 194.52,-123.16\"/>\n",
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+ "<g id=\"edge228\" class=\"edge\">\n",
+ "<title>9->13</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M106.41,-636.62C111.45,-564.57 111.65,-298.92 107.01,-211.45\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"110.51,-211.24 106.41,-201.47 103.52,-211.66 110.51,-211.24\"/>\n",
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+ "<g id=\"edge229\" class=\"edge\">\n",
+ "<title>9->12</title>\n",
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+ "<g id=\"edge230\" class=\"edge\">\n",
+ "<title>9->11</title>\n",
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+ "<g id=\"edge231\" class=\"edge\">\n",
+ "<title>9->10</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M99.8,-636.57C92.8,-618.41 78.95,-590.41 67.02,-569.75\"/>\n",
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+ "<g id=\"edge232\" class=\"edge\">\n",
+ "<title>9->9</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M130.84,-661.56C140.62,-661.7 148.6,-659.45 148.6,-654.81 148.6,-651.99 145.64,-650.05 141.11,-649\"/>\n",
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+ "<g id=\"edge233\" class=\"edge\">\n",
+ "<title>9->8</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M114.6,-673.12C125.6,-686.31 142.86,-704.16 158.36,-718.62\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"156.05,-721.26 165.79,-725.42 160.77,-716.09 156.05,-721.26\"/>\n",
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+ "<g id=\"edge234\" class=\"edge\">\n",
+ "<title>9->7</title>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"266.71,-783.99 277.01,-786.49 270.52,-778.11 266.71,-783.99\"/>\n",
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+ "<g id=\"edge235\" class=\"edge\">\n",
+ "<title>9->6</title>\n",
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+ "<g id=\"edge236\" class=\"edge\">\n",
+ "<title>9->5</title>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"514.4,-796.02 524.96,-795.27 516.23,-789.27 514.4,-796.02\"/>\n",
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+ "<g id=\"edge237\" class=\"edge\">\n",
+ "<title>9->4</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M130.89,-662.32C222.79,-681.6 521.16,-728.99 626.54,-741.24\"/>\n",
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+ "<g id=\"edge238\" class=\"edge\">\n",
+ "<title>9->3</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M130.66,-657.73C233.11,-662.69 597.39,-662.84 715.37,-658.17\"/>\n",
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+ "<g id=\"edge239\" class=\"edge\">\n",
+ "<title>9->2</title>\n",
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+ "<g id=\"edge240\" class=\"edge\">\n",
+ "<title>9->1</title>\n",
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+ "<title>8->20</title>\n",
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+ "<title>8->19</title>\n",
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+ "<title>8->18</title>\n",
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+ "<title>8->13</title>\n",
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+ "<title>8->12</title>\n",
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+ "<title>8->11</title>\n",
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+ "<title>8->10</title>\n",
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+ "<g id=\"edge252\" class=\"edge\">\n",
+ "<title>8->9</title>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"139.87,-677.1 130.13,-672.94 135.15,-682.27 139.87,-677.1\"/>\n",
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+ "<g id=\"edge253\" class=\"edge\">\n",
+ "<title>8->8</title>\n",
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+ "<g id=\"edge254\" class=\"edge\">\n",
+ "<title>8->7</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M215.08,-761.6C230.23,-770.59 250.42,-781.06 267.76,-788.99\"/>\n",
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+ "<title>8->6</title>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"390.5,-816.83 401.06,-815.99 392.27,-810.06 390.5,-816.83\"/>\n",
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+ "<title>8->5</title>\n",
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+ "<title>8->4</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M219.37,-746.9C299.27,-751.25 534.41,-751.42 626.36,-747.41\"/>\n",
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+ "<title>8->3</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M219.45,-742.38C311.2,-732.61 610.03,-685.45 715.37,-664.35\"/>\n",
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+ "<title>8->2</title>\n",
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+ "<title>8->1</title>\n",
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+ "<title>7->20</title>\n",
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+ "<title>7->16</title>\n",
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+ "<title>7->15</title>\n",
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+ "<g id=\"edge267\" class=\"edge\">\n",
+ "<title>7->14</title>\n",
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+ "<g id=\"edge268\" class=\"edge\">\n",
+ "<title>7->13</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M300.78,-782.28C278.12,-694.66 155.61,-316.93 115.81,-211.02\"/>\n",
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+ "<g id=\"edge269\" class=\"edge\">\n",
+ "<title>7->12</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M297.91,-782.5C265.16,-706.68 116.01,-413.43 64.22,-322.29\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"67.12,-320.31 59.1,-313.39 61.05,-323.81 67.12,-320.31\"/>\n",
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+ "<g id=\"edge270\" class=\"edge\">\n",
+ "<title>7->11</title>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"53.07,-442.66 44.07,-437.08 47.63,-447.06 53.07,-442.66\"/>\n",
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+ "<g id=\"edge271\" class=\"edge\">\n",
+ "<title>7->10</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M290.85,-782.35C251.37,-737.15 134.71,-620.03 78.07,-568.2\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"80.26,-565.46 70.5,-561.34 75.56,-570.65 80.26,-565.46\"/>\n",
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+ "<g id=\"edge272\" class=\"edge\">\n",
+ "<title>7->9</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M286.31,-782.28C253.32,-754.62 182.4,-702.74 139.23,-674.35\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"141.02,-671.34 130.72,-668.83 137.21,-677.21 141.02,-671.34\"/>\n",
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+ "<g id=\"edge273\" class=\"edge\">\n",
+ "<title>7->8</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M281.38,-782.45C266.23,-773.47 246.04,-763 228.7,-755.07\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"229.98,-751.8 219.42,-750.95 227.14,-758.2 229.98,-751.8\"/>\n",
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+ "<g id=\"edge274\" class=\"edge\">\n",
+ "<title>7->7</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M331.37,-807.25C341.16,-807.4 349.13,-805.15 349.13,-800.51 349.13,-797.68 346.17,-795.74 341.64,-794.69\"/>\n",
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+ "<g id=\"edge275\" class=\"edge\">\n",
+ "<title>7->6</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M331.36,-810.66C348.77,-814.57 371.6,-818.34 390.7,-820.51\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"390.57,-824.01 400.87,-821.54 391.28,-817.05 390.57,-824.01\"/>\n",
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+ "<g id=\"edge276\" class=\"edge\">\n",
+ "<title>7->5</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M331.61,-804.94C376.21,-807.83 464.42,-808.02 514.67,-805.5\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"515,-808.99 524.79,-804.93 514.61,-802 515,-808.99\"/>\n",
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+ "<g id=\"edge277\" class=\"edge\">\n",
+ "<title>7->4</title>\n",
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+ "<g id=\"edge278\" class=\"edge\">\n",
+ "<title>7->3</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M331.22,-795.26C408.23,-774.75 628.41,-703.42 715.99,-670.86\"/>\n",
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+ "<g id=\"edge279\" class=\"edge\">\n",
+ "<title>7->2</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M331.34,-790.23C416.41,-751.99 677.3,-619.27 773.45,-565.6\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"775.24,-568.62 782.23,-560.66 771.8,-562.52 775.24,-568.62\"/>\n",
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+ "<g id=\"edge280\" class=\"edge\">\n",
+ "<title>7->1</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M320.64,-782.2C391.4,-694.08 667.43,-314.31 736.76,-209.93\"/>\n",
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+ "<title>6->20</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M444.82,-801.84C508.27,-723.31 732.28,-415.15 793.85,-321.95\"/>\n",
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+ "<title>6->19</title>\n",
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+ "<title>6->18</title>\n",
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+ "<title>6->15</title>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"207.55,-120.97 200.74,-112.86 201.01,-123.45 207.55,-120.97\"/>\n",
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+ "<title>6->13</title>\n",
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+ "<title>6->12</title>\n",
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+ "<g id=\"edge290\" class=\"edge\">\n",
+ "<title>6->11</title>\n",
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+ "<g id=\"edge291\" class=\"edge\">\n",
+ "<title>6->10</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M408.7,-801.96C348.02,-752.53 161.38,-616.64 82.43,-564.04\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"84.07,-560.93 73.8,-558.34 80.21,-566.77 84.07,-560.93\"/>\n",
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+ "<g id=\"edge292\" class=\"edge\">\n",
+ "<title>6->9</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M401.39,-802.02C343.44,-768.55 206.11,-698.46 140.15,-668.54\"/>\n",
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+ "<g id=\"edge293\" class=\"edge\">\n",
+ "<title>6->8</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M401.05,-806.54C358.77,-789.92 276.94,-763.19 229.26,-750.3\"/>\n",
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+ "<g id=\"edge294\" class=\"edge\">\n",
+ "<title>6->7</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M400.85,-810C383.43,-806.08 360.61,-802.31 341.5,-800.14\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"341.63,-796.64 331.33,-799.11 340.93,-803.6 341.63,-796.64\"/>\n",
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+ "<g id=\"edge295\" class=\"edge\">\n",
+ "<title>6->6</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M455.31,-826.88C465.09,-827.03 473.07,-824.78 473.07,-820.14 473.07,-817.31 470.11,-815.37 465.58,-814.32\"/>\n",
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+ "<g id=\"edge296\" class=\"edge\">\n",
+ "<title>6->5</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M455.24,-821.54C472.77,-820.01 495.81,-816.58 515.01,-812.73\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"515.88,-816.12 524.94,-810.62 514.43,-809.27 515.88,-816.12\"/>\n",
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+ "<g id=\"edge297\" class=\"edge\">\n",
+ "<title>6->4</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M455.11,-815.98C497.44,-805.24 579.39,-778.87 627.04,-760.9\"/>\n",
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+ "<title>6->3</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M455.28,-810.59C513.86,-784.83 651.79,-714.8 717.1,-677.93\"/>\n",
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+ "<g id=\"edge299\" class=\"edge\">\n",
+ "<title>6->2</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M455.19,-804.84C525.33,-759.03 711.85,-623.62 782.2,-567.58\"/>\n",
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+ "<g id=\"edge300\" class=\"edge\">\n",
+ "<title>6->1</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M440.21,-801.88C492.08,-712.21 692.68,-318.77 741.79,-210.8\"/>\n",
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+ "<g id=\"edge301\" class=\"edge\">\n",
+ "<title>5->20</title>\n",
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+ "<title>5->19</title>\n",
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+ "<title>5->11</title>\n",
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+ "<title>5->10</title>\n",
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+ "<title>5->9</title>\n",
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+ "<title>5->8</title>\n",
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+ "<g id=\"edge314\" class=\"edge\">\n",
+ "<title>5->7</title>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"341.14,-792.03 331.35,-796.1 341.54,-799.02 341.14,-792.03\"/>\n",
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+ "<g id=\"edge315\" class=\"edge\">\n",
+ "<title>5->6</title>\n",
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+ "<g id=\"edge316\" class=\"edge\">\n",
+ "<title>5->5</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M579.25,-807.25C589.03,-807.4 597.01,-805.15 597.01,-800.51 597.01,-797.68 594.05,-795.74 589.52,-794.69\"/>\n",
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+ "<title>5->4</title>\n",
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+ "<title>5->1</title>\n",
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+ "<title>4->8</title>\n",
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+ "<title>4->6</title>\n",
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+ "<title>4->5</title>\n",
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+ "<title>4->4</title>\n",
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+ "<title>3->20</title>\n",
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+ "<title>3->13</title>\n",
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+ "<title>3->12</title>\n",
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+ "<title>3->11</title>\n",
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+ "<g id=\"edge351\" class=\"edge\">\n",
+ "<title>3->10</title>\n",
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+ "<g id=\"edge352\" class=\"edge\">\n",
+ "<title>3->9</title>\n",
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+ "<g id=\"edge353\" class=\"edge\">\n",
+ "<title>3->8</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M725.42,-655.98C633.67,-665.75 334.84,-712.91 229.5,-734.01\"/>\n",
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+ "<g id=\"edge354\" class=\"edge\">\n",
+ "<title>3->7</title>\n",
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+ "<g id=\"edge355\" class=\"edge\">\n",
+ "<title>3->6</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M725.34,-664.36C666.75,-690.13 528.83,-760.16 463.51,-797.02\"/>\n",
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+ "<g id=\"edge356\" class=\"edge\">\n",
+ "<title>3->5</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M725.24,-668.95C686.38,-693.78 615.79,-744.95 577.79,-775.73\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"575.25,-773.29 569.75,-782.34 579.7,-778.69 575.25,-773.29\"/>\n",
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+ "<g id=\"edge357\" class=\"edge\">\n",
+ "<title>3->4</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M725.94,-672.99C711.49,-685.92 694.08,-703.35 681.32,-717.7\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"678.64,-715.45 674.76,-725.31 683.94,-720.02 678.64,-715.45\"/>\n",
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+ "<g id=\"edge358\" class=\"edge\">\n",
+ "<title>3->3</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M779.78,-661.56C789.57,-661.7 797.54,-659.45 797.54,-654.81 797.54,-651.99 794.58,-650.05 790.05,-649\"/>\n",
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+ "<title>3->2</title>\n",
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+ "<g id=\"edge360\" class=\"edge\">\n",
+ "<title>3->1</title>\n",
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+ "<g id=\"edge361\" class=\"edge\">\n",
+ "<title>2->20</title>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"817.48,-323.1 813.25,-313.39 810.5,-323.62 817.48,-323.1\"/>\n",
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+ "<g id=\"edge362\" class=\"edge\">\n",
+ "<title>2->19</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M817.44,-524.97C822.74,-504.62 828.26,-471.27 830.44,-447.43\"/>\n",
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+ "<title>2->18</title>\n",
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"</g>\n",
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"</g>\n",
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+ "<title>1->11</title>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"62.45,-400.5 54.28,-407.24 64.87,-407.07 62.45,-400.5\"/>\n",
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"</g>\n",
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+ "<g id=\"edge391\" class=\"edge\">\n",
+ "<title>1->10</title>\n",
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"</g>\n",
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+ "<g id=\"edge392\" class=\"edge\">\n",
+ "<title>1->9</title>\n",
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"</g>\n",
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+ "<g id=\"edge393\" class=\"edge\">\n",
+ "<title>1->8</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M730.69,-201.35C641.05,-283.47 302.81,-621.61 213.83,-717.9\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"211.1,-715.7 206.93,-725.44 216.26,-720.43 211.1,-715.7\"/>\n",
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"</g>\n",
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+ "<g id=\"edge394\" class=\"edge\">\n",
+ "<title>1->7</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M736.04,-201.64C665.28,-289.76 389.24,-669.52 319.91,-773.91\"/>\n",
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"</g>\n",
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+ "<g id=\"edge395\" class=\"edge\">\n",
+ "<title>1->6</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M740.4,-201.59C688.53,-291.26 487.93,-684.7 438.82,-792.67\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"435.57,-791.37 434.67,-801.93 441.96,-794.23 435.57,-791.37\"/>\n",
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"</g>\n",
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+ "<g id=\"edge396\" class=\"edge\">\n",
+ "<title>1->5</title>\n",
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"</g>\n",
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+ "<g id=\"edge397\" class=\"edge\">\n",
+ "<title>1->4</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M746.99,-201.56C728.69,-283.16 675.91,-615.28 665.13,-715.06\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"661.62,-714.98 664.1,-725.28 668.58,-715.68 661.62,-714.98\"/>\n",
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"</g>\n",
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+ "<g id=\"edge398\" class=\"edge\">\n",
+ "<title>1->3</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M749.73,-201.52C744.69,-273.57 744.49,-539.22 749.13,-626.69\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"745.63,-626.9 749.73,-636.67 752.62,-626.48 745.63,-626.9\"/>\n",
+ "<text text-anchor=\"middle\" x=\"734.93\" y=\"-402.9\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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- "<text text-anchor=\"middle\" x=\"762.663\" y=\"-355.833\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<g id=\"edge399\" class=\"edge\">\n",
+ "<title>1->2</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M752.24,-201.41C756.87,-259.81 785.74,-443.67 801.09,-514.66\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"797.73,-515.68 803.34,-524.67 804.56,-514.14 797.73,-515.68\"/>\n",
+ "<text text-anchor=\"middle\" x=\"762.16\" y=\"-346.83\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 1->1 -->\n",
- "<g id=\"edge800\" class=\"edge\"><title>1->1</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M779.784,-190.071C789.567,-190.216 797.543,-187.968 797.543,-183.327 797.543,-180.499 794.581,-178.56 790.054,-177.509\"/>\n",
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+ "<g id=\"edge400\" class=\"edge\">\n",
+ "<title>1->1</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M779.78,-190.07C789.57,-190.22 797.54,-187.97 797.54,-183.33 797.54,-180.5 794.58,-178.56 790.05,-177.51\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"790.06,-174 779.78,-176.58 789.43,-180.97 790.06,-174\"/>\n",
+ "<text text-anchor=\"middle\" x=\"812.04\" y=\"-179.63\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"</g>\n",
"</svg>"
@@ -5911,7 +6516,7 @@
"<Dot visualization: expr_as_graph [(\"K20\",(1..20)*(1..20))]>"
]
},
- "execution_count": 96,
+ "execution_count": 76,
"metadata": {},
"output_type": "execute_result"
}
@@ -5922,7 +6527,7 @@
},
{
"cell_type": "code",
- "execution_count": 97,
+ "execution_count": 77,
"metadata": {},
"outputs": [
{
@@ -5931,7 +6536,7 @@
"Preference changed: DOT_ENGINE = dot\n"
]
},
- "execution_count": 97,
+ "execution_count": 77,
"metadata": {},
"output_type": "execute_result"
}
@@ -5956,7 +6561,7 @@
},
{
"cell_type": "code",
- "execution_count": 98,
+ "execution_count": 78,
"metadata": {},
"outputs": [
{
@@ -5968,7 +6573,7 @@
"{(2↦1),(4↦2),(6↦3),(8↦4),(10↦5)}"
]
},
- "execution_count": 98,
+ "execution_count": 78,
"metadata": {},
"output_type": "execute_result"
}
@@ -5979,7 +6584,7 @@
},
{
"cell_type": "code",
- "execution_count": 99,
+ "execution_count": 79,
"metadata": {},
"outputs": [
{
@@ -5991,7 +6596,7 @@
"{2,4,6,8,10}"
]
},
- "execution_count": 99,
+ "execution_count": 79,
"metadata": {},
"output_type": "execute_result"
}
@@ -6002,7 +6607,7 @@
},
{
"cell_type": "code",
- "execution_count": 100,
+ "execution_count": 80,
"metadata": {},
"outputs": [
{
@@ -6014,7 +6619,7 @@
"{1,2,3,4,5}"
]
},
- "execution_count": 100,
+ "execution_count": 80,
"metadata": {},
"output_type": "execute_result"
}
@@ -6025,7 +6630,7 @@
},
{
"cell_type": "code",
- "execution_count": 103,
+ "execution_count": 81,
"metadata": {},
"outputs": [
{
@@ -6037,7 +6642,7 @@
"{2,4,6,8,10}"
]
},
- "execution_count": 103,
+ "execution_count": 81,
"metadata": {},
"output_type": "execute_result"
}
@@ -6048,16 +6653,30 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 82,
"metadata": {},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{(2\\mapsto 1),(4\\mapsto 2),(6\\mapsto 3),(8\\mapsto 4),(10\\mapsto 5)\\}$"
+ ],
+ "text/plain": [
+ "{(2↦1),(4↦2),(6↦3),(8↦4),(10↦5)}"
+ ]
+ },
+ "execution_count": 82,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
":let h {a,b| a∈1..10 ∧ b∈1..10 & b*2=a}"
]
},
{
"cell_type": "code",
- "execution_count": 111,
+ "execution_count": 83,
"metadata": {},
"outputs": [
{
@@ -6066,92 +6685,105 @@
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"<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
" \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
- "<!-- Generated by graphviz version 2.28.0 (20110509.1545)\n",
+ "<!-- Generated by graphviz version 2.44.1 (0)\n",
" -->\n",
"<!-- Title: state Pages: 1 -->\n",
- "<svg width=\"206pt\" height=\"314pt\"\n",
- " viewBox=\"0.00 0.00 206.00 314.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
- "<g id=\"graph1\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 310)\">\n",
+ "<svg width=\"206pt\" height=\"305pt\"\n",
+ " viewBox=\"0.00 0.00 206.00 305.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
+ "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 301)\">\n",
"<title>state</title>\n",
- "<polygon fill=\"white\" stroke=\"white\" points=\"-4,5 -4,-310 203,-310 203,5 -4,5\"/>\n",
+ "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-301 202,-301 202,4 -4,4\"/>\n",
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- "<text text-anchor=\"middle\" x=\"27\" y=\"-283.8\" font-family=\"Times,serif\" font-size=\"14.00\">10</text>\n",
+ "<g id=\"node1\" class=\"node\">\n",
+ "<title>10</title>\n",
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+ "<text text-anchor=\"middle\" x=\"27\" y=\"-275.3\" font-family=\"Times,serif\" font-size=\"14.00\">10</text>\n",
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+ "<title>5</title>\n",
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+ "<text text-anchor=\"middle\" x=\"27\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">5</text>\n",
"</g>\n",
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- "<g id=\"edge2\" class=\"edge\"><title>10->5</title>\n",
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+ "<g id=\"edge1\" class=\"edge\">\n",
+ "<title>10->5</title>\n",
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"text/plain": [
- "<Dot visualization: expr_as_graph [dhd={(1,\"nein\"),(2,\"nein\"),(3,\"ja\"),(4,\"nein\"),(5,\"nein\"),(6,\"ja\"),(7,\"nein\")} & h={(2,1),(4,2),(6,3),(8,4),(10,5)}(\"h\",h)]>"
+ "<Dot visualization: expr_as_graph [hh={(2,1),(4,2),(6,3),(8,4),(10,5)}(\"h\",h)]>"
]
},
- "execution_count": 111,
+ "execution_count": 83,
"metadata": {},
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}
@@ -6162,7 +6794,7 @@
},
{
"cell_type": "code",
- "execution_count": 104,
+ "execution_count": 84,
"metadata": {},
"outputs": [
{
@@ -6174,7 +6806,7 @@
"{(1↦\"nein\"),(2↦\"nein\"),(3↦\"ja\"),(4↦\"nein\"),(5↦\"nein\"),(6↦\"ja\"),(7↦\"nein\")}"
]
},
- "execution_count": 104,
+ "execution_count": 84,
"metadata": {},
"output_type": "execute_result"
}
@@ -6185,7 +6817,7 @@
},
{
"cell_type": "code",
- "execution_count": 105,
+ "execution_count": 85,
"metadata": {},
"outputs": [
{
@@ -6197,7 +6829,7 @@
"{1,2,3,4,5,6,7}"
]
},
- "execution_count": 105,
+ "execution_count": 85,
"metadata": {},
"output_type": "execute_result"
}
@@ -6208,7 +6840,7 @@
},
{
"cell_type": "code",
- "execution_count": 106,
+ "execution_count": 86,
"metadata": {},
"outputs": [
{
@@ -6220,7 +6852,7 @@
"{\"ja\",\"nein\"}"
]
},
- "execution_count": 106,
+ "execution_count": 86,
"metadata": {},
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}
@@ -6231,7 +6863,7 @@
},
{
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- "execution_count": 107,
+ "execution_count": 87,
"metadata": {},
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"210.33,-38.15 199.94,-36.08 206.77,-44.18 210.33,-38.15\"/>\n",
+ "<text text-anchor=\"middle\" x=\"257\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">d</text>\n",
"</g>\n",
"</g>\n",
"</svg>"
@@ -6342,7 +6990,7 @@
"<Dot visualization: expr_as_graph [dhd={(1,\"nein\"),(2,\"nein\"),(3,\"ja\"),(4,\"nein\"),(5,\"nein\"),(6,\"ja\"),(7,\"nein\")} & h={(2,1),(4,2),(6,3),(8,4),(10,5)}(\"d\",d)]>"
]
},
- "execution_count": 107,
+ "execution_count": 87,
"metadata": {},
"output_type": "execute_result"
}
@@ -6368,7 +7016,7 @@
},
{
"cell_type": "code",
- "execution_count": 112,
+ "execution_count": 88,
"metadata": {},
"outputs": [
{
@@ -6380,7 +7028,7 @@
"{4,5}"
]
},
- "execution_count": 112,
+ "execution_count": 88,
"metadata": {},
"output_type": "execute_result"
}
@@ -6391,7 +7039,7 @@
},
{
"cell_type": "code",
- "execution_count": 110,
+ "execution_count": 89,
"metadata": {},
"outputs": [
{
@@ -6403,7 +7051,7 @@
"{\"ja\"}"
]
},
- "execution_count": 110,
+ "execution_count": 89,
"metadata": {},
"output_type": "execute_result"
}
@@ -6414,16 +7062,30 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 90,
"metadata": {},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\emptyset$"
+ ],
+ "text/plain": [
+ "∅"
+ ]
+ },
+ "execution_count": 90,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"d[{0}]"
]
},
{
"cell_type": "code",
- "execution_count": 113,
+ "execution_count": 91,
"metadata": {},
"outputs": [
{
@@ -6435,7 +7097,7 @@
"{(\"ja\"↦3),(\"ja\"↦6),(\"nein\"↦1),(\"nein\"↦2),(\"nein\"↦4),(\"nein\"↦5),(\"nein\"↦7)}"
]
},
- "execution_count": 113,
+ "execution_count": 91,
"metadata": {},
"output_type": "execute_result"
}
@@ -6446,7 +7108,7 @@
},
{
"cell_type": "code",
- "execution_count": 114,
+ "execution_count": 92,
"metadata": {},
"outputs": [
{
@@ -6455,100 +7117,116 @@
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- "<!-- Generated by graphviz version 2.28.0 (20110509.1545)\n",
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+ "<g id=\"edge7\" class=\"edge\">\n",
+ "<title>"ja"->3</title>\n",
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@@ -6557,7 +7235,7 @@
"<Dot visualization: expr_as_graph [dhd={(1,\"nein\"),(2,\"nein\"),(3,\"ja\"),(4,\"nein\"),(5,\"nein\"),(6,\"ja\"),(7,\"nein\")} & h={(2,1),(4,2),(6,3),(8,4),(10,5)}(\"d~\",d~)]>"
]
},
- "execution_count": 114,
+ "execution_count": 92,
"metadata": {},
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}
@@ -6575,7 +7253,7 @@
},
{
"cell_type": "code",
- "execution_count": 118,
+ "execution_count": 93,
"metadata": {},
"outputs": [
{
@@ -6587,7 +7265,7 @@
"{3,6}"
]
},
- "execution_count": 118,
+ "execution_count": 93,
"metadata": {},
"output_type": "execute_result"
}
@@ -6598,7 +7276,7 @@
},
{
"cell_type": "code",
- "execution_count": 119,
+ "execution_count": 94,
"metadata": {},
"outputs": [],
"source": [
@@ -6618,7 +7296,7 @@
},
{
"cell_type": "code",
- "execution_count": 121,
+ "execution_count": 95,
"metadata": {},
"outputs": [
{
@@ -6630,7 +7308,7 @@
"{(1↦4),(1↦8)}"
]
},
- "execution_count": 121,
+ "execution_count": 95,
"metadata": {},
"output_type": "execute_result"
}
@@ -6650,7 +7328,7 @@
},
{
"cell_type": "code",
- "execution_count": 123,
+ "execution_count": 96,
"metadata": {},
"outputs": [
{
@@ -6662,7 +7340,7 @@
"{(4↦1),(8↦2)}"
]
},
- "execution_count": 123,
+ "execution_count": 96,
"metadata": {},
"output_type": "execute_result"
}
@@ -6673,7 +7351,7 @@
},
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- "execution_count": 124,
+ "execution_count": 97,
"metadata": {},
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]
},
- "execution_count": 124,
+ "execution_count": 97,
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}
@@ -6790,18 +7483,191 @@
},
{
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+ "execution_count": 98,
"metadata": {},
- "outputs": [],
+ "outputs": [
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+ "execution_count": 98,
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]
},
{
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+ "execution_count": 99,
"metadata": {},
- "outputs": [],
+ "outputs": [
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+ "</g>\n",
+ "</g>\n",
+ "</svg>"
+ ],
+ "text/plain": [
+ "<Dot visualization: expr_as_graph [hh={(2,1),(4,2),(6,3),(8,4),(10,5)}(\"h\",h)]>"
+ ]
+ },
+ "execution_count": 99,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
":dot expr_as_graph (\"h\",(h))"
]
@@ -6821,7 +7687,7 @@
},
{
"cell_type": "code",
- "execution_count": 125,
+ "execution_count": 100,
"metadata": {},
"outputs": [
{
@@ -6833,7 +7699,7 @@
"{(4↦1),(8↦2)}"
]
},
- "execution_count": 125,
+ "execution_count": 100,
"metadata": {},
"output_type": "execute_result"
}
@@ -6844,7 +7710,7 @@
},
{
"cell_type": "code",
- "execution_count": 126,
+ "execution_count": 101,
"metadata": {},
"outputs": [
{
@@ -6856,7 +7722,7 @@
"{(8↦1)}"
]
},
- "execution_count": 126,
+ "execution_count": 101,
"metadata": {},
"output_type": "execute_result"
}
@@ -6867,7 +7733,7 @@
},
{
"cell_type": "code",
- "execution_count": 127,
+ "execution_count": 102,
"metadata": {},
"outputs": [
{
@@ -6879,7 +7745,7 @@
"∅"
]
},
- "execution_count": 127,
+ "execution_count": 102,
"metadata": {},
"output_type": "execute_result"
}
@@ -6897,7 +7763,7 @@
},
{
"cell_type": "code",
- "execution_count": 128,
+ "execution_count": 103,
"metadata": {},
"outputs": [
{
@@ -6909,7 +7775,7 @@
"{(2↦1),(4↦1),(4↦2),(6↦3),(8↦1),(8↦2),(8↦4),(10↦5)}"
]
},
- "execution_count": 128,
+ "execution_count": 103,
"metadata": {},
"output_type": "execute_result"
}
@@ -6920,7 +7786,7 @@
},
{
"cell_type": "code",
- "execution_count": 129,
+ "execution_count": 104,
"metadata": {},
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@@ -7032,7 +7914,7 @@
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]
},
- "execution_count": 129,
+ "execution_count": 104,
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}
@@ -7043,7 +7925,7 @@
},
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+ "execution_count": 105,
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"source": [
@@ -7059,7 +7941,7 @@
},
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- "execution_count": 131,
+ "execution_count": 106,
"metadata": {},
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{
@@ -7071,7 +7953,7 @@
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]
},
- "execution_count": 131,
+ "execution_count": 106,
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@@ -7082,7 +7964,7 @@
},
{
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- "execution_count": 132,
+ "execution_count": 107,
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{
@@ -7091,7 +7973,7 @@
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]
},
- "execution_count": 132,
+ "execution_count": 107,
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}
@@ -7102,7 +7984,7 @@
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+ "execution_count": 108,
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},
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}
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},
{
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+ "execution_count": 109,
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+ "<g id=\"edge4\" class=\"edge\">\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"59.92,-133.68 61.31,-123.18 53.67,-130.52 59.92,-133.68\"/>\n",
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"</g>\n",
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+ "<g id=\"edge6\" class=\"edge\">\n",
+ "<title>1->1</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M54.24,-199.87C64.02,-200.04 72,-197.41 72,-192 72,-188.7 69.04,-186.44 64.51,-185.21\"/>\n",
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@@ -7717,7 +8660,7 @@
"<Dot visualization: expr_as_graph [Arsub1A={1,2,3} & r={(1,2),(2,3)} & sub1={({},{1}),({},{2}),({},{3}),({1},{1,2}),({1},{1,3}),({1,2},{1,2,3}),({1,3},{1,2,3}),({2},{1,2}),({2},{2,3}),({2,3},{1,2,3}),({3},{1,3}),({3},{2,3})}(\"r*\",id(A)\\/r\\/(r;r)\\/((r;r);r)\\/(((r;r);r);r))]>"
]
},
- "execution_count": 141,
+ "execution_count": 116,
"metadata": {},
"output_type": "execute_result"
}
@@ -7735,18 +8678,97 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 117,
"metadata": {},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{(1\\mapsto 2),(1\\mapsto 3),(2\\mapsto 3)\\}$"
+ ],
+ "text/plain": [
+ "{(1↦2),(1↦3),(2↦3)}"
+ ]
+ },
+ "execution_count": 117,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"closure1(r)"
]
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 118,
"metadata": {},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "image/svg+xml": [
+ "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
+ "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
+ " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
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+ " viewBox=\"0.00 0.00 97.00 218.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
+ "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 214)\">\n",
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+ "<g id=\"node1\" class=\"node\">\n",
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+ "<text text-anchor=\"middle\" x=\"59\" y=\"-14.3\" font-family=\"Times,serif\" font-size=\"14.00\">3</text>\n",
+ "</g>\n",
+ "<!-- 2->3 -->\n",
+ "<g id=\"edge1\" class=\"edge\">\n",
+ "<title>2->3</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M33.48,-86.8C37.9,-75.05 43.85,-59.24 48.89,-45.84\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"52.29,-46.77 52.53,-36.18 45.74,-44.3 52.29,-46.77\"/>\n",
+ "<text text-anchor=\"middle\" x=\"53.5\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">r+</text>\n",
+ "</g>\n",
+ "<!-- 1 -->\n",
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+ "<title>1</title>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"86,-210 32,-210 32,-174 86,-174 86,-210\"/>\n",
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+ "</g>\n",
+ "<!-- 1->2 -->\n",
+ "<g id=\"edge3\" class=\"edge\">\n",
+ "<title>1->2</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M52.52,-173.8C48.1,-162.05 42.15,-146.24 37.11,-132.84\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"40.26,-131.3 33.47,-123.18 33.71,-133.77 40.26,-131.3\"/>\n",
+ "<text text-anchor=\"middle\" x=\"53.5\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">r+</text>\n",
+ "</g>\n",
+ "<!-- 1->3 -->\n",
+ "<g id=\"edge2\" class=\"edge\">\n",
+ "<title>1->3</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M63.77,-173.69C65.09,-168.09 66.34,-161.82 67,-156 72.08,-110.95 72.08,-99.05 67,-54 66.71,-51.45 66.31,-48.82 65.84,-46.2\"/>\n",
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+ "</g>\n",
+ "</g>\n",
+ "</svg>"
+ ],
+ "text/plain": [
+ "<Dot visualization: expr_as_graph [Arsub1A={1,2,3} & r={(1,2),(2,3)} & sub1={({},{1}),({},{2}),({},{3}),({1},{1,2}),({1},{1,3}),({1,2},{1,2,3}),({1,3},{1,2,3}),({2},{1,2}),({2},{2,3}),({2,3},{1,2,3}),({3},{1,3}),({3},{2,3})}(\"r+\",closure1(r))]>"
+ ]
+ },
+ "execution_count": 118,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
":dot expr_as_graph (\"r+\",closure1(r))"
]
@@ -7760,7 +8782,7 @@
},
{
"cell_type": "code",
- "execution_count": 142,
+ "execution_count": 119,
"metadata": {},
"outputs": [
{
@@ -7772,7 +8794,7 @@
"{1,2}"
]
},
- "execution_count": 142,
+ "execution_count": 119,
"metadata": {},
"output_type": "execute_result"
}
@@ -7790,16 +8812,30 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 120,
"metadata": {},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$/*@symbolic*/ \\{\\mathit{z\\_},\\mathit{z\\_\\_}\\mid \\mathit{z\\_} \\mapsto \\mathit{z\\_\\_} \\in \\{(1\\mapsto 2),(1\\mapsto 3),(2\\mapsto 3)\\} \\lor \\mathit{z\\_} = \\mathit{z\\_\\_}\\}$"
+ ],
+ "text/plain": [
+ "/*@symbolic*/ {z_,z__∣z_ ↦ z__ ∈ {(1↦2),(1↦3),(2↦3)} ∨ z_ = z__}"
+ ]
+ },
+ "execution_count": 120,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"closure(r)"
]
},
{
"cell_type": "code",
- "execution_count": 143,
+ "execution_count": 121,
"metadata": {},
"outputs": [
{
@@ -7811,7 +8847,7 @@
"{1,2,3}"
]
},
- "execution_count": 143,
+ "execution_count": 121,
"metadata": {},
"output_type": "execute_result"
}
@@ -7822,7 +8858,7 @@
},
{
"cell_type": "code",
- "execution_count": 144,
+ "execution_count": 122,
"metadata": {},
"outputs": [],
"source": [
@@ -7846,7 +8882,7 @@
},
{
"cell_type": "code",
- "execution_count": 145,
+ "execution_count": 123,
"metadata": {},
"outputs": [
{
@@ -7858,7 +8894,7 @@
"{(1↦2),(2↦3),(3↦4),(4↦5),(5↦6),(6↦7),(7↦8),(8↦9),(9↦10),(10↦11)}"
]
},
- "execution_count": 145,
+ "execution_count": 123,
"metadata": {},
"output_type": "execute_result"
}
@@ -7871,12 +8907,12 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "Eine Funktion kann also als Mengen von Paaren dargestellt werden. Es sind aber nicht alle Mengen an Paaren auch Funktionen. Zum Beispiel, unsere Relation sub1 von oben ist keine Funktion:"
+ "Eine Funktion kann also als Menge von Paaren dargestellt werden. Es sind aber nicht alle Mengen an Paaren auch Funktionen. Zum Beispiel, unsere Relation sub1 von oben ist keine Funktion:"
]
},
{
"cell_type": "code",
- "execution_count": 146,
+ "execution_count": 124,
"metadata": {},
"outputs": [
{
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- "<!-- Generated by graphviz version 2.28.0 (20110509.1545)\n",
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- "<g id=\"graph1\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 310)\">\n",
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"</g>\n",
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+ "<title>\\{2,3\\}</title>\n",
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+ "<g id=\"edge1\" class=\"edge\">\n",
+ "<title>\\{3\\}->\\{2,3\\}</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M58.53,-173.62C58.47,-163.17 60.11,-150.16 67.17,-141 78.24,-126.64 88.33,-130 108.99,-123\"/>\n",
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+ "<g id=\"edge12\" class=\"edge\">\n",
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@@ -8012,7 +9068,7 @@
"<Dot visualization: expr_as_graph [Asub1A={1,2,3} & sub1={({},{1}),({},{2}),({},{3}),({1},{1,2}),({1},{1,3}),({1,2},{1,2,3}),({1,3},{1,2,3}),({2},{1,2}),({2},{2,3}),({2,3},{1,2,3}),({3},{1,3}),({3},{2,3})}(\"sub1\",sub1)]>"
]
},
- "execution_count": 146,
+ "execution_count": 124,
"metadata": {},
"output_type": "execute_result"
}
@@ -8030,9 +9086,23 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 125,
"metadata": {},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\{(\\emptyset\\mapsto\\{1\\}),(\\emptyset\\mapsto\\{2\\}),(\\emptyset\\mapsto\\{3\\}),(\\{1\\}\\mapsto\\{1,2\\}),(\\{1\\}\\mapsto\\{1,3\\}),(\\{1,2\\}\\mapsto\\{1,2,3\\}),(\\{1,3\\}\\mapsto\\{1,2,3\\}),(\\{2\\}\\mapsto\\{1,2\\}),(\\{2\\}\\mapsto\\{2,3\\}),(\\{2,3\\}\\mapsto\\{1,2,3\\}),(\\{3\\}\\mapsto\\{1,3\\}),(\\{3\\}\\mapsto\\{2,3\\})\\}$"
+ ],
+ "text/plain": [
+ "{(∅↦{1}),(∅↦{2}),(∅↦{3}),({1}↦{1,2}),({1}↦{1,3}),({1,2}↦{1,2,3}),({1,3}↦{1,2,3}),({2}↦{1,2}),({2}↦{2,3}),({2,3}↦{1,2,3}),({3}↦{1,3}),({3}↦{2,3})}"
+ ]
+ },
+ "execution_count": 125,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
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]
@@ -8051,7 +9121,7 @@
},
{
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+ "execution_count": 126,
"metadata": {},
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@@ -8063,7 +9133,7 @@
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},
- "execution_count": 147,
+ "execution_count": 126,
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@@ -8074,7 +9144,7 @@
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+ "execution_count": 127,
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@@ -8086,7 +9156,7 @@
"FALSE"
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},
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@@ -8106,7 +9176,7 @@
},
{
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- "execution_count": 150,
+ "execution_count": 129,
"metadata": {},
"outputs": [
{
@@ -8118,7 +9188,7 @@
"{(1↦2),(2↦3),(3↦4),(4↦5),(5↦6),(6↦7),(7↦8),(8↦9),(9↦10),(10↦11)}"
]
},
- "execution_count": 150,
+ "execution_count": 129,
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@@ -8129,7 +9199,7 @@
},
{
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- "execution_count": 151,
+ "execution_count": 130,
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{
@@ -8149,7 +9219,7 @@
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]
},
- "execution_count": 151,
+ "execution_count": 130,
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@@ -8160,7 +9230,7 @@
},
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+ "execution_count": 131,
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{
@@ -8180,7 +9250,7 @@
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]
},
- "execution_count": 152,
+ "execution_count": 131,
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@@ -8198,7 +9268,7 @@
},
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- "execution_count": 153,
+ "execution_count": 132,
"metadata": {},
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{
@@ -8218,7 +9288,7 @@
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]
},
- "execution_count": 153,
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@@ -8236,7 +9306,7 @@
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+ "execution_count": 133,
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@@ -8248,7 +9318,7 @@
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+ "execution_count": 133,
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@@ -8259,7 +9329,7 @@
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- "execution_count": 155,
+ "execution_count": 134,
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@@ -8271,7 +9341,7 @@
"FALSE"
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+ "execution_count": 134,
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@@ -8282,7 +9352,7 @@
},
{
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- "execution_count": 156,
+ "execution_count": 135,
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{
@@ -8294,7 +9364,7 @@
"{1,2,3,4,5,6,7,8,9,10}"
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},
- "execution_count": 156,
+ "execution_count": 135,
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@@ -8312,7 +9382,7 @@
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+ "execution_count": 136,
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- "execution_count": 158,
+ "execution_count": 137,
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{
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"{1,2,3,4,5,6,7,8,9,10}"
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},
- "execution_count": 158,
+ "execution_count": 137,
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@@ -8368,7 +9438,7 @@
},
{
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- "execution_count": 159,
+ "execution_count": 138,
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{
@@ -8380,7 +9450,7 @@
"{2,3,4,5,6,7,8,9,10,11}"
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},
- "execution_count": 159,
+ "execution_count": 138,
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@@ -8391,7 +9461,7 @@
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+ "execution_count": 139,
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"{5,6}"
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},
- "execution_count": 160,
+ "execution_count": 139,
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@@ -8414,7 +9484,7 @@
},
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"{5}"
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},
- "execution_count": 167,
+ "execution_count": 140,
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@@ -8437,7 +9507,7 @@
},
{
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+ "execution_count": 141,
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"{3}"
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},
- "execution_count": 162,
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@@ -8460,7 +9530,7 @@
},
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+ "execution_count": 142,
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{
@@ -8472,7 +9542,7 @@
"{6}"
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},
- "execution_count": 163,
+ "execution_count": 142,
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}
@@ -8483,7 +9553,7 @@
},
{
"cell_type": "code",
- "execution_count": 164,
+ "execution_count": 143,
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{
@@ -8495,7 +9565,7 @@
"{5,6,7,8,9,10,11}"
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},
- "execution_count": 164,
+ "execution_count": 143,
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@@ -8506,7 +9576,7 @@
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+ "execution_count": 144,
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"{1,2,3}"
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@@ -8536,7 +9606,7 @@
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@@ -8559,7 +9629,7 @@
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@@ -8589,7 +9659,7 @@
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+ "execution_count": 147,
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"outputs": [
{
@@ -8601,7 +9671,7 @@
"∅"
]
},
- "execution_count": 172,
+ "execution_count": 147,
"metadata": {},
"output_type": "execute_result"
}
@@ -8612,7 +9682,7 @@
},
{
"cell_type": "code",
- "execution_count": 173,
+ "execution_count": 148,
"metadata": {},
"outputs": [
{
@@ -8624,7 +9694,7 @@
"{(1↦2),(2↦3),(3↦4),(4↦5),(5↦6),(6↦7),(7↦8),(8↦9),(9↦10),(10↦11)}"
]
},
- "execution_count": 173,
+ "execution_count": 148,
"metadata": {},
"output_type": "execute_result"
}
@@ -8635,17 +9705,17 @@
},
{
"cell_type": "code",
- "execution_count": 174,
+ "execution_count": 149,
"metadata": {},
"outputs": [
{
"ename": "CommandExecutionException",
- "evalue": ":eval: NOT-WELL-DEFINED: \nfunction applied outside of domain (#7): Function argument: 100, function value: {}\n ### Line: 1, Column: 99 until 107\n\n",
+ "evalue": ":eval: NOT-WELL-DEFINED: \nfunction applied outside of domain (#7): Function argument: 100, function value: {}\n ### Line: 2, Column: 0 until 8\n\n",
"output_type": "error",
"traceback": [
"\u001b[1m\u001b[31m:eval: NOT-WELL-DEFINED: \u001b[0m",
"\u001b[1m\u001b[31mfunction applied outside of domain (#7): Function argument: 100, function value: {}\u001b[0m",
- "\u001b[1m\u001b[31m ### Line: 1, Column: 99 until 107\u001b[0m"
+ "\u001b[1m\u001b[31m ### Line: 2, Column: 0 until 8\u001b[0m"
]
}
],
@@ -8662,7 +9732,7 @@
},
{
"cell_type": "code",
- "execution_count": 175,
+ "execution_count": 150,
"metadata": {},
"outputs": [
{
@@ -8674,7 +9744,7 @@
"{{(1↦1),(2↦1)},{(1↦1),(2↦2)},{(1↦2),(2↦1)},{(1↦2),(2↦2)}}"
]
},
- "execution_count": 175,
+ "execution_count": 150,
"metadata": {},
"output_type": "execute_result"
}
@@ -8698,7 +9768,7 @@
},
{
"cell_type": "code",
- "execution_count": 176,
+ "execution_count": 151,
"metadata": {},
"outputs": [
{
@@ -8710,7 +9780,7 @@
"{∅,{(1↦1)},{(1↦1),(2↦1)},{(1↦1),(2↦2)},{(1↦2)},{(1↦2),(2↦1)},{(1↦2),(2↦2)},{(2↦1)},{(2↦2)}}"
]
},
- "execution_count": 176,
+ "execution_count": 151,
"metadata": {},
"output_type": "execute_result"
}
@@ -8721,7 +9791,7 @@
},
{
"cell_type": "code",
- "execution_count": 177,
+ "execution_count": 152,
"metadata": {},
"outputs": [
{
@@ -8741,7 +9811,7 @@
"\ti10 = {(1↦2),(2↦3),(3↦4),(4↦5),(5↦6),(6↦7),(7↦8),(8↦9),(9↦10),(10↦11)}"
]
},
- "execution_count": 177,
+ "execution_count": 152,
"metadata": {},
"output_type": "execute_result"
}
@@ -8759,7 +9829,7 @@
},
{
"cell_type": "code",
- "execution_count": 178,
+ "execution_count": 153,
"metadata": {},
"outputs": [
{
@@ -8771,7 +9841,7 @@
"{∅,{(1↦1)},{(1↦2)},{(2↦1)},{(2↦2)}}"
]
},
- "execution_count": 178,
+ "execution_count": 153,
"metadata": {},
"output_type": "execute_result"
}
@@ -8789,7 +9859,7 @@
},
{
"cell_type": "code",
- "execution_count": 179,
+ "execution_count": 154,
"metadata": {},
"outputs": [
{
@@ -8801,7 +9871,7 @@
"{{(1↦1),(2↦2)},{(1↦2),(2↦1)}}"
]
},
- "execution_count": 179,
+ "execution_count": 154,
"metadata": {},
"output_type": "execute_result"
}
@@ -8819,7 +9889,7 @@
},
{
"cell_type": "code",
- "execution_count": 180,
+ "execution_count": 155,
"metadata": {},
"outputs": [
{
@@ -8831,7 +9901,7 @@
"{{(1↦1),(2↦2)},{(1↦2),(2↦1)}}"
]
},
- "execution_count": 180,
+ "execution_count": 155,
"metadata": {},
"output_type": "execute_result"
}
@@ -8842,7 +9912,7 @@
},
{
"cell_type": "code",
- "execution_count": 181,
+ "execution_count": 156,
"metadata": {},
"outputs": [
{
@@ -8854,7 +9924,7 @@
"{{(1↦1),(2↦2)},{(1↦2),(2↦1)}}"
]
},
- "execution_count": 181,
+ "execution_count": 156,
"metadata": {},
"output_type": "execute_result"
}
@@ -8865,7 +9935,7 @@
},
{
"cell_type": "code",
- "execution_count": 182,
+ "execution_count": 157,
"metadata": {},
"outputs": [],
"source": [
@@ -8916,7 +9986,7 @@
},
{
"cell_type": "code",
- "execution_count": 199,
+ "execution_count": 158,
"metadata": {},
"outputs": [
{
@@ -8928,7 +9998,7 @@
"{(1↦22),(2↦22),(3↦33)}"
]
},
- "execution_count": 199,
+ "execution_count": 158,
"metadata": {},
"output_type": "execute_result"
}
@@ -8939,7 +10009,7 @@
},
{
"cell_type": "code",
- "execution_count": 200,
+ "execution_count": 159,
"metadata": {},
"outputs": [
{
@@ -8951,7 +10021,7 @@
"{(1↦22),(2↦22),(3↦33)}"
]
},
- "execution_count": 200,
+ "execution_count": 159,
"metadata": {},
"output_type": "execute_result"
}
@@ -8962,7 +10032,7 @@
},
{
"cell_type": "code",
- "execution_count": 190,
+ "execution_count": 160,
"metadata": {},
"outputs": [
{
@@ -8971,56 +10041,64 @@
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"<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
" \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
- "<!-- Generated by graphviz version 2.28.0 (20110509.1545)\n",
+ "<!-- Generated by graphviz version 2.44.1 (0)\n",
" -->\n",
"<!-- Title: state Pages: 1 -->\n",
- "<svg width=\"206pt\" height=\"134pt\"\n",
- " viewBox=\"0.00 0.00 206.00 134.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
- "<g id=\"graph1\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 130)\">\n",
+ "<svg width=\"206pt\" height=\"131pt\"\n",
+ " viewBox=\"0.00 0.00 206.00 131.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
+ "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 127)\">\n",
"<title>state</title>\n",
- "<polygon fill=\"white\" stroke=\"white\" points=\"-4,5 -4,-130 203,-130 203,5 -4,5\"/>\n",
+ "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-127 202,-127 202,4 -4,4\"/>\n",
"<!-- 3 -->\n",
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- "<text text-anchor=\"middle\" x=\"27\" y=\"-103.8\" font-family=\"Times,serif\" font-size=\"14.00\">3</text>\n",
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+ "<title>33</title>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"54,-36 0,-36 0,0 54,0 54,-36\"/>\n",
+ "<text text-anchor=\"middle\" x=\"27\" y=\"-14.3\" font-family=\"Times,serif\" font-size=\"14.00\">33</text>\n",
"</g>\n",
"<!-- 3->33 -->\n",
- "<g id=\"edge2\" class=\"edge\"><title>3->33</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"30.5001,-46.0504 27,-36.0504 23.5001,-46.0504 30.5001,-46.0504\"/>\n",
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+ "<g id=\"edge1\" class=\"edge\">\n",
+ "<title>3->33</title>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"30.5,-46.18 27,-36.18 23.5,-46.18 30.5,-46.18\"/>\n",
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"</g>\n",
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"</g>\n",
"<!-- 2->22 -->\n",
- "<g id=\"edge4\" class=\"edge\"><title>2->22</title>\n",
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- "<text text-anchor=\"middle\" x=\"125.053\" y=\"-58.8\" font-family=\"Times,serif\" font-size=\"14.00\">G</text>\n",
+ "<g id=\"edge2\" class=\"edge\">\n",
+ "<title>2->22</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M106.29,-86.8C111.31,-74.93 118.09,-58.93 123.8,-45.45\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"127.05,-46.75 127.73,-36.18 120.6,-44.02 127.05,-46.75\"/>\n",
+ "<text text-anchor=\"middle\" x=\"125\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">G</text>\n",
"</g>\n",
"<!-- 1 -->\n",
- "<g id=\"node7\" class=\"node\"><title>1</title>\n",
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- "<text text-anchor=\"middle\" x=\"171\" y=\"-103.8\" font-family=\"Times,serif\" font-size=\"14.00\">1</text>\n",
+ "<g id=\"node5\" class=\"node\">\n",
+ "<title>1</title>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"198,-123 144,-123 144,-87 198,-87 198,-123\"/>\n",
+ "<text text-anchor=\"middle\" x=\"171\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">1</text>\n",
"</g>\n",
"<!-- 1->22 -->\n",
- "<g id=\"edge6\" class=\"edge\"><title>1->22</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M163.888,-89.614C158.726,-76.9977 151.652,-59.7052 145.796,-45.3907\"/>\n",
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- "<text text-anchor=\"middle\" x=\"161.053\" y=\"-58.8\" font-family=\"Times,serif\" font-size=\"14.00\">G</text>\n",
+ "<g id=\"edge3\" class=\"edge\">\n",
+ "<title>1->22</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M163.71,-86.8C158.69,-74.93 151.91,-58.93 146.2,-45.45\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"149.4,-44.02 142.27,-36.18 142.95,-46.75 149.4,-44.02\"/>\n",
+ "<text text-anchor=\"middle\" x=\"161\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">G</text>\n",
"</g>\n",
"</g>\n",
"</svg>"
@@ -9029,7 +10107,7 @@
"<Dot visualization: expr_as_graph [AGA={1,2,3} & G={(1,22),(2,22),(3,33)}(\"G\",G)]>"
]
},
- "execution_count": 190,
+ "execution_count": 160,
"metadata": {},
"output_type": "execute_result"
}
@@ -9053,7 +10131,7 @@
},
{
"cell_type": "code",
- "execution_count": 191,
+ "execution_count": 161,
"metadata": {},
"outputs": [
{
@@ -9065,7 +10143,7 @@
"22"
]
},
- "execution_count": 191,
+ "execution_count": 161,
"metadata": {},
"output_type": "execute_result"
}
@@ -9083,7 +10161,7 @@
},
{
"cell_type": "code",
- "execution_count": 192,
+ "execution_count": 162,
"metadata": {},
"outputs": [
{
@@ -9095,7 +10173,7 @@
"3"
]
},
- "execution_count": 192,
+ "execution_count": 162,
"metadata": {},
"output_type": "execute_result"
}
@@ -9106,7 +10184,7 @@
},
{
"cell_type": "code",
- "execution_count": 193,
+ "execution_count": 163,
"metadata": {},
"outputs": [
{
@@ -9118,7 +10196,7 @@
"3"
]
},
- "execution_count": 193,
+ "execution_count": 163,
"metadata": {},
"output_type": "execute_result"
}
@@ -9136,7 +10214,7 @@
},
{
"cell_type": "code",
- "execution_count": 194,
+ "execution_count": 164,
"metadata": {},
"outputs": [
{
@@ -9148,7 +10226,7 @@
"{1,2,3}"
]
},
- "execution_count": 194,
+ "execution_count": 164,
"metadata": {},
"output_type": "execute_result"
}
@@ -9159,7 +10237,7 @@
},
{
"cell_type": "code",
- "execution_count": 195,
+ "execution_count": 165,
"metadata": {},
"outputs": [
{
@@ -9171,7 +10249,7 @@
"{22,33}"
]
},
- "execution_count": 195,
+ "execution_count": 165,
"metadata": {},
"output_type": "execute_result"
}
@@ -9182,7 +10260,7 @@
},
{
"cell_type": "code",
- "execution_count": 196,
+ "execution_count": 166,
"metadata": {},
"outputs": [
{
@@ -9194,7 +10272,7 @@
"{1,2}"
]
},
- "execution_count": 196,
+ "execution_count": 166,
"metadata": {},
"output_type": "execute_result"
}
@@ -9205,7 +10283,7 @@
},
{
"cell_type": "code",
- "execution_count": 197,
+ "execution_count": 167,
"metadata": {},
"outputs": [],
"source": [
diff --git a/info4/kapitel-1/FormaleSprachen.ipynb b/info4/kapitel-1/FormaleSprachen.ipynb
index 5371160f86a09efa875eacfbcac3ec6896006d33..44a9b80a328235cea1b6d1629207f06bbb6e159b 100644
--- a/info4/kapitel-1/FormaleSprachen.ipynb
+++ b/info4/kapitel-1/FormaleSprachen.ipynb
@@ -11,7 +11,7 @@
},
{
"cell_type": "code",
- "execution_count": 67,
+ "execution_count": 1,
"metadata": {},
"outputs": [
{
@@ -20,7 +20,7 @@
"Loaded machine: Alphabet"
]
},
- "execution_count": 67,
+ "execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
@@ -47,7 +47,7 @@
},
{
"cell_type": "code",
- "execution_count": 70,
+ "execution_count": 2,
"metadata": {},
"outputs": [
{
@@ -59,7 +59,7 @@
"TRUE"
]
},
- "execution_count": 70,
+ "execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
@@ -70,7 +70,7 @@
},
{
"cell_type": "code",
- "execution_count": 71,
+ "execution_count": 3,
"metadata": {},
"outputs": [
{
@@ -82,7 +82,7 @@
"TRUE"
]
},
- "execution_count": 71,
+ "execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
@@ -100,7 +100,7 @@
},
{
"cell_type": "code",
- "execution_count": 72,
+ "execution_count": 4,
"metadata": {},
"outputs": [
{
@@ -112,7 +112,7 @@
"3"
]
},
- "execution_count": 72,
+ "execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
@@ -123,7 +123,7 @@
},
{
"cell_type": "code",
- "execution_count": 73,
+ "execution_count": 5,
"metadata": {},
"outputs": [
{
@@ -135,7 +135,7 @@
"4"
]
},
- "execution_count": 73,
+ "execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
@@ -153,7 +153,7 @@
},
{
"cell_type": "code",
- "execution_count": 74,
+ "execution_count": 6,
"metadata": {},
"outputs": [
{
@@ -171,7 +171,7 @@
"\tε = ∅"
]
},
- "execution_count": 74,
+ "execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
@@ -191,7 +191,7 @@
},
{
"cell_type": "code",
- "execution_count": 75,
+ "execution_count": 7,
"metadata": {},
"outputs": [
{
@@ -209,7 +209,7 @@
"\tSprachen = {L∣L ⊆ seq(Σ)}"
]
},
- "execution_count": 75,
+ "execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
@@ -227,7 +227,7 @@
},
{
"cell_type": "code",
- "execution_count": 77,
+ "execution_count": 8,
"metadata": {},
"outputs": [
{
@@ -236,7 +236,7 @@
"Loaded machine: Alphabet"
]
},
- "execution_count": 77,
+ "execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
@@ -258,7 +258,7 @@
},
{
"cell_type": "code",
- "execution_count": 78,
+ "execution_count": 9,
"metadata": {},
"outputs": [
{
@@ -267,7 +267,7 @@
"Machine constants set up using operation 0: $setup_constants()"
]
},
- "execution_count": 78,
+ "execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
@@ -278,7 +278,7 @@
},
{
"cell_type": "code",
- "execution_count": 79,
+ "execution_count": 10,
"metadata": {},
"outputs": [
{
@@ -290,7 +290,7 @@
"{(1↦a),(2↦a),(3↦b),(4↦c)}"
]
},
- "execution_count": 79,
+ "execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
@@ -308,7 +308,7 @@
},
{
"cell_type": "code",
- "execution_count": 80,
+ "execution_count": 11,
"metadata": {},
"outputs": [
{
@@ -317,7 +317,7 @@
"Preference changed: PP_SEQUENCES = TRUE\n"
]
},
- "execution_count": 80,
+ "execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
@@ -328,7 +328,7 @@
},
{
"cell_type": "code",
- "execution_count": 81,
+ "execution_count": 12,
"metadata": {},
"outputs": [
{
@@ -340,7 +340,7 @@
"[a,a,b,c]"
]
},
- "execution_count": 81,
+ "execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
@@ -358,7 +358,7 @@
},
{
"cell_type": "code",
- "execution_count": 82,
+ "execution_count": 13,
"metadata": {},
"outputs": [
{
@@ -370,7 +370,7 @@
"[c,c]"
]
},
- "execution_count": 82,
+ "execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
@@ -381,7 +381,7 @@
},
{
"cell_type": "code",
- "execution_count": 83,
+ "execution_count": 14,
"metadata": {},
"outputs": [
{
@@ -393,7 +393,7 @@
"{[c,c],[a,a,b,c]}"
]
},
- "execution_count": 83,
+ "execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
@@ -411,7 +411,7 @@
},
{
"cell_type": "code",
- "execution_count": 84,
+ "execution_count": 15,
"metadata": {},
"outputs": [
{
@@ -429,7 +429,7 @@
"\tLeereSprache = []"
]
},
- "execution_count": 84,
+ "execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
@@ -447,7 +447,7 @@
},
{
"cell_type": "code",
- "execution_count": 85,
+ "execution_count": 16,
"metadata": {},
"outputs": [
{
@@ -459,7 +459,7 @@
"2"
]
},
- "execution_count": 85,
+ "execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
@@ -470,7 +470,7 @@
},
{
"cell_type": "code",
- "execution_count": 86,
+ "execution_count": 17,
"metadata": {},
"outputs": [
{
@@ -482,7 +482,7 @@
"1"
]
},
- "execution_count": 86,
+ "execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
@@ -493,7 +493,7 @@
},
{
"cell_type": "code",
- "execution_count": 87,
+ "execution_count": 18,
"metadata": {},
"outputs": [
{
@@ -505,7 +505,7 @@
"0"
]
},
- "execution_count": 87,
+ "execution_count": 18,
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}
@@ -523,7 +523,7 @@
},
{
"cell_type": "code",
- "execution_count": 88,
+ "execution_count": 19,
"metadata": {},
"outputs": [
{
@@ -535,7 +535,7 @@
"{[c,c],[a,a,b,c]}"
]
},
- "execution_count": 88,
+ "execution_count": 19,
"metadata": {},
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}
@@ -546,7 +546,7 @@
},
{
"cell_type": "code",
- "execution_count": 90,
+ "execution_count": 20,
"metadata": {},
"outputs": [
{
@@ -558,7 +558,7 @@
"{[c,c],[a,a,a]}"
]
},
- "execution_count": 90,
+ "execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
@@ -569,7 +569,7 @@
},
{
"cell_type": "code",
- "execution_count": 91,
+ "execution_count": 21,
"metadata": {},
"outputs": [
{
@@ -581,7 +581,7 @@
"{[c,c],[a,a,a],[a,a,b,c]}"
]
},
- "execution_count": 91,
+ "execution_count": 21,
"metadata": {},
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}
@@ -592,7 +592,7 @@
},
{
"cell_type": "code",
- "execution_count": 92,
+ "execution_count": 22,
"metadata": {},
"outputs": [
{
@@ -604,7 +604,7 @@
"{[c,c],[a,a,a],[a,a,b,c]}"
]
},
- "execution_count": 92,
+ "execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
@@ -615,7 +615,7 @@
},
{
"cell_type": "code",
- "execution_count": 93,
+ "execution_count": 23,
"metadata": {},
"outputs": [
{
@@ -627,7 +627,7 @@
"{[c,c]}"
]
},
- "execution_count": 93,
+ "execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
@@ -638,7 +638,7 @@
},
{
"cell_type": "code",
- "execution_count": 94,
+ "execution_count": 24,
"metadata": {},
"outputs": [
{
@@ -650,7 +650,7 @@
"{[c,c]}"
]
},
- "execution_count": 94,
+ "execution_count": 24,
"metadata": {},
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}
@@ -661,7 +661,7 @@
},
{
"cell_type": "code",
- "execution_count": 95,
+ "execution_count": 25,
"metadata": {},
"outputs": [
{
@@ -673,7 +673,7 @@
"{[a,a,b,c]}"
]
},
- "execution_count": 95,
+ "execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
@@ -684,7 +684,7 @@
},
{
"cell_type": "code",
- "execution_count": 96,
+ "execution_count": 26,
"metadata": {},
"outputs": [
{
@@ -696,7 +696,7 @@
"{[a,a,a]}"
]
},
- "execution_count": 96,
+ "execution_count": 26,
"metadata": {},
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}
@@ -707,7 +707,7 @@
},
{
"cell_type": "code",
- "execution_count": 97,
+ "execution_count": 27,
"metadata": {},
"outputs": [
{
@@ -725,7 +725,7 @@
"\tKomplement = (seq(Σ) − {[c,c],[a,a,b,c]})"
]
},
- "execution_count": 97,
+ "execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
@@ -745,7 +745,7 @@
},
{
"cell_type": "code",
- "execution_count": 101,
+ "execution_count": 28,
"metadata": {},
"outputs": [
{
@@ -757,7 +757,7 @@
"[a,a,b,c]"
]
},
- "execution_count": 101,
+ "execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
@@ -768,7 +768,7 @@
},
{
"cell_type": "code",
- "execution_count": 102,
+ "execution_count": 29,
"metadata": {},
"outputs": [
{
@@ -780,7 +780,7 @@
"[a,a,b,c,a,a,b,c]"
]
},
- "execution_count": 102,
+ "execution_count": 29,
"metadata": {},
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}
@@ -791,7 +791,7 @@
},
{
"cell_type": "code",
- "execution_count": 103,
+ "execution_count": 30,
"metadata": {},
"outputs": [
{
@@ -803,7 +803,7 @@
"[a,a,b,c]"
]
},
- "execution_count": 103,
+ "execution_count": 30,
"metadata": {},
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}
@@ -814,7 +814,7 @@
},
{
"cell_type": "code",
- "execution_count": 104,
+ "execution_count": 31,
"metadata": {},
"outputs": [
{
@@ -826,7 +826,7 @@
"[a,a,b,c]"
]
},
- "execution_count": 104,
+ "execution_count": 31,
"metadata": {},
"output_type": "execute_result"
}
@@ -844,7 +844,7 @@
},
{
"cell_type": "code",
- "execution_count": 105,
+ "execution_count": 32,
"metadata": {},
"outputs": [
{
@@ -856,7 +856,7 @@
"{[c,c],[a,a,b,c]}"
]
},
- "execution_count": 105,
+ "execution_count": 32,
"metadata": {},
"output_type": "execute_result"
}
@@ -867,7 +867,7 @@
},
{
"cell_type": "code",
- "execution_count": 106,
+ "execution_count": 33,
"metadata": {},
"outputs": [
{
@@ -879,7 +879,7 @@
"{[c,c],[a,a,a]}"
]
},
- "execution_count": 106,
+ "execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
@@ -890,7 +890,7 @@
},
{
"cell_type": "code",
- "execution_count": 107,
+ "execution_count": 34,
"metadata": {},
"outputs": [
{
@@ -902,7 +902,7 @@
"{[c,c,c,c],[c,c,a,a,a],[a,a,b,c,c,c],[a,a,b,c,a,a,a]}"
]
},
- "execution_count": 107,
+ "execution_count": 34,
"metadata": {},
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}
@@ -921,7 +921,7 @@
},
{
"cell_type": "code",
- "execution_count": 114,
+ "execution_count": 35,
"metadata": {},
"outputs": [
{
@@ -933,7 +933,7 @@
"{[c,c,c,c],[c,c,a,a,b,c],[a,a,b,c,c,c],[a,a,b,c,a,a,b,c]}"
]
},
- "execution_count": 114,
+ "execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
@@ -952,7 +952,7 @@
},
{
"cell_type": "code",
- "execution_count": 118,
+ "execution_count": 36,
"metadata": {},
"outputs": [
{
@@ -964,7 +964,7 @@
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]
},
- "execution_count": 118,
+ "execution_count": 36,
"metadata": {},
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@@ -975,7 +975,7 @@
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- "execution_count": 119,
+ "execution_count": 37,
"metadata": {},
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{
@@ -987,7 +987,7 @@
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]
},
- "execution_count": 119,
+ "execution_count": 37,
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@@ -998,7 +998,7 @@
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@@ -1007,7 +1007,7 @@
},
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- "execution_count": 130,
+ "execution_count": 39,
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@@ -1016,7 +1016,7 @@
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+ "execution_count": 40,
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@@ -1032,7 +1032,7 @@
},
{
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- "execution_count": 132,
+ "execution_count": 41,
"metadata": {},
"outputs": [
{
@@ -1044,7 +1044,7 @@
"[c,b,a,a]"
]
},
- "execution_count": 132,
+ "execution_count": 41,
"metadata": {},
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@@ -1062,19 +1062,19 @@
},
{
"cell_type": "code",
- "execution_count": 133,
+ "execution_count": 42,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
- "$\\{[a,\\mathit{b},\\mathit{a},c],[c,\\mathit{b},\\mathit{a},a]\\}$"
+ "$\\{[c,c],[c,\\mathit{b},\\mathit{a},a]\\}$"
],
"text/plain": [
- "{[a,b,a,c],[c,b,a,a]}"
+ "{[c,c],[c,b,a,a]}"
]
},
- "execution_count": 133,
+ "execution_count": 42,
"metadata": {},
"output_type": "execute_result"
}
@@ -1092,7 +1092,7 @@
},
{
"cell_type": "code",
- "execution_count": 134,
+ "execution_count": 43,
"metadata": {},
"outputs": [
{
@@ -1101,7 +1101,7 @@
"Loaded machine: Alphabet"
]
},
- "execution_count": 134,
+ "execution_count": 43,
"metadata": {},
"output_type": "execute_result"
}
@@ -1128,7 +1128,7 @@
},
{
"cell_type": "code",
- "execution_count": 135,
+ "execution_count": 44,
"metadata": {},
"outputs": [
{
@@ -1137,7 +1137,7 @@
"Machine constants set up using operation 0: $setup_constants()"
]
},
- "execution_count": 135,
+ "execution_count": 44,
"metadata": {},
"output_type": "execute_result"
}
@@ -1148,7 +1148,7 @@
},
{
"cell_type": "code",
- "execution_count": 136,
+ "execution_count": 45,
"metadata": {},
"outputs": [
{
@@ -1160,7 +1160,7 @@
"TRUE"
]
},
- "execution_count": 136,
+ "execution_count": 45,
"metadata": {},
"output_type": "execute_result"
}
@@ -1171,7 +1171,7 @@
},
{
"cell_type": "code",
- "execution_count": 137,
+ "execution_count": 46,
"metadata": {},
"outputs": [
{
@@ -1183,7 +1183,7 @@
"FALSE"
]
},
- "execution_count": 137,
+ "execution_count": 46,
"metadata": {},
"output_type": "execute_result"
}
@@ -1194,7 +1194,7 @@
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{
"cell_type": "code",
- "execution_count": 138,
+ "execution_count": 47,
"metadata": {},
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{
@@ -1206,7 +1206,7 @@
"TRUE"
]
},
- "execution_count": 138,
+ "execution_count": 47,
"metadata": {},
"output_type": "execute_result"
}
@@ -1224,7 +1224,7 @@
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{
"cell_type": "code",
- "execution_count": 139,
+ "execution_count": 48,
"metadata": {},
"outputs": [
{
@@ -1236,7 +1236,7 @@
"{[],[c],[c,b],[c,b,a],[c,b,a,a],[c,b,a,a,b]}"
]
},
- "execution_count": 139,
+ "execution_count": 48,
"metadata": {},
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@@ -1247,7 +1247,7 @@
},
{
"cell_type": "code",
- "execution_count": 140,
+ "execution_count": 49,
"metadata": {},
"outputs": [
{
@@ -1259,7 +1259,7 @@
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]
},
- "execution_count": 140,
+ "execution_count": 49,
"metadata": {},
"output_type": "execute_result"
}
@@ -1270,7 +1270,7 @@
},
{
"cell_type": "code",
- "execution_count": 141,
+ "execution_count": 50,
"metadata": {},
"outputs": [
{
@@ -1282,7 +1282,7 @@
"{[],[a],[a,b],[b],[b,c],[c],[a,b,c]}"
]
},
- "execution_count": 141,
+ "execution_count": 50,
"metadata": {},
"output_type": "execute_result"
}
@@ -1293,7 +1293,7 @@
},
{
"cell_type": "code",
- "execution_count": 142,
+ "execution_count": 51,
"metadata": {},
"outputs": [
{
@@ -1302,7 +1302,7 @@
"Preference changed: DOT_DECOMPOSE_NODES = FALSE\n"
]
},
- "execution_count": 142,
+ "execution_count": 51,
"metadata": {},
"output_type": "execute_result"
}
@@ -1313,7 +1313,7 @@
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{
"cell_type": "code",
- "execution_count": 143,
+ "execution_count": 52,
"metadata": {},
"outputs": [
{
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- "<!-- Generated by graphviz version 2.28.0 (20110509.1545)\n",
+ "<!-- Generated by graphviz version 2.44.1 (0)\n",
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{
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"</g>\n",
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+ "<g id=\"edge11\" class=\"edge\">\n",
+ "<title>[]->[a,b,c]</title>\n",
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"</g>\n",
"<!-- []->[c] -->\n",
- "<g id=\"edge24\" class=\"edge\"><title>[]->[c]</title>\n",
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+ "<g id=\"edge12\" class=\"edge\">\n",
+ "<title>[]->[c]</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M181.28,-270.74C160.87,-264.76 132.67,-255.28 109.67,-243 95.87,-235.63 81.81,-225.6 70.15,-216.45\"/>\n",
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"</g>\n",
"<!-- []->[b,c] -->\n",
- "<g id=\"edge26\" class=\"edge\"><title>[]->[b,c]</title>\n",
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+ "<g id=\"edge13\" class=\"edge\">\n",
+ "<title>[]->[b,c]</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M197.35,-260.96C189.25,-247.7 179.04,-228.5 174.67,-210 168.59,-184.25 173.62,-154.19 179.09,-133.03\"/>\n",
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"</g>\n",
"<!-- []->[b] -->\n",
- "<g id=\"edge28\" class=\"edge\"><title>[]->[b]</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M195.117,-269.614C202.499,-256.876 212.643,-239.372 220.985,-224.979\"/>\n",
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+ "<g id=\"edge14\" class=\"edge\">\n",
+ "<title>[]->[b]</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M217.58,-260.8C223.78,-248.82 232.17,-232.62 239.19,-219.06\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"242.29,-220.67 243.78,-210.18 236.08,-217.45 242.29,-220.67\"/>\n",
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"</g>\n",
"<!-- []->[a,b] -->\n",
- "<g id=\"edge30\" class=\"edge\"><title>[]->[a,b]</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M212.058,-281C244.137,-272.383 296.724,-253.261 324.041,-216 340.867,-193.049 344.103,-159.922 343.928,-136.623\"/>\n",
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- "<text text-anchor=\"middle\" x=\"356.145\" y=\"-193.8\" font-family=\"Times,serif\" font-size=\"14.00\">präfix</text>\n",
+ "<g id=\"edge15\" class=\"edge\">\n",
+ "<title>[]->[a,b]</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M235.82,-272.63C268.07,-264.75 320.98,-246.81 348.67,-210 365.22,-188.01 368.55,-156.02 368.47,-133.33\"/>\n",
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+ "<text text-anchor=\"middle\" x=\"384.67\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">präfix</text>\n",
"</g>\n",
"<!-- []->[a] -->\n",
- "<g id=\"edge32\" class=\"edge\"><title>[]->[a]</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M212.234,-285.824C248.626,-283.354 314.445,-275.699 365.041,-252 380.147,-244.924 394.75,-233.568 406.275,-223.178\"/>\n",
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- "<text text-anchor=\"middle\" x=\"409.145\" y=\"-238.8\" font-family=\"Times,serif\" font-size=\"14.00\">präfix</text>\n",
+ "<g id=\"edge16\" class=\"edge\">\n",
+ "<title>[]->[a]</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M235.74,-276.22C272.01,-273.07 337.85,-264.65 389.67,-243 405.6,-236.34 421.61,-225.9 434.57,-216.27\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"436.8,-218.97 442.6,-210.11 432.54,-213.41 436.8,-218.97\"/>\n",
+ "<text text-anchor=\"middle\" x=\"437.67\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">präfix</text>\n",
"</g>\n",
"<!-- []->[] -->\n",
- "<g id=\"edge34\" class=\"edge\"><title>[]->[]</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M212.282,-296.242C222.065,-296.419 230.041,-293.672 230.041,-288 230.041,-284.544 227.079,-282.173 222.551,-280.889\"/>\n",
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- "<text text-anchor=\"middle\" x=\"247.145\" y=\"-283.8\" font-family=\"Times,serif\" font-size=\"14.00\">präfix</text>\n",
+ "<g id=\"edge17\" class=\"edge\">\n",
+ "<title>[]->[]</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M235.91,-286.87C245.7,-287.04 253.67,-284.41 253.67,-279 253.67,-275.7 250.71,-273.44 246.18,-272.21\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"246.23,-268.7 235.91,-271.13 245.49,-275.66 246.23,-268.7\"/>\n",
+ "<text text-anchor=\"middle\" x=\"274.67\" y=\"-275.3\" font-family=\"Times,serif\" font-size=\"14.00\">präfix</text>\n",
"</g>\n",
"</g>\n",
"</svg>"
@@ -1676,7 +1729,7 @@
"<Dot visualization: expr_as_graph [TWTW={[],[a],[a,b],[b],[b,c],[c],[a,b,c]}(\"präfix\",{x,y|(x,y):präfix & y:TW})]>"
]
},
- "execution_count": 144,
+ "execution_count": 53,
"metadata": {},
"output_type": "execute_result"
}
@@ -1687,7 +1740,7 @@
},
{
"cell_type": "code",
- "execution_count": 145,
+ "execution_count": 54,
"metadata": {},
"outputs": [],
"source": [
@@ -1696,10 +1749,157 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 63,
"metadata": {},
- "outputs": [],
- "source": []
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Loaded machine: FormaleSprachen"
+ ]
+ },
+ "execution_count": 63,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "MACHINE FormaleSprachen\n",
+ "/* Formulierung in B der Grundbegriffe aus dem Foliensatz 1 von Informatik 4 */\n",
+ "/* erstellt von Michael Leuschel, 2014 */\n",
+ "SETS\n",
+ " Alphabet /* Ein Alphabet ist eine endliche, nichtleere Menge von Buchstaben (oder Symbolen). */\n",
+ " = {a,b,c} /* Hier als Beispiel {a,b,c} */\n",
+ "ABSTRACT_CONSTANTS\n",
+ " kat, Worte, Sprachen, kleene, sp, teilwort, praefix\n",
+ "CONSTANTS\n",
+ " w1, w2, lambda,\n",
+ " L1, L2, LeereSprache, KomplementL1, L1L2\n",
+ "PROPERTIES\n",
+ " Worte = seq(Alphabet) /* Ein Wort ist eine endliche Folge von Elementen aus dem Alphabet */\n",
+ " &\n",
+ " /* Beispiele */\n",
+ " w1 = [a,a,b,c] & /* in den Folien schreiben wir aabc anstatt [a,a,b,c] */\n",
+ " w1 : Worte & /* w1 ist ein Wort */\n",
+ " w2 = [c,a,b,a] &\n",
+ " w2 : Worte &\n",
+ " /* Die Laenge eines Wortes ist die Anzahl der Symbole in w */\n",
+ " /* Beispiel: */\n",
+ " size(w1) = 4 &\n",
+ " \n",
+ "/* Das leere Wort ist das eindeutig bestimmte Wort der Laenge 0 */\n",
+ " lambda: Worte & size(lambda) = 0 &\n",
+ "\n",
+ "/* Die Menge aller Worte bezeichnen wir mit \\Sigma^*. Hier verwenden wir seq(Alphabet) */\n",
+ "\n",
+ " /* Eine formale Sprache ist eine jede Teilmenge von \\Sigma^*. */\n",
+ " Sprachen = {L | L <: Worte} &\n",
+ " /* Beispiel: */\n",
+ " L1 = {w1,w2} & L1 : Sprachen &\n",
+ " L2 = {w2, [a,a,a] } & L2 : Sprachen &\n",
+ "\n",
+ " /* Die Leere Sprache ist die Sprache die keine Wörter enthält */\n",
+ " LeereSprache = {} &\n",
+ " /* Beachte: */\n",
+ " LeereSprache /= {lambda} &\n",
+ "\n",
+ " /* Die Kardinalität einer Sprache L ist die Anzahl der Wörter von L. \n",
+ " Beispiel: */\n",
+ " card(L1) = 2 & card({lambda}) = 1 & card(LeereSprache) = 0 &\n",
+ "\n",
+ " /* Beispiele für die Vereinigung von Sprachen: */\n",
+ " L1 \\/ L2 = {x| x:Worte & (x:L1 or x:L2)} &\n",
+ "/* Beispiele für die Durchchnitt von Sprachen: */\n",
+ " L1 /\\ L2 = {x| x:Worte & x:L1 & x:L2} &\n",
+ "/* Beispiele für die Differenz von Sprachen: */\n",
+ " L1 - L2 = {x| x:Worte & x:L1 & x/: L2} &\n",
+ "/* Beispiele für das Komplement von Sprachen: */\n",
+ " KomplementL1 = Worte - L1 /* kann man auch so schreiben {x| x:Worte & x/: L1} */ &\n",
+ " w1 /: KomplementL1 & [a,a,a] : KomplementL1 & lambda : KomplementL1 &\n",
+ "\n",
+ " /* Konkatenation von Wörtern wird in B mit ^ geschrieben: */\n",
+ " w1^w2 = [a,a,b,c,c,a,b,a] &\n",
+ " w1^lambda = w1 & lambda^w1 = w1 &\n",
+ "\n",
+ " /* Verkettung von Sprachen */\n",
+ " L1L2 = {w | #(a,b).(a:L1 & b:L2 & w = a^b)}\n",
+ "\n",
+ " & kat = %(x,y).(x:Worte & y:Worte | x^y) /* Die Konkatenierung als explizite Funktion */\n",
+ " & kat(w1,w2) = [a,a,b,c,c,a,b,a] \n",
+ " & L1L2 = kat[L1*L2]\n",
+ "\n",
+ " /* Eine rekursive Definition von A^n: */\n",
+ " & kleene = %(A,n).(A : Sprachen & n=0|{lambda}) \\/\n",
+ " %(A,n).(A : Sprachen & n>0| kat[A*kleene(A,n-1)])\n",
+ " & kleene(L1,1) = L1\n",
+ " & kleene(L2,1) = L2\n",
+ "/* A^* ist momentan schwieriger darstellbar, zumindest so, dass man mit ProB damit\n",
+ " arbeiten könnte */\n",
+ "\n",
+ " & /* Die Spiegelbildoperation */\n",
+ " sp = %u.(u:Worte| %i.(i:dom(u)|u(size(u)+1-i)))\n",
+ " & sp(w1) = [c,b,a,a]\n",
+ " & sp(sp(w1)) = w1\n",
+ " & sp(lambda) = lambda\n",
+ "\n",
+ " /* Die Spiegelung einer Sprache bekommt man einfach durch das relational image: */\n",
+ " & sp[L1] = {sp(w1),sp(w2)}\n",
+ " & sp[sp[L1]] = L1\n",
+ " & sp[sp[L2]] = L2\n",
+ "\n",
+ " /* Die Teilwortrelation */\n",
+ " & teilwort = {u,v | u:Worte & v:Worte & #(v1,v2).(v1^u^v2 = v)}\n",
+ " & [a,a] |-> [c,b,a,a,b] : teilwort\n",
+ " & [a,a,a] |-> [c,b,a,a,b] /: teilwort\n",
+ " /* Wir können alle Teilworte durch das relational image der Umkehrrelation berechnen: */\n",
+ " & teilwort~[{[a,a,b]}] = { lambda, [a], [b], [a,a],[a,b], [a,a,b] }\n",
+ "\n",
+ " /* Die Anfangswortrelation */\n",
+ " & praefix = {u,v | u:Worte & v:Worte & #(w).(u^w = v)}\n",
+ " & [a,a] |-> [c,b,a,a,b] /: praefix\n",
+ " & [c,b] |-> [c,b,a,a,b] : praefix\n",
+ " /* Wir können alle Präfixe durch das relational image der Umkehrrelation berechnen: */\n",
+ " & praefix~[{[a,a,b]}] = { lambda, [a], [a,a], [a,a,b] }\n",
+ "ASSERTIONS\n",
+ " card(L1L2) = 4\n",
+ "END"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 66,
+ "metadata": {},
+ "outputs": [
+ {
+ "ename": "IllegalArgumentException",
+ "evalue": "Executing operation $initialise_machine with predicate 1=1 produced errors: Could not execute operation INITIALISATION in state root",
+ "output_type": "error",
+ "traceback": [
+ "\u001b[1m\u001b[31mjava.lang.IllegalArgumentException: Executing operation $initialise_machine with predicate 1=1 produced errors: Could not execute operation INITIALISATION in state root\u001b[0m"
+ ]
+ }
+ ],
+ "source": [
+ ":init"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 67,
+ "metadata": {},
+ "outputs": [
+ {
+ "ename": "CommandExecutionException",
+ "evalue": ":eval: NOT-INITIALISED",
+ "output_type": "error",
+ "traceback": [
+ "\u001b[1m\u001b[31m:eval: NOT-INITIALISED\u001b[0m"
+ ]
+ }
+ ],
+ "source": [
+ "kleene(L1, 2)"
+ ]
}
],
"metadata": {
diff --git a/info4/kapitel-1/Grammatiken.ipynb b/info4/kapitel-1/Grammatiken.ipynb
index ea71fb552165b1ee6382e718cffff4a86f5ce71e..b7f35c32954122df213dfe0f30879693743d4be0 100644
--- a/info4/kapitel-1/Grammatiken.ipynb
+++ b/info4/kapitel-1/Grammatiken.ipynb
@@ -6,7 +6,7 @@
"source": [
"## Grammatiken und Chomsky-Hierarchie\n",
"\n",
- "Dieses Notebook begleitet Teil 2 der Vorlesung 3 und führt am Ende die Chomsky-Hierarchie ein"
+ "Dieses Notebook begleitet Teil 2 der Vorlesung 3 und führt am Ende die Chomsky-Hierarchie ein."
]
},
{
@@ -29,7 +29,7 @@
},
{
"cell_type": "code",
- "execution_count": 4,
+ "execution_count": 1,
"metadata": {},
"outputs": [
{
@@ -38,7 +38,7 @@
"Loaded machine: Grammatik"
]
},
- "execution_count": 4,
+ "execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
@@ -65,7 +65,7 @@
},
{
"cell_type": "code",
- "execution_count": 5,
+ "execution_count": 2,
"metadata": {},
"outputs": [
{
@@ -74,7 +74,7 @@
"Machine constants set up using operation 0: $setup_constants()"
]
},
- "execution_count": 5,
+ "execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
@@ -85,7 +85,7 @@
},
{
"cell_type": "code",
- "execution_count": 6,
+ "execution_count": 3,
"metadata": {},
"outputs": [
{
@@ -103,7 +103,7 @@
"\tRHS = []"
]
},
- "execution_count": 6,
+ "execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
@@ -121,7 +121,7 @@
},
{
"cell_type": "code",
- "execution_count": 7,
+ "execution_count": 4,
"metadata": {},
"outputs": [
{
@@ -133,7 +133,7 @@
"{[],[a,S,b]}"
]
},
- "execution_count": 7,
+ "execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
@@ -159,7 +159,7 @@
},
{
"cell_type": "code",
- "execution_count": 8,
+ "execution_count": 5,
"metadata": {},
"outputs": [
{
@@ -168,7 +168,7 @@
"Loaded machine: Grammatik"
]
},
- "execution_count": 8,
+ "execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
@@ -199,7 +199,7 @@
},
{
"cell_type": "code",
- "execution_count": 9,
+ "execution_count": 6,
"metadata": {},
"outputs": [
{
@@ -208,7 +208,7 @@
"Machine constants set up using operation 0: $setup_constants()"
]
},
- "execution_count": 9,
+ "execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
@@ -226,7 +226,7 @@
},
{
"cell_type": "code",
- "execution_count": 10,
+ "execution_count": 7,
"metadata": {},
"outputs": [
{
@@ -238,7 +238,7 @@
"/*@symbolic*/ {u,v∣∃(x,z,p,q)·(p ↦ q ∈ {([S]↦[]),([S]↦[a,S,b])} ∧ u = x ⌒ p ⌒ z ∧ v = x ⌒ q ⌒ z)}"
]
},
- "execution_count": 10,
+ "execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
@@ -249,7 +249,7 @@
},
{
"cell_type": "code",
- "execution_count": 14,
+ "execution_count": 8,
"metadata": {},
"outputs": [
{
@@ -261,7 +261,7 @@
"TRUE"
]
},
- "execution_count": 14,
+ "execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
@@ -272,7 +272,7 @@
},
{
"cell_type": "code",
- "execution_count": 12,
+ "execution_count": 9,
"metadata": {},
"outputs": [
{
@@ -284,7 +284,7 @@
"FALSE"
]
},
- "execution_count": 12,
+ "execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
@@ -295,7 +295,7 @@
},
{
"cell_type": "code",
- "execution_count": 15,
+ "execution_count": 10,
"metadata": {},
"outputs": [
{
@@ -313,7 +313,7 @@
"\tx = []"
]
},
- "execution_count": 15,
+ "execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
@@ -331,7 +331,7 @@
},
{
"cell_type": "code",
- "execution_count": 16,
+ "execution_count": 11,
"metadata": {},
"outputs": [
{
@@ -343,7 +343,7 @@
"{[],[a,S,b]}"
]
},
- "execution_count": 16,
+ "execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
@@ -354,7 +354,7 @@
},
{
"cell_type": "code",
- "execution_count": 18,
+ "execution_count": 12,
"metadata": {},
"outputs": [
{
@@ -366,7 +366,7 @@
"{[a,b,S],[a,S,b],[a,S,b,a,S,b],[a,a,S,b,b,S]}"
]
},
- "execution_count": 18,
+ "execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
@@ -384,7 +384,7 @@
},
{
"cell_type": "code",
- "execution_count": 19,
+ "execution_count": 13,
"metadata": {},
"outputs": [
{
@@ -396,7 +396,7 @@
"{[a,b],[a,a,S,b,b]}"
]
},
- "execution_count": 19,
+ "execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
@@ -407,7 +407,7 @@
},
{
"cell_type": "code",
- "execution_count": 20,
+ "execution_count": 14,
"metadata": {},
"outputs": [
{
@@ -419,7 +419,7 @@
"{[a,a,b,b],[a,a,a,S,b,b,b]}"
]
},
- "execution_count": 20,
+ "execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
@@ -443,7 +443,7 @@
},
{
"cell_type": "code",
- "execution_count": 23,
+ "execution_count": 15,
"metadata": {},
"outputs": [
{
@@ -455,7 +455,7 @@
"TRUE"
]
},
- "execution_count": 23,
+ "execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
@@ -473,7 +473,7 @@
},
{
"cell_type": "code",
- "execution_count": 24,
+ "execution_count": 16,
"metadata": {},
"outputs": [
{
@@ -485,7 +485,7 @@
"{[S]}"
]
},
- "execution_count": 24,
+ "execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
@@ -496,7 +496,7 @@
},
{
"cell_type": "code",
- "execution_count": 25,
+ "execution_count": 17,
"metadata": {},
"outputs": [
{
@@ -508,7 +508,7 @@
"{[],[a,S,b]}"
]
},
- "execution_count": 25,
+ "execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
@@ -519,7 +519,7 @@
},
{
"cell_type": "code",
- "execution_count": 26,
+ "execution_count": 18,
"metadata": {},
"outputs": [
{
@@ -531,7 +531,7 @@
"{[a,b],[a,a,S,b,b]}"
]
},
- "execution_count": 26,
+ "execution_count": 18,
"metadata": {},
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}
@@ -542,7 +542,7 @@
},
{
"cell_type": "code",
- "execution_count": 27,
+ "execution_count": 19,
"metadata": {},
"outputs": [
{
@@ -554,7 +554,7 @@
"{[a,a,b,b],[a,a,a,S,b,b,b]}"
]
},
- "execution_count": 27,
+ "execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
@@ -565,7 +565,7 @@
},
{
"cell_type": "code",
- "execution_count": 28,
+ "execution_count": 20,
"metadata": {},
"outputs": [
{
@@ -577,7 +577,7 @@
"{[a,a,a,b,b,b],[a,a,a,a,S,b,b,b,b]}"
]
},
- "execution_count": 28,
+ "execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
@@ -600,7 +600,7 @@
},
{
"cell_type": "code",
- "execution_count": 29,
+ "execution_count": 21,
"metadata": {},
"outputs": [
{
@@ -612,7 +612,7 @@
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]
},
- "execution_count": 29,
+ "execution_count": 21,
"metadata": {},
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}
@@ -625,12 +625,12 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "Wenn wir mit $\\Sigma^*$ schneiden bekommen wir Wörter der von der Grammatik generierten Sprache:"
+ "Wenn wir mit $\\Sigma^*$ schneiden, bekommen wir Wörter der von der Grammatik generierten Sprache:"
]
},
{
"cell_type": "code",
- "execution_count": 30,
+ "execution_count": 22,
"metadata": {},
"outputs": [
{
@@ -642,7 +642,7 @@
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]
},
- "execution_count": 30,
+ "execution_count": 22,
"metadata": {},
"output_type": "execute_result"
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@@ -660,7 +660,7 @@
},
{
"cell_type": "code",
- "execution_count": 31,
+ "execution_count": 23,
"metadata": {},
"outputs": [
{
@@ -672,7 +672,7 @@
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},
- "execution_count": 31,
+ "execution_count": 23,
"metadata": {},
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}
@@ -683,7 +683,7 @@
},
{
"cell_type": "code",
- "execution_count": 32,
+ "execution_count": 24,
"metadata": {},
"outputs": [
{
@@ -692,7 +692,7 @@
"Preference changed: DOT_DECOMPOSE_NODES = FALSE\n"
]
},
- "execution_count": 32,
+ "execution_count": 24,
"metadata": {},
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}
@@ -703,7 +703,7 @@
},
{
"cell_type": "code",
- "execution_count": 33,
+ "execution_count": 25,
"metadata": {},
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@@ -820,7 +837,7 @@
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]
},
- "execution_count": 33,
+ "execution_count": 25,
"metadata": {},
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}
@@ -838,7 +855,7 @@
},
{
"cell_type": "code",
- "execution_count": 34,
+ "execution_count": 26,
"metadata": {},
"outputs": [
{
@@ -847,7 +864,7 @@
"Loaded machine: Grammatik2"
]
},
- "execution_count": 34,
+ "execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
@@ -883,7 +900,7 @@
},
{
"cell_type": "code",
- "execution_count": 35,
+ "execution_count": 27,
"metadata": {},
"outputs": [
{
@@ -892,7 +909,7 @@
"Machine constants set up using operation 0: $setup_constants()"
]
},
- "execution_count": 35,
+ "execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
@@ -903,7 +920,7 @@
},
{
"cell_type": "code",
- "execution_count": 36,
+ "execution_count": 28,
"metadata": {},
"outputs": [
{
@@ -915,7 +932,7 @@
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]
},
- "execution_count": 36,
+ "execution_count": 28,
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}
@@ -926,7 +943,7 @@
},
{
"cell_type": "code",
- "execution_count": 37,
+ "execution_count": 29,
"metadata": {},
"outputs": [
{
@@ -938,7 +955,7 @@
"{[a,b,c]}"
]
},
- "execution_count": 37,
+ "execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
@@ -949,7 +966,7 @@
},
{
"cell_type": "code",
- "execution_count": 38,
+ "execution_count": 30,
"metadata": {},
"outputs": [
{
@@ -958,7 +975,7 @@
"Preference changed: DOT_DECOMPOSE_NODES = FALSE\n"
]
},
- "execution_count": 38,
+ "execution_count": 30,
"metadata": {},
"output_type": "execute_result"
}
@@ -969,7 +986,7 @@
},
{
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- "execution_count": 39,
+ "execution_count": 31,
"metadata": {},
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@@ -1236,7 +1297,7 @@
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]
},
- "execution_count": 39,
+ "execution_count": 31,
"metadata": {},
"output_type": "execute_result"
}
@@ -1247,7 +1308,7 @@
},
{
"cell_type": "code",
- "execution_count": 40,
+ "execution_count": 32,
"metadata": {},
"outputs": [],
"source": [
@@ -1289,9 +1350,23 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 33,
"metadata": {},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{([a,B]\\mapsto [a,b]),([b,B]\\mapsto [b,b]),([b,C]\\mapsto [b,c]),([c,C]\\mapsto [c,c]),([S]\\mapsto [a,\\mathit{S},\\mathit{B},C]),([S]\\mapsto [a,\\mathit{B},C]),([C,B]\\mapsto [B,C])\\}$"
+ ],
+ "text/plain": [
+ "{([a,B]↦[a,b]),([b,B]↦[b,b]),([b,C]↦[b,c]),([c,C]↦[c,c]),([S]↦[a,S,B,C]),([S]↦[a,B,C]),([C,B]↦[B,C])}"
+ ]
+ },
+ "execution_count": 33,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"P"
]
@@ -1306,7 +1381,7 @@
},
{
"cell_type": "code",
- "execution_count": 42,
+ "execution_count": 34,
"metadata": {},
"outputs": [
{
@@ -1318,7 +1393,7 @@
"TRUE"
]
},
- "execution_count": 42,
+ "execution_count": 34,
"metadata": {},
"output_type": "execute_result"
}
@@ -1329,7 +1404,7 @@
},
{
"cell_type": "code",
- "execution_count": 41,
+ "execution_count": 35,
"metadata": {},
"outputs": [
{
@@ -1341,7 +1416,7 @@
"{([a,B]↦[a,b]),([b,B]↦[b,b]),([b,C]↦[b,c]),([c,C]↦[c,c]),([S]↦[a,S,B,C]),([S]↦[a,B,C]),([C,B]↦[B,C])}"
]
},
- "execution_count": 41,
+ "execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
@@ -1359,7 +1434,7 @@
},
{
"cell_type": "code",
- "execution_count": 43,
+ "execution_count": 36,
"metadata": {},
"outputs": [
{
@@ -1371,7 +1446,7 @@
"FALSE"
]
},
- "execution_count": 43,
+ "execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
@@ -1390,7 +1465,7 @@
},
{
"cell_type": "code",
- "execution_count": 44,
+ "execution_count": 37,
"metadata": {},
"outputs": [
{
@@ -1417,7 +1492,7 @@
},
{
"cell_type": "code",
- "execution_count": 45,
+ "execution_count": 38,
"metadata": {},
"outputs": [
{
@@ -1429,7 +1504,7 @@
"{([S]↦[]),([S]↦[a,S,b])}"
]
},
- "execution_count": 45,
+ "execution_count": 38,
"metadata": {},
"output_type": "execute_result"
}
@@ -1440,7 +1515,7 @@
},
{
"cell_type": "code",
- "execution_count": 46,
+ "execution_count": 39,
"metadata": {},
"outputs": [
{
@@ -1458,7 +1533,7 @@
"\tP1 = {([S]↦[]),([S]↦[a,S,b])}"
]
},
- "execution_count": 46,
+ "execution_count": 39,
"metadata": {},
"output_type": "execute_result"
}
@@ -1476,7 +1551,7 @@
},
{
"cell_type": "code",
- "execution_count": 47,
+ "execution_count": 40,
"metadata": {},
"outputs": [
{
@@ -1488,7 +1563,7 @@
"FALSE"
]
},
- "execution_count": 47,
+ "execution_count": 40,
"metadata": {},
"output_type": "execute_result"
}
@@ -1499,7 +1574,7 @@
},
{
"cell_type": "code",
- "execution_count": 48,
+ "execution_count": 41,
"metadata": {},
"outputs": [
{
@@ -1521,7 +1596,7 @@
"\tP1 = {([S]↦[]),([S]↦[a,S,b])}"
]
},
- "execution_count": 48,
+ "execution_count": 41,
"metadata": {},
"output_type": "execute_result"
}
@@ -1560,6 +1635,178 @@
"* $REG \\subseteq CF \\subseteq CS \\subseteq \\mathfrak{L}_0.$"
]
},
+ {
+ "cell_type": "code",
+ "execution_count": 42,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Loaded machine: Chomsky"
+ ]
+ },
+ "execution_count": 42,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "::load\n",
+ "MACHINE Chomsky\n",
+ "SETS ΣN = {a, b, c, S, A, B, C}\n",
+ "CONSTANTS Σ, N, P0, P1, P2, P3, P4, Grammatiken\n",
+ "PROPERTIES\n",
+ " Σ = {a, b, c} ∧\n",
+ " N = {S, A, B, C} ∧\n",
+ " Σ ∩ N = {} ∧\n",
+ " Σ ∪ N = ΣN ∧\n",
+ " //L0 = {x∈{a, b}^* | x enthält gleich viele as und bs}\n",
+ " //Ohne Sonderrregelung für lambda ist L0 von Typ 0.\n",
+ " P0 = {([S] ↦ [a, b]), ([S] ↦ [S, S]), ([S] ↦ []), ([a, b] ↦ [b, a]), ([b, a] ↦ [a, b])} ∧\n",
+ " //L1 = {a^mb^mc^m | m≥1}\n",
+ " P1 = {([S] ↦ [S, S]), ([S] ↦ [A, B, C]), ([B, A] ↦ [A, B]), ([C, A] ↦ [A, C]), ([C, B] ↦ [B, C]),\n",
+ " ([A] ↦ [a]), ([a, B] ↦ [a, b]), ([b, B] ↦ [b, b]), ([b, C] ↦ [b, c]), ([c, C] ↦ [c, c])} ∧\n",
+ " //L2 = {x∈{a, b}^+ | x enthält gleich viele as und bs}\n",
+ " P2 = {([S] ↦ [A, B]), ([S] ↦ [B, A]),\n",
+ " ([A] ↦ [A, A, B]), ([A] ↦ [A, B, A]), ([A] ↦ [B, A, A]), ([A] ↦ [a]),\n",
+ " ([B] ↦ [B, B, A]), ([B] ↦ [B, A, B]), ([B] ↦ [A, B, B]), ([B] ↦ [b])} ∧\n",
+ " //L3 = L4 = {a, b, c}\n",
+ " P3 = {([S] ↦ [a]), ([S] ↦ [b]), ([S] ↦ [c])} ∧\n",
+ " P4 = {([S] ↦ [A, A]), ([A, A] ↦ [a]), ([S] ↦ [b]), ([S] ↦ [c])} ∧\n",
+ " Grammatiken = [P0, P1, P2, P3, P4]\n",
+ " \n",
+ "DEFINITIONS\n",
+ " SET_PREF_PP_SEQUENCES == TRUE;\n",
+ " L0(R) == TRUE; //Jede Grammatik ist vom Typ 0\n",
+ " L1(R) == IF ∀(p,q).( p↦q ∈ R ⇒ size(p) ≤ size(q)) THEN L0(R) ELSE FALSE END;\n",
+ " L2(R) == IF ∀(p,q).( p↦q ∈ R ⇒ size(p)=1 & p∈seq(N)) THEN L1(R) ELSE FALSE END;\n",
+ " L3(R) == IF ∀(p,q).( p↦q ∈ R ⇒ size(q)>0 & size(q) ≤ 2 & first(q)∈Σ & (size(q)≤1 or first(tail(q))∈N)) THEN L2(R) ELSE FALSE END;\n",
+ " \n",
+ " \"LibraryStrings.def\";\n",
+ " ANIMATION_FUNCTION_DEFAULT == {(0,0,\"P\"), (1,0,\"P0\"), (2,0,\"P1\"), (3,0,\"P2\"), (4,0,\"P3\"), (5,0,\"P4\")};\n",
+ " ANIMATION_FUNCTION1 == {c,r,i| r=1 ∧ c∈dom(Grammatiken) ∧ i=TO_STRING(L0(Grammatiken(c)))} ∪ {(0,1,\"L0(P)\")};\n",
+ " ANIMATION_FUNCTION2 == {c,r,i| r=2 ∧ c∈dom(Grammatiken) ∧ i=TO_STRING(L1(Grammatiken(c)))} ∪ {(0,2,\"L1(P)\")};\n",
+ " ANIMATION_FUNCTION3 == {c,r,i| r=3 ∧ c∈dom(Grammatiken) ∧ i=TO_STRING(L2(Grammatiken(c)))} ∪ {(0,3,\"L2(P)\")};\n",
+ " ANIMATION_FUNCTION4 == {c,r,i| r=4 ∧ c∈dom(Grammatiken) ∧ i=TO_STRING(L3(Grammatiken(c)))} ∪ {(0,4,\"L3(P)\")};\n",
+ "END"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 43,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Machine constants set up using operation 0: $setup_constants()"
+ ]
+ },
+ "execution_count": 43,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":constants"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 44,
+ "metadata": {
+ "scrolled": true
+ },
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Machine initialised using operation 1: $initialise_machine()"
+ ]
+ },
+ "execution_count": 44,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":init"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 45,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "<table style=\"font-family:monospace\"><tbody>\n",
+ "<tr>\n",
+ "<td style=\"padding:10px\">P</td>\n",
+ "<td style=\"padding:10px\">L0(P)</td>\n",
+ "<td style=\"padding:10px\">L1(P)</td>\n",
+ "<td style=\"padding:10px\">L2(P)</td>\n",
+ "<td style=\"padding:10px\">L3(P)</td>\n",
+ "</tr>\n",
+ "<tr>\n",
+ "<td style=\"padding:10px\">P0</td>\n",
+ "<td style=\"padding:10px\">TRUE</td>\n",
+ "<td style=\"padding:10px\">FALSE</td>\n",
+ "<td style=\"padding:10px\">FALSE</td>\n",
+ "<td style=\"padding:10px\">FALSE</td>\n",
+ "</tr>\n",
+ "<tr>\n",
+ "<td style=\"padding:10px\">P1</td>\n",
+ "<td style=\"padding:10px\">TRUE</td>\n",
+ "<td style=\"padding:10px\">TRUE</td>\n",
+ "<td style=\"padding:10px\">FALSE</td>\n",
+ "<td style=\"padding:10px\">FALSE</td>\n",
+ "</tr>\n",
+ "<tr>\n",
+ "<td style=\"padding:10px\">P2</td>\n",
+ "<td style=\"padding:10px\">TRUE</td>\n",
+ "<td style=\"padding:10px\">TRUE</td>\n",
+ "<td style=\"padding:10px\">TRUE</td>\n",
+ "<td style=\"padding:10px\">FALSE</td>\n",
+ "</tr>\n",
+ "<tr>\n",
+ "<td style=\"padding:10px\">P3</td>\n",
+ "<td style=\"padding:10px\">TRUE</td>\n",
+ "<td style=\"padding:10px\">TRUE</td>\n",
+ "<td style=\"padding:10px\">TRUE</td>\n",
+ "<td style=\"padding:10px\">TRUE</td>\n",
+ "</tr>\n",
+ "<tr>\n",
+ "<td style=\"padding:10px\">P4</td>\n",
+ "<td style=\"padding:10px\">TRUE</td>\n",
+ "<td style=\"padding:10px\">FALSE</td>\n",
+ "<td style=\"padding:10px\">FALSE</td>\n",
+ "<td style=\"padding:10px\">FALSE</td>\n",
+ "</tr>\n",
+ "</tbody></table>"
+ ],
+ "text/plain": [
+ "<Animation function visualisation>"
+ ]
+ },
+ "execution_count": 45,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":show"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Die obige Tabelle zeigt, in welche Sprachklasse die entsprechenden Grammatiken fallen. Dabei ist zu beachten, dass L3 = L4 ist, aber P3 und P4 unterschiedliche Typen haben. Die Sprache fällt allerdings immer in die größte Spracheklasse zu der es eine Grammatik gibt, die diese Sprache erzeugt. Also gilt $L4 ∈ \\mathfrak{L}_3 = REG$. Daher reicht die Angabe einer Grammatik nicht aus, um zu zeigen, dass eine Sprache höchstens in einer der Klassen liegt. Um dies zu zeigen, gibt es verschiedene Methoden, wie z.B. das Pumping-Lemma oder die Myhill-Nerode Relation."
+ ]
+ },
{
"cell_type": "code",
"execution_count": null,
diff --git a/info4/kapitel-10/Das Halteproblem.ipynb b/info4/kapitel-10/Das Halteproblem.ipynb
new file mode 100644
index 0000000000000000000000000000000000000000..60fbb3f3e86c0e67ef68aa593db917b280fba772
--- /dev/null
+++ b/info4/kapitel-10/Das Halteproblem.ipynb
@@ -0,0 +1,83 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Das (spezielle) Halteproblem\n",
+ "\n",
+ "Im Skript wird das Halteproblem formal definiert. Hier wollen wir das Halteproblem einmal intuitiv betrachten.\n",
+ "\n",
+ "Intuitiv gesehen stellt man beim speziellen Halteproblem die Frage, ob eine Turingmaschiene hält, wenn Sie sich selbst als eine Eingabe erhält.\n",
+ "Um Turingmaschienen als Eingaben nutzen zu können benötigen wir eine Darstellung als Wörter über einem Alphabet. Als Alphabet wählen wir $\\Sigma$ = {0, 1} und als Möglichkeit eine TM über $\\Sigma$ darzustellen wählen wir die Gödelisierung. Die genau Definition finden Sie im Skript auf Seite 153 (zwischen Korrolar 10.18 und Beispiel 10.19). Für unsere Zwecke reicht es allerdings aus zu verstehen, dass mit der Gödelisierung jeder TM ein Wort aus $\\Sigma^*$ zugeordet wird. Auf Grund der Art der Gödelisierung gilt die Umkehrung allerdings nicht. Um die Umkehrabbildung angeben zu können sei $M_0$ eine beliebige feste TM. Dann ist für jedes Wort $w\\in\\Sigma^*$ eine TM $M_w$ definiert durch:\n",
+ "\n",
+ "$M_w$ = M, falls w = code(M)\n",
+ "\n",
+ "$M_w$ = $M_0$, sonst\n",
+ "\n",
+ "Mit dieser Definition lässt sich das spezielle Halteproblem definieren als:\n",
+ "$K=\\{w\\in\\{0,1\\}^* | \\text{$M_w$ angesetzt auf w hält nach endlich vielen Schritten}\\}$\n",
+ "\n",
+ "Nun zeigen wir, dass K nicht entscheidbar ist, es also keine TM gibt, die für jedes $x\\in\\Sigma^*$ entscheidet, ob es in K liegt oder nicht. Für einen Wiederspruchsbeweis nehmen wir an, dass K entscheidbar wäre.\n",
+ "\n",
+ "Sei H dann die TM, die für alle x entscheided, ob x in K liegt oder nicht.\n",
+ "Wenn also die TM M mit code(M)=x hält, dann gibt H bei der Eingabe x 1 und sonst 0 aus.\n",
+ "\n",
+ "Sei nun N eine TM, die bei der Eingabe 1 in eine Endlosschleife läuft und bei der Eingabe 0 eine 1 ausgibt.\n",
+ "\n",
+ "Betrachten wir nun die TM C, die aus der Hintereinanderschaltung von H und N entsteht.\n",
+ "Wenn wir C auf die Gödelisierung einer TM anwenden, die mit sich selbst als Eingabe hält, dann gibt H eine 1 aus und somit hält N nicht an. Damit hält auch C nicht.\n",
+ "Und wenn wir C auf die Gödelisierung einer TM anwenden, die in eine Endlosschleife läuft, wenn Sie sich selbst als Eingabe erhält, dann gibt H eine O aus und somit N eine 1 und C hält.\n",
+ "\n",
+ "Wenn wir nun C auf sich selbst anwenden, was geschieht dann?\n",
+ "Angenommen C hält, dann gibt H eine 1 aus und N läuft in eine Endlosschleife. Daher hält C nicht.\n",
+ "Wenn C allerdings in eine Endlosschleife läuft, dann gibt H eine 0 aus und somit N eine 1. Damit hält C.\n",
+ "Dies ist ein Wiederspruch dazu, dass K entscheidbar ist."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "<table align=\"LEFT\">\n",
+ " <tr>\n",
+ " <td width=50%><center>Die Turingmaschiene $H$</center></td>\n",
+ " <td width=50%><center>Die Turingmaschiene $N$</center></td>\n",
+ " </tr>\n",
+ " <tr>\n",
+ " <td width=50% align=\"CENTER\"><img src=\"./img/TM H.svg\" width=100%></td>\n",
+ " <td width=50%><img src=\"./img/TM N.svg\" width=100%></td>\n",
+ " </tr>\n",
+ " <tr>\n",
+ " <td width=100% colspan=2><center>Die Turingmaschiene $C$</center></td>\n",
+ " </tr>\n",
+ " <tr>\n",
+ " <td width=100% colspan=2><img src=\"./img/TM C2.svg\" width=100%></td>\n",
+ " </tr>\n",
+ "</table>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "ProB 2",
+ "language": "prob",
+ "name": "prob2"
+ },
+ "language_info": {
+ "codemirror_mode": "prob2_jupyter_repl",
+ "file_extension": ".prob",
+ "mimetype": "text/x-prob2-jupyter-repl",
+ "name": "prob"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/info4/kapitel-10/img/TM C.svg b/info4/kapitel-10/img/TM C.svg
new file mode 100644
index 0000000000000000000000000000000000000000..c6eefeb43ea254c58e02a00d20d6ded2f7110b21
--- /dev/null
+++ b/info4/kapitel-10/img/TM C.svg
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diff --git a/info4/kapitel-10/img/TM C2.svg b/info4/kapitel-10/img/TM C2.svg
new file mode 100644
index 0000000000000000000000000000000000000000..00d1eef43bf588f7de9f46c9676114c1ed004478
--- /dev/null
+++ b/info4/kapitel-10/img/TM C2.svg
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+ <g id="bullet-char-template-61548" transform="scale(0.00048828125,-0.00048828125)">
+ <path d="M 173,740 C 173,903 231,1043 346,1159 462,1274 601,1332 765,1332 928,1332 1067,1274 1183,1159 1299,1043 1357,903 1357,740 1357,577 1299,437 1183,322 1067,206 928,148 765,148 601,148 462,206 346,322 231,437 173,577 173,740 Z"/>
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+ </defs>
+ <g>
+ <g id="id2" class="Master_Slide">
+ <g id="bg-id2" class="Background"/>
+ <g id="bo-id2" class="BackgroundObjects"/>
+ </g>
+ </g>
+ <g class="SlideGroup">
+ <g>
+ <g id="container-id1">
+ <g id="id1" class="Slide" clip-path="url(#presentation_clip_path)">
+ <g class="Page">
+ <g class="Graphic">
+ <g id="id3">
+ <rect class="BoundingBox" stroke="none" fill="none" x="4700" y="1092" width="21693" height="5225"/>
+ <defs>
+ <clipPath id="clip_path_1" clipPathUnits="userSpaceOnUse">
+ <path d="M 4728,1117 L 26143,1117 26143,6307 4728,6307 4728,1117 Z"/>
+ </clipPath>
+ </defs>
+ <g clip-path="url(#clip_path_1)">
+ <path fill="none" stroke="rgb(0,0,0)" stroke-width="25" stroke-linejoin="miter" d="M 10949,6002 L 8448,6002 8448,1406 13451,1406 13451,6002 10949,6002 Z"/>
+ <path fill="rgb(0,0,0)" stroke="none" d="M 6463,3745 L 8168,3725 8169,3799 8389,3686 8166,3577 8167,3651 6463,3671 6463,3745 Z"/>
+ <text class="TextShape"><tspan class="TextParagraph" font-family="Calibri, sans-serif" font-size="295px" font-weight="400"><tspan class="TextPosition" x="6787" y="3612"><tspan fill="rgb(0,0,0)" stroke="none">code(M) für </tspan></tspan></tspan></text>
+ <text class="TextShape"><tspan class="TextParagraph" font-family="Calibri, sans-serif" font-size="295px" font-weight="400"><tspan class="TextPosition" x="6787" y="3962"><tspan fill="rgb(0,0,0)" stroke="none">eine TM M</tspan></tspan></tspan></text>
+ <path fill="rgb(0,0,0)" stroke="none" d="M 13455,2636 L 15968,3606 15995,3537 16162,3720 15915,3744 15942,3675 13428,2705 13455,2636 Z"/>
+ <path fill="rgb(0,0,0)" stroke="none" d="M 13454,4686 L 15965,3794 15990,3864 16162,3686 15916,3655 15940,3725 13429,4617 13454,4686 Z"/>
+ <text class="TextShape"><tspan class="TextParagraph" font-family="Calibri, sans-serif" font-size="277px" font-weight="400"><tspan class="TextPosition" x="13720" y="1932"><tspan fill="rgb(0,0,0)" stroke="none">1, falls M bei </tspan></tspan></tspan></text>
+ <text class="TextShape"><tspan class="TextParagraph" font-family="Calibri, sans-serif" font-size="277px" font-weight="400"><tspan class="TextPosition" x="13720" y="2246"><tspan fill="rgb(0,0,0)" stroke="none">Eingabe code(M) </tspan></tspan></tspan></text>
+ <text class="TextShape"><tspan class="TextParagraph" font-family="Calibri, sans-serif" font-size="277px" font-weight="400"><tspan class="TextPosition" x="13720" y="2578"><tspan fill="rgb(0,0,0)" stroke="none">hält </tspan></tspan></tspan></text>
+ <text class="TextShape"><tspan class="TextParagraph" font-family="Calibri, sans-serif" font-size="295px" font-weight="400"><tspan class="TextPosition" x="13768" y="4830"><tspan fill="rgb(0,0,0)" stroke="none">0, sonst </tspan></tspan></tspan></text>
+ <text class="TextShape"><tspan class="TextParagraph" font-family="Calibri, sans-serif" font-size="443px" font-weight="400"><tspan class="TextPosition" x="8753" y="2153"><tspan fill="rgb(0,0,0)" stroke="none">Die TM H</tspan></tspan></tspan></text>
+ <path fill="none" stroke="rgb(0,0,0)" stroke-width="74" stroke-linejoin="miter" d="M 21121,5338 C 21121,4976 21316,4682 21555,4682 21795,4682 21989,4976 21989,5338 21989,5700 21795,5993 21555,5993 21316,5993 21121,5700 21121,5338 Z"/>
+ <path fill="rgb(255,255,255)" stroke="none" d="M 18703,5928 L 16146,5928 16146,1664 21260,1664 21260,5928 18703,5928 Z"/>
+ <path fill="none" stroke="rgb(0,0,0)" stroke-width="25" stroke-linejoin="miter" d="M 18703,5928 L 16146,5928 16146,1664 21260,1664 21260,5928 18703,5928 Z"/>
+ <path fill="rgb(0,0,0)" stroke="none" d="M 21251,2799 L 24059,2799 24059,2726 24280,2836 24059,2947 24059,2873 21251,2873 21251,2799 Z"/>
+ <text class="TextShape"><tspan class="TextParagraph" font-family="Calibri, sans-serif" font-size="295px" font-weight="400"><tspan class="TextPosition" x="21569" y="2596"><tspan fill="rgb(0,0,0)" stroke="none">1, falls die </tspan></tspan></tspan></text>
+ <text class="TextShape"><tspan class="TextParagraph" font-family="Calibri, sans-serif" font-size="295px" font-weight="400"><tspan class="TextPosition" x="21569" y="3298"><tspan fill="rgb(0,0,0)" stroke="none">Eingabe 0 ist </tspan></tspan></tspan></text>
+ <text class="TextShape"><tspan class="TextParagraph" font-family="Calibri, sans-serif" font-size="295px" font-weight="400"><tspan class="TextPosition" x="22159" y="5218"><tspan fill="rgb(0,0,0)" stroke="none">Endlosschleife, </tspan></tspan></tspan></text>
+ <text class="TextShape"><tspan class="TextParagraph" font-family="Calibri, sans-serif" font-size="295px" font-weight="400"><tspan class="TextPosition" x="22159" y="5568"><tspan fill="rgb(0,0,0)" stroke="none">sonst </tspan></tspan></tspan></text>
+ <text class="TextShape"><tspan class="TextParagraph" font-family="Calibri, sans-serif" font-size="443px" font-weight="400"><tspan class="TextPosition" x="16457" y="2393"><tspan fill="rgb(0,0,0)" stroke="none">Die TM N</tspan></tspan></tspan></text>
+ <path fill="rgb(0,0,0)" stroke="none" d="M 21272,6018 L 21251,5771 21461,5902 21272,6018 Z"/>
+ <path fill="none" stroke="rgb(47,82,143)" stroke-width="25" stroke-linejoin="miter" d="M 15444,6298 L 6602,6298 6602,1129 24287,1129 24287,6298 15444,6298 Z"/>
+ <path fill="rgb(0,0,0)" stroke="none" d="M 4728,3667 L 6371,3667 6371,3593 6593,3704 6371,3815 6371,3741 4728,3741 4728,3667 Z"/>
+ <path fill="rgb(0,0,0)" stroke="none" d="M 24278,2799 L 25922,2799 25922,2726 26143,2836 25922,2947 25922,2873 24278,2873 24278,2799 Z"/>
+ </g>
+ </g>
+ </g>
+ </g>
+ </g>
+ </g>
+ </g>
+ </g>
+</svg>
\ No newline at end of file
diff --git a/info4/kapitel-10/img/TM H.svg b/info4/kapitel-10/img/TM H.svg
new file mode 100644
index 0000000000000000000000000000000000000000..b6857edeadf72276b3fed8427f9cc3971c0a87b1
--- /dev/null
+++ b/info4/kapitel-10/img/TM H.svg
@@ -0,0 +1 @@
+<svg width="853" height="234" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" overflow="hidden"><defs><clipPath id="clip0"><rect x="162" y="141" width="853" height="234"/></clipPath></defs><g clip-path="url(#clip0)" transform="translate(-162 -141)"><rect x="319.5" y="143.5" width="443" height="230" stroke="#000000" stroke-width="1.33333" stroke-miterlimit="8" fill="none"/><path d="M168 256 309.486 256 309.486 260 168 260ZM307.486 252 319.486 258 307.486 264Z"/><text font-family="Calibri,Calibri_MSFontService,sans-serif" font-weight="400" font-size="24" transform="translate(177.101 236)">code(M) für<tspan x="0" y="58">eine TM M</tspan></text><path d="M762 205 992.359 205 992.359 209 762 209ZM990.359 201 1002.36 207 990.359 213Z"/><path d="M762 304 992.359 304 992.359 308 762 308ZM990.359 300 1002.36 306 990.359 312Z"/><text font-family="Calibri,Calibri_MSFontService,sans-serif" font-weight="400" font-size="24" transform="translate(783.855 185)">1, falls M bei Eingabe<tspan x="0" y="58">code(M) hält</tspan><tspan x="6.10352e-05" y="101">0, sonst</tspan><tspan x="-443.153" y="-2">Die TM H</tspan></text></g></svg>
\ No newline at end of file
diff --git a/info4/kapitel-10/img/TM H2.svg b/info4/kapitel-10/img/TM H2.svg
new file mode 100644
index 0000000000000000000000000000000000000000..9f9a681bb10e369021cfb2ebdb2c4c8a9d4ab3b3
--- /dev/null
+++ b/info4/kapitel-10/img/TM H2.svg
@@ -0,0 +1 @@
+<svg width="1160" height="282" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" overflow="hidden"><defs><clipPath id="clip0"><rect x="59" y="219" width="1160" height="282"/></clipPath></defs><g clip-path="url(#clip0)" transform="translate(-59 -219)"><rect x="261.5" y="235.5" width="271" height="249" stroke="#000000" stroke-width="1.33333" stroke-miterlimit="8" fill="none"/><path d="M0.0236198-1.99986 94.3379-0.885938 94.2907 3.11378-0.0236198 1.99986ZM92.3853-4.90928 104.314 1.23202 92.2436 7.08988Z" transform="matrix(1 0 0 -1 161 360.232)"/><text font-family="Calibri,Calibri_MSFontService,sans-serif" font-weight="400" font-size="16" transform="translate(178.04 355)">code(M) für<tspan x="0" y="19">eine TM M</tspan></text><path d="M532.72 302.134 670.737 355.422 669.296 359.154 531.28 305.866ZM670.312 350.97 679.345 360.89 665.99 362.165Z"/><path d="M0.669392-1.88465 138.591 47.1025 137.252 50.8718-0.669392 1.88465ZM138.045 42.6639 147.345 52.3342 134.029 53.9718Z" transform="matrix(1 0 0 -1 532 411.334)"/><text font-family="Calibri,Calibri_MSFontService,sans-serif" font-weight="400" font-size="15" transform="translate(549.699 270)">1, falls M bei <tspan x="0" y="17">Eingabe code(M) </tspan><tspan x="0" y="35">hält</tspan><tspan font-size="16" x="-0.000183105" y="151">0, sonst</tspan><tspan font-size="24" x="-271.662" y="6">Die TM H</tspan></text><path d="M948 448.5C948 428.894 958.521 413 971.5 413 984.479 413 995 428.894 995 448.5 995 468.106 984.479 484 971.5 484 958.521 484 948 468.106 948 448.5Z" stroke="#000000" stroke-width="4" stroke-miterlimit="8" fill="none" fill-rule="evenodd"/><rect x="678.5" y="249.5" width="277" height="231" stroke="#000000" stroke-width="1.33333" stroke-miterlimit="8" fill="#FFFFFF"/><path d="M955 311 1094.83 311 1094.83 315 955 315ZM1092.83 307 1104.83 313 1092.83 319Z"/><text font-family="Calibri,Calibri_MSFontService,sans-serif" font-weight="400" font-size="16" transform="translate(972.277 300)">1, falls die <tspan x="0" y="38">Eingabe 0 ist</tspan><tspan x="31.9123" y="142">Endlosschleife, </tspan><tspan x="31.9123" y="161">sonst</tspan><tspan font-size="24" x="-276.951" y="-11">Die TM N</tspan></text><path d="M956.183 485.364 955 472 966.401 479.072Z"/><rect x="161.5" y="220.5" width="944" height="280" stroke="#2F528F" stroke-width="1.33333" stroke-miterlimit="8" fill="none"/><path d="M60 358 151.027 358 151.027 362 60 362ZM149.027 354 161.027 360 149.027 366Z"/><path d="M1105 313 1196.03 313 1196.03 317 1105 317ZM1194.03 309 1206.03 315 1194.03 321Z"/></g></svg>
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diff --git a/info4/kapitel-10/img/TM N.svg b/info4/kapitel-10/img/TM N.svg
new file mode 100644
index 0000000000000000000000000000000000000000..6883d36ac620ce97a1bd5d28d5035e39ce531cd0
--- /dev/null
+++ b/info4/kapitel-10/img/TM N.svg
@@ -0,0 +1 @@
+<svg width="838" height="243" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" overflow="hidden"><defs><clipPath id="clip0"><rect x="162" y="141" width="838" height="243"/></clipPath></defs><g clip-path="url(#clip0)" transform="translate(-162 -141)"><path d="M750 342C750 322.67 766.789 307 787.5 307 808.211 307 825 322.67 825 342 825 361.33 808.211 377 787.5 377 766.789 377 750 361.33 750 342Z" stroke="#000000" stroke-width="4" stroke-miterlimit="8" fill="none" fill-rule="evenodd"/><rect x="319.5" y="143.5" width="443" height="230" stroke="#000000" stroke-width="1.33333" stroke-miterlimit="8" fill="#FFFFFF"/><path d="M0.0195984-1.9999 141.506-0.613381 141.467 3.38643-0.0195984 1.9999ZM139.546-4.63279 151.486 1.48451 139.428 7.36664Z" transform="matrix(1 0 0 -1 168 259.485)"/><text font-family="Calibri,Calibri_MSFontService,sans-serif" font-weight="400" font-size="24" transform="translate(177.101 236)">0 oder 1</text><path d="M1.05382e-06-2 189.249-1.9999 189.249 2.0001-1.05382e-06 2ZM187.249-5.9999 199.249 0.000104987 187.249 6.0001Z" transform="matrix(1 0 0 -1 762 207)"/><text font-family="Calibri,Calibri_MSFontService,sans-serif" font-weight="400" font-size="24" transform="translate(783.855 185)">1, falls die <tspan x="0" y="58">Eingabe 0 ist</tspan><tspan x="50.6421" y="149">Endlosschleife, </tspan><tspan x="50.6421" y="178">sonst</tspan><tspan x="-443.153" y="-2">Die TM N</tspan></text><path d="M766.143 377.761 762 365 774.694 369.342Z"/></g></svg>
\ No newline at end of file
diff --git a/info4/kapitel-11/PCP.ipynb b/info4/kapitel-11/PCP.ipynb
index 848d689913ad559f027581b011ece86eab95d4dd..35951dcc3b2658fa60fac8511a76d56dc30b9a58 100644
--- a/info4/kapitel-11/PCP.ipynb
+++ b/info4/kapitel-11/PCP.ipynb
@@ -33,7 +33,7 @@
},
{
"cell_type": "code",
- "execution_count": 190,
+ "execution_count": 1,
"metadata": {},
"outputs": [
{
@@ -42,7 +42,7 @@
"Loaded machine: PostCorrespondence_MC"
]
},
- "execution_count": 190,
+ "execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
@@ -85,7 +85,7 @@
},
{
"cell_type": "code",
- "execution_count": 191,
+ "execution_count": 2,
"metadata": {},
"outputs": [
{
@@ -94,7 +94,7 @@
"Machine constants set up using operation 0: $setup_constants()"
]
},
- "execution_count": 191,
+ "execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
@@ -105,7 +105,7 @@
},
{
"cell_type": "code",
- "execution_count": 192,
+ "execution_count": 3,
"metadata": {},
"outputs": [
{
@@ -114,7 +114,7 @@
"Machine initialised using operation 1: $initialise_machine()"
]
},
- "execution_count": 192,
+ "execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
@@ -125,7 +125,7 @@
},
{
"cell_type": "code",
- "execution_count": 193,
+ "execution_count": 4,
"metadata": {},
"outputs": [
{
@@ -137,7 +137,7 @@
"[([1]↦[1,0]),([0,1]↦[1]),([0,1,0]↦[1,0,0])]"
]
},
- "execution_count": 193,
+ "execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
@@ -148,7 +148,7 @@
},
{
"cell_type": "code",
- "execution_count": 194,
+ "execution_count": 5,
"metadata": {},
"outputs": [
{
@@ -167,7 +167,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 194,
+ "execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
@@ -178,7 +178,7 @@
},
{
"cell_type": "code",
- "execution_count": 195,
+ "execution_count": 6,
"metadata": {},
"outputs": [
{
@@ -192,7 +192,7 @@
"Schritt(1,[1],[1,0])"
]
},
- "execution_count": 195,
+ "execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
@@ -210,7 +210,7 @@
},
{
"cell_type": "code",
- "execution_count": 196,
+ "execution_count": 7,
"metadata": {},
"outputs": [
{
@@ -219,7 +219,7 @@
"Executed operation: Schritt(1,[1],[1,0])"
]
},
- "execution_count": 196,
+ "execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
@@ -230,7 +230,7 @@
},
{
"cell_type": "code",
- "execution_count": 197,
+ "execution_count": 8,
"metadata": {},
"outputs": [
{
@@ -253,7 +253,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 197,
+ "execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
@@ -264,7 +264,7 @@
},
{
"cell_type": "code",
- "execution_count": 198,
+ "execution_count": 9,
"metadata": {},
"outputs": [
{
@@ -276,7 +276,7 @@
"[1]"
]
},
- "execution_count": 198,
+ "execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
@@ -287,7 +287,7 @@
},
{
"cell_type": "code",
- "execution_count": 199,
+ "execution_count": 10,
"metadata": {},
"outputs": [
{
@@ -299,7 +299,7 @@
"[1,0]"
]
},
- "execution_count": 199,
+ "execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
@@ -310,7 +310,7 @@
},
{
"cell_type": "code",
- "execution_count": 200,
+ "execution_count": 11,
"metadata": {},
"outputs": [
{
@@ -325,7 +325,7 @@
"Schritt(3,[0,1,0],[1,0,0])"
]
},
- "execution_count": 200,
+ "execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
@@ -343,7 +343,7 @@
},
{
"cell_type": "code",
- "execution_count": 201,
+ "execution_count": 12,
"metadata": {},
"outputs": [
{
@@ -352,7 +352,7 @@
"Executed operation: Schritt(2,[0,1],[1])"
]
},
- "execution_count": 201,
+ "execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
@@ -363,7 +363,7 @@
},
{
"cell_type": "code",
- "execution_count": 202,
+ "execution_count": 13,
"metadata": {},
"outputs": [
{
@@ -388,7 +388,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 202,
+ "execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
@@ -399,7 +399,7 @@
},
{
"cell_type": "code",
- "execution_count": 203,
+ "execution_count": 14,
"metadata": {},
"outputs": [
{
@@ -408,7 +408,7 @@
"Executed operation: Lösung()"
]
},
- "execution_count": 203,
+ "execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
@@ -427,7 +427,7 @@
},
{
"cell_type": "code",
- "execution_count": 204,
+ "execution_count": 15,
"metadata": {},
"outputs": [
{
@@ -449,7 +449,7 @@
"4: Lösung() (current)"
]
},
- "execution_count": 204,
+ "execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
@@ -460,7 +460,7 @@
},
{
"cell_type": "code",
- "execution_count": 205,
+ "execution_count": 16,
"metadata": {},
"outputs": [
{
@@ -469,7 +469,7 @@
"Changed to state with index 2"
]
},
- "execution_count": 205,
+ "execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
@@ -480,7 +480,7 @@
},
{
"cell_type": "code",
- "execution_count": 206,
+ "execution_count": 17,
"metadata": {},
"outputs": [
{
@@ -503,7 +503,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 206,
+ "execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
@@ -514,7 +514,7 @@
},
{
"cell_type": "code",
- "execution_count": 207,
+ "execution_count": 18,
"metadata": {},
"outputs": [
{
@@ -523,7 +523,7 @@
"Executed operation: Schritt(3,[0,1,0],[1,0,0])"
]
},
- "execution_count": 207,
+ "execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
@@ -534,7 +534,7 @@
},
{
"cell_type": "code",
- "execution_count": 208,
+ "execution_count": 19,
"metadata": {},
"outputs": [
{
@@ -563,7 +563,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 208,
+ "execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
@@ -574,7 +574,7 @@
},
{
"cell_type": "code",
- "execution_count": 209,
+ "execution_count": 20,
"metadata": {},
"outputs": [
{
@@ -583,7 +583,7 @@
"Executed operation: Schritt(3,[0,1,0],[1,0,0])"
]
},
- "execution_count": 209,
+ "execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
@@ -594,7 +594,7 @@
},
{
"cell_type": "code",
- "execution_count": 210,
+ "execution_count": 21,
"metadata": {},
"outputs": [
{
@@ -629,7 +629,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 210,
+ "execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
@@ -640,7 +640,7 @@
},
{
"cell_type": "code",
- "execution_count": 211,
+ "execution_count": 22,
"metadata": {},
"outputs": [
{
@@ -649,7 +649,7 @@
"Executed operation: Schritt(2,[0,1],[1])"
]
},
- "execution_count": 211,
+ "execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
@@ -660,7 +660,7 @@
},
{
"cell_type": "code",
- "execution_count": 212,
+ "execution_count": 23,
"metadata": {},
"outputs": [
{
@@ -697,7 +697,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 212,
+ "execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
@@ -708,7 +708,7 @@
},
{
"cell_type": "code",
- "execution_count": 213,
+ "execution_count": 24,
"metadata": {},
"outputs": [
{
@@ -717,7 +717,7 @@
"Executed operation: Lösung()"
]
},
- "execution_count": 213,
+ "execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
@@ -728,7 +728,7 @@
},
{
"cell_type": "code",
- "execution_count": 214,
+ "execution_count": 25,
"metadata": {},
"outputs": [
{
@@ -754,7 +754,7 @@
"6: Lösung() (current)"
]
},
- "execution_count": 214,
+ "execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
@@ -770,7 +770,7 @@
"Sie können das B Modell oben mit anderen Werten für K ausprobieren:\n",
"- K = [ [0,1]↦[1], [0,1,0]↦[1,0,0], [1]↦[1,0] ] Problem von Folie 36 */\n",
"- K = [ [1]↦[1,0,1], [1,0]↦[0,0], [0,1,1]↦[1,1] ] \n",
- "- K = [ [0,0,1]↦[0], [0,1]↦[0,1,1], [0,1]↦[1,0,1], [1,0] ↦ [0,0,1] ], Problem von Folie 32; kleinse Lösung hat 66 Elemente\n",
+ "- K = [ [0,0,1]↦[0], [0,1]↦[0,1,1], [0,1]↦[1,0,1], [1,0] ↦ [0,0,1] ], Problem von Folie 32; kleinste Lösung hat 66 Elemente\n",
"\n",
"Hier erst einmal Illustration der Lösung von Folie 36.\n",
"Folgende Maschine hat keinen vordefinierten Wert von K; man muss diesen der :constants Anweisung übergeben."
@@ -778,7 +778,7 @@
},
{
"cell_type": "code",
- "execution_count": 215,
+ "execution_count": 26,
"metadata": {},
"outputs": [
{
@@ -787,7 +787,7 @@
"Loaded machine: PostCorrespondence_MC"
]
},
- "execution_count": 215,
+ "execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
@@ -830,7 +830,7 @@
},
{
"cell_type": "code",
- "execution_count": 216,
+ "execution_count": 27,
"metadata": {},
"outputs": [
{
@@ -839,7 +839,7 @@
"Machine constants set up using operation 0: $setup_constants()"
]
},
- "execution_count": 216,
+ "execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
@@ -850,7 +850,7 @@
},
{
"cell_type": "code",
- "execution_count": 217,
+ "execution_count": 28,
"metadata": {},
"outputs": [
{
@@ -859,7 +859,7 @@
"Machine initialised using operation 1: $initialise_machine()"
]
},
- "execution_count": 217,
+ "execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
@@ -870,7 +870,7 @@
},
{
"cell_type": "code",
- "execution_count": 218,
+ "execution_count": 29,
"metadata": {},
"outputs": [
{
@@ -879,7 +879,7 @@
"Executed operation: Schritt(3,[1],[1,0])"
]
},
- "execution_count": 218,
+ "execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
@@ -890,7 +890,7 @@
},
{
"cell_type": "code",
- "execution_count": 219,
+ "execution_count": 30,
"metadata": {},
"outputs": [
{
@@ -913,7 +913,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 219,
+ "execution_count": 30,
"metadata": {},
"output_type": "execute_result"
}
@@ -924,7 +924,7 @@
},
{
"cell_type": "code",
- "execution_count": 220,
+ "execution_count": 31,
"metadata": {},
"outputs": [
{
@@ -933,7 +933,7 @@
"Executed operation: Schritt(2,[0,1,0],[1,0,0])"
]
},
- "execution_count": 220,
+ "execution_count": 31,
"metadata": {},
"output_type": "execute_result"
}
@@ -944,7 +944,7 @@
},
{
"cell_type": "code",
- "execution_count": 221,
+ "execution_count": 32,
"metadata": {},
"outputs": [
{
@@ -973,7 +973,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 221,
+ "execution_count": 32,
"metadata": {},
"output_type": "execute_result"
}
@@ -984,7 +984,7 @@
},
{
"cell_type": "code",
- "execution_count": 222,
+ "execution_count": 33,
"metadata": {},
"outputs": [
{
@@ -993,7 +993,7 @@
"Executed operation: Schritt(2,[0,1,0],[1,0,0])"
]
},
- "execution_count": 222,
+ "execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
@@ -1004,7 +1004,7 @@
},
{
"cell_type": "code",
- "execution_count": 223,
+ "execution_count": 34,
"metadata": {},
"outputs": [
{
@@ -1039,7 +1039,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 223,
+ "execution_count": 34,
"metadata": {},
"output_type": "execute_result"
}
@@ -1050,7 +1050,7 @@
},
{
"cell_type": "code",
- "execution_count": 224,
+ "execution_count": 35,
"metadata": {},
"outputs": [
{
@@ -1059,7 +1059,7 @@
"Executed operation: Schritt(1,[0,1],[1])"
]
},
- "execution_count": 224,
+ "execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
@@ -1070,7 +1070,7 @@
},
{
"cell_type": "code",
- "execution_count": 225,
+ "execution_count": 36,
"metadata": {},
"outputs": [
{
@@ -1107,7 +1107,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 225,
+ "execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
@@ -1125,7 +1125,7 @@
},
{
"cell_type": "code",
- "execution_count": 226,
+ "execution_count": 37,
"metadata": {},
"outputs": [
{
@@ -1134,7 +1134,7 @@
"Changed to state with index -1"
]
},
- "execution_count": 226,
+ "execution_count": 37,
"metadata": {},
"output_type": "execute_result"
}
@@ -1145,7 +1145,7 @@
},
{
"cell_type": "code",
- "execution_count": 227,
+ "execution_count": 38,
"metadata": {},
"outputs": [
{
@@ -1154,7 +1154,7 @@
"Machine constants set up using operation 11: $setup_constants()"
]
},
- "execution_count": 227,
+ "execution_count": 38,
"metadata": {},
"output_type": "execute_result"
}
@@ -1165,7 +1165,7 @@
},
{
"cell_type": "code",
- "execution_count": 228,
+ "execution_count": 39,
"metadata": {},
"outputs": [
{
@@ -1174,7 +1174,7 @@
"Machine initialised using operation 12: $initialise_machine()"
]
},
- "execution_count": 228,
+ "execution_count": 39,
"metadata": {},
"output_type": "execute_result"
}
@@ -1185,7 +1185,7 @@
},
{
"cell_type": "code",
- "execution_count": 229,
+ "execution_count": 40,
"metadata": {},
"outputs": [
{
@@ -1200,7 +1200,7 @@
"Schritt(2,[0,1],[0,1,1])"
]
},
- "execution_count": 229,
+ "execution_count": 40,
"metadata": {},
"output_type": "execute_result"
}
@@ -1211,7 +1211,7 @@
},
{
"cell_type": "code",
- "execution_count": 230,
+ "execution_count": 41,
"metadata": {},
"outputs": [
{
@@ -1220,7 +1220,7 @@
"Executed operation: Schritt(2,[0,1],[0,1,1])"
]
},
- "execution_count": 230,
+ "execution_count": 41,
"metadata": {},
"output_type": "execute_result"
}
@@ -1231,7 +1231,7 @@
},
{
"cell_type": "code",
- "execution_count": 231,
+ "execution_count": 42,
"metadata": {},
"outputs": [
{
@@ -1240,7 +1240,7 @@
"Executed operation: Schritt(4,[1,0],[0,0,1])"
]
},
- "execution_count": 231,
+ "execution_count": 42,
"metadata": {},
"output_type": "execute_result"
}
@@ -1251,7 +1251,7 @@
},
{
"cell_type": "code",
- "execution_count": 232,
+ "execution_count": 43,
"metadata": {},
"outputs": [
{
@@ -1282,7 +1282,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 232,
+ "execution_count": 43,
"metadata": {},
"output_type": "execute_result"
}
@@ -1300,7 +1300,7 @@
},
{
"cell_type": "code",
- "execution_count": 233,
+ "execution_count": 44,
"metadata": {},
"outputs": [
{
@@ -1309,7 +1309,7 @@
"Executed operation: Schritt(3,[0,1],[1,0,1])"
]
},
- "execution_count": 233,
+ "execution_count": 44,
"metadata": {},
"output_type": "execute_result"
}
@@ -1320,7 +1320,7 @@
},
{
"cell_type": "code",
- "execution_count": 234,
+ "execution_count": 45,
"metadata": {},
"outputs": [
{
@@ -1329,7 +1329,7 @@
"Executed operation: Schritt(4,[1,0],[0,0,1])"
]
},
- "execution_count": 234,
+ "execution_count": 45,
"metadata": {},
"output_type": "execute_result"
}
@@ -1340,7 +1340,7 @@
},
{
"cell_type": "code",
- "execution_count": 235,
+ "execution_count": 46,
"metadata": {},
"outputs": [
{
@@ -1349,7 +1349,7 @@
"Executed operation: Schritt(4,[1,0],[0,0,1])"
]
},
- "execution_count": 235,
+ "execution_count": 46,
"metadata": {},
"output_type": "execute_result"
}
@@ -1360,7 +1360,7 @@
},
{
"cell_type": "code",
- "execution_count": 236,
+ "execution_count": 47,
"metadata": {},
"outputs": [
{
@@ -1369,7 +1369,7 @@
"Executed operation: Schritt(2,[0,1],[0,1,1])"
]
},
- "execution_count": 236,
+ "execution_count": 47,
"metadata": {},
"output_type": "execute_result"
}
@@ -1380,7 +1380,7 @@
},
{
"cell_type": "code",
- "execution_count": 237,
+ "execution_count": 48,
"metadata": {},
"outputs": [
{
@@ -1435,7 +1435,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 237,
+ "execution_count": 48,
"metadata": {},
"output_type": "execute_result"
}
@@ -1526,15 +1526,15 @@
"## Kodierung des Halteproblems als MPCP Problem\n",
"\n",
"Folgende B Maschine kodiert das Halteproblem einer Turingmaschine als MPCP Problem.\n",
- "Der erste Schritt zu einer Lösung wird in der Initialisierung kodiert; dort wir erzwungen, dass wir das erste Paar aus dem Skript $(\\#,\\# z_0 w)$ benutzen.\n",
+ "Der erste Schritt zu einer Lösung wird in der Initialisierung kodiert; dort wird erzwungen, dass wir das erste Paar aus dem Skript $(\\#,\\# z_0 w \\#)$ benutzen.\n",
"Die Turing Maschine hat 3 Zustände: z0,z1,z2.\n",
"z2 ist der Endzustand.\n",
- "z0 akzeptiert a and wechselt bei einem b zu z1, z1 akzeptiert b's und geht zu z2 bei einem Blank. \n"
+ "z0 akzeptiert a and wechselt bei einem b zu z1, z1 akzeptiert b's und geht zu z2 bei einem Blank. "
]
},
{
"cell_type": "code",
- "execution_count": 238,
+ "execution_count": 49,
"metadata": {},
"outputs": [
{
@@ -1543,7 +1543,7 @@
"Loaded machine: PostCorrespondence_Turing_MC"
]
},
- "execution_count": 238,
+ "execution_count": 49,
"metadata": {},
"output_type": "execute_result"
}
@@ -1593,7 +1593,7 @@
},
{
"cell_type": "code",
- "execution_count": 239,
+ "execution_count": 50,
"metadata": {},
"outputs": [
{
@@ -1602,7 +1602,7 @@
"Machine constants set up using operation 0: $setup_constants()"
]
},
- "execution_count": 239,
+ "execution_count": 50,
"metadata": {},
"output_type": "execute_result"
}
@@ -1613,7 +1613,7 @@
},
{
"cell_type": "code",
- "execution_count": 240,
+ "execution_count": 51,
"metadata": {},
"outputs": [
{
@@ -1622,7 +1622,7 @@
"Machine initialised using operation 1: $initialise_machine()"
]
},
- "execution_count": 240,
+ "execution_count": 51,
"metadata": {},
"output_type": "execute_result"
}
@@ -1633,7 +1633,7 @@
},
{
"cell_type": "code",
- "execution_count": 241,
+ "execution_count": 52,
"metadata": {},
"outputs": [
{
@@ -1662,7 +1662,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 241,
+ "execution_count": 52,
"metadata": {},
"output_type": "execute_result"
}
@@ -1673,7 +1673,7 @@
},
{
"cell_type": "code",
- "execution_count": 242,
+ "execution_count": 53,
"metadata": {},
"outputs": [
{
@@ -1687,7 +1687,7 @@
"Schritt(4,[z0,a],[a,z0])"
]
},
- "execution_count": 242,
+ "execution_count": 53,
"metadata": {},
"output_type": "execute_result"
}
@@ -1698,7 +1698,7 @@
},
{
"cell_type": "code",
- "execution_count": 243,
+ "execution_count": 54,
"metadata": {},
"outputs": [
{
@@ -1707,7 +1707,7 @@
"Executed operation: Schritt(4,[z0,a],[a,z0])"
]
},
- "execution_count": 243,
+ "execution_count": 54,
"metadata": {},
"output_type": "execute_result"
}
@@ -1718,7 +1718,7 @@
},
{
"cell_type": "code",
- "execution_count": 244,
+ "execution_count": 55,
"metadata": {},
"outputs": [
{
@@ -1751,7 +1751,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 244,
+ "execution_count": 55,
"metadata": {},
"output_type": "execute_result"
}
@@ -1770,7 +1770,7 @@
},
{
"cell_type": "code",
- "execution_count": 245,
+ "execution_count": 56,
"metadata": {},
"outputs": [
{
@@ -1784,7 +1784,7 @@
"Schritt(2,[b],[b])"
]
},
- "execution_count": 245,
+ "execution_count": 56,
"metadata": {},
"output_type": "execute_result"
}
@@ -1795,7 +1795,7 @@
},
{
"cell_type": "code",
- "execution_count": 246,
+ "execution_count": 57,
"metadata": {},
"outputs": [
{
@@ -1804,7 +1804,7 @@
"Executed operation: Schritt(2,[b],[b])"
]
},
- "execution_count": 246,
+ "execution_count": 57,
"metadata": {},
"output_type": "execute_result"
}
@@ -1815,7 +1815,7 @@
},
{
"cell_type": "code",
- "execution_count": 247,
+ "execution_count": 58,
"metadata": {},
"outputs": [
{
@@ -1829,7 +1829,7 @@
"Schritt(3,[Hash],[Hash])"
]
},
- "execution_count": 247,
+ "execution_count": 58,
"metadata": {},
"output_type": "execute_result"
}
@@ -1840,7 +1840,7 @@
},
{
"cell_type": "code",
- "execution_count": 248,
+ "execution_count": 59,
"metadata": {},
"outputs": [
{
@@ -1849,7 +1849,7 @@
"Executed operation: Schritt(3,[Hash],[Hash])"
]
},
- "execution_count": 248,
+ "execution_count": 59,
"metadata": {},
"output_type": "execute_result"
}
@@ -1860,7 +1860,7 @@
},
{
"cell_type": "code",
- "execution_count": 249,
+ "execution_count": 60,
"metadata": {},
"outputs": [
{
@@ -1897,7 +1897,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 249,
+ "execution_count": 60,
"metadata": {},
"output_type": "execute_result"
}
@@ -1908,7 +1908,7 @@
},
{
"cell_type": "code",
- "execution_count": 250,
+ "execution_count": 61,
"metadata": {},
"outputs": [
{
@@ -1922,7 +1922,7 @@
"Schritt(1,[a],[a])"
]
},
- "execution_count": 250,
+ "execution_count": 61,
"metadata": {},
"output_type": "execute_result"
}
@@ -1940,7 +1940,7 @@
},
{
"cell_type": "code",
- "execution_count": 251,
+ "execution_count": 62,
"metadata": {},
"outputs": [
{
@@ -1949,7 +1949,7 @@
"Executed operation: Schritt(1,[a],[a])"
]
},
- "execution_count": 251,
+ "execution_count": 62,
"metadata": {},
"output_type": "execute_result"
}
@@ -1960,7 +1960,7 @@
},
{
"cell_type": "code",
- "execution_count": 252,
+ "execution_count": 63,
"metadata": {},
"outputs": [
{
@@ -1999,7 +1999,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 252,
+ "execution_count": 63,
"metadata": {},
"output_type": "execute_result"
}
@@ -2010,7 +2010,7 @@
},
{
"cell_type": "code",
- "execution_count": 253,
+ "execution_count": 64,
"metadata": {},
"outputs": [
{
@@ -2024,7 +2024,7 @@
"Schritt(5,[z0,b],[b,z1])"
]
},
- "execution_count": 253,
+ "execution_count": 64,
"metadata": {},
"output_type": "execute_result"
}
@@ -2035,7 +2035,7 @@
},
{
"cell_type": "code",
- "execution_count": 254,
+ "execution_count": 65,
"metadata": {},
"outputs": [
{
@@ -2044,7 +2044,7 @@
"Executed operation: Schritt(5,[z0,b],[b,z1])"
]
},
- "execution_count": 254,
+ "execution_count": 65,
"metadata": {},
"output_type": "execute_result"
}
@@ -2055,7 +2055,7 @@
},
{
"cell_type": "code",
- "execution_count": 255,
+ "execution_count": 66,
"metadata": {},
"outputs": [
{
@@ -2098,7 +2098,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 255,
+ "execution_count": 66,
"metadata": {},
"output_type": "execute_result"
}
@@ -2116,7 +2116,7 @@
},
{
"cell_type": "code",
- "execution_count": 256,
+ "execution_count": 67,
"metadata": {},
"outputs": [
{
@@ -2125,7 +2125,7 @@
"Executed operation: Schritt(3,[Hash],[Hash])"
]
},
- "execution_count": 256,
+ "execution_count": 67,
"metadata": {},
"output_type": "execute_result"
}
@@ -2136,7 +2136,7 @@
},
{
"cell_type": "code",
- "execution_count": 257,
+ "execution_count": 68,
"metadata": {},
"outputs": [
{
@@ -2145,7 +2145,7 @@
"Executed operation: Schritt(1,[a],[a])"
]
},
- "execution_count": 257,
+ "execution_count": 68,
"metadata": {},
"output_type": "execute_result"
}
@@ -2156,7 +2156,7 @@
},
{
"cell_type": "code",
- "execution_count": 258,
+ "execution_count": 69,
"metadata": {},
"outputs": [
{
@@ -2203,7 +2203,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 258,
+ "execution_count": 69,
"metadata": {},
"output_type": "execute_result"
}
@@ -2214,7 +2214,7 @@
},
{
"cell_type": "code",
- "execution_count": 259,
+ "execution_count": 70,
"metadata": {},
"outputs": [
{
@@ -2223,7 +2223,7 @@
"Executed operation: Schritt(2,[b],[b])"
]
},
- "execution_count": 259,
+ "execution_count": 70,
"metadata": {},
"output_type": "execute_result"
}
@@ -2234,7 +2234,7 @@
},
{
"cell_type": "code",
- "execution_count": 260,
+ "execution_count": 71,
"metadata": {},
"outputs": [
{
@@ -2283,7 +2283,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 260,
+ "execution_count": 71,
"metadata": {},
"output_type": "execute_result"
}
@@ -2294,7 +2294,7 @@
},
{
"cell_type": "code",
- "execution_count": 261,
+ "execution_count": 72,
"metadata": {},
"outputs": [
{
@@ -2308,7 +2308,7 @@
"Schritt(7,[z1,Hash],[z2,Hash])"
]
},
- "execution_count": 261,
+ "execution_count": 72,
"metadata": {},
"output_type": "execute_result"
}
@@ -2319,7 +2319,7 @@
},
{
"cell_type": "code",
- "execution_count": 262,
+ "execution_count": 73,
"metadata": {},
"outputs": [
{
@@ -2328,7 +2328,7 @@
"Executed operation: Schritt(7,[z1,Hash],[z2,Hash])"
]
},
- "execution_count": 262,
+ "execution_count": 73,
"metadata": {},
"output_type": "execute_result"
}
@@ -2339,7 +2339,7 @@
},
{
"cell_type": "code",
- "execution_count": 263,
+ "execution_count": 74,
"metadata": {},
"outputs": [
{
@@ -2392,7 +2392,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 263,
+ "execution_count": 74,
"metadata": {},
"output_type": "execute_result"
}
@@ -2410,7 +2410,7 @@
},
{
"cell_type": "code",
- "execution_count": 264,
+ "execution_count": 75,
"metadata": {},
"outputs": [
{
@@ -2424,7 +2424,7 @@
"Schritt(1,[a],[a])"
]
},
- "execution_count": 264,
+ "execution_count": 75,
"metadata": {},
"output_type": "execute_result"
}
@@ -2435,7 +2435,7 @@
},
{
"cell_type": "code",
- "execution_count": 265,
+ "execution_count": 76,
"metadata": {},
"outputs": [
{
@@ -2444,7 +2444,7 @@
"Executed operation: Schritt(1,[a],[a])"
]
},
- "execution_count": 265,
+ "execution_count": 76,
"metadata": {},
"output_type": "execute_result"
}
@@ -2455,7 +2455,7 @@
},
{
"cell_type": "code",
- "execution_count": 266,
+ "execution_count": 77,
"metadata": {},
"outputs": [
{
@@ -2510,7 +2510,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 266,
+ "execution_count": 77,
"metadata": {},
"output_type": "execute_result"
}
@@ -2521,7 +2521,7 @@
},
{
"cell_type": "code",
- "execution_count": 267,
+ "execution_count": 78,
"metadata": {},
"outputs": [
{
@@ -2536,7 +2536,7 @@
"Schritt(9,[b,z2],[z2])"
]
},
- "execution_count": 267,
+ "execution_count": 78,
"metadata": {},
"output_type": "execute_result"
}
@@ -2554,7 +2554,7 @@
},
{
"cell_type": "code",
- "execution_count": 268,
+ "execution_count": 79,
"metadata": {},
"outputs": [
{
@@ -2563,7 +2563,7 @@
"Executed operation: Schritt(9,[b,z2],[z2])"
]
},
- "execution_count": 268,
+ "execution_count": 79,
"metadata": {},
"output_type": "execute_result"
}
@@ -2574,7 +2574,7 @@
},
{
"cell_type": "code",
- "execution_count": 269,
+ "execution_count": 80,
"metadata": {},
"outputs": [
{
@@ -2631,7 +2631,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 269,
+ "execution_count": 80,
"metadata": {},
"output_type": "execute_result"
}
@@ -2642,7 +2642,7 @@
},
{
"cell_type": "code",
- "execution_count": 270,
+ "execution_count": 81,
"metadata": {},
"outputs": [
{
@@ -2656,7 +2656,7 @@
"Schritt(3,[Hash],[Hash])"
]
},
- "execution_count": 270,
+ "execution_count": 81,
"metadata": {},
"output_type": "execute_result"
}
@@ -2667,7 +2667,7 @@
},
{
"cell_type": "code",
- "execution_count": 271,
+ "execution_count": 82,
"metadata": {},
"outputs": [
{
@@ -2676,7 +2676,7 @@
"Executed operation: Schritt(3,[Hash],[Hash])"
]
},
- "execution_count": 271,
+ "execution_count": 82,
"metadata": {},
"output_type": "execute_result"
}
@@ -2687,7 +2687,7 @@
},
{
"cell_type": "code",
- "execution_count": 272,
+ "execution_count": 83,
"metadata": {},
"outputs": [
{
@@ -2746,7 +2746,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 272,
+ "execution_count": 83,
"metadata": {},
"output_type": "execute_result"
}
@@ -2757,7 +2757,7 @@
},
{
"cell_type": "code",
- "execution_count": 273,
+ "execution_count": 84,
"metadata": {},
"outputs": [
{
@@ -2772,7 +2772,7 @@
"Schritt(8,[a,z2],[z2])"
]
},
- "execution_count": 273,
+ "execution_count": 84,
"metadata": {},
"output_type": "execute_result"
}
@@ -2790,7 +2790,7 @@
},
{
"cell_type": "code",
- "execution_count": 274,
+ "execution_count": 85,
"metadata": {},
"outputs": [
{
@@ -2799,7 +2799,7 @@
"Executed operation: Schritt(8,[a,z2],[z2])"
]
},
- "execution_count": 274,
+ "execution_count": 85,
"metadata": {},
"output_type": "execute_result"
}
@@ -2810,7 +2810,7 @@
},
{
"cell_type": "code",
- "execution_count": 275,
+ "execution_count": 86,
"metadata": {},
"outputs": [
{
@@ -2871,7 +2871,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 275,
+ "execution_count": 86,
"metadata": {},
"output_type": "execute_result"
}
@@ -2882,7 +2882,7 @@
},
{
"cell_type": "code",
- "execution_count": 276,
+ "execution_count": 87,
"metadata": {},
"outputs": [
{
@@ -2891,7 +2891,7 @@
"Executed operation: Schritt(3,[Hash],[Hash])"
]
},
- "execution_count": 276,
+ "execution_count": 87,
"metadata": {},
"output_type": "execute_result"
}
@@ -2902,7 +2902,7 @@
},
{
"cell_type": "code",
- "execution_count": 277,
+ "execution_count": 88,
"metadata": {},
"outputs": [
{
@@ -2965,7 +2965,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 277,
+ "execution_count": 88,
"metadata": {},
"output_type": "execute_result"
}
@@ -2976,7 +2976,7 @@
},
{
"cell_type": "code",
- "execution_count": 278,
+ "execution_count": 89,
"metadata": {},
"outputs": [
{
@@ -2990,7 +2990,7 @@
"Schritt(10,[z2,Hash,Hash],[Hash])"
]
},
- "execution_count": 278,
+ "execution_count": 89,
"metadata": {},
"output_type": "execute_result"
}
@@ -3008,7 +3008,7 @@
},
{
"cell_type": "code",
- "execution_count": 279,
+ "execution_count": 90,
"metadata": {},
"outputs": [
{
@@ -3017,7 +3017,7 @@
"Executed operation: Schritt(10,[z2,Hash,Hash],[Hash])"
]
},
- "execution_count": 279,
+ "execution_count": 90,
"metadata": {},
"output_type": "execute_result"
}
@@ -3028,7 +3028,7 @@
},
{
"cell_type": "code",
- "execution_count": 280,
+ "execution_count": 91,
"metadata": {},
"outputs": [
{
@@ -3093,7 +3093,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 280,
+ "execution_count": 91,
"metadata": {},
"output_type": "execute_result"
}
@@ -3104,7 +3104,7 @@
},
{
"cell_type": "code",
- "execution_count": 281,
+ "execution_count": 92,
"metadata": {},
"outputs": [
{
@@ -3113,7 +3113,7 @@
"Executed operation: Lösung()"
]
},
- "execution_count": 281,
+ "execution_count": 92,
"metadata": {},
"output_type": "execute_result"
}
diff --git a/info4/kapitel-2/Abschlusseigenschaften.ipynb b/info4/kapitel-2/Abschlusseigenschaften.ipynb
new file mode 100644
index 0000000000000000000000000000000000000000..971401daafeae9593dc7b667acf3e2985416f5eb
--- /dev/null
+++ b/info4/kapitel-2/Abschlusseigenschaften.ipynb
@@ -0,0 +1,1980 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Abschlusseigenschaften regulärer Sprachen\n",
+ "\n",
+ "Abschlusseigenschaften sind ein weiteres nützliches Werkzeug um verschiedene Sprachklassen zu charakterisieren.\n",
+ "Damit kann man untersuchen, ob eine Sprache zu einer Sprachklasse gehört oder nicht.\n",
+ "\n",
+ "Die regulären Sprachen sind abgeschlossen unter:\n",
+ "* Vereinigung A ∪ B\n",
+ "* Komplement A̅\n",
+ "* Schnitt A ∩ B\n",
+ "* Differenz A/B\n",
+ "* Konkatenation AB\n",
+ "* Iteration A*\n",
+ "* Spiegelung sp(A)\n",
+ "\n",
+ "In den Folien wird diese Aussage zwar aufgestellt, aber nicht bewiesen.\n",
+ "Hier schauen wir uns für einige Operationen Beweisideen an konkreten Beispielen an."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Zuerst betrachten wir die Vereinigung.\n",
+ "\n",
+ "Zu jeder regulären Sprache kann man auch einen NFA angeben.\n",
+ "Wenn $M_1=(Σ, Z_1, δ_1, S_1, F_1)$ und $M_2=(Σ, Z_2, δ_2, S_2, F_2)$ zwei NFAs sind, dann lässt sich ein Vereinigungs-NFA $M=(Σ, Z_1 ∪ Z_2, δ_1∪δ_2, S_1∪S_2, F_1∪F_2)$ erstellen, so dass $L(M) = L(M_1) ∪ L(M_2)$. Man erhält M, indem man alle Mengen der Tupel von $M_1$ und $M_2$ paarweise vereinigt (analog zum Beweis des Satzes von Kleene Folie 52)."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Loaded machine: NFA"
+ ]
+ },
+ "execution_count": 1,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "::load\n",
+ "MACHINE NFA\n",
+ "SETS\n",
+ " Z = {z10,z11,z12,z13, z20,z21,z22}\n",
+ "CONSTANTS Σ, M, δ, S, F, \n",
+ " M1, Z1, δ1, S1, F1,\n",
+ " M2, Z2, δ2, S2, F2\n",
+ " \n",
+ "PROPERTIES\n",
+ " Σ = {0,1} ∧\n",
+ " Z1 ∩ Z2 = ∅\n",
+ " ∧\n",
+ " // Der Automat M1 von Folie 21 (L(M1)={u1v|u∈{0,1}* ∧ v∈{0,1}}):\n",
+ " M1=(Σ, Z1, δ1, S1, F1) ∧\n",
+ " Z1 ⊆ Z ∧\n",
+ " δ1 ∈ (Z1×Σ) → ℙ(Z1) ∧\n",
+ " S1 ⊆ Z1 ∧ F1 ⊆ Z1\n",
+ " ∧\n",
+ " Z1 = {z10,z11,z12,z13} ∧\n",
+ " S1 = {z10} ∧ F1 = {z12} ∧\n",
+ " δ1 = { (z10,0)↦{z10}, (z10,1)↦{z10,z11},\n",
+ " (z11,0)↦{z12}, (z11,1)↦{z12},\n",
+ " (z12,0)↦{z13}, (z12,1)↦{z13},\n",
+ " (z13,0)↦{z13}, (z13,1)↦{z13} }\n",
+ " \n",
+ " ∧\n",
+ " // Der Automat M2 von Folie 28 (L(M2)={u1|u∈{0,1}*}):\n",
+ " M2=(Σ, Z2, δ2, S2, F2) ∧\n",
+ " Z2 ⊆ Z ∧\n",
+ " δ2 ∈ (Z2×Σ) → ℙ(Z2) ∧\n",
+ " S2 ⊆ Z2 ∧ F2 ⊆ Z2\n",
+ " ∧\n",
+ " Z2 = {z20,z21,z22} ∧\n",
+ " S2 = {z20} ∧ F2 = {z21} ∧\n",
+ " δ2 = { (z20,0)↦{z20}, (z20,1)↦{z20,z21},\n",
+ " (z21,0)↦{z22}, (z21,1)↦{z22},\n",
+ " (z22,0)↦{z22}, (z22,1)↦{z22}}\n",
+ " \n",
+ " ∧\n",
+ " //Der Vereinigungsautomat M:\n",
+ " M=(Σ, Z, δ, S, F) ∧\n",
+ " Z= Z1 ∪ Z2 ∧\n",
+ " δ = δ1 ∪ δ2 ∧\n",
+ " S = S1 ∪ S2 ∧\n",
+ " F = F1 ∪ F2\n",
+ " \n",
+ " \n",
+ "DEFINITIONS // Für den Zustandsgraphen:\n",
+ " CUSTOM_GRAPH_NODES1 == rec(shape:\"doublecircle\",nodes∈F); // Endzustände\n",
+ " CUSTOM_GRAPH_NODES2 == rec(shape:\"circle\",nodes∈Z\\F); // andere Zustände\n",
+ " CUSTOM_GRAPH_NODES3 == rec(shape:\"none\",color:\"white\",style:\"none\",nodes:{\"\"});\n",
+ " CUSTOM_GRAPH_EDGES1 == rec(color:\"green\",label:\"0\",edges:{x,y|y∈δ(x,0) ∧ y∉δ(x,1)}); \n",
+ " CUSTOM_GRAPH_EDGES2 == rec(color:\"green\",label:\"1\",edges:{x,y|y∈δ(x,1) ∧ y∉δ(x,0)});\n",
+ " CUSTOM_GRAPH_EDGES3 == rec(color:\"green\",label:\"0, 1\",edges:{x,y|y∈δ(x,0) ∧ y∈δ(x,1)});\n",
+ " CUSTOM_GRAPH_EDGES4 == rec(color:\"black\",label:\"\",edges:{\"\"}*S) // Kanten für Startknoten\n",
+ "END"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "metadata": {
+ "scrolled": true
+ },
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Machine constants set up using operation 0: $setup_constants()"
+ ]
+ },
+ "execution_count": 2,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":constants"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Hier sieht man auf der linken Seite $M_1$, auf der rechten $M_2$ und das gesamte Bild stellt den Vereinigungsautomaten $M$ dar."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "image/svg+xml": [
+ "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
+ "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
+ " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
+ "<!-- Generated by graphviz version 2.44.1 (0)\n",
+ " -->\n",
+ "<!-- Title: prob_graph Pages: 1 -->\n",
+ "<svg width=\"540pt\" height=\"715pt\"\n",
+ " viewBox=\"0.00 0.00 540.00 714.80\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
+ "<g id=\"graph0\" class=\"graph\" transform=\"scale(0.98 0.98) rotate(0) translate(4 724.24)\">\n",
+ "<title>prob_graph</title>\n",
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+ "<!-- 0 -->\n",
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+ "<text text-anchor=\"middle\" x=\"74.9\" y=\"-206.9\" font-family=\"Times,serif\" font-size=\"12.00\">z12</text>\n",
+ "</g>\n",
+ "<!-- 4 -->\n",
+ "<g id=\"node5\" class=\"node\">\n",
+ "<title>4</title>\n",
+ "<ellipse fill=\"lightgrey\" stroke=\"black\" cx=\"74.9\" cy=\"-40\" rx=\"23.29\" ry=\"23.29\"/>\n",
+ "<text text-anchor=\"middle\" x=\"74.9\" y=\"-36.9\" font-family=\"Times,serif\" font-size=\"12.00\">z13</text>\n",
+ "</g>\n",
+ "<!-- 0->4 -->\n",
+ "<g id=\"edge7\" class=\"edge\">\n",
+ "<title>0->4</title>\n",
+ "<path fill=\"none\" stroke=\"green\" d=\"M74.9,-182.38C74.9,-152.87 74.9,-105.26 74.9,-73.56\"/>\n",
+ "<polygon fill=\"green\" stroke=\"green\" points=\"78.4,-73.46 74.9,-63.46 71.4,-73.46 78.4,-73.46\"/>\n",
+ "<text text-anchor=\"middle\" x=\"86.4\" y=\"-117.9\" font-family=\"Times,serif\" font-size=\"12.00\">0, 1</text>\n",
+ "</g>\n",
+ "<!-- 1 -->\n",
+ "<g id=\"node2\" class=\"node\">\n",
+ "<title>1</title>\n",
+ "<ellipse fill=\"lightgrey\" stroke=\"black\" cx=\"365.9\" cy=\"-386\" rx=\"23.3\" ry=\"23.3\"/>\n",
+ "<ellipse fill=\"none\" stroke=\"black\" cx=\"365.9\" cy=\"-386\" rx=\"27.29\" ry=\"27.29\"/>\n",
+ "<text text-anchor=\"middle\" x=\"365.9\" y=\"-382.9\" font-family=\"Times,serif\" font-size=\"12.00\">z21</text>\n",
+ "</g>\n",
+ "<!-- 6 -->\n",
+ "<g id=\"node7\" class=\"node\">\n",
+ "<title>6</title>\n",
+ "<ellipse fill=\"lightgrey\" stroke=\"black\" cx=\"365.9\" cy=\"-210\" rx=\"23.29\" ry=\"23.29\"/>\n",
+ "<text text-anchor=\"middle\" x=\"365.9\" y=\"-206.9\" font-family=\"Times,serif\" font-size=\"12.00\">z22</text>\n",
+ "</g>\n",
+ "<!-- 1->6 -->\n",
+ "<g id=\"edge10\" class=\"edge\">\n",
+ "<title>1->6</title>\n",
+ "<path fill=\"none\" stroke=\"green\" d=\"M365.9,-358.57C365.9,-327.78 365.9,-276.87 365.9,-243.64\"/>\n",
+ "<polygon fill=\"green\" stroke=\"green\" points=\"369.4,-243.51 365.9,-233.51 362.4,-243.51 369.4,-243.51\"/>\n",
+ "<text text-anchor=\"middle\" x=\"377.4\" y=\"-294.9\" font-family=\"Times,serif\" font-size=\"12.00\">0, 1</text>\n",
+ "</g>\n",
+ "<!-- 2 -->\n",
+ "<g id=\"node3\" class=\"node\">\n",
+ "<title>2</title>\n",
+ "<ellipse fill=\"lightgrey\" stroke=\"black\" cx=\"74.9\" cy=\"-556\" rx=\"23.29\" ry=\"23.29\"/>\n",
+ "<text text-anchor=\"middle\" x=\"74.9\" y=\"-552.9\" font-family=\"Times,serif\" font-size=\"12.00\">z10</text>\n",
+ "</g>\n",
+ "<!-- 2->2 -->\n",
+ "<g id=\"edge5\" class=\"edge\">\n",
+ "<title>2->2</title>\n",
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+ }
+ ],
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+ ":dot custom_graph"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Eine weitere Eigenschaft, die man leicht zeigen kann ist, dass das Komplement einer Sprache $L$ regulär ist. Sei $M=(Σ, Z, δ, z_0, F)$ ein DFA mit $L(M)=L$. Dann gilt für $M_2=(Σ, Z, δ, z_0, Z/F)$, dass $L(M_2)= \\overline{L}$. Dies kann man auch mit folgender Machiene darstellen:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 4,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Loaded machine: DFA"
+ ]
+ },
+ "execution_count": 4,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "::load\n",
+ "MACHINE DFA\n",
+ "SETS\n",
+ " Z = {z0,z1,z2,z3}\n",
+ "CONSTANTS Σ, M, δ, F, M2, z_start\n",
+ "PROPERTIES\n",
+ " Σ = {0,1}\n",
+ " ∧\n",
+ " M = (Σ, Z, δ, z_start, F) ∧\n",
+ " M2 = (Σ, Z, δ, z_start, Z\\F) ∧\n",
+ " δ ∈ (Z×Σ) → Z ∧\n",
+ " z_start ∈ Z ∧\n",
+ " F ⊆ Z\n",
+ "DEFINITIONS // Für den Zustandsgraphen:\n",
+ " CUSTOM_GRAPH_NODES1 == rec(shape:\"doublecircle\",nodes∈Z\\F); // Endzustände\n",
+ " CUSTOM_GRAPH_NODES2 == rec(shape:\"circle\",nodes∈F); // andere Zustände\n",
+ " CUSTOM_GRAPH_NODES3 == rec(shape:\"none\",color:\"white\",style:\"none\",nodes:{\"\"});\n",
+ " CUSTOM_GRAPH_EDGES1 == rec(color:\"green\",label:\"0\",edges:{x,y|y=δ(x,0) ∧ y≠δ(x,1)}); \n",
+ " CUSTOM_GRAPH_EDGES2 == rec(color:\"green\",label:\"1\",edges:{x,y|y=δ(x,1) ∧ y≠ δ(x,0)});\n",
+ " CUSTOM_GRAPH_EDGES3 == rec(color:\"green\",label:\"0, 1\",edges:{x,y|y=δ(x,0) ∧ y=δ(x,1)});\n",
+ " CUSTOM_GRAPH_EDGES4 == rec(color:\"black\",label:\"\",edges:{\"\" ↦ z_start}) // Kante für den Startknoten\n",
+ "END"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Diese Maschiene hat die feste Zustandsmenge Z={z0,z1,z2,z3} und das Alphabet Σ = {0,1}, aber die restlichen Elemente werden der Anweisung ```:constants``` übergeben. Somit kann das Komplement eines beliebigen DFAs erzeugt werden."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 5,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Machine constants set up using operation 0: $setup_constants()"
+ ]
+ },
+ "execution_count": 5,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":constants M=({0,1}, //Σ\n",
+ " {z0,z1,z2,z3}, //Z\n",
+ " {(z0,0)↦z1, (z0,1)↦z3,\n",
+ " (z1,0)↦z3, (z1,1)↦z2,\n",
+ " (z2,0)↦z2, (z2,1)↦z2,\n",
+ " (z3,0)↦z3, (z3,1)↦z3 }, //δ\n",
+ " z0, {z0, z2}) //z0, F"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Hier sieht man $M_2$, der aus $M$ erzeugt wurde, indem man alle das Komplement der Endzustandsmenge als neue Endzustandsmenge wählt."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 6,
+ "metadata": {},
+ "outputs": [
+ {
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+ ":dot custom_graph"
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+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Als nächstes befassen wir uns mit der Konkatenation zweier Sprachen. Um dies zu erreichen schaltet man 2 NFAs $M_1=(Σ, Z_1, δ_1, S_1, F_1)$ udnd $M_2=(Σ, Z_2, δ_2, S_2, F_2)$ hintereinander. Dies erreicht man, indem man jedes Mal, wenn ein Zustand in $F_1$ erreicht wird zusätzlich auch alle Zustände in $S_2$ erreicht werden."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 7,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Loaded machine: NFA"
+ ]
+ },
+ "execution_count": 7,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "::load\n",
+ "MACHINE NFA\n",
+ "SETS\n",
+ " Z = {z10,z11, z12, z20,z21,z22}\n",
+ "CONSTANTS Σ, M, δ, S, F, \n",
+ " M1, Z1, δ1, S1, F1, \n",
+ " M2, Z2, δ2, S2, F2\n",
+ " \n",
+ "PROPERTIES\n",
+ " Σ = {0,1} ∧\n",
+ " Z1 ∩ Z2 = ∅\n",
+ " ∧\n",
+ " // Der Automat M1 (L(M1)={w| |w|=0 ∨ w=u0 mit u∈{0,1}*}):\n",
+ " M1=(Σ, Z1, δ1, S1, F1) ∧\n",
+ " Z1 ⊆ Z ∧ Z1 = {z10,z11,z12} ∧\n",
+ " δ1 ∈ (Z1×Σ) → ℙ(Z1) ∧\n",
+ " S1 ⊆ Z1 ∧ F1 ⊆ Z1\n",
+ " ∧\n",
+ " S1 = {z10} ∧ F1 = {z10, z12} ∧\n",
+ " δ1 = { (z10,0)↦{z12}, (z10,1)↦{z11},\n",
+ " (z11,0)↦{z12}, (z11,1)↦{z11},\n",
+ " (z12,0)↦{z12}, (z12,1)↦{z11}} ∧\n",
+ " \n",
+ " \n",
+ " // Der Automat M2 (L(M2)={1^n|n∈ℕ_0}):\n",
+ " M2 = (Σ, Z2, δ2, S2, F2) ∧\n",
+ " Z2 ⊆ Z ∧ Z2 = {z20,z21,z22} ∧\n",
+ " δ2 ∈ (Z2×Σ) → ℙ(Z2) ∧\n",
+ " S2 ⊆ Z2 ∧ F2 ⊆ Z2\n",
+ " ∧\n",
+ " S2 = {z20} ∧ F2 = {z21} ∧\n",
+ " δ2 = { (z20,0)↦{z22}, (z20,1)↦{z21},\n",
+ " (z21,0)↦{z22}, (z21,1)↦{z21},\n",
+ " (z22,0)↦{z22}, (z22,1)↦{z22}} ∧\n",
+ "\n",
+ " //Die Regeln von Folie 51:\n",
+ " M = (Σ, Z, δ, S, F) ∧\n",
+ " Z = Z1 ∪ Z2 ∧\n",
+ " S = S1 ∪ ran((S1∩F1)*S2) ∧\n",
+ " F = F2 ∪ ran((S2∩F2)*F1) ∧\n",
+ " δ = {x | x∈(Z*INTEGER)*POW(Z) ∧\n",
+ " ∃zustand,symbol,menge.(x=(zustand, symbol)↦menge ∧ (\n",
+ " (zustand∈Z1 ∧ menge = δ1(zustand,symbol) ∧ δ1(zustand,symbol)∩F1= ∅) ∨\n",
+ " (zustand∈Z1 ∧ menge = δ1(zustand,symbol) ∪ S2 ∧ δ1(zustand,symbol)∩F1 ≠ ∅) ∨\n",
+ " (zustand∈Z2 ∧ menge = δ2(zustand,symbol))))}\n",
+ " \n",
+ "DEFINITIONS // Für den Zustandsgraphen:\n",
+ " CUSTOM_GRAPH_NODES1 == rec(shape:\"doublecircle\",nodes∈F); // Endzustände\n",
+ " CUSTOM_GRAPH_NODES2 == rec(shape:\"circle\",nodes∈Z\\F); // andere Zustände\n",
+ " CUSTOM_GRAPH_NODES3 == rec(shape:\"none\",color:\"white\",style:\"none\",nodes:{\"\"});\n",
+ " CUSTOM_GRAPH_EDGES1 == rec(color:\"green\",label:\"0\",edges:{x,y|y∈δ(x,0) ∧ y∉δ(x,1)}); \n",
+ " CUSTOM_GRAPH_EDGES2 == rec(color:\"green\",label:\"1\",edges:{x,y|y∈δ(x,1) ∧ y∉δ(x,0)});\n",
+ " CUSTOM_GRAPH_EDGES3 == rec(color:\"green\",label:\"0, 1\",edges:{x,y|y∈δ(x,0) ∧ y∈δ(x,1)});\n",
+ " CUSTOM_GRAPH_EDGES4 == rec(color:\"black\",label:\"\",edges:{\"\"}*S) // Kanten für Startknoten\n",
+ "END"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 8,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Machine constants set up using operation 0: $setup_constants()"
+ ]
+ },
+ "execution_count": 8,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":constants"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "<table align=\"LEFT\">\n",
+ " <tr>\n",
+ " <td width=50%>Der Automat $M_1$</td>\n",
+ " <td width=50%>Der Automat $M_2$</td>\n",
+ " </tr>\n",
+ " <tr>\n",
+ " <td width=50%><img src=\"./img/konkatenation_M1.svg\" width=100%></td>\n",
+ " <td width=50%><img src=\"./img/konkatenation_M2.svg\" width=100%></td>\n",
+ " </tr>\n",
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+ "source": [
+ "Um zu zeigen, dass die Iteration einer regulären Sprache wieder regulär ist, bilden wir zu einem NFA $M=(Σ, Z_1, δ_1, S_1, F_1)$ einen Rückkopplungsautomaten $M_2$. Wir starten damit den $M$ mit sich selbst zu verketten (analog zur Konkatenation). Dann müssen wir nur noch sichergehen, dass das leere Wort akzeptiert wird. Dafür fügen wir (falls notwendig) einen Zustand ein, der sowohl Start- als auch Endzustand ist."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 10,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Loaded machine: NFA"
+ ]
+ },
+ "execution_count": 10,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "::load\n",
+ "MACHINE NFA\n",
+ "SETS\n",
+ " Z = {z10,z11, z12, z}\n",
+ "CONSTANTS Σ, M, δ, \n",
+ " M1, Z1, δ1, S1, F1,\n",
+ " M2, Z2, δ2, S2, F2\n",
+ " \n",
+ "PROPERTIES\n",
+ " Σ = {0,1}\n",
+ " ∧\n",
+ " // Der Automat M1 (L(M1)={w| w=u0 mit u∈{0,1}*}):\n",
+ " M1 = (Σ, Z1, δ1, S1, F1) ∧\n",
+ " Z1 ⊆ Z ∧ Z1 = {z10,z11,z12} ∧\n",
+ " δ1 ∈ (Z1×Σ) → ℙ(Z1) ∧\n",
+ " S1 ⊆ Z1 ∧ F1 ⊆ Z1\n",
+ " ∧\n",
+ " S1 = {z10} ∧ F1 = {z12} ∧\n",
+ " δ1 = { (z10,0)↦{z12}, (z10,1)↦{z11},\n",
+ " (z11,0)↦{z12}, (z11,1)↦{z11},\n",
+ " (z12,0)↦{z12}, (z12,1)↦{z11}} ∧\n",
+ " \n",
+ " //Die Regeln von Folie 53\n",
+ " M2 = (Σ, Z2, δ2, S2, F2) ∧\n",
+ " Z2 ⊆ Z ∧\n",
+ " δ2 ∈ (Z2×Σ) +-> ℙ(Z2) ∧\n",
+ " S2 ⊆ Z2 ∧ F2 ⊆ Z2\n",
+ " ∧\n",
+ " Z2 = IF S1∩F1 ≠ ∅ THEN Z1 ELSE Z1∪{z} END ∧\n",
+ " δ2 = δ1 ∧\n",
+ " S2 = IF S1∩F1 ≠ ∅ THEN S1 ELSE S1∪{z} END ∧\n",
+ " F2 = IF S1∩F1 ≠ ∅ THEN F1 ELSE F1∪{z} END ∧\n",
+ " \n",
+ " \n",
+ " //Die Regeln von Folie 54:\n",
+ " M = (Σ, Z2, δ, S2, F2) ∧\n",
+ " δ = {x | x∈(Z*INTEGER)*POW(Z) ∧\n",
+ " ∃zustand,symbol,menge.(x=(zustand, symbol)↦menge ∧ (\n",
+ " (zustand∈Z2 ∧ menge = δ2(zustand,symbol) ∧ δ2(zustand,symbol)∩F2= ∅) ∨\n",
+ " (zustand∈Z2 ∧ menge = δ2(zustand,symbol) ∪ S2 ∧ δ2(zustand,symbol)∩F2 ≠ ∅)))}\n",
+ " \n",
+ "DEFINITIONS // Für den Zustandsgraphen:\n",
+ " CUSTOM_GRAPH_NODES1 == rec(shape:\"doublecircle\",nodes∈F2); // Endzustände\n",
+ " CUSTOM_GRAPH_NODES2 == rec(shape:\"circle\",nodes∈Z2\\F2); // andere Zustände\n",
+ " CUSTOM_GRAPH_NODES3 == rec(shape:\"none\",color:\"white\",style:\"none\",nodes:{\"\"});\n",
+ " CUSTOM_GRAPH_EDGES1 == rec(color:\"green\",label:\"0\",edges:{x,y|{(x,0),(x,1)} ⊆dom(δ) ∧ y∈δ(x,0) ∧ y∉δ(x,1)}); \n",
+ " CUSTOM_GRAPH_EDGES2 == rec(color:\"green\",label:\"1\",edges:{x,y|{(x,0),(x,1)} ⊆dom(δ) ∧ y∈δ(x,1) ∧ y∉δ(x,0)});\n",
+ " CUSTOM_GRAPH_EDGES3 == rec(color:\"green\",label:\"0, 1\",edges:{x,y|{(x,0),(x,1)} ⊆dom(δ) ∧ y∈δ(x,0) ∧ y∈δ(x,1)});\n",
+ " CUSTOM_GRAPH_EDGES4 == rec(color:\"black\",label:\"\",edges:{\"\"}*S2) // Kanten für Startknoten\n",
+ "END"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 11,
+ "metadata": {},
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+ "data": {
+ "text/plain": [
+ "Machine constants set up using operation 0: $setup_constants()"
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+ "<Dot visualization: custom_graph []>"
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+ "execution_count": 12,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
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+ "source": [
+ ":dot custom_graph"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Im folgenden schauen wir uns die Differenz zweier Sprachen $L1-L2$ genauer an.\n",
+ "Das Vorgehen (aus dem Buch [Grundkurs Theoretische Informatik](https://link.springer.com/content/pdf/10.1007/978-3-8348-2202-4.pdf \"Vossen, G., & Witt, K. U. (2016). Grundkurs Theoretische Informatik. Springer Fachmedien Wiesbaden.\") von G. Vossen & K. U. Witt, Seite 99f.) ist hierbei, dass man zwei NFAs $M_1=(Σ, Z_1, δ_1, S_1, F_1)$ udnd $M_2=(Σ, Z_2, δ_2, S_2, F_2)$ parallel ausführt und die Endzustandsmenge so wählt, dass man in einem Zustand aus $F_1$, aber nicht gleichzeitig in einem aus $F_2$ landet."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 13,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Loaded machine: NFA"
+ ]
+ },
+ "execution_count": 13,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "::load\n",
+ "MACHINE NFA\n",
+ "SETS\n",
+ " Z = {z10,z11, z12, z20,z21,z22}\n",
+ "CONSTANTS Σ, M, Z_gesamt, δ, S, F,\n",
+ " M1, Z1, δ1, S1, F1,\n",
+ " M2, Z2, δ2, S2, F2\n",
+ " \n",
+ "PROPERTIES\n",
+ " Σ = {0,1} ∧\n",
+ " Z1 ∩ Z2 = ∅ ∧\n",
+ " Z= Z1 ∪ Z2\n",
+ "\n",
+ " ∧\n",
+ " // Der Automat M1 (L(M1)={w| |w|=0 ∨ w=u0 mit u∈{0,1}*}):\n",
+ " M1 = (Σ, Z1, δ1, S1, F1) ∧\n",
+ " Z1 ⊆ Z ∧\n",
+ " δ1 ∈ (Z1×Σ) → ℙ(Z1) ∧\n",
+ " S1 ⊆ Z1 ∧ F1 ⊆ Z1\n",
+ " ∧\n",
+ " Z1 = {z10,z11,z12} ∧\n",
+ " S1 = {z10} ∧ F1 = {z10, z12} ∧\n",
+ " δ1 = { (z10,0)↦{z12}, (z10,1)↦{z11},\n",
+ " (z11,0)↦{z12}, (z11,1)↦{z11},\n",
+ " (z12,0)↦{z12}, (z12,1)↦{z11}} ∧\n",
+ " \n",
+ " // Der Automat M2 (L(M2)={0^n|n∈ℕ_0}):\n",
+ " M2 = (Σ, Z2, δ2, S2, F2) ∧\n",
+ " Z2 ⊆ Z ∧\n",
+ " δ2 ∈ (Z2×Σ) → ℙ(Z2) ∧\n",
+ " S2 ⊆ Z2 ∧ F2 ⊆ Z2\n",
+ " ∧\n",
+ " Z2 = {z20,z21,z22} ∧\n",
+ " S2 = {z20} ∧ F2 = {z20, z21} ∧\n",
+ " δ2 = { (z20,0)↦{z21}, (z20,1)↦{z22},\n",
+ " (z21,0)↦{z21}, (z21,1)↦{z22},\n",
+ " (z22,0)↦{z22}, (z22,1)↦{z22}} ∧\n",
+ "\n",
+ " //Der Automat nach \"Grundkurs Theoretische Informatik\"\n",
+ " M = (Σ, Z, δ, S, F) ∧\n",
+ " Z_gesamt = Z1*Z2 ∧\n",
+ " S = S1*S2 ∧\n",
+ " F = F1*(Z2\\F2) ∧\n",
+ " δ = {x | x∈((Z1*Z2)*Σ)*ℙ(Z_gesamt) ∧\n",
+ " ∃zustand1,zustand2,symbol,menge.(\n",
+ " x=((zustand1, zustand2), symbol)↦menge ∧\n",
+ " zustand1∈Z1 ∧ zustand2∈Z2 ∧ symbol∈Σ ∧\n",
+ " menge = δ1(zustand1, symbol)*δ2(zustand2, symbol))}\n",
+ "\n",
+ "DEFINITIONS // Für den Zustandsgraphen:\n",
+ " \"LibraryStrings.def\";\n",
+ " CUSTOM_GRAPH_NODES1 == rec(shape:\"doublecircle\",nodes:{x | ∃y.(x=TO_STRING(y) ∧ y∈F)}); //Endzustände\n",
+ " CUSTOM_GRAPH_NODES2 == rec(shape:\"circle\",nodes:{x | ∃y.(x=TO_STRING(y) ∧ y∈Z_gesamt\\F)}); // andere Zustände\n",
+ " CUSTOM_GRAPH_NODES3 == rec(shape:\"none\",color:\"white\",style:\"none\",nodes:{\"\"});\n",
+ " CUSTOM_GRAPH_EDGES1 == rec(color:\"black\",label:\"\",edges:{\"\"}*{x | ∃y.(x=TO_STRING(y) ∧ y∈S)}); // Kanten für Startknoten\n",
+ " CUSTOM_GRAPH_EDGES2 == rec(color:\"green\",label:\"0\",edges:{a,b|∃x,y.({(x,0),(x,1)} ⊆dom(δ) ∧ y∈δ(x,0) ∧ y∉δ(x,1) ∧ a=TO_STRING(x) ∧ b=TO_STRING(y))}); \n",
+ " CUSTOM_GRAPH_EDGES3 == rec(color:\"green\",label:\"1\",edges:{a,b|∃x,y.({(x,0),(x,1)} ⊆dom(δ) ∧ y∈δ(x,1) ∧ y∉δ(x,0) ∧ a=TO_STRING(x) ∧ b=TO_STRING(y))});\n",
+ " CUSTOM_GRAPH_EDGES4 == rec(color:\"green\",label:\"0, 1\",edges:{a,b|∃x,y.({(x,0),(x,1)} ⊆dom(δ) ∧ y∈δ(x,0) ∧ y∈δ(x,1)∧ a=TO_STRING(x) ∧ b=TO_STRING(y))})\n",
+ "END"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 14,
+ "metadata": {
+ "scrolled": true
+ },
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Machine constants set up using operation 0: $setup_constants()"
+ ]
+ },
+ "execution_count": 14,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":constants"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Bei dieser Konstruktion können hier viele Zustände nicht erreicht werden. Dennoch akzeptiert der Automat die gewünsche Sprache."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 15,
+ "metadata": {},
+ "outputs": [
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+ ":dot custom_graph"
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+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Indem man die Endzustände in der obigen Konstruktion so wählt, dass man in einem Zustand in $F_1$ und $F_2$ landet, kann man auch zeigen, dass der Schnitt zweier regulärer Sprachen regulär ist."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 16,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Loaded machine: NFA"
+ ]
+ },
+ "execution_count": 16,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "::load\n",
+ "MACHINE NFA\n",
+ "SETS\n",
+ " Z = {z10,z11, z12, z20,z21,z22}\n",
+ "CONSTANTS Σ, M, Z_gesamt, δ, S, F,\n",
+ " M1, Z1, δ1, S1, F1,\n",
+ " M2, Z2, δ2, S2, F2\n",
+ " \n",
+ "PROPERTIES\n",
+ " Σ = {0,1}∧\n",
+ " Z1 ∩ Z2 = ∅ ∧\n",
+ " Z= Z1 ∪ Z2\n",
+ " \n",
+ " ∧\n",
+ " // Der Automat M1 (L(M1)={w| |w|=0 ∨ w=u0 mit u∈{0,1}*}):\n",
+ " M1 = (Σ, Z1, δ1, S1, F1) ∧\n",
+ " Z1 ⊆ Z ∧ Z1 = {z10,z11,z12} ∧\n",
+ " F1 ⊆ Z1 ∧ S1 ⊆ Z1 ∧\n",
+ " δ1 ∈ (Z1×Σ) → ℙ(Z1)\n",
+ " ∧\n",
+ " S1 = {z10} ∧ F1 = {z10, z12} ∧\n",
+ " δ1 = { (z10,0)↦{z12}, (z10,1)↦{z11},\n",
+ " (z11,0)↦{z12}, (z11,1)↦{z11},\n",
+ " (z12,0)↦{z12}, (z12,1)↦{z11}} ∧\n",
+ "\n",
+ " // Der Automat M2 (L(M2)={0^n|n∈ℕ_0}):\n",
+ " M2 = (Σ, Z2, δ2, S2, F2) ∧\n",
+ " Z2 ⊆ Z ∧\n",
+ " δ2 ∈ (Z2×Σ) → ℙ(Z2) ∧\n",
+ " S2 ⊆ Z2 ∧ F2 ⊆ Z2 \n",
+ " ∧\n",
+ " Z2 = {z20,z21,z22} ∧\n",
+ " S2 = {z20} ∧ F2 = {z20, z21} ∧\n",
+ " δ2 = { (z20,0)↦{z21}, (z20,1)↦{z22},\n",
+ " (z21,0)↦{z21}, (z21,1)↦{z22},\n",
+ " (z22,0)↦{z22}, (z22,1)↦{z22}} ∧\n",
+ "\n",
+ " //Eigene Konstruktion analog zu \"Grundkurs Theoretische Informatik\" (s.o.)\n",
+ " M = (Σ, Z_gesamt, δ, S, F) ∧\n",
+ " Z_gesamt = Z1*Z2 ∧\n",
+ " S = S1*S2 ∧\n",
+ " F = F1*F2 ∧\n",
+ " δ = {x | x∈((Z*Z)*Σ)*ℙ(Z_gesamt) ∧\n",
+ " ∃zustand1,zustand2,symbol,menge.(\n",
+ " x=((zustand1, zustand2), symbol)↦menge ∧\n",
+ " zustand1∈Z1 ∧ zustand2∈Z2 ∧ symbol∈Σ ∧\n",
+ " menge = δ1(zustand1, symbol)*δ2(zustand2, symbol))}\n",
+ " \n",
+ "DEFINITIONS // Für den Zustandsgraphen:\n",
+ " \"LibraryStrings.def\";\n",
+ " CUSTOM_GRAPH_NODES1 == rec(shape:\"doublecircle\",nodes:{x | ∃y.(x=TO_STRING(y) ∧ y∈F)}); //Endzustände\n",
+ " CUSTOM_GRAPH_NODES2 == rec(shape:\"circle\",nodes:{x | ∃y.(x=TO_STRING(y) ∧ y∈Z_gesamt\\F)}); // andere Zustände\n",
+ " CUSTOM_GRAPH_NODES3 == rec(shape:\"none\",color:\"white\",style:\"none\",nodes:{\"\"});\n",
+ " CUSTOM_GRAPH_EDGES1 == rec(color:\"black\",label:\"\",edges:{\"\"}*{x | ∃y.(x=TO_STRING(y) ∧ y∈S)}); // Kanten für Startknoten\n",
+ " CUSTOM_GRAPH_EDGES2 == rec(color:\"green\",label:\"0\",edges:{a,b|∃x,y.({(x,0),(x,1)} ⊆dom(δ) ∧ y∈δ(x,0) ∧ y∉δ(x,1) ∧ a=TO_STRING(x) ∧ b=TO_STRING(y))}); \n",
+ " CUSTOM_GRAPH_EDGES3 == rec(color:\"green\",label:\"1\",edges:{a,b|∃x,y.({(x,0),(x,1)} ⊆dom(δ) ∧ y∈δ(x,1) ∧ y∉δ(x,0) ∧ a=TO_STRING(x) ∧ b=TO_STRING(y))});\n",
+ " CUSTOM_GRAPH_EDGES4 == rec(color:\"green\",label:\"0, 1\",edges:{a,b|∃x,y.({(x,0),(x,1)} ⊆dom(δ) ∧ y∈δ(x,0) ∧ y∈δ(x,1)∧ a=TO_STRING(x) ∧ b=TO_STRING(y))})\n",
+ "END"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 17,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Machine constants set up using operation 0: $setup_constants()"
+ ]
+ },
+ "execution_count": 17,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":constants"
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+ "<title>9</title>\n",
+ "</g>\n",
+ "<!-- 9->0 -->\n",
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+ "<title>9->0</title>\n",
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+ "</g>\n",
+ "</svg>"
+ ],
+ "text/plain": [
+ "<Dot visualization: custom_graph []>"
+ ]
+ },
+ "execution_count": 18,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":dot custom_graph"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Zuletzt bleibt noch die Spiegelung einer Sprache. Um dies zu erreichen erstellen wir aus einem DFA $M$ einen NFA $M_2$ (Vorgehen nach [Grundkurs Theoretische Informatik](https://link.springer.com/content/pdf/10.1007/978-3-8348-2202-4.pdf \"Vossen, G., & Witt, K. U. (2016). Grundkurs Theoretische Informatik. Springer Fachmedien Wiesbaden.\"), Seite 101). $M_2$ hat als Startzustände die Endzustände von $M$ und als einzigen Endzustand den Startzustand von $M$. Die Übergänge von $M_2$ erhält man, indem man alle Übergange von $M$ umdreht. Dadurch ist der Automat ggf. auch nicht deterministisch."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 19,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Loaded machine: DFA"
+ ]
+ },
+ "execution_count": 19,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "::load\n",
+ "MACHINE DFA\n",
+ "SETS\n",
+ " Z = {z0,z1,z2,z3}\n",
+ "CONSTANTS Σ, M, δ, δ2, F, M2, z_start\n",
+ "PROPERTIES\n",
+ " Σ = {0,1}\n",
+ " ∧\n",
+ " M = (Σ, Z, δ, z_start, F) ∧\n",
+ " δ ∈ (Z×Σ) → Z ∧\n",
+ " z_start ∈ Z ∧\n",
+ " F ⊆ Z ∧\n",
+ " \n",
+ " M2 = (Σ, Z, δ2, F, {z_start}) ∧\n",
+ " δ2 ∈ (Z×Σ) <-> ℙ(Z) ∧\n",
+ " δ2 = {x |∃a,s,s2.(x=((s2, a)↦{s}) ∧ a∈Σ ∧ s∈Z ∧ s2∈Z ∧ δ(s,a) = s2)}\n",
+ " \n",
+ "DEFINITIONS // Für den Zustandsgraphen:\n",
+ " CUSTOM_GRAPH_NODES1 == rec(shape:\"doublecircle\",nodes:{z_start}); // Endzustände\n",
+ " CUSTOM_GRAPH_NODES2 == rec(shape:\"circle\",nodes:Z\\{z_start}); // andere Zustände\n",
+ " CUSTOM_GRAPH_NODES3 == rec(shape:\"none\",color:\"white\",style:\"none\",nodes:{\"\"});\n",
+ " CUSTOM_GRAPH_EDGES1 == rec(color:\"green\",label:\"0\",edges:{x,z| ∃y.(y∈δ2[{(x,0)}] ∧ y∉δ2[{(x,1)}] ∧ z∈y)}); \n",
+ " CUSTOM_GRAPH_EDGES2 == rec(color:\"green\",label:\"1\",edges:{x,z| ∃y.(y∈δ2[{(x,1)}] ∧ y∉δ2[{(x,0)}] ∧ z∈y)});\n",
+ " CUSTOM_GRAPH_EDGES3 == rec(color:\"green\",label:\"0, 1\",edges:{x,z| ∃y.(y∈δ2[{(x,0)}] ∧ y∈δ2[{(x,1)}] ∧ z∈y)});\n",
+ " CUSTOM_GRAPH_EDGES4 == rec(color:\"black\",label:\"\",edges:{\"\"}*F) // Kanten für Startknoten\n",
+ "END"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 20,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Machine constants set up using operation 0: $setup_constants()"
+ ]
+ },
+ "execution_count": 20,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":constants M=({0,1}, //Σ\n",
+ " {z0,z1,z2,z3}, //Z\n",
+ " {(z0,0)↦z1, (z0,1)↦z3,\n",
+ " (z1,0)↦z3, (z1,1)↦z2,\n",
+ " (z2,0)↦z2, (z2,1)↦z2,\n",
+ " (z3,0)↦z3, (z3,1)↦z3 }, //δ\n",
+ " z0, {z0, z2}) //z0, F"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 21,
+ "metadata": {
+ "scrolled": false
+ },
+ "outputs": [
+ {
+ "data": {
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+ "<!-- 1->0 -->\n",
+ "<g id=\"edge3\" class=\"edge\">\n",
+ "<title>1->0</title>\n",
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+ ],
+ "text/plain": [
+ "<Dot visualization: custom_graph []>"
+ ]
+ },
+ "execution_count": 21,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":dot custom_graph"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Nun haben wir bewiesen, dass die regulären Sprachen abgeschlossen über \n",
+ "* Vereinigung A ∪ B\n",
+ "* Komplement A̅\n",
+ "* Schnitt A ∩ B\n",
+ "* Differenz A/B\n",
+ "* Konkatenation AB\n",
+ "* Iteration A*\n",
+ "* Spiegelung sp(A)\n",
+ "\n",
+ "sind. Aber was kann man eigentlich damit machen?"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Mit Hilfe der Abschlusseigenschaften für reguläre Sprachen kann man zeigen, dass eine Sprache\n",
+ "1. regulär ist\n",
+ "2. nicht regulär ist (über Beweis mit Widerspruch)\n",
+ "\n",
+ "Um zu zeigen, dass eine Sprache regulär ist, findet man zwei (oder eine) reguläre Sprachen, die mit Hilfe einer oder mehrerer der oben angegebenen Operationen die gewünschte Sprache ergeben.\n",
+ "\n",
+ "Will man zum Beispiel zeigen, dass die Sprache $L=\\{0^m1^n\\}$ regulär ist, dann wählt man die Sprachen $L_1=\\{0^m\\}$ und $L_2=\\{1^n\\}$. Diese sind aus der Vorlesung als regulär bekannt. Damit gilt dann, dass $L=L_1L_2$ und somit ist L regulär.\n",
+ "\n",
+ "Dabei muss man allerdings beachten, dass $L_1$ und $L_2$ unabhängig voneinander sind. Man kann nicht $L_1=\\{0^n\\}$ und $L_2=\\{1^n\\}$ wählen und damit zeigen, dass $L_3=L_1L_2=\\{0^n1^n\\}$ regulär ist. Dabei wurde die Konkatenation falsch verwendet. In diesem Fall wäre $L_3=L_1L_2=\\{0^m1^n\\}=L$, was wie oben beschrieben regulär ist.\n",
+ "\n",
+ "Um zu zeigen, dass eine Sprache $L$ nicht regulär ist, nimmt man für einen Widerspruchsbeweis an, dass $L$ regulär sei. Dann wählt man eine nicht reguläre Sprache $L_1$ und ggf. eine reguläre Sprache $L_2$ und führt mit Hilfe der Operationen $L$ und ggf. $L_2$ auf $L_1$ zurück. Man zeigt also z.B. dass $L_2∩L=L_1$ gilt. Da $L_2$ regulär ist und die regulären Sprachen abgeschlossen bezüglich des Schnittes zweier Sprachen sind, müsste $L_1$ auch reulär sein, wenn $L$ regulär wäre. Da $L_1$ aber nach Voraussetzung nicht regulär ist, kann $L$ nicht regulär gewesen sein. Das ist ein Wiederspruch zur Annahme und somit ist $L$ nicht regulär. Ein konkretes Beispiel hierfür ist $L=\\{x∈\\{0,1\\}^*|\\text{x hat gleich viele 0en und 1en}\\}$, $L_1=\\{0^n1^n\\}$ und $L_2=L(0^*1^*)$. Aus der Vorlesung ist bekannt, dass $L_1$ nicht regulär und $L_2$ regulär ist. Außerdem gilt $L_2∩L=L_1$. Daher kann L nicht regulär sein."
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "ProB 2",
+ "language": "prob",
+ "name": "prob2"
+ },
+ "language_info": {
+ "codemirror_mode": "prob2_jupyter_repl",
+ "file_extension": ".prob",
+ "mimetype": "text/x-prob2-jupyter-repl",
+ "name": "prob"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/info4/kapitel-2/DFA.ipynb b/info4/kapitel-2/DFA.ipynb
index dc0fca0ed30292f548b7644c6c939b34ce199ce5..1182c2660b38bacda49b1cae10aa21f42526a555 100644
--- a/info4/kapitel-2/DFA.ipynb
+++ b/info4/kapitel-2/DFA.ipynb
@@ -37,7 +37,7 @@
},
{
"cell_type": "code",
- "execution_count": 2,
+ "execution_count": 1,
"metadata": {},
"outputs": [
{
@@ -46,7 +46,7 @@
"Loaded machine: DFA"
]
},
- "execution_count": 2,
+ "execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
@@ -80,7 +80,7 @@
},
{
"cell_type": "code",
- "execution_count": 3,
+ "execution_count": 2,
"metadata": {},
"outputs": [
{
@@ -89,7 +89,7 @@
"Machine constants set up using operation 0: $setup_constants()"
]
},
- "execution_count": 3,
+ "execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
@@ -109,7 +109,7 @@
},
{
"cell_type": "code",
- "execution_count": 4,
+ "execution_count": 3,
"metadata": {},
"outputs": [
{
@@ -121,7 +121,7 @@
"z1"
]
},
- "execution_count": 4,
+ "execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
@@ -139,7 +139,7 @@
},
{
"cell_type": "code",
- "execution_count": 5,
+ "execution_count": 4,
"metadata": {},
"outputs": [
{
@@ -151,7 +151,7 @@
"z2"
]
},
- "execution_count": 5,
+ "execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
@@ -194,7 +194,7 @@
},
{
"cell_type": "code",
- "execution_count": 6,
+ "execution_count": 5,
"metadata": {},
"outputs": [
{
@@ -206,7 +206,7 @@
"FALSE"
]
},
- "execution_count": 6,
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+ "execution_count": 8,
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+ "execution_count": 9,
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]
},
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+ "execution_count": 9,
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diff --git a/info4/kapitel-2/Minimalautomat.ipynb b/info4/kapitel-2/Minimalautomat.ipynb
index f776b2a719edf1ebe59cf03a114be7a1b50b97f4..7d1b0b0e83d32e216f7321514e9635e4509cdb0e 100644
--- a/info4/kapitel-2/Minimalautomat.ipynb
+++ b/info4/kapitel-2/Minimalautomat.ipynb
@@ -6,7 +6,7 @@
"source": [
"## Algorithmus Minimalautomat\n",
"\n",
- "Dieses Notebook begleitet die Vorlesung zur Myhill-Nerode Äquivalenzrelation und dem Minimalautomaten.\n",
+ "Dieses Notebook begleitet die Vorlesung zu dem Minimalautomaten.\n",
"\n",
"Hier ist der Beispiel DFA aus den Folien zur Illustration der Konstruktion des Minimalautomaten.\n"
]
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},
{
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+ "execution_count": 10,
"metadata": {},
"outputs": [
{
@@ -590,7 +636,7 @@
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]
},
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+ "execution_count": 10,
"metadata": {},
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@@ -601,7 +647,7 @@
},
{
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+ "execution_count": 11,
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+ "<g id=\"node6\" class=\"node\">\n",
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@@ -804,7 +881,7 @@
"<Dot visualization: custom_graph []>"
]
},
- "execution_count": 10,
+ "execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
@@ -815,7 +892,7 @@
},
{
"cell_type": "code",
- "execution_count": 11,
+ "execution_count": 12,
"metadata": {},
"outputs": [
{
@@ -824,7 +901,7 @@
"Executed operation: Markiere(z2,z0,0)"
]
},
- "execution_count": 11,
+ "execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
@@ -835,7 +912,7 @@
},
{
"cell_type": "code",
- "execution_count": 12,
+ "execution_count": 13,
"metadata": {},
"outputs": [
{
@@ -844,204 +921,237 @@
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},
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+ "<g id=\"node3\" class=\"node\">\n",
+ "<title>2</title>\n",
+ "<ellipse fill=\"lightgrey\" stroke=\"black\" cx=\"325.23\" cy=\"-360\" rx=\"18.44\" ry=\"18.44\"/>\n",
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- "<text text-anchor=\"middle\" x=\"125.206\" y=\"-14.4\" font-family=\"Times,serif\" font-size=\"12.00\">z1</text>\n",
+ "<g id=\"node4\" class=\"node\">\n",
+ "<title>3</title>\n",
+ "<ellipse fill=\"lightgrey\" stroke=\"black\" cx=\"215.63\" cy=\"-62.15\" rx=\"18.44\" ry=\"18.44\"/>\n",
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"</g>\n",
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+ "<g id=\"edge11\" class=\"edge\">\n",
+ "<title>3->4</title>\n",
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"</g>\n",
"<!-- 5 -->\n",
- "<g id=\"node6\" class=\"node\"><title>5</title>\n",
+ "<g id=\"node6\" class=\"node\">\n",
+ "<title>5</title>\n",
"</g>\n",
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- "<g id=\"edge30\" class=\"edge\"><title>5->2</title>\n",
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@@ -1231,7 +1362,7 @@
"<Dot visualization: custom_graph []>"
]
},
- "execution_count": 14,
+ "execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
@@ -1286,9 +1417,23 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 16,
"metadata": {},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{\\{\\mathit{z0}\\},\\{\\mathit{z1},\\mathit{z2}\\},\\{\\mathit{z3},\\mathit{z4}\\}\\}$"
+ ],
+ "text/plain": [
+ "{{z0},{z1,z2},{z3,z4}}"
+ ]
+ },
+ "execution_count": 16,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
":let ZF {zs| #x.(x:Z & closure(fusioniert)[{x}] = zs)}"
]
@@ -1302,7 +1447,7 @@
},
{
"cell_type": "code",
- "execution_count": 32,
+ "execution_count": 17,
"metadata": {},
"outputs": [
{
@@ -1314,7 +1459,7 @@
"{({z0}↦0↦{z1,z2}),({z0}↦1↦{z1,z2}),({z1,z2}↦0↦{z3,z4}),({z1,z2}↦1↦{z3,z4}),({z3,z4}↦0↦{z3,z4}),({z3,z4}↦1↦{z3,z4})}"
]
},
- "execution_count": 32,
+ "execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
@@ -1325,7 +1470,7 @@
},
{
"cell_type": "code",
- "execution_count": 33,
+ "execution_count": 18,
"metadata": {},
"outputs": [
{
@@ -1335,7 +1480,7 @@
"Preference changed: DOT_ENGINE = dot\n"
]
},
- "execution_count": 33,
+ "execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
@@ -1346,7 +1491,7 @@
},
{
"cell_type": "code",
- "execution_count": 34,
+ "execution_count": 19,
"metadata": {},
"outputs": [
{
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+ "<!-- Generated by graphviz version 2.44.1 (0)\n",
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+ "<svg width=\"136pt\" height=\"218pt\"\n",
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+ "<g id=\"edge6\" class=\"edge\">\n",
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@@ -1421,7 +1575,7 @@
"<Dot visualization: expr_as_graph [δFZFδF={(({z0},0),{z1,z2}),(({z0},1),{z1,z2}),(({z1,z2},0),{z3,z4}),(({z1,z2},1),{z3,z4}),(({z3,z4},0),{z3,z4}),(({z3,z4},1),{z3,z4})} & ZF={{z0},{z1,z2},{z3,z4}}(\"0\",{a,b|a:ZF & b=δF(a, 0)})]>"
]
},
- "execution_count": 34,
+ "execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
@@ -1432,7 +1586,7 @@
},
{
"cell_type": "code",
- "execution_count": 35,
+ "execution_count": 20,
"metadata": {},
"outputs": [
{
@@ -1457,7 +1611,7 @@
"{z3,z4}\t1\t{z3,z4}\n"
]
},
- "execution_count": 35,
+ "execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
@@ -1466,31 +1620,6 @@
":table δF"
]
},
- {
- "cell_type": "code",
- "execution_count": 36,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/plain": [
- "ProB 2 Jupyter kernel: 1.1.1-SNAPSHOT (e52e0cac2096ac02eac65e556edce9dcd142d920)\n",
- "ProB 2: 3.10.0 (0ee5d5eea6894b2899690565dfc3d9042098ce89)\n",
- "ProB CLI:\n",
- "\t1.10.0-nightly (05b24c45455b15cf94ea1daf18d2189efafa22fa)\n",
- "\tLast changed: Tue May 19 11:43:24 2020 +0200\n",
- "\tProlog: SICStus 4.5.1 (x86_64-darwin-17.7.0): Tue Apr 2 15:34:32 CEST 2019"
- ]
- },
- "execution_count": 36,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- ":version"
- ]
- },
{
"cell_type": "code",
"execution_count": null,
diff --git a/info4/kapitel-2/Myhill_Nerode.ipynb b/info4/kapitel-2/Myhill_Nerode.ipynb
new file mode 100644
index 0000000000000000000000000000000000000000..d8861a307139f188c851f1b396681ba49bc788c9
--- /dev/null
+++ b/info4/kapitel-2/Myhill_Nerode.ipynb
@@ -0,0 +1,471 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Der Satz von Myhill und Nerode\n",
+ "Wir betrachten die Myhill-Nerode-Äquivalenzrelation anhand der Sprache L = {a, b, aa, bb, aac, bbc, ccc}."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Loaded machine: EquivalenceRelation"
+ ]
+ },
+ "execution_count": 1,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "MACHINE EquivalenceRelation\n",
+ "/* Ein Modell der Myhill-Nerode Äquivalenzrelation R_L,\n",
+ " der entsprechenden Äquivalenzklassen und dem Index der Sprache.*/\n",
+ "SETS\n",
+ " Alphabet = {a,b,c}\n",
+ "CONSTANTS L, RL, maxsize, All, Classes, index\n",
+ "DEFINITIONS\n",
+ " class(x) == {y | x↦y : RL} ;\n",
+ " ANIMATION_FUNCTION1 == {r,c,i |r=1 ∧ c∈ dom(word) ∧ i=word(c)};\n",
+ " ANIMATION_FUNCTION2 == {r,c,i |r=2 ∧ c=1 ∧ i=z};\n",
+ " ANIMATION_FUNCTION3 == {(1, 0, \"Wort:\"), (2, 0, \"Äquivalenzklasse:\")};\n",
+ " \n",
+ "PROPERTIES\n",
+ " L ⊆ seq(Alphabet) ∧\n",
+ " \n",
+ " // All = {z | z∈seq(Alphabet) ∧ size(z)<=maxsz} & /* beschränkt auf endliche Folgen */\n",
+ " All = UNION(ii).(ii:0..maxsize| (1..ii) --> Alphabet) ∧\n",
+ "\n",
+ " RL = ({x,y| x∈All ∧ y∈All ∧ ∀z.(z∈All ⇒ ( x^z ∈ L ⇔ y^z ∈ L))}) ∧\n",
+ "\n",
+ " L = {[a],[b],[a,a],[b,b],[a,a,c],[b,b,c],[c,c,c]} ∧ maxsize = 3 ∧\n",
+ "\n",
+ " Classes = ran( %x.(x∈All|class(x))) ∧ /* Menge der Äquivalenzklassen {class(x)|x∈All} */\n",
+ " index = card( Classes ) \n",
+ "\n",
+ "ASSERTIONS\n",
+ " /* Test ob wir eine Äquivalenzrelation haben: */\n",
+ " ∀x.(x∈All ⇒ x↦x ∈ RL); /* Reflexivität */\n",
+ " ∀(x,y).(x↦y ∈ RL ⇒ y↦x ∈ RL); /* Symetrie */\n",
+ " ∀(x,y,z).(x↦y ∈ RL ∧ y↦z ∈ RL ⇒ x↦z ∈ RL); /* Transitivität */\n",
+ "\n",
+ " /* Einige Beispiele : */\n",
+ " [a,a] ↦ [b,b] ∈ RL;\n",
+ " [a,a] ↦ [c,c] ∉ RL;\n",
+ " [b,b,c] ↦ [c,c,c] ∈ RL;\n",
+ " class([a,a]) = {[a,a],[b,b]};\n",
+ " class([c,c,c]) = {[a,a,c],[b,b,c],[c,c,c]}\n",
+ "\n",
+ "/* Der durch die Äquivalenzklassen induzierte DFA: */\n",
+ "VARIABLES z, word\n",
+ "INVARIANT z ⊆ All ∧ word ∈ seq(Alphabet)\n",
+ "INITIALISATION z := class([]); word := []\n",
+ "OPERATIONS\n",
+ " Delta(terminal) = PRE terminal∈Alphabet THEN\n",
+ " ANY x WHERE x∈z ∧ ∀x2.(x2∈z ⇒ size(x2)≥size(x)) THEN\n",
+ " z := class(x^[terminal]);\n",
+ " word := word^[terminal]\n",
+ " END\n",
+ " END;\n",
+ " Final = SELECT z ∩ L ≠ {} THEN skip END\n",
+ "END"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Machine constants set up using operation 0: $setup_constants()"
+ ]
+ },
+ "execution_count": 2,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":constants"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Machine initialised using operation 1: $initialise_machine()"
+ ]
+ },
+ "execution_count": 3,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":init"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 4,
+ "metadata": {
+ "scrolled": true
+ },
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$8$"
+ ],
+ "text/plain": [
+ "8"
+ ]
+ },
+ "execution_count": 4,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "index"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Da der $Index(L)=8<\\infty$ ist die gegebene Sprache regulär."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Durch die Äquivalenzklassen wird ein Minimalautomat induziert.\n",
+ "Die Menge der Zustände ist gleich der Menge der Äquivalenzklassen.\n",
+ "Und nach dem Einlesen eines Wortes $w∈Σ^*$ landet man in dem Zustand, der der Äquivalenzklasse von $w$ bezüglich $R_L$ entspricht."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 5,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Machine: EquivalenceRelation\n",
+ "Sets: Alphabet\n",
+ "Constants: L, RL, maxsize, All, Classes, index\n",
+ "Variables: z, word\n",
+ "Operations: \n",
+ "Delta(a)\n",
+ "Delta(b)\n",
+ "Delta(c)"
+ ]
+ },
+ "execution_count": 5,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":browse"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 6,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Executed operation: Delta(a)"
+ ]
+ },
+ "execution_count": 6,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":exec Delta terminal=a"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 7,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Executed operation: Delta(a)"
+ ]
+ },
+ "execution_count": 7,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":exec Delta terminal=a"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 8,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "<table style=\"font-family:monospace\"><tbody>\n",
+ "<tr>\n",
+ "<td style=\"padding:10px\">Wort:</td>\n",
+ "<td style=\"padding:10px\">a</td>\n",
+ "<td style=\"padding:10px\">a</td>\n",
+ "</tr>\n",
+ "<tr>\n",
+ "<td style=\"padding:10px\">Äquivalenzklasse:</td>\n",
+ "<td style=\"padding:10px\">{{(1|->a),(2|->a)},{(1|->b),(2|->b)}}</td>\n",
+ "<td style=\"padding:0px\"></td>\n",
+ "</tr>\n",
+ "</tbody></table>"
+ ],
+ "text/plain": [
+ "<Animation function visualisation>"
+ ]
+ },
+ "execution_count": 8,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":show"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Ein weiteres Beispiel ist die Sprache $L=\\{a^n | n\\geq0\\}$.\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 9,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Loaded machine: EquivalenceRelation2"
+ ]
+ },
+ "execution_count": 9,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "MACHINE EquivalenceRelation2\n",
+ "SETS\n",
+ " Alphabet = {a,b,c}\n",
+ "CONSTANTS L, RL, maxsize, All, Classes, index\n",
+ "DEFINITIONS\n",
+ " class(x) == {y | x↦y : RL} ;\n",
+ " \n",
+ "PROPERTIES\n",
+ " L ⊆ seq(Alphabet) ∧\n",
+ " \n",
+ " // All = {z | z∈seq(Alphabet) ∧ size(z)≤maxsize} & /* Beschränkt auf endliche Folgen */\n",
+ " All = UNION(ii).(ii:0..maxsize| (1..ii) --> Alphabet) ∧\n",
+ "\n",
+ " RL = ({x,y| x∈All ∧ y∈All ∧ ∀z.(z∈All ⇒ ( x^z ∈ L ⇔ y^z ∈ L))}) ∧\n",
+ "\n",
+ " /*L = {x | x=a^n ∧ size(x)≤maxsize} Beschränkt auf endliche Folgen*/ \n",
+ " L = UNION(ii).(ii:0..maxsize| (1..ii) --> {a}) ∧ maxsize = 3 ∧\n",
+ "\n",
+ " Classes = ran( %x.(x∈All|class(x))) ∧ /* Menge der Äquivalenzklassen {class(x)|x∈All} */\n",
+ " index = card( Classes )\n",
+ "END"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 10,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Machine constants set up using operation 0: $setup_constants()"
+ ]
+ },
+ "execution_count": 10,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":constants"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 11,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Machine initialised using operation 1: $initialise_machine()"
+ ]
+ },
+ "execution_count": 11,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":init"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 12,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$5$"
+ ],
+ "text/plain": [
+ "5"
+ ]
+ },
+ "execution_count": 12,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "index"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Auch diese Sprache ist offensichtlich regulär.\n",
+ "Hier könnte man davon ausgehen, dass dies aus der Beschränkung auf endliche Folgen folgt.\n",
+ "Denn je länger man die zugelassenen Folgen macht, desdo mehr Äquivalenzklassen gibt es.\n",
+ "Es gibt eine Äquivalenzklasse für jedes $a^m$ mit $0≤m≤n$ und eine für Wörter, die nicht in der Sprache liegen, also insgesamt $n+2$."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 13,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\{\\emptyset,\\{(1\\mapsto \\mathit{a})\\},\\{(1\\mapsto \\mathit{a}),(2\\mapsto \\mathit{a})\\},\\{(1\\mapsto \\mathit{a}),(2\\mapsto \\mathit{a}),(3\\mapsto \\mathit{a})\\}\\}$"
+ ],
+ "text/plain": [
+ "{∅,{(1↦a)},{(1↦a),(2↦a)},{(1↦a),(2↦a),(3↦a)}}"
+ ]
+ },
+ "execution_count": 13,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "L"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 14,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\mathit{TRUE}$\n",
+ "\n",
+ "**Solution:**\n",
+ "* $\\mathit{x} = \\{(1\\mapsto \\mathit{a})\\}$\n",
+ "* $\\mathit{y} = \\{(1\\mapsto \\mathit{a}),(2\\mapsto \\mathit{a})\\}$\n",
+ "* $\\mathit{z} = \\{(1\\mapsto \\mathit{a}),(2\\mapsto \\mathit{a})\\}$"
+ ],
+ "text/plain": [
+ "TRUE\n",
+ "\n",
+ "Solution:\n",
+ "\tx = {(1↦a)}\n",
+ "\ty = {(1↦a),(2↦a)}\n",
+ "\tz = {(1↦a),(2↦a)}"
+ ]
+ },
+ "execution_count": 14,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "x=[a] ∧ y=[a, a] ∧ ∃z.(z∈All ∧ not(x^z ∈ L ⇔ y^z ∈ L)) /*Gegenbeispiel für [a] ↦ [a,a] ∈ R_L*/"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Betrachtet man allerdings den unendlichen Fall, so fallen die Wörter $a^m$ alle in eine Klasse.\n",
+ "Dies gilt, da es kein \"Ende\" der Folgen mehr gibt und somit kein Gegenbeispiel wie oben.\n",
+ "Insgesamt gibt es dann 2 Äquivalenzklassen ([a], [b]) und somit bleibt die Sprache dennoch regulär."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "ProB 2",
+ "language": "prob",
+ "name": "prob2"
+ },
+ "language_info": {
+ "codemirror_mode": "prob2_jupyter_repl",
+ "file_extension": ".prob",
+ "mimetype": "text/x-prob2-jupyter-repl",
+ "name": "prob"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/info4/kapitel-2/NFA_nach_DFA.ipynb b/info4/kapitel-2/NFA_nach_DFA.ipynb
index 55f2a267a5c4933c7b0c25393fb50a2e754fbe89..9735e0c4795eb6880dde63d60e5bb616b9acb1fe 100644
--- a/info4/kapitel-2/NFA_nach_DFA.ipynb
+++ b/info4/kapitel-2/NFA_nach_DFA.ipynb
@@ -100,7 +100,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "Die Übergangsfunktion $\\delta$ gibt us für einen Zustand und ein Symbol die möglichen nächsten Zustände an."
+ "Die Überführungsfunktion $\\delta$ gibt uns für einen Zustand und ein Symbol die möglichen nächsten Zustände an."
]
},
{
diff --git a/info4/kapitel-2/img/konkatenation_M1.svg b/info4/kapitel-2/img/konkatenation_M1.svg
new file mode 100644
index 0000000000000000000000000000000000000000..e0944989c3ad4409f62c8b2cd5d394d7a4aed252
--- /dev/null
+++ b/info4/kapitel-2/img/konkatenation_M1.svg
@@ -0,0 +1 @@
+<svg width="720" height="713pt" viewBox="0 0 540 712.99" xmlns="http://www.w3.org/2000/svg" style="width:100%;height:100%"><g class="graph" transform="matrix(.98 0 0 .98 3.92 710.284)"><path fill="#fff" stroke="transparent" d="M-4 4v-728.78h551.96V4H-4z"/><g class="node"><circle fill="#d3d3d3" stroke="#000" cx="360.3" cy="-500" r="23.3"/><circle fill="none" stroke="#000" cx="360.3" cy="-500" r="27.29"/><text text-anchor="middle" x="360.3" y="-496.9" font-family="Times,serif" font-size="12">z10</text></g><g class="node"><circle fill="#d3d3d3" stroke="#000" cx="138.3" cy="-271" r="23.3"/><circle fill="none" stroke="#000" cx="138.3" cy="-271" r="27.29"/><text text-anchor="middle" x="138.3" y="-267.9" font-family="Times,serif" font-size="12">z12</text></g><g class="edge"><path fill="none" stroke="green" d="M341.37-479.65C301.64-439.02 209.5-344.8 164.28-298.56"/><path fill="green" stroke="green" d="M166.67-296.01l-9.49 4.71 4.49-9.6 5 4.89z"/><text text-anchor="middle" x="253.3" y="-381.9" font-family="Times,serif" font-size="12">0</text></g><g class="node"><circle fill="#d3d3d3" stroke="#000" cx="294.3" cy="-52" r="23.29"/><text text-anchor="middle" x="294.3" y="-48.9" font-family="Times,serif" font-size="12">z11</text></g><g class="edge"><path fill="none" stroke="green" d="M356.42-472.77c-11.41 77.09-44.81 302.78-57.32 387.31"/><path fill="green" stroke="green" d="M302.54-84.79l-4.93 9.38-1.99-10.41 6.92 1.03z"/><text text-anchor="middle" x="335.3" y="-267.9" font-family="Times,serif" font-size="12">1</text></g><g class="edge"><path fill="none" stroke="green" d="M157.89-290.49c13.08-5.49 25.8 1.01 25.8 19.49 0 13.72-7.01 20.83-15.97 21.34"/><path fill="green" stroke="green" d="M167.07-246.22l-9.18-5.29 10.48-1.58-1.3 6.87z"/><text text-anchor="middle" x="187.69" y="-267.9" font-family="Times,serif" font-size="12">0</text></g><g class="edge"><path fill="none" stroke="green" d="M153.83-248.4c28.69 39.92 89.94 125.12 121.22 168.63"/><path fill="green" stroke="green" d="M278.07-81.56l3 10.16-8.68-6.08 5.68-4.08z"/><text text-anchor="middle" x="225.3" y="-153.9" font-family="Times,serif" font-size="12">1</text></g><g class="edge"><path fill="none" stroke="green" d="M284.58-73.78C273.5-96.62 254.2-134 233.3-163.5c-20.8-29.35-48.56-59.66-68.7-80.4"/><path fill="green" stroke="green" d="M162.06-241.5l-4.5-9.59 9.5 4.69-5 4.9z"/><text text-anchor="middle" x="245.3" y="-153.9" font-family="Times,serif" font-size="12">0</text></g><g class="edge"><path fill="none" stroke="green" d="M310.44-68.98c12.5-6.35 25.25-.69 25.25 16.98 0 12.98-6.88 19.48-15.48 19.5"/><path fill="green" stroke="green" d="M319.25-29.13l-8.81-5.89 10.56-.89-1.75 6.78z"/><text text-anchor="middle" x="339.69" y="-48.9" font-family="Times,serif" font-size="12">1</text></g><g class="edge"><path fill="none" stroke="#000" d="M360.3-662.84v125.01"/><path stroke="#000" d="M363.8-537.61l-3.5 10-3.5-10h7z"/></g></g></svg>
\ No newline at end of file
diff --git a/info4/kapitel-2/img/konkatenation_M2.svg b/info4/kapitel-2/img/konkatenation_M2.svg
new file mode 100644
index 0000000000000000000000000000000000000000..98312078750a113efcd1a1e1d8b427a91f1340d6
--- /dev/null
+++ b/info4/kapitel-2/img/konkatenation_M2.svg
@@ -0,0 +1 @@
+<svg width="720" height="952" viewBox="0 0 540 714.07" xmlns="http://www.w3.org/2000/svg" style="width:100%;height:100%"><g class="graph" transform="matrix(.98 0 0 .98 3.92 710.304)"><path fill="#fff" stroke="transparent" d="M-4 4v-728.8h551.14V4H-4z"/><g class="node"><circle fill="#d3d3d3" stroke="#000" cx="109.57" cy="-278" r="23.3"/><circle fill="none" stroke="#000" cx="109.57" cy="-278" r="27.29"/><text text-anchor="middle" x="109.57" y="-274.9" font-family="Times,serif" font-size="12">z21</text></g><g class="edge"><path fill="none" stroke="green" d="M128.69-297.63c13.24-5.94 26.27.6 26.27 19.63 0 14.12-7.18 21.37-16.29 21.74"/><path fill="green" stroke="green" d="M137.75-252.88l-9.06-5.49 10.51-1.36-1.45 6.85z"/><text text-anchor="middle" x="158.96" y="-274.9" font-family="Times,serif" font-size="12">1</text></g><g class="node"><circle fill="#d3d3d3" stroke="#000" cx="285.57" cy="-53" r="23.29"/><text text-anchor="middle" x="285.57" y="-49.9" font-family="Times,serif" font-size="12">z22</text></g><g class="edge"><path fill="none" stroke="green" d="M126.13-256.02c32.31 40.95 103.71 131.4 138.94 176.05"/><path fill="green" stroke="green" d="M267.97-81.95l3.45 10.02-8.95-5.68 5.5-4.34z"/><text text-anchor="middle" x="207.57" y="-157.9" font-family="Times,serif" font-size="12">0</text></g><g class="node"><circle fill="#d3d3d3" stroke="#000" cx="285.57" cy="-503" r="23.29"/><text text-anchor="middle" x="285.57" y="-499.9" font-family="Times,serif" font-size="12">z20</text></g><g class="edge"><path fill="none" stroke="green" d="M271.46-484.13c-30.31 38.42-101.93 129.16-138.91 176.01"/><path fill="green" stroke="green" d="M135.22-305.86l-8.94 5.68 3.45-10.01 5.49 4.33z"/><text text-anchor="middle" x="207.57" y="-391.9" font-family="Times,serif" font-size="12">1</text></g><g class="edge"><path fill="none" stroke="green" d="M285.57-479.48v392.9"/><path fill="green" stroke="green" d="M289.07-86.57l-3.5 10-3.5-10h7z"/><text text-anchor="middle" x="289.57" y="-274.9" font-family="Times,serif" font-size="12">0</text></g><g class="edge"><path fill="none" stroke="green" d="M301.28-70.4c12.64-6.96 25.68-1.16 25.68 17.4 0 13.78-7.18 20.52-16.05 20.24"/><path fill="green" stroke="green" d="M309.88-29.41l-8.6-6.19 10.59-.52-1.99 6.71z"/><text text-anchor="middle" x="338.46" y="-49.9" font-family="Times,serif" font-size="12">0, 1</text></g><g class="edge"><path fill="none" stroke="#000" d="M285.57-661.9v124.95"/><path stroke="#000" d="M289.07-536.64l-3.5 10-3.5-10h7z"/></g></g></svg>
\ No newline at end of file
diff --git a/info4/kapitel-3/CYK_Algorithmus.ipynb b/info4/kapitel-3/CYK_Algorithmus.ipynb
index 79917a0d46eb43a2b5b0bce6d2178d87be24db1a..d83a945820f7dc73e07cc04a00340a21e41f4dff 100644
--- a/info4/kapitel-3/CYK_Algorithmus.ipynb
+++ b/info4/kapitel-3/CYK_Algorithmus.ipynb
@@ -9,7 +9,7 @@
},
{
"cell_type": "code",
- "execution_count": 74,
+ "execution_count": 1,
"metadata": {},
"outputs": [
{
@@ -18,7 +18,7 @@
"Loaded machine: CYK"
]
},
- "execution_count": 74,
+ "execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
@@ -79,7 +79,7 @@
},
{
"cell_type": "code",
- "execution_count": 75,
+ "execution_count": 2,
"metadata": {},
"outputs": [
{
@@ -88,7 +88,7 @@
"Machine constants set up using operation 0: $setup_constants()"
]
},
- "execution_count": 75,
+ "execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
@@ -99,7 +99,7 @@
},
{
"cell_type": "code",
- "execution_count": 76,
+ "execution_count": 3,
"metadata": {},
"outputs": [
{
@@ -108,7 +108,7 @@
"Machine initialised using operation 1: $initialise_machine()"
]
},
- "execution_count": 76,
+ "execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
@@ -126,7 +126,7 @@
},
{
"cell_type": "code",
- "execution_count": 77,
+ "execution_count": 4,
"metadata": {},
"outputs": [
{
@@ -138,7 +138,7 @@
"{(1↦b),(2↦a),(3↦a),(4↦b),(5↦a)}"
]
},
- "execution_count": 77,
+ "execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
@@ -152,12 +152,12 @@
"metadata": {},
"source": [
"Am Anfang werden die Werte für T(i,0) in der INITIALISATION der Maschine berechnet:\n",
- "* for(i =1,2,...,n){T(i,0)={A∈N | A→x(i) ist Regel in P};"
+ "* for(i =1,2,...,n){T(i,0)={A∈N | A→x(i) ist Regel in P};}"
]
},
{
"cell_type": "code",
- "execution_count": 78,
+ "execution_count": 5,
"metadata": {},
"outputs": [
{
@@ -184,7 +184,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 78,
+ "execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
@@ -195,7 +195,7 @@
},
{
"cell_type": "code",
- "execution_count": 79,
+ "execution_count": 6,
"metadata": {},
"outputs": [
{
@@ -207,7 +207,7 @@
"{S,A,B,C}"
]
},
- "execution_count": 79,
+ "execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
@@ -218,7 +218,7 @@
},
{
"cell_type": "code",
- "execution_count": 59,
+ "execution_count": 7,
"metadata": {},
"outputs": [
{
@@ -230,7 +230,7 @@
"{({(1↦S)}↦{(1↦A),(2↦B)}),({(1↦S)}↦{(1↦B),(2↦C)}),({(1↦A)}↦{(1↦a)}),({(1↦A)}↦{(1↦B),(2↦A)}),({(1↦B)}↦{(1↦b)}),({(1↦B)}↦{(1↦C),(2↦C)}),({(1↦C)}↦{(1↦a)}),({(1↦C)}↦{(1↦A),(2↦B)})}"
]
},
- "execution_count": 59,
+ "execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
@@ -241,7 +241,7 @@
},
{
"cell_type": "code",
- "execution_count": 56,
+ "execution_count": 8,
"metadata": {},
"outputs": [
{
@@ -253,7 +253,7 @@
"(1↦1)"
]
},
- "execution_count": 56,
+ "execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
@@ -264,7 +264,7 @@
},
{
"cell_type": "code",
- "execution_count": 22,
+ "execution_count": 9,
"metadata": {},
"outputs": [
{
@@ -273,7 +273,7 @@
"Executed operation: For_k_loop(1,1,{S,A})"
]
},
- "execution_count": 22,
+ "execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
@@ -284,7 +284,7 @@
},
{
"cell_type": "code",
- "execution_count": 23,
+ "execution_count": 10,
"metadata": {},
"outputs": [
{
@@ -296,7 +296,7 @@
"(2↦1)"
]
},
- "execution_count": 23,
+ "execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
@@ -307,7 +307,7 @@
},
{
"cell_type": "code",
- "execution_count": 24,
+ "execution_count": 11,
"metadata": {},
"outputs": [
{
@@ -341,7 +341,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 24,
+ "execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
@@ -352,7 +352,7 @@
},
{
"cell_type": "code",
- "execution_count": 25,
+ "execution_count": 12,
"metadata": {},
"outputs": [
{
@@ -361,7 +361,7 @@
"Executed operation: For_k_loop(2,1,{B})"
]
},
- "execution_count": 25,
+ "execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
@@ -372,7 +372,7 @@
},
{
"cell_type": "code",
- "execution_count": 26,
+ "execution_count": 13,
"metadata": {},
"outputs": [
{
@@ -384,7 +384,7 @@
"(3↦1)"
]
},
- "execution_count": 26,
+ "execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
@@ -395,7 +395,7 @@
},
{
"cell_type": "code",
- "execution_count": 27,
+ "execution_count": 14,
"metadata": {},
"outputs": [
{
@@ -429,7 +429,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 27,
+ "execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
@@ -440,7 +440,7 @@
},
{
"cell_type": "code",
- "execution_count": 28,
+ "execution_count": 15,
"metadata": {},
"outputs": [
{
@@ -452,7 +452,7 @@
"{(1↦0↦{B}),(1↦1↦{S,A}),(2↦0↦{A,C}),(2↦1↦{B}),(3↦0↦{A,C}),(4↦0↦{B}),(5↦0↦{A,C})}"
]
},
- "execution_count": 28,
+ "execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
@@ -463,7 +463,7 @@
},
{
"cell_type": "code",
- "execution_count": 29,
+ "execution_count": 16,
"metadata": {},
"outputs": [
{
@@ -472,7 +472,7 @@
"Executed operation: For_k_loop(3,1,{S,C})"
]
},
- "execution_count": 29,
+ "execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
@@ -483,7 +483,7 @@
},
{
"cell_type": "code",
- "execution_count": 30,
+ "execution_count": 17,
"metadata": {},
"outputs": [
{
@@ -492,7 +492,7 @@
"Executed operation: For_k_loop(4,1,{S,A})"
]
},
- "execution_count": 30,
+ "execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
@@ -503,7 +503,7 @@
},
{
"cell_type": "code",
- "execution_count": 31,
+ "execution_count": 18,
"metadata": {},
"outputs": [
{
@@ -537,7 +537,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 31,
+ "execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
@@ -548,7 +548,7 @@
},
{
"cell_type": "code",
- "execution_count": 32,
+ "execution_count": 19,
"metadata": {},
"outputs": [
{
@@ -557,7 +557,7 @@
"Executed operation: For_k_loop(1,2,{})"
]
},
- "execution_count": 32,
+ "execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
@@ -568,7 +568,7 @@
},
{
"cell_type": "code",
- "execution_count": 33,
+ "execution_count": 20,
"metadata": {},
"outputs": [
{
@@ -580,7 +580,7 @@
"(2↦2)"
]
},
- "execution_count": 33,
+ "execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
@@ -591,7 +591,7 @@
},
{
"cell_type": "code",
- "execution_count": 34,
+ "execution_count": 21,
"metadata": {},
"outputs": [
{
@@ -632,7 +632,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 34,
+ "execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
@@ -643,7 +643,7 @@
},
{
"cell_type": "code",
- "execution_count": 35,
+ "execution_count": 22,
"metadata": {},
"outputs": [
{
@@ -652,7 +652,7 @@
"Executed operation: For_k_loop(2,2,{B})"
]
},
- "execution_count": 35,
+ "execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
@@ -663,7 +663,7 @@
},
{
"cell_type": "code",
- "execution_count": 36,
+ "execution_count": 23,
"metadata": {},
"outputs": [
{
@@ -672,7 +672,7 @@
"Executed operation: For_k_loop(3,2,{B})"
]
},
- "execution_count": 36,
+ "execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
@@ -683,7 +683,7 @@
},
{
"cell_type": "code",
- "execution_count": 37,
+ "execution_count": 24,
"metadata": {},
"outputs": [
{
@@ -724,7 +724,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 37,
+ "execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
@@ -735,7 +735,7 @@
},
{
"cell_type": "code",
- "execution_count": 38,
+ "execution_count": 25,
"metadata": {},
"outputs": [
{
@@ -744,7 +744,7 @@
"Executed operation: For_k_loop(1,3,{})"
]
},
- "execution_count": 38,
+ "execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
@@ -755,7 +755,7 @@
},
{
"cell_type": "code",
- "execution_count": 39,
+ "execution_count": 26,
"metadata": {},
"outputs": [
{
@@ -803,7 +803,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 39,
+ "execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
@@ -814,7 +814,7 @@
},
{
"cell_type": "code",
- "execution_count": 40,
+ "execution_count": 27,
"metadata": {},
"outputs": [
{
@@ -823,7 +823,7 @@
"Executed operation: For_k_loop(2,3,{S,A,C})"
]
},
- "execution_count": 40,
+ "execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
@@ -834,7 +834,7 @@
},
{
"cell_type": "code",
- "execution_count": 41,
+ "execution_count": 28,
"metadata": {},
"outputs": [
{
@@ -882,7 +882,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 41,
+ "execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
@@ -893,7 +893,7 @@
},
{
"cell_type": "code",
- "execution_count": 42,
+ "execution_count": 29,
"metadata": {},
"outputs": [
{
@@ -902,7 +902,7 @@
"Executed operation: For_k_loop(1,4,{S,A,C})"
]
},
- "execution_count": 42,
+ "execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
@@ -913,7 +913,7 @@
},
{
"cell_type": "code",
- "execution_count": 43,
+ "execution_count": 30,
"metadata": {},
"outputs": [
{
@@ -968,7 +968,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 43,
+ "execution_count": 30,
"metadata": {},
"output_type": "execute_result"
}
@@ -979,7 +979,7 @@
},
{
"cell_type": "code",
- "execution_count": 46,
+ "execution_count": 31,
"metadata": {},
"outputs": [
{
@@ -991,7 +991,7 @@
"TRUE"
]
},
- "execution_count": 46,
+ "execution_count": 31,
"metadata": {},
"output_type": "execute_result"
}
@@ -1002,7 +1002,7 @@
},
{
"cell_type": "code",
- "execution_count": 47,
+ "execution_count": 32,
"metadata": {},
"outputs": [
{
@@ -1011,7 +1011,7 @@
"Executed operation: TRUE <-- Accept()"
]
},
- "execution_count": 47,
+ "execution_count": 32,
"metadata": {},
"output_type": "execute_result"
}
@@ -1022,7 +1022,7 @@
},
{
"cell_type": "code",
- "execution_count": 48,
+ "execution_count": 33,
"metadata": {},
"outputs": [
{
@@ -1034,7 +1034,7 @@
"(1↦5↦5)"
]
},
- "execution_count": 48,
+ "execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
diff --git a/info4/kapitel-3/PDA-Kellerautomaten.ipynb b/info4/kapitel-3/PDA-Kellerautomaten.ipynb
index d2f8a3452066adbea33a42fd926cb9de0dc380ce..27ccb389590d60227542e83247123599296d760e 100644
--- a/info4/kapitel-3/PDA-Kellerautomaten.ipynb
+++ b/info4/kapitel-3/PDA-Kellerautomaten.ipynb
@@ -191,7 +191,7 @@
},
{
"cell_type": "code",
- "execution_count": 63,
+ "execution_count": 5,
"metadata": {},
"outputs": [
{
@@ -199,13 +199,13 @@
"text/plain": [
"Machine: PDA\n",
"Sets: Z, SYMBOLE\n",
- "Constants: δ\n",
+ "Constants: δ, Σ, Γ\n",
"Variables: z, α, γ\n",
"Operations: \n",
"Schritt(z0,[A,BOT])"
]
},
- "execution_count": 63,
+ "execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
@@ -216,7 +216,7 @@
},
{
"cell_type": "code",
- "execution_count": 64,
+ "execution_count": 6,
"metadata": {},
"outputs": [
{
@@ -225,7 +225,7 @@
"Executed operation: Schritt(z0,[A,BOT])"
]
},
- "execution_count": 64,
+ "execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
@@ -236,7 +236,7 @@
},
{
"cell_type": "code",
- "execution_count": 65,
+ "execution_count": 7,
"metadata": {},
"outputs": [
{
@@ -267,7 +267,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 65,
+ "execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
@@ -278,7 +278,7 @@
},
{
"cell_type": "code",
- "execution_count": 66,
+ "execution_count": 8,
"metadata": {},
"outputs": [
{
@@ -290,7 +290,7 @@
"(z0↦[a,b,b]↦[A,BOT])"
]
},
- "execution_count": 66,
+ "execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
@@ -301,7 +301,7 @@
},
{
"cell_type": "code",
- "execution_count": 67,
+ "execution_count": 9,
"metadata": {},
"outputs": [
{
@@ -309,13 +309,13 @@
"text/plain": [
"Machine: PDA\n",
"Sets: Z, SYMBOLE\n",
- "Constants: δ\n",
+ "Constants: δ, Σ, Γ\n",
"Variables: z, α, γ\n",
"Operations: \n",
"Schritt(z0,[A,A])"
]
},
- "execution_count": 67,
+ "execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
@@ -326,7 +326,7 @@
},
{
"cell_type": "code",
- "execution_count": 68,
+ "execution_count": 10,
"metadata": {},
"outputs": [
{
@@ -335,7 +335,7 @@
"Executed operation: Schritt(z0,[A,A])"
]
},
- "execution_count": 68,
+ "execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
@@ -346,7 +346,7 @@
},
{
"cell_type": "code",
- "execution_count": 69,
+ "execution_count": 11,
"metadata": {},
"outputs": [
{
@@ -377,7 +377,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 69,
+ "execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
@@ -388,7 +388,7 @@
},
{
"cell_type": "code",
- "execution_count": 70,
+ "execution_count": 12,
"metadata": {},
"outputs": [
{
@@ -396,13 +396,13 @@
"text/plain": [
"Machine: PDA\n",
"Sets: Z, SYMBOLE\n",
- "Constants: δ\n",
+ "Constants: δ, Σ, Γ\n",
"Variables: z, α, γ\n",
"Operations: \n",
"Schritt(z1,[])"
]
},
- "execution_count": 70,
+ "execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
@@ -413,7 +413,7 @@
},
{
"cell_type": "code",
- "execution_count": 71,
+ "execution_count": 13,
"metadata": {},
"outputs": [
{
@@ -422,7 +422,7 @@
"Executed operation: Schritt(z1,[])"
]
},
- "execution_count": 71,
+ "execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
@@ -433,7 +433,7 @@
},
{
"cell_type": "code",
- "execution_count": 72,
+ "execution_count": 14,
"metadata": {},
"outputs": [
{
@@ -461,7 +461,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 72,
+ "execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
@@ -472,7 +472,7 @@
},
{
"cell_type": "code",
- "execution_count": 73,
+ "execution_count": 15,
"metadata": {},
"outputs": [
{
@@ -480,13 +480,13 @@
"text/plain": [
"Machine: PDA\n",
"Sets: Z, SYMBOLE\n",
- "Constants: δ\n",
+ "Constants: δ, Σ, Γ\n",
"Variables: z, α, γ\n",
"Operations: \n",
"Schritt(z1,[])"
]
},
- "execution_count": 73,
+ "execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
@@ -497,7 +497,7 @@
},
{
"cell_type": "code",
- "execution_count": 74,
+ "execution_count": 16,
"metadata": {},
"outputs": [
{
@@ -506,7 +506,7 @@
"Executed operation: Schritt(z1,[])"
]
},
- "execution_count": 74,
+ "execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
@@ -517,7 +517,7 @@
},
{
"cell_type": "code",
- "execution_count": 75,
+ "execution_count": 17,
"metadata": {},
"outputs": [
{
@@ -542,7 +542,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 75,
+ "execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
@@ -553,7 +553,7 @@
},
{
"cell_type": "code",
- "execution_count": 76,
+ "execution_count": 18,
"metadata": {},
"outputs": [
{
@@ -561,13 +561,13 @@
"text/plain": [
"Machine: PDA\n",
"Sets: Z, SYMBOLE\n",
- "Constants: δ\n",
+ "Constants: δ, Σ, Γ\n",
"Variables: z, α, γ\n",
"Operations: \n",
"LambdaSchritt(z1,[])"
]
},
- "execution_count": 76,
+ "execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
@@ -578,7 +578,7 @@
},
{
"cell_type": "code",
- "execution_count": 77,
+ "execution_count": 19,
"metadata": {},
"outputs": [
{
@@ -587,7 +587,7 @@
"Executed operation: LambdaSchritt(z1,[])"
]
},
- "execution_count": 77,
+ "execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
@@ -598,7 +598,7 @@
},
{
"cell_type": "code",
- "execution_count": 78,
+ "execution_count": 20,
"metadata": {},
"outputs": [
{
@@ -623,7 +623,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 78,
+ "execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
@@ -634,7 +634,7 @@
},
{
"cell_type": "code",
- "execution_count": 79,
+ "execution_count": 21,
"metadata": {},
"outputs": [
{
@@ -642,13 +642,13 @@
"text/plain": [
"Machine: PDA\n",
"Sets: Z, SYMBOLE\n",
- "Constants: δ\n",
+ "Constants: δ, Σ, Γ\n",
"Variables: z, α, γ\n",
"Operations: \n",
"Akzeptieren()"
]
},
- "execution_count": 79,
+ "execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
@@ -659,7 +659,7 @@
},
{
"cell_type": "code",
- "execution_count": 80,
+ "execution_count": 22,
"metadata": {},
"outputs": [
{
@@ -668,7 +668,7 @@
"Executed operation: Akzeptieren()"
]
},
- "execution_count": 80,
+ "execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
@@ -695,7 +695,7 @@
},
{
"cell_type": "code",
- "execution_count": 34,
+ "execution_count": 23,
"metadata": {},
"outputs": [
{
@@ -704,7 +704,7 @@
"Loaded machine: PDA_für_kfG"
]
},
- "execution_count": 34,
+ "execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
@@ -775,7 +775,7 @@
},
{
"cell_type": "code",
- "execution_count": 35,
+ "execution_count": 24,
"metadata": {},
"outputs": [
{
@@ -784,7 +784,7 @@
"Machine constants set up using operation 0: $setup_constants()"
]
},
- "execution_count": 35,
+ "execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
@@ -795,7 +795,7 @@
},
{
"cell_type": "code",
- "execution_count": 36,
+ "execution_count": 25,
"metadata": {},
"outputs": [
{
@@ -804,7 +804,7 @@
"Machine initialised using operation 1: $initialise_machine()"
]
},
- "execution_count": 36,
+ "execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
@@ -815,7 +815,7 @@
},
{
"cell_type": "code",
- "execution_count": 37,
+ "execution_count": 26,
"metadata": {},
"outputs": [
{
@@ -844,7 +844,7 @@
"z0\tlambda\tlambda\tz0\t[]\n"
]
},
- "execution_count": 37,
+ "execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
@@ -855,7 +855,7 @@
},
{
"cell_type": "code",
- "execution_count": 38,
+ "execution_count": 27,
"metadata": {},
"outputs": [
{
@@ -889,7 +889,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 38,
+ "execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
@@ -900,7 +900,7 @@
},
{
"cell_type": "code",
- "execution_count": 39,
+ "execution_count": 28,
"metadata": {},
"outputs": [
{
@@ -915,7 +915,7 @@
"LambdaSchritt(z0,[a,S,b])"
]
},
- "execution_count": 39,
+ "execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
@@ -926,7 +926,7 @@
},
{
"cell_type": "code",
- "execution_count": 40,
+ "execution_count": 29,
"metadata": {},
"outputs": [
{
@@ -935,7 +935,7 @@
"Executed operation: LambdaSchritt(z0,[a,S,b])"
]
},
- "execution_count": 40,
+ "execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
@@ -946,7 +946,7 @@
},
{
"cell_type": "code",
- "execution_count": 41,
+ "execution_count": 30,
"metadata": {},
"outputs": [
{
@@ -980,7 +980,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 41,
+ "execution_count": 30,
"metadata": {},
"output_type": "execute_result"
}
@@ -991,7 +991,7 @@
},
{
"cell_type": "code",
- "execution_count": 42,
+ "execution_count": 31,
"metadata": {},
"outputs": [
{
@@ -1005,7 +1005,7 @@
"Schritt(z0,[])"
]
},
- "execution_count": 42,
+ "execution_count": 31,
"metadata": {},
"output_type": "execute_result"
}
@@ -1016,7 +1016,7 @@
},
{
"cell_type": "code",
- "execution_count": 43,
+ "execution_count": 32,
"metadata": {},
"outputs": [
{
@@ -1025,7 +1025,7 @@
"Executed operation: Schritt(z0,[])"
]
},
- "execution_count": 43,
+ "execution_count": 32,
"metadata": {},
"output_type": "execute_result"
}
@@ -1036,7 +1036,7 @@
},
{
"cell_type": "code",
- "execution_count": 44,
+ "execution_count": 33,
"metadata": {},
"outputs": [
{
@@ -1067,7 +1067,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 44,
+ "execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
@@ -1078,7 +1078,7 @@
},
{
"cell_type": "code",
- "execution_count": 45,
+ "execution_count": 34,
"metadata": {},
"outputs": [
{
@@ -1093,7 +1093,7 @@
"LambdaSchritt(z0,[a,S,b])"
]
},
- "execution_count": 45,
+ "execution_count": 34,
"metadata": {},
"output_type": "execute_result"
}
@@ -1104,7 +1104,7 @@
},
{
"cell_type": "code",
- "execution_count": 46,
+ "execution_count": 35,
"metadata": {},
"outputs": [
{
@@ -1113,7 +1113,7 @@
"Executed operation: LambdaSchritt(z0,[C])"
]
},
- "execution_count": 46,
+ "execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
@@ -1124,7 +1124,7 @@
},
{
"cell_type": "code",
- "execution_count": 47,
+ "execution_count": 36,
"metadata": {},
"outputs": [
{
@@ -1155,7 +1155,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 47,
+ "execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
@@ -1166,7 +1166,7 @@
},
{
"cell_type": "code",
- "execution_count": 48,
+ "execution_count": 37,
"metadata": {},
"outputs": [
{
@@ -1180,7 +1180,7 @@
"LambdaSchritt(z0,[a,b])"
]
},
- "execution_count": 48,
+ "execution_count": 37,
"metadata": {},
"output_type": "execute_result"
}
@@ -1191,7 +1191,7 @@
},
{
"cell_type": "code",
- "execution_count": 49,
+ "execution_count": 38,
"metadata": {},
"outputs": [
{
@@ -1200,7 +1200,7 @@
"Executed operation: LambdaSchritt(z0,[a,b])"
]
},
- "execution_count": 49,
+ "execution_count": 38,
"metadata": {},
"output_type": "execute_result"
}
@@ -1211,7 +1211,7 @@
},
{
"cell_type": "code",
- "execution_count": 50,
+ "execution_count": 39,
"metadata": {},
"outputs": [
{
@@ -1242,7 +1242,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 50,
+ "execution_count": 39,
"metadata": {},
"output_type": "execute_result"
}
@@ -1253,7 +1253,7 @@
},
{
"cell_type": "code",
- "execution_count": 51,
+ "execution_count": 40,
"metadata": {},
"outputs": [
{
@@ -1267,7 +1267,7 @@
"Schritt(z0,[])"
]
},
- "execution_count": 51,
+ "execution_count": 40,
"metadata": {},
"output_type": "execute_result"
}
@@ -1278,7 +1278,7 @@
},
{
"cell_type": "code",
- "execution_count": 52,
+ "execution_count": 41,
"metadata": {},
"outputs": [
{
@@ -1287,7 +1287,7 @@
"Executed operation: Schritt(z0,[])"
]
},
- "execution_count": 52,
+ "execution_count": 41,
"metadata": {},
"output_type": "execute_result"
}
@@ -1298,7 +1298,7 @@
},
{
"cell_type": "code",
- "execution_count": 53,
+ "execution_count": 42,
"metadata": {},
"outputs": [
{
@@ -1326,7 +1326,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 53,
+ "execution_count": 42,
"metadata": {},
"output_type": "execute_result"
}
@@ -1337,7 +1337,7 @@
},
{
"cell_type": "code",
- "execution_count": 54,
+ "execution_count": 43,
"metadata": {},
"outputs": [
{
@@ -1346,7 +1346,7 @@
"Executed operation: Schritt(z0,[])"
]
},
- "execution_count": 54,
+ "execution_count": 43,
"metadata": {},
"output_type": "execute_result"
}
@@ -1357,7 +1357,7 @@
},
{
"cell_type": "code",
- "execution_count": 55,
+ "execution_count": 44,
"metadata": {},
"outputs": [
{
@@ -1382,7 +1382,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 55,
+ "execution_count": 44,
"metadata": {},
"output_type": "execute_result"
}
@@ -1393,7 +1393,7 @@
},
{
"cell_type": "code",
- "execution_count": 56,
+ "execution_count": 45,
"metadata": {},
"outputs": [
{
@@ -1402,7 +1402,7 @@
"Executed operation: Schritt(z0,[])"
]
},
- "execution_count": 56,
+ "execution_count": 45,
"metadata": {},
"output_type": "execute_result"
}
@@ -1413,7 +1413,7 @@
},
{
"cell_type": "code",
- "execution_count": 57,
+ "execution_count": 46,
"metadata": {},
"outputs": [
{
@@ -1438,7 +1438,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 57,
+ "execution_count": 46,
"metadata": {},
"output_type": "execute_result"
}
@@ -1449,7 +1449,7 @@
},
{
"cell_type": "code",
- "execution_count": 58,
+ "execution_count": 47,
"metadata": {},
"outputs": [
{
@@ -1458,7 +1458,7 @@
"Executed operation: Akzeptieren()"
]
},
- "execution_count": 58,
+ "execution_count": 47,
"metadata": {},
"output_type": "execute_result"
}
diff --git a/info4/kapitel-3/PDA_nach_kfG.ipynb b/info4/kapitel-3/PDA_nach_kfG.ipynb
index bd2e7f740be66b4903ab1337bc310b9669767aaa..d21054f911344d2bcac989bef5a65be320b1a51f 100644
--- a/info4/kapitel-3/PDA_nach_kfG.ipynb
+++ b/info4/kapitel-3/PDA_nach_kfG.ipynb
@@ -4,12 +4,14 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "# PDA nach kfG\n"
+ "# PDA nach kfG\n",
+ "\n",
+ "Aus einem gegebenen PDA kann man auch eine kontextfreie Grammatik generieren, die die gleiche Sprache akzeptiert."
]
},
{
"cell_type": "code",
- "execution_count": 41,
+ "execution_count": 1,
"metadata": {},
"outputs": [
{
@@ -18,7 +20,7 @@
"Loaded machine: PDA_to_CFG"
]
},
- "execution_count": 41,
+ "execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
@@ -41,12 +43,14 @@
" SYM(s) == (symbol,s,symbol); // Darstellung eines Symbols als Tripel für die Grammatik\n",
" kfG_TERMINALE == {x|∃t.(t∈Σ ∧ x=SYM(t))};\n",
" \n",
- " ANIMATION_FUNCTION1 == {r,c,i| r=1 ∧ c∈dom(cur) ∧ i=prj1(Z,SYMBOLE)(prj1(Z*SYMBOLE,Z)(cur(c)))};\n",
- " ANIMATION_FUNCTION2 == {r,c,i| r=2 ∧ c∈dom(cur) ∧ i=prj2(Z,SYMBOLE)(prj1(Z*SYMBOLE,Z)(cur(c)))};\n",
- " ANIMATION_FUNCTION3 == {r,c,i| r=3 ∧ c∈dom(cur) ∧ i=prj2(Z*SYMBOLE,Z)(cur(c))};\n",
+ " \"LibraryStrings.def\";\n",
+ " ANIMATION_FUNCTION1 == {r,c,i| r=1 ∧ c∈dom(cur) ∧\n",
+ " i= IF ∃t.(t∈SYMBOLE ∧ cur(c) = (symbol |->t|-> symbol))\n",
+ " THEN TO_STRING(prj2(Z,SYMBOLE)(prj1((Z*SYMBOLE),Z)(cur(c))))\n",
+ " ELSE TO_STRING(cur(c)) END};\n",
" ANIMATION_STR_JUSTIFY_LEFT == TRUE;\n",
" SET_PREF_PP_SEQUENCES == TRUE\n",
- "CONSTANTS δ, Regeln\n",
+ "CONSTANTS δ, Regeln, RegelnP1, RegelnP2, RegelnP3, RegelnP4\n",
"PROPERTIES\n",
" /* Ein PDA für {a^m b^m| m≥1} ; Beispiel von Info 4 (Folie 95ff) */\n",
" δ = { (z0,a,BOT) ↦ (z0,[A,BOT]),\n",
@@ -54,25 +58,23 @@
" (z0,b,A) ↦ (z1,[]),\n",
" (z1,lambda,BOT) ↦ (z1,[]),\n",
" (z1,b,A) ↦ (z1,[]) } ∧\n",
- "\n",
- "\n",
- " Regeln = /* Punkt 1 Folie 109 */\n",
- " { lhs,rhs | ∃z.(z∈PDA_Zustände ∧ lhs=SYM(S) ∧ rhs = [(z0,BOT,z)])}\n",
- " ∪\n",
- " /* Punkt 2 Folie 109 */\n",
- " { lhs,rhs | ∃(z,a,A,z2).((z,a,A)↦(z2,[])∈δ ∧\n",
- " lhs=(z,A,z2) ∧ rhs = SYMS(a)) }\n",
- " ∪\n",
- " /* Punkt 3 Folie 110 */\n",
- " { lhs,rhs | ∃(z,a,A,B,z1,z2).((z,a,A)↦(z1,[B])∈δ ∧ z2∈PDA_Zustände ∧\n",
- " lhs=(z,A,z2) ∧ rhs = SYMS(a)^[(z1,B,z2)]) }\n",
- " ∪\n",
- " /* Punkt 4 Folie 110 */\n",
- " { lhs,rhs | ∃(z,a,A,B,C,z1,z2,z3).((z,a,A)↦(z1,[B,C])∈δ ∧ \n",
- " z2∈PDA_Zustände ∧ \n",
- " z3∈PDA_Zustände ∧\n",
+ " \n",
+ " /*Die Regeln der kfG zum PDA*/\n",
+ " /* Punkt 1 Folie 109 */\n",
+ " RegelnP1 = { lhs,rhs | ∃z.(z∈PDA_Zustände ∧ lhs=SYM(S) ∧ rhs = [(z0,BOT,z)])} ∧\n",
+ " /* Punkt 2 Folie 109 */\n",
+ " RegelnP2 = { lhs,rhs | ∃(z,a,A,z2).((z,a,A)↦(z2,[])∈δ ∧ lhs=(z,A,z2) ∧ rhs = SYMS(a)) } ∧\n",
+ " /* Punkt 3 Folie 110 */\n",
+ " RegelnP3 = { lhs,rhs | ∃(z,a,A,B,z1,z2).((z,a,A)↦(z1,[B])∈δ ∧ z2∈PDA_Zustände ∧\n",
+ " lhs=(z,A,z2) ∧ rhs = SYMS(a)^[(z1,B,z2)]) } ∧\n",
+ " /* Punkt 4 Folie 110 */\n",
+ " RegelnP4 = { lhs,rhs | ∃(z,a,A,B,C,z1,z2,z3).((z,a,A)↦(z1,[B,C])∈δ ∧ \n",
+ " z2∈PDA_Zustände ∧ z3∈PDA_Zustände ∧\n",
" lhs=(z,A,z3) ∧ \n",
- " rhs = SYMS(a)^[(z1,B,z2),(z2,C,z3)]) }\n",
+ " rhs = SYMS(a)^[(z1,B,z2),(z2,C,z3)]) } ∧\n",
+ " \n",
+ " Regeln = RegelnP1 ∪ RegelnP2 ∪ RegelnP3 ∪ RegelnP4\n",
+ " \n",
"VARIABLES cur\n",
"INVARIANT\n",
" cur ∈ seq(kfG_Alphabet)\n",
@@ -82,7 +84,7 @@
" // Anwendung einer Grammatikregel\n",
" ApplyRule(LHS,RHS,pre,post) = PRE LHS↦RHS ∈ Regeln ∧\n",
" cur = pre^[LHS]^post ∧\n",
- " ran(pre) ⊆ kfG_TERMINALE /* Links Ableitung */\n",
+ " ran(pre) ⊆ kfG_TERMINALE /* Linksableitung */\n",
" THEN\n",
" cur := pre^RHS^post\n",
" END\n",
@@ -91,7 +93,7 @@
},
{
"cell_type": "code",
- "execution_count": 42,
+ "execution_count": 2,
"metadata": {},
"outputs": [
{
@@ -100,7 +102,7 @@
"Machine constants set up using operation 0: $setup_constants()"
]
},
- "execution_count": 42,
+ "execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
@@ -111,7 +113,7 @@
},
{
"cell_type": "code",
- "execution_count": 43,
+ "execution_count": 3,
"metadata": {},
"outputs": [
{
@@ -120,7 +122,7 @@
"Machine initialised using operation 1: $initialise_machine()"
]
},
- "execution_count": 43,
+ "execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
@@ -131,7 +133,7 @@
},
{
"cell_type": "code",
- "execution_count": 45,
+ "execution_count": 4,
"metadata": {},
"outputs": [
{
@@ -143,7 +145,7 @@
"{(z0↦a↦A↦(z0↦[A,A])),(z0↦a↦BOT↦(z0↦[A,BOT])),(z0↦b↦A↦(z1↦[])),(z1↦b↦A↦(z1↦[])),(z1↦lambda↦BOT↦(z1↦[]))}"
]
},
- "execution_count": 45,
+ "execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
@@ -152,9 +154,16 @@
"δ"
]
},
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Die Grammatik startet mit dem Startsymbol und simuliert die Ausführung eines entsprechenden PDAs. Dabei gibt es teils mehrere Wege, aber das Wort $ab$ wird von mindestens einem erreicht."
+ ]
+ },
{
"cell_type": "code",
- "execution_count": 46,
+ "execution_count": 5,
"metadata": {},
"outputs": [
{
@@ -162,21 +171,15 @@
"text/markdown": [
"<table style=\"font-family:monospace\"><tbody>\n",
"<tr>\n",
- "<td style=\"padding:10px\">symbol</td>\n",
- "</tr>\n",
- "<tr>\n",
"<td style=\"padding:10px\">S</td>\n",
"</tr>\n",
- "<tr>\n",
- "<td style=\"padding:10px\">symbol</td>\n",
- "</tr>\n",
"</tbody></table>"
],
"text/plain": [
"<Animation function visualisation>"
]
},
- "execution_count": 46,
+ "execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
@@ -187,22 +190,22 @@
},
{
"cell_type": "code",
- "execution_count": 20,
+ "execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Machine: PDA_to_CFG\n",
- "Sets: STATES, SYMBOLS\n",
- "Constants: delta, Productions\n",
+ "Sets: Z, SYMBOLE\n",
+ "Constants: δ, Regeln, RegelnP1, RegelnP2, RegelnP3, RegelnP4\n",
"Variables: cur\n",
"Operations: \n",
"ApplyRule((symbol|->S|->symbol),[(z0|->BOT|->z0)],[],[])\n",
"ApplyRule((symbol|->S|->symbol),[(z0|->BOT|->z1)],[],[])"
]
},
- "execution_count": 20,
+ "execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
@@ -213,7 +216,7 @@
},
{
"cell_type": "code",
- "execution_count": 21,
+ "execution_count": 7,
"metadata": {},
"outputs": [
{
@@ -222,7 +225,7 @@
"Executed operation: ApplyRule((symbol|->S|->symbol),[(z0|->BOT|->z1)],[],[])"
]
},
- "execution_count": 21,
+ "execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
@@ -233,7 +236,7 @@
},
{
"cell_type": "code",
- "execution_count": 22,
+ "execution_count": 8,
"metadata": {},
"outputs": [
{
@@ -241,13 +244,7 @@
"text/markdown": [
"<table style=\"font-family:monospace\"><tbody>\n",
"<tr>\n",
- "<td style=\"padding:10px\">z0</td>\n",
- "</tr>\n",
- "<tr>\n",
- "<td style=\"padding:10px\">BOT</td>\n",
- "</tr>\n",
- "<tr>\n",
- "<td style=\"padding:10px\">z1</td>\n",
+ "<td style=\"padding:10px\">(z0|->BOT|->z1)</td>\n",
"</tr>\n",
"</tbody></table>"
],
@@ -255,7 +252,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 22,
+ "execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
@@ -266,33 +263,7 @@
},
{
"cell_type": "code",
- "execution_count": 23,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/plain": [
- "Machine: PDA_to_CFG\n",
- "Sets: STATES, SYMBOLS\n",
- "Constants: delta, Productions\n",
- "Variables: cur\n",
- "Operations: \n",
- "ApplyRule((z0|->BOT|->z1),[(symbol|->a|->symbol),(z0|->A|->z0),(z0|->BOT|->z1)],[],[])\n",
- "ApplyRule((z0|->BOT|->z1),[(symbol|->a|->symbol),(z0|->A|->z1),(z1|->BOT|->z1)],[],[])"
- ]
- },
- "execution_count": 23,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- ":browse"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 24,
+ "execution_count": 9,
"metadata": {},
"outputs": [
{
@@ -301,7 +272,7 @@
"Executed operation: ApplyRule((z0|->BOT|->z1),[(symbol|->a|->symbol),(z0|->A|->z1),(z1|->BOT|->z1)],[],[])"
]
},
- "execution_count": 24,
+ "execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
@@ -312,7 +283,7 @@
},
{
"cell_type": "code",
- "execution_count": 25,
+ "execution_count": 10,
"metadata": {},
"outputs": [
{
@@ -320,19 +291,9 @@
"text/markdown": [
"<table style=\"font-family:monospace\"><tbody>\n",
"<tr>\n",
- "<td style=\"padding:10px\">symbol</td>\n",
- "<td style=\"padding:10px\">z0</td>\n",
- "<td style=\"padding:10px\">z1</td>\n",
- "</tr>\n",
- "<tr>\n",
"<td style=\"padding:10px\">a</td>\n",
- "<td style=\"padding:10px\">A</td>\n",
- "<td style=\"padding:10px\">BOT</td>\n",
- "</tr>\n",
- "<tr>\n",
- "<td style=\"padding:10px\">symbol</td>\n",
- "<td style=\"padding:10px\">z1</td>\n",
- "<td style=\"padding:10px\">z1</td>\n",
+ "<td style=\"padding:10px\">(z0|->A|->z1)</td>\n",
+ "<td style=\"padding:10px\">(z1|->BOT|->z1)</td>\n",
"</tr>\n",
"</tbody></table>"
],
@@ -340,7 +301,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 25,
+ "execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
@@ -351,34 +312,7 @@
},
{
"cell_type": "code",
- "execution_count": 26,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/plain": [
- "Machine: PDA_to_CFG\n",
- "Sets: STATES, SYMBOLS\n",
- "Constants: delta, Productions\n",
- "Variables: cur\n",
- "Operations: \n",
- "ApplyRule((z0|->A|->z1),[(symbol|->b|->symbol)],[(symbol|->a|->symbol)],[(z1|->BOT|->z1)])\n",
- "ApplyRule((z0|->A|->z1),[(symbol|->a|->symbol),(z0|->A|->z0),(z0|->A|->z1)],[(symbol|->a|->symbol)],[(z1|->BOT|->z1)])\n",
- "ApplyRule((z0|->A|->z1),[(symbol|->a|->symbol),(z0|->A|->z1),(z1|->A|->z1)],[(symbol|->a|->symbol)],[(z1|->BOT|->z1)])"
- ]
- },
- "execution_count": 26,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- ":browse"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 27,
+ "execution_count": 11,
"metadata": {},
"outputs": [
{
@@ -387,7 +321,7 @@
"Executed operation: ApplyRule((z0|->A|->z1),[(symbol|->b|->symbol)],[(symbol|->a|->symbol)],[(z1|->BOT|->z1)])"
]
},
- "execution_count": 27,
+ "execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
@@ -398,7 +332,7 @@
},
{
"cell_type": "code",
- "execution_count": 28,
+ "execution_count": 12,
"metadata": {},
"outputs": [
{
@@ -406,19 +340,9 @@
"text/markdown": [
"<table style=\"font-family:monospace\"><tbody>\n",
"<tr>\n",
- "<td style=\"padding:10px\">symbol</td>\n",
- "<td style=\"padding:10px\">symbol</td>\n",
- "<td style=\"padding:10px\">z1</td>\n",
- "</tr>\n",
- "<tr>\n",
"<td style=\"padding:10px\">a</td>\n",
"<td style=\"padding:10px\">b</td>\n",
- "<td style=\"padding:10px\">BOT</td>\n",
- "</tr>\n",
- "<tr>\n",
- "<td style=\"padding:10px\">symbol</td>\n",
- "<td style=\"padding:10px\">symbol</td>\n",
- "<td style=\"padding:10px\">z1</td>\n",
+ "<td style=\"padding:10px\">(z1|->BOT|->z1)</td>\n",
"</tr>\n",
"</tbody></table>"
],
@@ -426,7 +350,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 28,
+ "execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
@@ -437,32 +361,7 @@
},
{
"cell_type": "code",
- "execution_count": 29,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/plain": [
- "Machine: PDA_to_CFG\n",
- "Sets: STATES, SYMBOLS\n",
- "Constants: delta, Productions\n",
- "Variables: cur\n",
- "Operations: \n",
- "ApplyRule((z1|->BOT|->z1),[],[(symbol|->a|->symbol),(symbol|->b|->symbol)],[])"
- ]
- },
- "execution_count": 29,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- ":browse"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 31,
+ "execution_count": 13,
"metadata": {},
"outputs": [
{
@@ -471,7 +370,7 @@
"Executed operation: ApplyRule((z1|->BOT|->z1),[],[(symbol|->a|->symbol),(symbol|->b|->symbol)],[])"
]
},
- "execution_count": 31,
+ "execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
@@ -482,7 +381,7 @@
},
{
"cell_type": "code",
- "execution_count": 32,
+ "execution_count": 14,
"metadata": {},
"outputs": [
{
@@ -490,24 +389,16 @@
"text/markdown": [
"<table style=\"font-family:monospace\"><tbody>\n",
"<tr>\n",
- "<td style=\"padding:10px\">symbol</td>\n",
- "<td style=\"padding:10px\">symbol</td>\n",
- "</tr>\n",
- "<tr>\n",
"<td style=\"padding:10px\">a</td>\n",
"<td style=\"padding:10px\">b</td>\n",
"</tr>\n",
- "<tr>\n",
- "<td style=\"padding:10px\">symbol</td>\n",
- "<td style=\"padding:10px\">symbol</td>\n",
- "</tr>\n",
"</tbody></table>"
],
"text/plain": [
"<Animation function visualisation>"
]
},
- "execution_count": 32,
+ "execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
@@ -518,20 +409,20 @@
},
{
"cell_type": "code",
- "execution_count": 33,
+ "execution_count": 15,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Machine: PDA_to_CFG\n",
- "Sets: STATES, SYMBOLS\n",
- "Constants: delta, Productions\n",
+ "Sets: Z, SYMBOLE\n",
+ "Constants: δ, Regeln, RegelnP1, RegelnP2, RegelnP3, RegelnP4\n",
"Variables: cur\n",
"Operations: "
]
},
- "execution_count": 33,
+ "execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
@@ -540,6 +431,13 @@
":browse"
]
},
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Da das Wort fertig abgeleitet wurde ist keine Regel mehr ausführbar."
+ ]
+ },
{
"cell_type": "code",
"execution_count": null,
diff --git a/info4/kapitel-5/DFA_PDA_TM.ipynb b/info4/kapitel-5/DFA_PDA_TM.ipynb
new file mode 100644
index 0000000000000000000000000000000000000000..14ef7aff976fa905875d143fb3dcfd39192cdd0f
--- /dev/null
+++ b/info4/kapitel-5/DFA_PDA_TM.ipynb
@@ -0,0 +1,407 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# DFAs, PDAs und TM\n",
+ "Im folgenden werden wir uns damit befassen wie die verschiedenen Automatenmodelle in Verbinndung stehen und wie man sie ineinander übersetzt.\n",
+ "\n",
+ "Wie bereits aus der Vorlesung bekannt ist, akzeptieren DFAs reguläre, PDAs kontextfreie und Turingmaschienen $\\mathbb{L}_0$ Sprachen. Da die $REG\\subseteq CF \\subseteq \\mathbb{L}_0$ kann jeder DFA durch einen PDA und jeder PDA durch eine TM dargestellt werden. Folgende Maschiene demonstriert eine Möglichkeit dies zu tun:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Loaded machine: DFA_PDA_TM"
+ ]
+ },
+ "execution_count": 1,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "MACHINE DFA_PDA_TM\n",
+ "\n",
+ "SETS\n",
+ " Alphabet={a,b,Blank, Bottom, lambda};\n",
+ " States = {z_0,z_1,z_2,z_3,z_4,z_5,z_6,z_end};\n",
+ " Direction = {L,R,N}\n",
+ "\n",
+ "CONSTANTS DFA, Z_DFA, δ_DFA, z_start, F_DFA,\n",
+ " PDA, Γ_PDA, Z_PDA, δ_PDA,\n",
+ " TM, Γ_TM, Z_TM, δ_TM, F_TM\n",
+ "PROPERTIES\n",
+ " //Allgemeine Regeln\n",
+ " Σ ⊆ Alphabet ∧\n",
+ " z_end ∉ Z_DFA ∪ Z_PDA ∧\n",
+ " lambda ∉ Σ ∪ Γ_PDA ∪ Γ_TM ∧\n",
+ " \n",
+ " //Definition des DFAs\n",
+ " DFA = (Σ, Z_DFA, δ_DFA, z_start, F_DFA) ∧\n",
+ " Z_DFA ⊆ States ∧\n",
+ " δ_DFA ∈ (Z_DFA×Σ)→Z_DFA ∧\n",
+ " z_start ∈ Z_DFA ∧\n",
+ " F_DFA ⊆ Z_DFA ∧\n",
+ "\n",
+ " //Definition eines PDAs zum DFA\n",
+ " PDA = (Σ, Γ_PDA, Z_PDA, δ_PDA, z_start, Bottom) ∧\n",
+ " Γ_PDA ⊆ Alphabet ∧\n",
+ " Z_PDA ⊆ States ∧\n",
+ " δ_PDA ∈ (Z_PDA×(Σ∪{lambda})×Γ_PDA)⇸(Z_PDA×seq(Γ_PDA)) ∧\n",
+ " z_start ∈ Z_PDA ∧ \n",
+ " Bottom ∈ Γ_PDA ∧\n",
+ " \n",
+ " Γ_PDA = Σ∪{Bottom} ∧\n",
+ " Z_PDA = Z_DFA ∧\n",
+ " δ_PDA = {regel|∃zustand1,zustand2,terminal.((zustand1,terminal)↦zustand2 ∈ δ_DFA ∧\n",
+ " (regel = (zustand1, terminal, Bottom) ↦ (zustand2, [Bottom])))}\n",
+ " ∪\n",
+ " {regel | ∃zustand.(zustand ∈ F_DFA ∧\n",
+ " regel = (zustand, lambda, Bottom) ↦ (zustand, []))} ∧\n",
+ " \n",
+ " //Definition einer TM zum DFA\n",
+ " TM = (Σ, Γ_TM, Z_TM, δ_TM, z_start, Blank, F_TM) ∧\n",
+ " Γ_TM ⊆ Alphabet ∧\n",
+ " Z_TM ⊆ States ∧\n",
+ " δ_TM ∈ (Z_TM×Γ_TM)→(Z_TM×Γ_TM×Direction) ∧\n",
+ " z_start ∈ Z_TM ∧\n",
+ " Blank ∈ Γ_TM ∧\n",
+ " F_TM ⊆ Z_TM ∧\n",
+ " \n",
+ " Γ_TM = Σ ∪ {Blank} ∧\n",
+ " Z_TM = Z_DFA ∪ {z_end} ∧\n",
+ " δ_TM = {regel|∃zustand1,zustand2,terminal.((zustand1,terminal)↦zustand2 ∈ δ_DFA ∧\n",
+ " (regel = (zustand1, terminal) ↦ (zustand2, Blank, R)))}\n",
+ " ∪\n",
+ " {regel | ∃zustand.(zustand ∈ Z_DFA\\F_DFA ∧\n",
+ " regel = (zustand, Blank) ↦ (zustand, Blank, N))}\n",
+ " ∪\n",
+ " {regel | ∃zustand.(zustand ∈ F_DFA ∧\n",
+ " regel = (zustand, Blank) ↦ (z_end, Blank, N))}\n",
+ " ∪\n",
+ " {regel | ∃terminal.(terminal ∈ Γ_TM ∧ regel = (z_end, terminal) ↦ (z_end, terminal, N))} ∧\n",
+ " F_TM = {z_end}\n",
+ "\n",
+ "DEFINITIONS\n",
+ " Σ == {a, b};\n",
+ " \n",
+ " //Animation des Zustandsgraphen für den DFA\n",
+ " CUSTOM_GRAPH_NODES1 == rec(shape:\"doublecircle\",nodes∈F_DFA); // Endzustände\n",
+ " CUSTOM_GRAPH_NODES2 == rec(shape:\"circle\",nodes∈Z_DFA\\F_DFA); // andere Zustände\n",
+ " CUSTOM_GRAPH_NODES3 == rec(shape:\"none\",color:\"white\",style:\"none\",nodes:{\"\"});\n",
+ " CUSTOM_GRAPH_EDGES1 == rec(color:\"green\",label:\"a\",edges:{x,y|x∈Z_DFA ∧ y=δ_DFA(x,a) ∧ y≠δ_DFA(x,b)}); \n",
+ " CUSTOM_GRAPH_EDGES2 == rec(color:\"green\",label:\"b\",edges:{x,y|x∈Z_DFA ∧ y=δ_DFA(x,b) ∧ y≠δ_DFA(x,a)});\n",
+ " CUSTOM_GRAPH_EDGES3 == rec(color:\"green\",label:\"a, b\",edges:{x,y|x∈Z_DFA ∧ y=δ_DFA(x,a) ∧ y=δ_DFA(x,b)});\n",
+ " CUSTOM_GRAPH_EDGES4 == rec(color:\"black\",label:\"\",edges:{\"\"}*{z_start}); // Kanten für Startknoten\n",
+ "END"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Machine constants set up using operation 0: $setup_constants()"
+ ]
+ },
+ "execution_count": 2,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":constants DFA=({a, b}, //Σ\n",
+ " {z_0,z_1,z_2,z_3}, //Z\n",
+ " {(z_0,a)↦z_1, (z_0,b)↦z_3,\n",
+ " (z_1,a)↦z_3, (z_1,b)↦z_2,\n",
+ " (z_2,a)↦z_2, (z_2,b)↦z_2,\n",
+ " (z_3,a)↦z_3, (z_3,b)↦z_3 }, //δ\n",
+ " z_0, {z_0, z_2}) //z0, F"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Der Zustandsgraph des DFA sieht wie folgt aus:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "image/svg+xml": [
+ "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
+ "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
+ " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
+ "<!-- Generated by graphviz version 2.44.1 (0)\n",
+ " -->\n",
+ "<!-- Title: prob_graph Pages: 1 -->\n",
+ "<svg width=\"540pt\" height=\"718pt\"\n",
+ " viewBox=\"0.00 0.00 540.00 718.36\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
+ "<g id=\"graph0\" class=\"graph\" transform=\"scale(0.98 0.98) rotate(0) translate(4 726.18)\">\n",
+ "<title>prob_graph</title>\n",
+ "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-726.18 544.89,-726.18 544.89,4 -4,4\"/>\n",
+ "<!-- 0 -->\n",
+ "<g id=\"node1\" class=\"node\">\n",
+ "<title>0</title>\n",
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+ "<text text-anchor=\"middle\" x=\"286.97\" y=\"-494.9\" font-family=\"Times,serif\" font-size=\"12.00\">z_0</text>\n",
+ "</g>\n",
+ "<!-- 2 -->\n",
+ "<g id=\"node3\" class=\"node\">\n",
+ "<title>2</title>\n",
+ "<ellipse fill=\"lightgrey\" stroke=\"black\" cx=\"206.97\" cy=\"-279\" rx=\"22.32\" ry=\"22.32\"/>\n",
+ "<text text-anchor=\"middle\" x=\"206.97\" y=\"-275.9\" font-family=\"Times,serif\" font-size=\"12.00\">z_1</text>\n",
+ "</g>\n",
+ "<!-- 0->2 -->\n",
+ "<g id=\"edge3\" class=\"edge\">\n",
+ "<title>0->2</title>\n",
+ "<path fill=\"none\" stroke=\"green\" d=\"M278.11,-472.95C263.35,-432.92 233.8,-352.76 217.82,-309.43\"/>\n",
+ "<polygon fill=\"green\" stroke=\"green\" points=\"221.09,-308.18 214.35,-300.01 214.53,-310.6 221.09,-308.18\"/>\n",
+ "<text text-anchor=\"middle\" x=\"250.97\" y=\"-380.9\" font-family=\"Times,serif\" font-size=\"12.00\">a</text>\n",
+ "</g>\n",
+ "<!-- 3 -->\n",
+ "<g id=\"node4\" class=\"node\">\n",
+ "<title>3</title>\n",
+ "<ellipse fill=\"lightgrey\" stroke=\"black\" cx=\"366.97\" cy=\"-59\" rx=\"22.32\" ry=\"22.32\"/>\n",
+ "<text text-anchor=\"middle\" x=\"366.97\" y=\"-55.9\" font-family=\"Times,serif\" font-size=\"12.00\">z_3</text>\n",
+ "</g>\n",
+ "<!-- 0->3 -->\n",
+ "<g id=\"edge1\" class=\"edge\">\n",
+ "<title>0->3</title>\n",
+ "<path fill=\"none\" stroke=\"green\" d=\"M291.56,-471.94C305.3,-396.88 346.1,-174.01 361.24,-91.33\"/>\n",
+ "<polygon fill=\"green\" stroke=\"green\" points=\"364.72,-91.76 363.08,-81.29 357.83,-90.5 364.72,-91.76\"/>\n",
+ "<text text-anchor=\"middle\" x=\"332.97\" y=\"-275.9\" font-family=\"Times,serif\" font-size=\"12.00\">b</text>\n",
+ "</g>\n",
+ "<!-- 1 -->\n",
+ "<g id=\"node2\" class=\"node\">\n",
+ "<title>1</title>\n",
+ "<ellipse fill=\"lightgrey\" stroke=\"black\" cx=\"71.97\" cy=\"-59\" rx=\"22.3\" ry=\"22.3\"/>\n",
+ "<ellipse fill=\"none\" stroke=\"black\" cx=\"71.97\" cy=\"-59\" rx=\"26.32\" ry=\"26.32\"/>\n",
+ "<text text-anchor=\"middle\" x=\"71.97\" y=\"-55.9\" font-family=\"Times,serif\" font-size=\"12.00\">z_2</text>\n",
+ "</g>\n",
+ "<!-- 1->1 -->\n",
+ "<g id=\"edge6\" class=\"edge\">\n",
+ "<title>1->1</title>\n",
+ "<path fill=\"none\" stroke=\"green\" d=\"M90.12,-78.07C103.15,-84.28 116.14,-77.93 116.14,-59 116.14,-44.95 108.98,-37.83 99.97,-37.63\"/>\n",
+ "<polygon fill=\"green\" stroke=\"green\" points=\"99.06,-34.25 90.12,-39.93 100.65,-41.06 99.06,-34.25\"/>\n",
+ "<text text-anchor=\"middle\" x=\"127.64\" y=\"-55.9\" font-family=\"Times,serif\" font-size=\"12.00\">a, b</text>\n",
+ "</g>\n",
+ "<!-- 2->1 -->\n",
+ "<g id=\"edge2\" class=\"edge\">\n",
+ "<title>2->1</title>\n",
+ "<path fill=\"none\" stroke=\"green\" d=\"M195.7,-259.79C172.53,-222.37 119.36,-136.51 90.85,-90.48\"/>\n",
+ "<polygon fill=\"green\" stroke=\"green\" points=\"93.73,-88.49 85.49,-81.83 87.78,-92.17 93.73,-88.49\"/>\n",
+ "<text text-anchor=\"middle\" x=\"147.97\" y=\"-169.9\" font-family=\"Times,serif\" font-size=\"12.00\">b</text>\n",
+ "</g>\n",
+ "<!-- 2->3 -->\n",
+ "<g id=\"edge4\" class=\"edge\">\n",
+ "<title>2->3</title>\n",
+ "<path fill=\"none\" stroke=\"green\" d=\"M219.8,-260.53C247.95,-222.17 315.29,-130.42 348.21,-85.57\"/>\n",
+ "<polygon fill=\"green\" stroke=\"green\" points=\"351.22,-87.38 354.31,-77.25 345.57,-83.24 351.22,-87.38\"/>\n",
+ "<text text-anchor=\"middle\" x=\"290.97\" y=\"-169.9\" font-family=\"Times,serif\" font-size=\"12.00\">a</text>\n",
+ "</g>\n",
+ "<!-- 3->3 -->\n",
+ "<g id=\"edge7\" class=\"edge\">\n",
+ "<title>3->3</title>\n",
+ "<path fill=\"none\" stroke=\"green\" d=\"M380.97,-76.57C393.6,-85.12 407.14,-79.26 407.14,-59 407.14,-43.8 399.52,-36.71 390.39,-37.72\"/>\n",
+ "<polygon fill=\"green\" stroke=\"green\" points=\"388.99,-34.51 380.97,-41.43 391.56,-41.02 388.99,-34.51\"/>\n",
+ "<text text-anchor=\"middle\" x=\"418.64\" y=\"-55.9\" font-family=\"Times,serif\" font-size=\"12.00\">a, b</text>\n",
+ "</g>\n",
+ "<!-- 4 -->\n",
+ "<g id=\"node5\" class=\"node\">\n",
+ "<title>4</title>\n",
+ "</g>\n",
+ "<!-- 4->0 -->\n",
+ "<g id=\"edge5\" class=\"edge\">\n",
+ "<title>4->0</title>\n",
+ "<path fill=\"none\" stroke=\"black\" d=\"M286.97,-663.88C286.97,-634.44 286.97,-573.63 286.97,-534.66\"/>\n",
+ "<polygon fill=\"black\" stroke=\"black\" points=\"290.47,-534.29 286.97,-524.29 283.47,-534.29 290.47,-534.29\"/>\n",
+ "</g>\n",
+ "</g>\n",
+ "</svg>"
+ ],
+ "text/plain": [
+ "<Dot visualization: custom_graph []>"
+ ]
+ },
+ "execution_count": 3,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":dot custom_graph"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 4,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Machine initialised using operation 1: $initialise_machine()"
+ ]
+ },
+ "execution_count": 4,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":init"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Der entstandene PDA hat die gleiche Zustandsmenge, wie der gegebene DFA.\n",
+ "Die Überführungen erfolgen ebenfalls analog.\n",
+ "Damit der PDA allerdings ein Wort akzeptiert muss der Keller leer sein. Dafür werden lambda-Übergänge eingefügt, mit denen man in einem Endzustand des DFA den Keller leeren kann.\n",
+ "Die vollständige Überführungsfunktion des PDA ist durch die nachfolgende Tabelle gegeben. Dabei ist die Spalte **keller_symbole** als aneinanderkettung der Symbole in der Menge zu verstehen. Also bei {(1↦Bottom)} wird Bottom und bei ∅ wird kein Symbol wieder auf den Keller gelegt."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 5,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "|zustand|symbol|keller_symbol|neuer_zustand|keller_symbole|\n",
+ "|---|---|---|---|---|\n",
+ "|$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\mathit{z\\_0}$|$\\mathit{a}$|$\\mathit{Bottom}$|$\\mathit{z\\_1}$|$\\{(1\\mapsto \\mathit{Bottom})\\}$|\n",
+ "|$\\mathit{z\\_0}$|$\\mathit{b}$|$\\mathit{Bottom}$|$\\mathit{z\\_3}$|$\\{(1\\mapsto \\mathit{Bottom})\\}$|\n",
+ "|$\\mathit{z\\_0}$|$\\mathit{lambda}$|$\\mathit{Bottom}$|$\\mathit{z\\_0}$|$\\emptyset$|\n",
+ "|$\\mathit{z\\_1}$|$\\mathit{a}$|$\\mathit{Bottom}$|$\\mathit{z\\_3}$|$\\{(1\\mapsto \\mathit{Bottom})\\}$|\n",
+ "|$\\mathit{z\\_1}$|$\\mathit{b}$|$\\mathit{Bottom}$|$\\mathit{z\\_2}$|$\\{(1\\mapsto \\mathit{Bottom})\\}$|\n",
+ "|$\\mathit{z\\_2}$|$\\mathit{a}$|$\\mathit{Bottom}$|$\\mathit{z\\_2}$|$\\{(1\\mapsto \\mathit{Bottom})\\}$|\n",
+ "|$\\mathit{z\\_2}$|$\\mathit{b}$|$\\mathit{Bottom}$|$\\mathit{z\\_2}$|$\\{(1\\mapsto \\mathit{Bottom})\\}$|\n",
+ "|$\\mathit{z\\_2}$|$\\mathit{lambda}$|$\\mathit{Bottom}$|$\\mathit{z\\_2}$|$\\emptyset$|\n",
+ "|$\\mathit{z\\_3}$|$\\mathit{a}$|$\\mathit{Bottom}$|$\\mathit{z\\_3}$|$\\{(1\\mapsto \\mathit{Bottom})\\}$|\n",
+ "|$\\mathit{z\\_3}$|$\\mathit{b}$|$\\mathit{Bottom}$|$\\mathit{z\\_3}$|$\\{(1\\mapsto \\mathit{Bottom})\\}$|\n"
+ ],
+ "text/plain": [
+ "zustand\tsymbol\tkeller_symbol\tneuer_zustand\tkeller_symbole\n",
+ "z_0\ta\tBottom\tz_1\t{(1|->Bottom)}\n",
+ "z_0\tb\tBottom\tz_3\t{(1|->Bottom)}\n",
+ "z_0\tlambda\tBottom\tz_0\t{}\n",
+ "z_1\ta\tBottom\tz_3\t{(1|->Bottom)}\n",
+ "z_1\tb\tBottom\tz_2\t{(1|->Bottom)}\n",
+ "z_2\ta\tBottom\tz_2\t{(1|->Bottom)}\n",
+ "z_2\tb\tBottom\tz_2\t{(1|->Bottom)}\n",
+ "z_2\tlambda\tBottom\tz_2\t{}\n",
+ "z_3\ta\tBottom\tz_3\t{(1|->Bottom)}\n",
+ "z_3\tb\tBottom\tz_3\t{(1|->Bottom)}\n"
+ ]
+ },
+ "execution_count": 5,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":table {zustand,symbol,keller_symbol,neuer_zustand,keller_symbole| ((zustand,symbol,keller_symbol)↦(neuer_zustand,keller_symbole) : δ_PDA)}"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Im Gegensatz zum PDA hat die TM einen zusätzlichen Zustand.\n",
+ "Dieser wird benötigt, da eine TM sofort hält, wenn sie einen Endzustand erreicht.\n",
+ "Wenn das Wort also komplett gelesen wurde und die TM in einem Endzustand des DFA ist, darf in den Endzustand der TM übergegangen werden.\n",
+ "Ansonsten sind die Übergänge analog zum DFA.\n",
+ "Die Überführungsfunktion der TM ist durch folgende Tabelle gegeben:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 6,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "|zustand|symbol|neuer_zustand|geschriebenes_symbol|Richtung|\n",
+ "|---|---|---|---|---|\n",
+ "|$\\mathit{z\\_0}$|$\\mathit{a}$|$\\mathit{z\\_1}$|$\\mathit{Blank}$|$\\mathit{R}$|\n",
+ "|$\\mathit{z\\_0}$|$\\mathit{b}$|$\\mathit{z\\_3}$|$\\mathit{Blank}$|$\\mathit{R}$|\n",
+ "|$\\mathit{z\\_0}$|$\\mathit{Blank}$|$\\mathit{z\\_end}$|$\\mathit{Blank}$|$\\mathit{N}$|\n",
+ "|$\\mathit{z\\_1}$|$\\mathit{a}$|$\\mathit{z\\_3}$|$\\mathit{Blank}$|$\\mathit{R}$|\n",
+ "|$\\mathit{z\\_1}$|$\\mathit{b}$|$\\mathit{z\\_2}$|$\\mathit{Blank}$|$\\mathit{R}$|\n",
+ "|$\\mathit{z\\_1}$|$\\mathit{Blank}$|$\\mathit{z\\_1}$|$\\mathit{Blank}$|$\\mathit{N}$|\n",
+ "|$\\mathit{z\\_2}$|$\\mathit{a}$|$\\mathit{z\\_2}$|$\\mathit{Blank}$|$\\mathit{R}$|\n",
+ "|$\\mathit{z\\_2}$|$\\mathit{b}$|$\\mathit{z\\_2}$|$\\mathit{Blank}$|$\\mathit{R}$|\n",
+ "|$\\mathit{z\\_2}$|$\\mathit{Blank}$|$\\mathit{z\\_end}$|$\\mathit{Blank}$|$\\mathit{N}$|\n",
+ "|$\\mathit{z\\_3}$|$\\mathit{a}$|$\\mathit{z\\_3}$|$\\mathit{Blank}$|$\\mathit{R}$|\n",
+ "|$\\mathit{z\\_3}$|$\\mathit{b}$|$\\mathit{z\\_3}$|$\\mathit{Blank}$|$\\mathit{R}$|\n",
+ "|$\\mathit{z\\_3}$|$\\mathit{Blank}$|$\\mathit{z\\_3}$|$\\mathit{Blank}$|$\\mathit{N}$|\n",
+ "|$\\mathit{z\\_end}$|$\\mathit{a}$|$\\mathit{z\\_end}$|$\\mathit{a}$|$\\mathit{N}$|\n",
+ "|$\\mathit{z\\_end}$|$\\mathit{b}$|$\\mathit{z\\_end}$|$\\mathit{b}$|$\\mathit{N}$|\n",
+ "|$\\mathit{z\\_end}$|$\\mathit{Blank}$|$\\mathit{z\\_end}$|$\\mathit{Blank}$|$\\mathit{N}$|\n"
+ ],
+ "text/plain": [
+ "zustand\tsymbol\tneuer_zustand\tgeschriebenes_symbol\tRichtung\n",
+ "z_0\ta\tz_1\tBlank\tR\n",
+ "z_0\tb\tz_3\tBlank\tR\n",
+ "z_0\tBlank\tz_end\tBlank\tN\n",
+ "z_1\ta\tz_3\tBlank\tR\n",
+ "z_1\tb\tz_2\tBlank\tR\n",
+ "z_1\tBlank\tz_1\tBlank\tN\n",
+ "z_2\ta\tz_2\tBlank\tR\n",
+ "z_2\tb\tz_2\tBlank\tR\n",
+ "z_2\tBlank\tz_end\tBlank\tN\n",
+ "z_3\ta\tz_3\tBlank\tR\n",
+ "z_3\tb\tz_3\tBlank\tR\n",
+ "z_3\tBlank\tz_3\tBlank\tN\n",
+ "z_end\ta\tz_end\ta\tN\n",
+ "z_end\tb\tz_end\tb\tN\n",
+ "z_end\tBlank\tz_end\tBlank\tN\n"
+ ]
+ },
+ "execution_count": 6,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":table {zustand,symbol,neuer_zustand,geschriebenes_symbol,Richtung| ((zustand,symbol)↦(neuer_zustand,geschriebenes_symbol,Richtung)) : δ_TM}"
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "ProB 2",
+ "language": "prob",
+ "name": "prob2"
+ },
+ "language_info": {
+ "codemirror_mode": "prob2_jupyter_repl",
+ "file_extension": ".prob",
+ "mimetype": "text/x-prob2-jupyter-repl",
+ "name": "prob"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/info4/kapitel-7/TuringMaschineBerechenbarkeit.ipynb b/info4/kapitel-7/TuringMaschineBerechenbarkeit.ipynb
index 6f7bacbe4308cb6c6b1ed3111b8c03e256d8ce7d..29af86cbcd2e4e90a4cd906ab3e9c10e8bac4dc6 100644
--- a/info4/kapitel-7/TuringMaschineBerechenbarkeit.ipynb
+++ b/info4/kapitel-7/TuringMaschineBerechenbarkeit.ipynb
@@ -39,7 +39,7 @@
},
{
"cell_type": "code",
- "execution_count": 13,
+ "execution_count": 1,
"metadata": {},
"outputs": [
{
@@ -48,7 +48,7 @@
"Loaded machine: TuringMachine_2"
]
},
- "execution_count": 13,
+ "execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
@@ -113,7 +113,7 @@
},
{
"cell_type": "code",
- "execution_count": 14,
+ "execution_count": 2,
"metadata": {},
"outputs": [
{
@@ -122,7 +122,7 @@
"Machine constants set up using operation 0: $setup_constants()"
]
},
- "execution_count": 14,
+ "execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
@@ -133,7 +133,7 @@
},
{
"cell_type": "code",
- "execution_count": 15,
+ "execution_count": 3,
"metadata": {},
"outputs": [
{
@@ -142,7 +142,7 @@
"Machine initialised using operation 1: $initialise_machine()"
]
},
- "execution_count": 15,
+ "execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
@@ -160,7 +160,7 @@
},
{
"cell_type": "code",
- "execution_count": 16,
+ "execution_count": 4,
"metadata": {},
"outputs": [
{
@@ -181,7 +181,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 16,
+ "execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
@@ -192,7 +192,7 @@
},
{
"cell_type": "code",
- "execution_count": 17,
+ "execution_count": 5,
"metadata": {},
"outputs": [
{
@@ -201,7 +201,7 @@
"Executed operation: GoTo(z0,I,z0,I,R)"
]
},
- "execution_count": 17,
+ "execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
@@ -212,7 +212,7 @@
},
{
"cell_type": "code",
- "execution_count": 18,
+ "execution_count": 6,
"metadata": {},
"outputs": [
{
@@ -233,7 +233,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 18,
+ "execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
@@ -244,7 +244,7 @@
},
{
"cell_type": "code",
- "execution_count": 19,
+ "execution_count": 7,
"metadata": {},
"outputs": [
{
@@ -253,7 +253,7 @@
"Executed operation: GoTo(z0,I,z0,I,R)"
]
},
- "execution_count": 19,
+ "execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
@@ -264,7 +264,7 @@
},
{
"cell_type": "code",
- "execution_count": 20,
+ "execution_count": 8,
"metadata": {},
"outputs": [
{
@@ -285,7 +285,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 20,
+ "execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
@@ -296,7 +296,7 @@
},
{
"cell_type": "code",
- "execution_count": 21,
+ "execution_count": 9,
"metadata": {},
"outputs": [
{
@@ -305,7 +305,7 @@
"Executed operation: GoTo(z0,I,z0,I,R)"
]
},
- "execution_count": 21,
+ "execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
@@ -316,7 +316,7 @@
},
{
"cell_type": "code",
- "execution_count": 22,
+ "execution_count": 10,
"metadata": {},
"outputs": [
{
@@ -337,7 +337,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 22,
+ "execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
@@ -348,7 +348,7 @@
},
{
"cell_type": "code",
- "execution_count": 23,
+ "execution_count": 11,
"metadata": {},
"outputs": [
{
@@ -357,7 +357,7 @@
"Executed operation: GoTo(z0,I,z0,I,R)"
]
},
- "execution_count": 23,
+ "execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
@@ -368,7 +368,7 @@
},
{
"cell_type": "code",
- "execution_count": 24,
+ "execution_count": 12,
"metadata": {},
"outputs": [
{
@@ -389,7 +389,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 24,
+ "execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
@@ -400,7 +400,7 @@
},
{
"cell_type": "code",
- "execution_count": 25,
+ "execution_count": 13,
"metadata": {},
"outputs": [
{
@@ -409,7 +409,7 @@
"Executed operation: GoTo(z0,I,z0,I,R)"
]
},
- "execution_count": 25,
+ "execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
@@ -420,7 +420,7 @@
},
{
"cell_type": "code",
- "execution_count": 26,
+ "execution_count": 14,
"metadata": {},
"outputs": [
{
@@ -441,7 +441,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 26,
+ "execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
@@ -452,7 +452,7 @@
},
{
"cell_type": "code",
- "execution_count": 28,
+ "execution_count": 15,
"metadata": {},
"outputs": [
{
@@ -461,7 +461,7 @@
"Executed operation: GoTo(z0,Blank,z1,Blank,L)"
]
},
- "execution_count": 28,
+ "execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
@@ -472,7 +472,7 @@
},
{
"cell_type": "code",
- "execution_count": 29,
+ "execution_count": 16,
"metadata": {},
"outputs": [
{
@@ -494,7 +494,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 29,
+ "execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
@@ -505,7 +505,7 @@
},
{
"cell_type": "code",
- "execution_count": 30,
+ "execution_count": 17,
"metadata": {},
"outputs": [
{
@@ -514,7 +514,7 @@
"Executed operation: GoTo(z1,I,z1,O,L)"
]
},
- "execution_count": 30,
+ "execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
@@ -525,7 +525,7 @@
},
{
"cell_type": "code",
- "execution_count": 31,
+ "execution_count": 18,
"metadata": {},
"outputs": [
{
@@ -547,7 +547,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 31,
+ "execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
@@ -558,7 +558,7 @@
},
{
"cell_type": "code",
- "execution_count": 33,
+ "execution_count": 19,
"metadata": {},
"outputs": [
{
@@ -567,7 +567,7 @@
"Executed operation: GoTo(z1,I,z1,O,L)"
]
},
- "execution_count": 33,
+ "execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
@@ -578,7 +578,7 @@
},
{
"cell_type": "code",
- "execution_count": 34,
+ "execution_count": 20,
"metadata": {},
"outputs": [
{
@@ -600,7 +600,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 34,
+ "execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
@@ -611,7 +611,7 @@
},
{
"cell_type": "code",
- "execution_count": 35,
+ "execution_count": 21,
"metadata": {},
"outputs": [
{
@@ -620,7 +620,7 @@
"Executed operation: GoTo(z1,I,z1,O,L)"
]
},
- "execution_count": 35,
+ "execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
@@ -631,7 +631,7 @@
},
{
"cell_type": "code",
- "execution_count": 36,
+ "execution_count": 22,
"metadata": {},
"outputs": [
{
@@ -653,7 +653,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 36,
+ "execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
@@ -664,7 +664,7 @@
},
{
"cell_type": "code",
- "execution_count": 37,
+ "execution_count": 23,
"metadata": {},
"outputs": [
{
@@ -673,7 +673,7 @@
"Executed operation: GoTo(z1,I,z1,O,L)"
]
},
- "execution_count": 37,
+ "execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
@@ -684,7 +684,7 @@
},
{
"cell_type": "code",
- "execution_count": 38,
+ "execution_count": 24,
"metadata": {},
"outputs": [
{
@@ -706,7 +706,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 38,
+ "execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
@@ -717,7 +717,7 @@
},
{
"cell_type": "code",
- "execution_count": 40,
+ "execution_count": 25,
"metadata": {},
"outputs": [
{
@@ -726,7 +726,7 @@
"Executed operation: GoTo(z1,I,z1,O,L)"
]
},
- "execution_count": 40,
+ "execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
@@ -737,7 +737,7 @@
},
{
"cell_type": "code",
- "execution_count": 41,
+ "execution_count": 26,
"metadata": {},
"outputs": [
{
@@ -760,7 +760,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 41,
+ "execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
@@ -771,7 +771,7 @@
},
{
"cell_type": "code",
- "execution_count": 42,
+ "execution_count": 27,
"metadata": {},
"outputs": [
{
@@ -780,7 +780,7 @@
"Executed operation: GoTo(z1,Blank,ze,I,N)"
]
},
- "execution_count": 42,
+ "execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
@@ -791,7 +791,7 @@
},
{
"cell_type": "code",
- "execution_count": 43,
+ "execution_count": 28,
"metadata": {},
"outputs": [
{
@@ -814,7 +814,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 43,
+ "execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
@@ -825,7 +825,7 @@
},
{
"cell_type": "code",
- "execution_count": 44,
+ "execution_count": 29,
"metadata": {},
"outputs": [
{
@@ -834,7 +834,7 @@
"Executed operation: Accept()"
]
},
- "execution_count": 44,
+ "execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
@@ -852,7 +852,7 @@
},
{
"cell_type": "code",
- "execution_count": 46,
+ "execution_count": 30,
"metadata": {},
"outputs": [
{
@@ -875,7 +875,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 46,
+ "execution_count": 30,
"metadata": {},
"output_type": "execute_result"
}
@@ -886,7 +886,7 @@
},
{
"cell_type": "code",
- "execution_count": 47,
+ "execution_count": 31,
"metadata": {},
"outputs": [
{
@@ -895,7 +895,7 @@
"Executed operation: GoTo(z0,I,z0,I,R)"
]
},
- "execution_count": 47,
+ "execution_count": 31,
"metadata": {},
"output_type": "execute_result"
}
@@ -906,7 +906,7 @@
},
{
"cell_type": "code",
- "execution_count": 49,
+ "execution_count": 32,
"metadata": {},
"outputs": [
{
@@ -915,7 +915,7 @@
"Executed operation: GoTo(z0,O,z0,O,R)"
]
},
- "execution_count": 49,
+ "execution_count": 32,
"metadata": {},
"output_type": "execute_result"
}
@@ -926,7 +926,7 @@
},
{
"cell_type": "code",
- "execution_count": 50,
+ "execution_count": 33,
"metadata": {},
"outputs": [
{
@@ -949,7 +949,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 50,
+ "execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
@@ -960,7 +960,7 @@
},
{
"cell_type": "code",
- "execution_count": 51,
+ "execution_count": 34,
"metadata": {},
"outputs": [
{
@@ -969,7 +969,7 @@
"Executed operation: GoTo(z0,O,z0,O,R)"
]
},
- "execution_count": 51,
+ "execution_count": 34,
"metadata": {},
"output_type": "execute_result"
}
@@ -980,7 +980,7 @@
},
{
"cell_type": "code",
- "execution_count": 52,
+ "execution_count": 35,
"metadata": {},
"outputs": [
{
@@ -989,7 +989,7 @@
"Executed operation: GoTo(z0,O,z0,O,R)"
]
},
- "execution_count": 52,
+ "execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
@@ -1000,7 +1000,7 @@
},
{
"cell_type": "code",
- "execution_count": 53,
+ "execution_count": 36,
"metadata": {},
"outputs": [
{
@@ -1009,7 +1009,7 @@
"Executed operation: GoTo(z0,O,z0,O,R)"
]
},
- "execution_count": 53,
+ "execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
@@ -1020,7 +1020,7 @@
},
{
"cell_type": "code",
- "execution_count": 54,
+ "execution_count": 37,
"metadata": {},
"outputs": [
{
@@ -1029,7 +1029,7 @@
"Executed operation: GoTo(z0,O,z0,O,R)"
]
},
- "execution_count": 54,
+ "execution_count": 37,
"metadata": {},
"output_type": "execute_result"
}
@@ -1040,7 +1040,7 @@
},
{
"cell_type": "code",
- "execution_count": 55,
+ "execution_count": 38,
"metadata": {},
"outputs": [
{
@@ -1049,7 +1049,7 @@
"Executed operation: GoTo(z0,Blank,z1,Blank,L)"
]
},
- "execution_count": 55,
+ "execution_count": 38,
"metadata": {},
"output_type": "execute_result"
}
@@ -1060,7 +1060,7 @@
},
{
"cell_type": "code",
- "execution_count": 56,
+ "execution_count": 39,
"metadata": {},
"outputs": [
{
@@ -1083,7 +1083,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 56,
+ "execution_count": 39,
"metadata": {},
"output_type": "execute_result"
}
@@ -1094,7 +1094,7 @@
},
{
"cell_type": "code",
- "execution_count": 58,
+ "execution_count": 40,
"metadata": {},
"outputs": [
{
@@ -1103,7 +1103,7 @@
"Executed operation: GoTo(z1,O,z2,I,L)"
]
},
- "execution_count": 58,
+ "execution_count": 40,
"metadata": {},
"output_type": "execute_result"
}
@@ -1114,7 +1114,7 @@
},
{
"cell_type": "code",
- "execution_count": 59,
+ "execution_count": 41,
"metadata": {},
"outputs": [
{
@@ -1137,7 +1137,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 59,
+ "execution_count": 41,
"metadata": {},
"output_type": "execute_result"
}
@@ -1148,7 +1148,7 @@
},
{
"cell_type": "code",
- "execution_count": 60,
+ "execution_count": 42,
"metadata": {},
"outputs": [
{
@@ -1157,7 +1157,7 @@
"Executed operation: GoTo(z2,O,z2,O,L)"
]
},
- "execution_count": 60,
+ "execution_count": 42,
"metadata": {},
"output_type": "execute_result"
}
@@ -1168,7 +1168,7 @@
},
{
"cell_type": "code",
- "execution_count": 61,
+ "execution_count": 43,
"metadata": {},
"outputs": [
{
@@ -1177,7 +1177,7 @@
"Executed operation: GoTo(z2,O,z2,O,L)"
]
},
- "execution_count": 61,
+ "execution_count": 43,
"metadata": {},
"output_type": "execute_result"
}
@@ -1188,7 +1188,7 @@
},
{
"cell_type": "code",
- "execution_count": 62,
+ "execution_count": 44,
"metadata": {},
"outputs": [
{
@@ -1197,7 +1197,7 @@
"Executed operation: GoTo(z2,O,z2,O,L)"
]
},
- "execution_count": 62,
+ "execution_count": 44,
"metadata": {},
"output_type": "execute_result"
}
@@ -1208,7 +1208,7 @@
},
{
"cell_type": "code",
- "execution_count": 63,
+ "execution_count": 45,
"metadata": {},
"outputs": [
{
@@ -1217,7 +1217,7 @@
"Executed operation: GoTo(z2,O,z2,O,L)"
]
},
- "execution_count": 63,
+ "execution_count": 45,
"metadata": {},
"output_type": "execute_result"
}
@@ -1228,7 +1228,7 @@
},
{
"cell_type": "code",
- "execution_count": 64,
+ "execution_count": 46,
"metadata": {},
"outputs": [
{
@@ -1237,7 +1237,7 @@
"Executed operation: GoTo(z2,I,z2,I,L)"
]
},
- "execution_count": 64,
+ "execution_count": 46,
"metadata": {},
"output_type": "execute_result"
}
@@ -1248,7 +1248,7 @@
},
{
"cell_type": "code",
- "execution_count": 65,
+ "execution_count": 47,
"metadata": {},
"outputs": [
{
@@ -1272,7 +1272,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 65,
+ "execution_count": 47,
"metadata": {},
"output_type": "execute_result"
}
@@ -1283,7 +1283,7 @@
},
{
"cell_type": "code",
- "execution_count": 66,
+ "execution_count": 48,
"metadata": {},
"outputs": [
{
@@ -1292,7 +1292,7 @@
"Executed operation: GoTo(z2,Blank,ze,Blank,R)"
]
},
- "execution_count": 66,
+ "execution_count": 48,
"metadata": {},
"output_type": "execute_result"
}
@@ -1303,7 +1303,7 @@
},
{
"cell_type": "code",
- "execution_count": 67,
+ "execution_count": 49,
"metadata": {},
"outputs": [
{
@@ -1327,7 +1327,7 @@
"<Animation function visualisation>"
]
},
- "execution_count": 67,
+ "execution_count": 49,
"metadata": {},
"output_type": "execute_result"
}
@@ -1340,12 +1340,12 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "Die Turing Maschine at 33 als Ergebnis zurückgegeben."
+ "Die Turing Maschine hat 33 als Ergebnis zurückgegeben."
]
},
{
"cell_type": "code",
- "execution_count": 68,
+ "execution_count": 50,
"metadata": {},
"outputs": [
{
@@ -1354,7 +1354,7 @@
"Executed operation: Accept()"
]
},
- "execution_count": 68,
+ "execution_count": 50,
"metadata": {},
"output_type": "execute_result"
}
diff --git a/info4/kapitel-8/GOTO-Programme.ipynb b/info4/kapitel-8/GOTO-Programme.ipynb
new file mode 100644
index 0000000000000000000000000000000000000000..a966f2c69713bd956b3f13c1b6e1bd124869ab77
--- /dev/null
+++ b/info4/kapitel-8/GOTO-Programme.ipynb
@@ -0,0 +1,168 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# GOTO-Programme\n",
+ "\n",
+ "Die letzte Programmiersprache, die wir betrachten ist GOTO.\n",
+ "Die Syntax eines GOTO-Programms ist wie folgt definiert:\n",
+ "* Ein GOTO-Programm besteht aus Markierten Anweisungen $M_1: A_1; M_2: A_2; ... M_m: A_m;$\n",
+ "* Die Markierungen beginnen bei 1 und steigen immer um 1 an.\n",
+ "* Jede Anweisung wird durch 1 Semicolon abgeschlossen (ein GOTO-Programm endet immer mit einem Semicolon)\n",
+ "* Anweisungen $A_i$ dürfen dabei sein:\n",
+ " * Zuweisung $x_i:=x_j+c$, $x_i:=x_j-c$ und $x_i:=c$ für $c\\in\\mathbb{N}$\n",
+ " * Unbedingter Sprung $GOTO\\ M_i$\n",
+ " * Bedingter Sprung $IF\\ x_i=c\\ THEN\\ GOTO\\ M_j$\n",
+ " * Abbruchanweisung $HALT$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Die Semantik eines GOTO-Programms lässt sich wie folgt verstehen:\n",
+ "Das Programm startet an der mit $M_1$ markierten Anweisung und führt diese aus.\n",
+ "Ist diese eine Zuweisung oder ein bedingter Sprung, dessen Bedingung nicht erfüllt ist, so wird mit der nächsten Anweisung weiter gemacht.\n",
+ "Wenn ein unbedingter Sprung oder ein bedingter Sprung miterfüllter Bedingung ausgeführt wird, wird an der entsprechenden Markierung fortgefahren.\n",
+ "Bei einer Abbruchanweisung hält das Programm sofort an und der Wert von $x_0$ wird zurück gegeben.\n",
+ "\n",
+ "Da ein GOTO-Programm stets von einem $HALT$ beendet wird, ist die letze Anweisung in einem GOTO-Programm immer $HALT$ oder $GOTO$."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Eine Funktion heißt GOTO-berechenbar, falls es ein GOTO-Programm gibt, das mit $n_1,...,n_k$ in den Variablen $x_1,...,x_k$ gestartet wird und mit $f(n_1,n_2,...,n_k)$ in der Variablen $x_0$ endet, falls $f(n_1,n_2,...,n_k)$ definiert ist. Andernfalls stoppt das GOTO-Programm nicht.\n",
+ "Ein Beispiel dafür ist die 2er-Potenz $f(n) = 2^n$:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Help on function interpret in module __main__:\n",
+ "\n",
+ "interpret(program, value_list=None, timeout=60)\n",
+ " Funktion zum Ausführen eines GOTO-Programms.\n",
+ " :param program: GOTO-Programm als String 'M1: A1; M2: A2; ... Mn: An;'\n",
+ " :param value_list: Array von Integern ([v1, ..., vn]).\n",
+ " Das GOTO-Programm startet mit diesen Werten in x1, ..., xn.\n",
+ " :param timeout: Zeit nach der die Ausführung eines Programms pausiert wird.\n",
+ " Gibt eine Möglichkeit zum Abbrechen bei einer Endlosschleife.\n",
+ " Ein Wert von 0 deaktiviert den Timeout.\n",
+ " :returns integer: Gibt bei Abbruch -1 und sonst den Wert von x0 nach dem GOTO-Programm zurück.\n",
+ " :usage interpret('M1: x0 := x0 + 1; M2: IF x0 = 10 THEN GOTO M4; M3: GOTO M1; M4: HALT;')\n",
+ "\n"
+ ]
+ }
+ ],
+ "source": [
+ "%run ./Interpreter/gotointerpreter.py"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "256"
+ ]
+ },
+ "execution_count": 2,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "interpret('''\n",
+ "M1:x0:=1;\n",
+ "M2: IF x1=0 THEN GOTO M10;\n",
+ "M3:x2:=x0+0;\n",
+ "M4:IF x2=0 THEN GOTO M8;\n",
+ "M5:x0:=x0 + 1;\n",
+ "M6:x2:=x2 − 1;\n",
+ "M7:GOTO M4;\n",
+ "M8: x1:=x1-1;\n",
+ "M9:GOTO M2;\n",
+ "M10: HALT;''', [8])"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Mit GOTO- und WHILE-Programmen kann man auch partielle Funktionen berechnen.\n",
+ "das folgende GOTO-Programm berechnet die Funktion f(x)=x, falls x>5 und undefiniert sonst:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "metadata": {
+ "scrolled": true
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Die Funktion rechnet relativ lange.\n",
+ "Vielleicht liegt eine Endlosschleife vor.\n",
+ "Möchten sie abbrechen? [J,n]:Ja\n",
+ "Die Ausführung des Programms wurde unterbrochen.\n",
+ "Daher ist der Rückgabewert des Programms nicht definiert.\n"
+ ]
+ },
+ {
+ "data": {
+ "text/plain": [
+ "-1"
+ ]
+ },
+ "execution_count": 3,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "interpret('''\n",
+ "M1: x2 := x1 -5;\n",
+ "M2: IF x2 = 0 THEN GOTO M2;\n",
+ "M3: x0:=x1+0;\n",
+ "M4: HALT;''', [5])"
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.8.6"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/info4/kapitel-8/Interpreter/gotointerpreter.py b/info4/kapitel-8/Interpreter/gotointerpreter.py
new file mode 100644
index 0000000000000000000000000000000000000000..5ef4fb659037a880a800a41aa23e171f79edbacc
--- /dev/null
+++ b/info4/kapitel-8/Interpreter/gotointerpreter.py
@@ -0,0 +1,320 @@
+import lexer
+from loopinterpreter import ErrorHandler
+from whileinterpreter import Timeout
+from whileinterpreter import State
+import re
+import operator
+
+
+class GOTOInterpreter:
+ def __init__(self, timeout=60):
+ self.values = {}
+ self.lex = None
+ self.error_handler = None
+ self.regex_to_token = [(re.compile(r'\d+'), 'NUMBER'),
+ (re.compile(r'x\d+'), 'IDENTIFIER'),
+ (re.compile(r'M\d+'), 'MARKER'),
+ (re.compile(r'\+'), 'PLUS'),
+ (re.compile(r'[−-]'), 'MINUS'),
+ (re.compile(r':=|≔'), 'ALLOCATION'),
+ (re.compile(r'='), 'EQUALS'),
+ (re.compile(r'/=|≠|!='), 'NOTEQUALS'),
+ (re.compile(r'IF'), 'IF'),
+ (re.compile(r'THEN'), 'THEN'),
+ (re.compile(r'GOTO'), 'GOTO'),
+ (re.compile(r'HALT'), 'HALT'),
+ (re.compile(r';'), 'SEMICOLON'),
+ (re.compile(r':'), 'COLON'),
+ (re.compile(r'BREAK'), 'BREAK'),
+ (re.compile(r'\s+', re.MULTILINE), 'WHITESPACE'),
+ (re.compile(r'[^\n]*'), 'UNKNOWN')]
+ self.marker_to_position = {}
+ self.marker_to_line = {}
+ self.timeout = timeout
+ self.halted = False
+
+ def next_token(self):
+ new_token = self.lex.next()
+ if new_token is None:
+ return None
+ elif new_token.k == 'BREAK':
+ self.error_handler.handle_break()
+ return self.next_token()
+ elif new_token.k == 'WHITESPACE':
+ if new_token.v.count('\n') > 0:
+ self.error_handler.increase_line(new_token.v.count('\n'))
+ return self.next_token()
+ else:
+ return new_token
+
+ def next_nonempty_token(self, current_function, expected_token):
+ token = self.next_token()
+ if token is None:
+ self.error_handler.handle_error(
+ 'Frühzeitiges Ende von ' + current_function + '\n' + 'Erwartet: ' + expected_token)
+ return token
+
+ def process_assignment(self, identifier_token_1):
+ identifier_1 = identifier_token_1.v
+ if not self.next_nonempty_token('Zuweisung', ':=').k == 'ALLOCATION':
+ self.error_handler.handle_error(':= in Zuweisung erwartet.')
+ identifier_token_2 = self.next_nonempty_token('Zuweisung', 'IDENTIFIER (x0, x1, ...) oder NUMBER')
+ if identifier_token_2.k == 'NUMBER':
+ value_1 = int(identifier_token_2.v)
+ self.values.update({identifier_token_1.v: value_1})
+ if not self.next_nonempty_token('Zuweisung', 'SEMICOLON').k == 'SEMICOLON':
+ self.error_handler.handle_error('SEMICOLON zum Abschluss einer Zuweisung erwartet.')
+ return self.next_token()
+ if not identifier_token_2.k == 'IDENTIFIER':
+ self.error_handler.handle_error('IDENTIFIER in Zuweisung erwartet.')
+ identifier_2 = identifier_token_2.v
+ if identifier_2 in self.values:
+ value_2 = self.values.get(identifier_2)
+ else:
+ value_2 = 0
+ operator_token = self.next_nonempty_token('Zuweisung', '+ oder -')
+ op = None
+ if operator_token.k == 'PLUS':
+ op = operator.__add__
+ elif operator_token.k == 'MINUS':
+ op = operator.__sub__
+ else:
+ self.error_handler.handle_error('+ oder - in Zuweisung erwartet.')
+ number_token = self.next_nonempty_token('Zuweisung', 'NUMBER')
+ if not number_token.k == 'NUMBER':
+ self.error_handler.handle_error('NUMBER in Zuweisung erwartet.')
+ value_1 = max(0, op(value_2, int(number_token.v)))
+ self.values.update({identifier_1: value_1})
+ if not self.next_nonempty_token('Zuweisung', 'SEMICOLON').k == 'SEMICOLON':
+ self.error_handler.handle_error('SEMICOLON zum Abschluss einer Zuweisung erwartet.')
+ return self.next_token()
+
+ def verify_assignment(self, identifier_token_1):
+ if not self.next_nonempty_token('Zuweisung', ':=').k == 'ALLOCATION':
+ self.error_handler.handle_error(':= in Zuweisung erwartet.')
+
+ identifier_token_2 = self.next_nonempty_token('Zuweisung', 'IDENTIFIER (x0, x1, ...) oder NUMBER')
+ if identifier_token_2.k == 'NUMBER':
+ if not self.next_nonempty_token('Zuweisung', 'SEMICOLON').k == 'SEMICOLON':
+ self.error_handler.handle_error('SEMICOLON zum Abschluss einer Zuweisung erwartet.')
+ return self.next_token()
+
+ if not identifier_token_2.k == 'IDENTIFIER':
+ self.error_handler.handle_error('IDENTIFIER in Zuweisung erwartet.')
+ if self.next_nonempty_token('Zuweisung', '+ oder -').k not in ['PLUS', 'MINUS']:
+ self.error_handler.handle_error('+ oder - in Zuweisung erwartet.')
+ if not self.next_nonempty_token('Zuweisung', 'NUMBER').k == 'NUMBER':
+ self.error_handler.handle_error('NUMBER in Zuweisung erwartet.')
+ if not self.next_nonempty_token('Zuweisung', 'SEMICOLON').k == 'SEMICOLON':
+ self.error_handler.handle_error('SEMICOLON zum Abschluss einer Zuweisung erwartet.')
+ return self.next_token()
+
+ def process_goto(self, goto_token):
+ marker_token = self.next_nonempty_token('GOTO', 'MARKER (M1, M2, ...)')
+ if not marker_token.k == 'MARKER':
+ self.error_handler.handle_error('MARKER (M1, M2, ...) in GOTO erwartet.')
+ if not self.next_nonempty_token('GOTO', 'SEMICOLON').k == 'SEMICOLON':
+ self.error_handler.handle_error('SEMICOLON zum Abschluss eines GOTO erwartet')
+ marker_number = int(marker_token.v[1:])
+ saved_position = marker_token.p
+ saved_line = self.error_handler.line_number
+ current_token = self.next_token()
+ while current_token is not None and max(self.marker_to_position.keys()) < int(marker_number):
+ current_token = self.verify_line(current_token)
+ if marker_number not in self.marker_to_position.keys():
+ self.lex.current_position = saved_position
+ self.error_handler.line_number = saved_line
+ self.error_handler.handle_error('GOTO zu nicht vorhandener Markierung')
+ self.lex.current_position = self.marker_to_position.get(marker_number)
+ self.error_handler.line_number = self.marker_to_line.get(marker_number)
+ if self.test_homeomorphic_embedding:
+ new_state = State(self.lex.current_position, self.values)
+ for state in self.state_list:
+
+ if state.less_or_equal(new_state):
+ self.error_handler.handle_endless_loop()
+ self.state_list.append(new_state)
+ return self.next_token()
+
+ def verify_goto(self, goto_token):
+ marker_token = self.next_nonempty_token('GOTO', 'MARKER (M1, M2, ...)')
+ if not marker_token.k == 'MARKER':
+ self.error_handler.handle_error('MARKER (M1, M2, ...) in GOTO erwartet.')
+ if not self.next_nonempty_token('GOTO', 'SEMICOLON').k == 'SEMICOLON':
+ self.error_handler.handle_error('SEMICOLON zum Abschluss eines GOTO erwartet')
+ marker_number = int(marker_token.v[1:])
+ saved_position = self.lex.current_position
+ saved_line = self.error_handler.line_number
+ current_token = self.next_token()
+ while current_token is not None and max(self.marker_to_position.keys()) < int(marker_number):
+ current_token = self.verify_line(current_token)
+ self.lex.current_position = saved_position
+ self.error_handler.line_number = saved_line
+ if marker_number not in self.marker_to_position.keys():
+ self.error_handler.handle_error('GOTO zu nicht vorhandener Markierung')
+ return self.next_token()
+
+ def process_if(self, if_token):
+ identifier_token = self.next_nonempty_token('IF', 'IDENTIFIER')
+ if not identifier_token.k == 'IDENTIFIER':
+ self.error_handler.handle_error('IDENTIFIER in IF erwartet')
+ if not self.next_nonempty_token('IF', '=').k == 'EQUALS':
+ self.error_handler.handle_error('= in IF erwartet')
+ number_token = self.next_nonempty_token('IF', 'NUMBER')
+ if not number_token.k == 'NUMBER':
+ self.error_handler.handle_error('NUMBER in IF erwartet')
+
+ if not self.next_nonempty_token('IF', 'THEN').k == 'THEN':
+ self.error_handler.handle_error('THEN in IF erwartet')
+ current_token = self.next_nonempty_token('IF', 'GOTO')
+ if not current_token.k == 'GOTO':
+ self.error_handler.handle_error('GOTO in IF erwartet')
+ if identifier_token.v in self.values.keys():
+ identifier_value = self.values.get(identifier_token.v)
+ else:
+ identifier_value = 0
+ if identifier_value == int(number_token.v):
+ return self.process_goto(current_token)
+ else:
+ return self.verify_goto(current_token)
+
+ def verify_if(self, if_token):
+ if not self.next_nonempty_token('IF', 'IDENTIFIER').k == 'IDENTIFIER':
+ self.error_handler.handle_error('IDENTIFIER in IF erwartet')
+ if not self.next_nonempty_token('IF', '=').k == 'EQUALS':
+ self.error_handler.handle_error('= in IF erwartet')
+ if not self.next_nonempty_token('IF', 'NUMBER').k == 'NUMBER':
+ self.error_handler.handle_error('NUMBER in IF erwartet')
+ if not self.next_nonempty_token('IF', 'THEN').k == 'THEN':
+ self.error_handler.handle_error('THEN in IF erwartet')
+ current_token = self.next_nonempty_token('IF', 'GOTO')
+ if not current_token.k == 'GOTO':
+ self.error_handler.handle_error('GOTO in IF erwartet')
+ return self.verify_goto(current_token)
+
+ def process_halt(self, halt_token):
+ if not self.next_nonempty_token('HALT', 'SEMICOLON').k == 'SEMICOLON':
+ self.error_handler.handle_error('SEMICOLON zum Abschluss eines HALT erwartet')
+ self.halted = True
+ current_token = self.next_token()
+ while current_token is not None:
+ current_token = self.verify_line(current_token)
+ return current_token
+
+ def verify_halt(self, halt_token):
+ if not self.next_nonempty_token('HALT', 'SEMICOLON').k == 'SEMICOLON':
+ self.error_handler.handle_error('SEMICOLON zum Abschluss eines HALT erwartet')
+ return self.next_token()
+
+ def next_marker_number(self):
+ if self.marker_to_position.keys():
+ return max(self.marker_to_position.keys()) + 1
+ else:
+ return 1
+
+ def process_line(self, current_token):
+ if current_token is None or not current_token.k == 'MARKER':
+ self.error_handler.handle_error('Keine passende Anweisung gefunden\n' +
+ 'Erwartet: MARKER (M1, M2, ...)')
+ marker_number = int(current_token.v[1:])
+ if not (marker_number == self.next_marker_number() or
+ (marker_number in self.marker_to_position.keys() and
+ current_token.p == self.marker_to_position.get(marker_number))):
+ self.error_handler.handle_error('Die Nummern der Anweisungen starten bei 1 und steigen fortlaufend.')
+ self.marker_to_position.update({marker_number: current_token.p})
+ self.marker_to_line.update({marker_number: self.error_handler.line_number})
+ colon_token = self.next_nonempty_token('LINE', 'COLON')
+ if not colon_token.k == 'COLON':
+ self.error_handler.handle_error('DOPPELPUNKT nach MARKER (M1, M2, ...) erwartet.')
+
+ current_token = self.next_nonempty_token('LINE', 'IDENTIFIER (x0, x1, ...), IF, GOTO oder HALT')
+ if current_token.k not in ['IDENTIFIER', 'IF', 'GOTO', 'HALT']:
+ self.error_handler.handle_error('Keine passende Anweisung gefunden\n' +
+ 'Erwartet: IDENTIFIER (x0, x1, ...), IF, GOTO oder HALT')
+ elif current_token.k == 'IDENTIFIER':
+ current_token = self.process_assignment(current_token)
+ elif current_token.k == 'GOTO':
+ current_token = self.process_goto(current_token)
+ elif current_token.k == 'IF':
+ current_token = self.process_if(current_token)
+ elif current_token.k == 'HALT':
+ current_token = self.process_halt(current_token)
+ return current_token
+
+ def verify_line(self, current_token):
+ if current_token is None or not current_token.k == 'MARKER':
+ self.error_handler.handle_error('Keine passende Anweisung gefunden\n' +
+ 'Erwartet: MARKER (M1, M2, ...)')
+ marker_number = int(current_token.v[1:])
+ if not (marker_number == self.next_marker_number() or
+ (marker_number in self.marker_to_position.keys() and
+ current_token.p == self.marker_to_position.get(marker_number))):
+ self.error_handler.handle_error('Die Nummern der Anweisungen starten bei 1 und steigen fortlaufend.')
+ self.marker_to_position.update({marker_number: current_token.p})
+ self.marker_to_line.update({marker_number: self.error_handler.line_number})
+ if not self.next_nonempty_token('LINE', 'COLON').k == 'COLON':
+ self.error_handler.handle_error('DOPPELPUNKT nach MARKER (M1, M2, ...) erwartet.')
+
+ current_token = self.next_nonempty_token('LINE', 'IDENTIFIER (x0, x1, ...), IF, GOTO oder HALT')
+ if current_token.k not in ['IDENTIFIER', 'IF', 'GOTO', 'HALT']:
+ self.error_handler.handle_error('Keine passende Anweisung gefunden\n' +
+ 'Erwartet: IDENTIFIER (x0, x1, ...), IF, GOTO oder HALT')
+ elif current_token.k == 'IDENTIFIER':
+ current_token = self.verify_assignment(current_token)
+ elif current_token.k == 'GOTO':
+ current_token = self.verify_goto(current_token)
+ elif current_token.k == 'IF':
+ current_token = self.verify_if(current_token)
+ elif current_token.k == 'HALT':
+ current_token = self.verify_halt(current_token)
+ return current_token
+
+ def interpret(self, program, values=None, test_homeomorphic_embedding=False):
+ try:
+ with Timeout(self.timeout):
+ self.halted = False
+ self.lex = lexer.Lexer(self.regex_to_token, program)
+ self.error_handler = ErrorHandler(program, self)
+ self.state_list = []
+ self.test_homeomorphic_embedding = test_homeomorphic_embedding
+ self.values=values
+ if values is None:
+ self.values = {}
+ current_token = self.next_token()
+ while current_token is not None:
+ current_token = self.process_line(current_token)
+ if not self.halted:
+ self.error_handler.handle_error('Ende des Programms ohne HALT erreicht.')
+ if 'x0' in self.values:
+ return self.values.get('x0')
+ return 0
+ except KeyboardInterrupt:
+ print('Die Ausführung des Programms wurde unterbrochen.\n' +
+ 'Daher ist der Rückgabewert des Programms nicht definiert.')
+ return -1
+
+
+def interpret(program, value_list=None, timeout=60, debug=False):
+ """Funktion zum Ausführen eines GOTO-Programms.
+ :param program: GOTO-Programm als String 'M1: A1; M2: A2; ... Mn: An;'
+ :param value_list: Array von Integern ([v1, ..., vn]).
+ Das GOTO-Programm startet mit diesen Werten in x1, ..., xn.
+ :param timeout: Zeit nach der die Ausführung eines Programms pausiert wird.
+ Gibt eine Möglichkeit zum Abbrechen bei einer Endlosschleife.
+ Ein Wert von 0 deaktiviert den Timeout.
+ :param debug Boolean der angibt ob mit Homeomorphic Embedding auf mögliche Endlosschleifen geprüft werden soll.
+ :returns integer: Gibt bei Abbruch -1 und sonst den Wert von x0 nach dem GOTO-Programm zurück.
+ :usage interpret('M1: x0 := x0 + 1; M2: IF x0 = 10 THEN GOTO M4; M3: GOTO M1; M4: HALT;')"""
+ interpreter = GOTOInterpreter(timeout)
+ values = None
+ if value_list is not None:
+ values = {}
+ for index, value in enumerate(value_list):
+ if not isinstance(value, int) or value < 0 or not int(value) == value:
+ raise ValueError("Variablen können nur natürliche Zahlen zugewiesen bekommen.")
+ values.update({'x' + str(index + 1): value})
+ return interpreter.interpret(program, values, debug)
+
+
+if __name__ == '__main__':
+ help(interpret)
diff --git a/info4/kapitel-8/Interpreter/lexer.py b/info4/kapitel-8/Interpreter/lexer.py
new file mode 100644
index 0000000000000000000000000000000000000000..a7fe3c16d952f89ed00d280b6df989e7dd759c2c
--- /dev/null
+++ b/info4/kapitel-8/Interpreter/lexer.py
@@ -0,0 +1,32 @@
+import re
+
+
+class Token:
+ def __init__(self, key, value, position):
+ self.k = key
+ self.v = value
+ self.p = position
+
+
+class Lexer:
+ def __init__(self, regex_to_token, program):
+ self.regex_to_token = regex_to_token
+ self.program = program
+ self.current_position = 0
+
+ def next(self):
+ new_position = 0
+ next_token = None
+ if self.current_position < len(self.program):
+ for pattern, value in self.regex_to_token:
+ match = pattern.match(self.program, self.current_position)
+ if match:
+ next_token = Token(value, match.group(), self.current_position)
+ new_position = match.span()[1]
+ break
+ if self.current_position == new_position:
+ msg = ['Fehler in Zeile ' + str(self.program.count('\n', 0, self.current_position) + 1) + ':',
+ 'Unbekannter String: ' + re.compile(r'[^\n]*').match(self.program, self.current_position).group()]
+ raise SyntaxError('\n'.join(msg))
+ self.current_position = new_position
+ return next_token
diff --git a/info4/kapitel-8/Interpreter/loopinterpreter.py b/info4/kapitel-8/Interpreter/loopinterpreter.py
new file mode 100644
index 0000000000000000000000000000000000000000..68823fef09fbf20c46c2a0393aa4cd0d113aa2cc
--- /dev/null
+++ b/info4/kapitel-8/Interpreter/loopinterpreter.py
@@ -0,0 +1,277 @@
+import lexer
+import sys
+import operator
+import re
+from IPython.display import clear_output
+
+
+class ErrorHandler:
+ def __init__(self, program, interpreter):
+ sys.tracebacklimit = 0
+ self.program = program
+ self.line_number = 0
+ self.interpreter = interpreter
+
+ def handle_error(self, message):
+ msg = ['Fehler in Zeile ' + str(self.line_number + 1),
+ self.program.split('\n')[self.line_number],
+ message]
+ raise SyntaxError('\n'.join(msg)) from None
+
+ def increase_line(self, value):
+ self.line_number += value
+
+ def handle_break(self):
+ clear_output(wait=True)
+ print('BREAK in Zeile ' + str(self.line_number))
+ print('Aktueller Zustand:')
+ for k, v in self.interpreter.values.items():
+ print('Variable ' + k + ': ' + str(v))
+ user_input = input('Drücke ENTER zum Fotfahren oder schreibe EXIT zum Beenden:')
+ if user_input.lower() == 'exit':
+ raise KeyboardInterrupt
+
+ def handle_endless_loop(self):
+ self.handle_error("Mögliche Endlosschleife gefunden.")
+
+
+class LOOPInterpreter:
+ def __init__(self):
+ self.regex_to_token = [(re.compile(r'\d+'), 'NUMBER'),
+ (re.compile(r'x\d+'), 'IDENTIFIER'),
+ (re.compile(r'\+'), 'PLUS'),
+ (re.compile(r'[−-]'), 'MINUS'),
+ (re.compile(r':=|≔'), 'ALLOCATION'),
+ (re.compile(r'LOOP'), 'LOOP'),
+ (re.compile(r'DO'), 'DO'),
+ (re.compile(r'END'), 'END'),
+ (re.compile(r';'), 'SEMICOLON'),
+ (re.compile(r'BREAK'), 'BREAK'),
+ (re.compile(r'\s+', re.MULTILINE), 'WHITESPACE'),
+ (re.compile(r'[^\n]*'), 'UNKNOWN')]
+ self.values = {}
+ self.lex = None
+ self.error_handler = None
+
+ def process_assignment(self, forbidden_identifiers, identifier_token_1):
+ identifier_1 = identifier_token_1.v
+ if identifier_1 in forbidden_identifiers:
+ self.error_handler.handle_error('Identifier ' + identifier_1 +
+ ' ist bereits in Loop vorhanden und darf nicht verwendet werden.')
+ if not self.next_nonempty_token('Zuweisung', ':=').k == 'ALLOCATION':
+ self.error_handler.handle_error(':= in Zuweisung erwartet.')
+ identifier_token_2 = self.next_nonempty_token('Zuweisung', 'IDENTIFIER (x0, x1, ...) oder NUMBER')
+ if identifier_token_2.k == 'NUMBER':
+ value_1 = int(identifier_token_2.v)
+ self.values.update({identifier_token_1.v: value_1})
+ return self.next_token()
+ if not identifier_token_2.k == 'IDENTIFIER':
+ self.error_handler.handle_error('IDENTIFIER in Zuweisung erwartet.')
+ identifier_2 = identifier_token_2.v
+ if identifier_2 in forbidden_identifiers:
+ self.error_handler.handle_error('Identifier ' + identifier_2 +
+ ' ist bereits in Loop vorhanden und darf nicht verwendet werden.')
+ if identifier_2 in self.values:
+ value_2 = self.values.get(identifier_2)
+ else:
+ value_2 = 0
+ operator_token = self.next_nonempty_token('Zuweisung', '+ oder -')
+ op = None
+ if operator_token.k == 'PLUS':
+ op = operator.__add__
+ elif operator_token.k == 'MINUS':
+ op = operator.__sub__
+ else:
+ self.error_handler.handle_error('+ oder - in Zuweisung erwartet.')
+ number_token = self.next_nonempty_token('Zuweisung', 'NUMBER')
+ if not number_token.k == 'NUMBER':
+ self.error_handler.handle_error('NUMBER in Zuweisung erwartet.')
+ value_1 = max(0, op(value_2, int(number_token.v)))
+ self.values.update({identifier_1: value_1})
+ return self.next_token()
+
+ def verify_assignment(self, forbidden_identifiers, identifier_token_1):
+ identifier_1 = identifier_token_1.v
+ if identifier_1 in forbidden_identifiers:
+ self.error_handler.handle_error('Identifier ' + identifier_1 +
+ ' ist bereits in Loop vorhanden und darf nicht verwendet werden.')
+ if not self.next_nonempty_token('Zuweisung', ':=').k == 'ALLOCATION':
+ self.error_handler.handle_error(':= in Zuweisung erwartet.')
+
+ identifier_token_2 = self.next_nonempty_token('Zuweisung', 'IDENTIFIER (x0, x1, ...) oder NUMBER')
+ if identifier_token_2.k == 'NUMBER':
+ return self.next_token()
+
+ if not identifier_token_2.k == 'IDENTIFIER':
+ self.error_handler.handle_error('IDENTIFIER in Zuweisung erwartet.')
+ identifier_2 = identifier_token_2.v
+ if identifier_2 in forbidden_identifiers:
+ self.error_handler.handle_error('Identifier ' + identifier_2 +
+ ' ist bereits in Loop vorhanden und darf nicht verwendet werden.')
+
+ if self.next_nonempty_token('Zuweisung', '+ oder -').k not in ['PLUS', 'MINUS']:
+ self.error_handler.handle_error('+ oder - in Zuweisung erwartet.')
+ if not self.next_nonempty_token('Zuweisung', 'NUMBER').k == 'NUMBER':
+ self.error_handler.handle_error('NUMBER in Zuweisung erwartet.')
+
+ return self.next_token()
+
+ def process_loop(self, forbidden_identifiers, loop_token):
+ identifier_token = self.next_nonempty_token('LOOP', 'IDENTIFIER (x0, x1, ...)')
+ if not identifier_token.k == 'IDENTIFIER':
+ self.error_handler.handle_error('IDENTIFIER in LOOP erwartet.')
+ if identifier_token.v in forbidden_identifiers:
+ self.error_handler.handle_error('Identifier ' + identifier_token.v +
+ ' ist bereits in Loop vorhanden und darf nicht verwendet werden.')
+ if not self.next_nonempty_token('LOOP', 'DO').k == 'DO':
+ self.error_handler.handle_error('DO in LOOP erwartet.')
+
+ if identifier_token.v in self.values:
+ number_of_loops = int(self.values.get(identifier_token.v))
+ else:
+ number_of_loops = 0
+
+ saved_position = self.lex.current_position
+ saved_line = self.error_handler.line_number
+ forbidden_identifiers.append(identifier_token.v)
+
+ if number_of_loops == 0:
+ end_found = False
+ while not end_found:
+ token = self.verify_program(forbidden_identifiers, self.next_token())
+ if token is None or token.k not in ['SEMICOLON', 'END']:
+ self.error_handler.handle_error('SEMICOLON oder END in LOOP erwartet.')
+ elif token.k == 'SEMICOLON':
+ continue
+ elif token.k == 'END':
+ end_found = True
+
+ for index in range(number_of_loops):
+ self.lex.current_position = saved_position
+ self.error_handler.line_number = saved_line
+ end_found = False
+ while not end_found:
+ token = self.process_program(forbidden_identifiers, self.next_token())
+ if token is None or token.k not in ['SEMICOLON', 'END']:
+ self.error_handler.handle_error('SEMICOLON oder END in LOOP erwartet.')
+ elif token.k == 'SEMICOLON':
+ continue
+ elif token.k == 'END':
+ end_found = True
+
+ forbidden_identifiers.remove(identifier_token.v)
+ return self.next_token()
+
+ def verify_loop(self, forbidden_identifiers, loop_token):
+ identifier_token = self.next_nonempty_token('LOOP', 'IDENTIFIER')
+ if not identifier_token.k == 'IDENTIFIER':
+ self.error_handler.handle_error('IDENTIFIER in LOOP erwartet.')
+ if identifier_token.v in forbidden_identifiers:
+ self.error_handler.handle_error('Identifier ' + identifier_token.v +
+ ' ist bereits in Loop vorhanden und darf nicht verwendet werden.')
+ if not self.next_nonempty_token('LOOP', 'DO').k == 'DO':
+ self.error_handler.handle_error('DO in LOOP erwartet.')
+
+ forbidden_identifiers.append(identifier_token.v)
+
+ end_found = False
+ while not end_found:
+ token = self.verify_program(forbidden_identifiers, self.next_token())
+ if token is None or token.k not in ['SEMICOLON', 'END']:
+ self.error_handler.handle_error('SEMICOLON oder END in LOOP erwartet.')
+ elif token.k == 'SEMICOLON':
+ continue
+ elif token.k == 'END':
+ end_found = True
+
+ forbidden_identifiers.remove(identifier_token.v)
+ return self.next_token()
+
+ def process_program(self, forbidden_identifiers, current_token):
+ if current_token is None or current_token.k not in ['IDENTIFIER', 'LOOP']:
+ self.error_handler.handle_error('Keine passende Anweisung gefunden\n' +
+ 'Erwartet: IDENTIFIER (x0, x1, ...) oder LOOP')
+ elif current_token.k == 'IDENTIFIER':
+ current_token = self.process_assignment(forbidden_identifiers, current_token)
+ elif current_token.k == 'LOOP':
+ current_token = self.process_loop(forbidden_identifiers, current_token)
+ return current_token
+
+ def verify_program(self, forbidden_identifiers, current_token):
+ if current_token is None or current_token.k not in ['IDENTIFIER', 'LOOP']:
+ self.error_handler.handle_error('Keine passende Anweisung gefunden\n' +
+ 'Erwartet: IDENTIFIER (x0, x1, ...) oder LOOP')
+ elif current_token.k == 'IDENTIFIER':
+ current_token = self.verify_assignment(forbidden_identifiers, current_token)
+ elif current_token.k == 'LOOP':
+ current_token = self.verify_loop(forbidden_identifiers, current_token)
+ return current_token
+
+ def next_token(self):
+ new_token = self.lex.next()
+ if new_token is None:
+ return None
+ elif new_token.k == 'BREAK':
+ self.error_handler.handle_break()
+ return self.next_token()
+ elif new_token.k == 'WHITESPACE':
+ if new_token.v.count('\n') > 0:
+ self.error_handler.increase_line(new_token.v.count('\n'))
+ return self.next_token()
+ else:
+ return new_token
+
+ def next_nonempty_token(self, current_function, expected_token):
+ token = self.next_token()
+ if token is None:
+ self.error_handler.handle_error(
+ 'Frühzeitiges Ende von ' + current_function + '\n' + 'Erwartet: ' + expected_token)
+ return token
+
+ def interpret(self, program, values=None):
+ try:
+ self.lex = lexer.Lexer(self.regex_to_token, program)
+ self.error_handler = ErrorHandler(program, self)
+ self.values = values
+ if values is None:
+ self.values = {}
+ forbidden_identifiers = []
+ current_token = self.next_token()
+ while current_token is not None:
+ current_token = self.process_program(forbidden_identifiers, current_token)
+ if current_token is not None:
+ if not current_token.k == 'SEMICOLON':
+ self.error_handler.handle_error('Semicolon erwartet')
+ current_token = self.next_token()
+ if current_token is None:
+ self.error_handler.handle_error('Semikolons werden nur zur Trennung und nicht zum ' +
+ 'Abschluss von Programmen verwendet')
+ if 'x0' in self.values:
+ return self.values.get('x0')
+ return 0
+ except KeyboardInterrupt:
+ print('Die Ausführung des Programms wurde unterbrochen.\n' +
+ 'Daher ist der Rückgabewert des Programms nicht definiert.')
+ return -1
+
+
+def interpret(program, value_list=None):
+ """Funktion zum Ausführen eines LOOP-Programms.
+ :param program: LOOP-Programm als String 'x1 := 4; LOOP x1 DO x0:=x0+1 END'
+ :param value_list: Array von Integern ([v1, ..., vn]).
+ Das LOOP-Programm startet mit diesen Werten in x1, ..., xn.
+ :returns integer: Gibt bei Abbruch -1 und sonst den Wert von x0 nach dem LOOP-Programm zurück.
+ :usage interpret('x1:=10; x2:=8; x0:=x2+0; LOOP x1 DO x0:=x0+1 END')"""
+ interpreter = LOOPInterpreter()
+ values = None
+ if value_list is not None:
+ values = {}
+ for index, value in enumerate(value_list):
+ if not isinstance(value, int) or value < 0 or not int(value) == value:
+ raise ValueError("Variablen können nur natürliche Zahlen zugewiesen bekommen.")
+ values.update({'x' + str(index + 1): value})
+ return interpreter.interpret(program, values)
+
+
+if __name__ == '__main__':
+ help(interpret)
diff --git a/info4/kapitel-8/Interpreter/test_goto_interpreter.py b/info4/kapitel-8/Interpreter/test_goto_interpreter.py
new file mode 100644
index 0000000000000000000000000000000000000000..8c11924c9d5bc1e1c511d73962c1315b9fa52644
--- /dev/null
+++ b/info4/kapitel-8/Interpreter/test_goto_interpreter.py
@@ -0,0 +1,216 @@
+from gotointerpreter import interpret
+from unittest import mock
+from unittest import TestCase
+
+
+def input_continue(prompt):
+ return ''
+
+
+def str_yes(prompt):
+ return 'J'
+
+
+def input_exit(prompt):
+ return 'EXIT'
+
+
+class GOTOInterpreterTest(TestCase):
+ def test_assignment_default_zero(self):
+ self.assertEqual(0, interpret('M1:x0:=x0 + 0;M2:HALT;'))
+ self.assertEqual(0, interpret('M1:x0:=x1 + 0;M2:HALT;'))
+ self.assertEqual(0, interpret('M1:x0:=x2 + 0;M2:HALT;'))
+
+ def test_assignment_non_negative(self):
+ self.assertEqual(0, interpret('M1:x0:=x0-1;M2:HALT;'))
+ self.assertEqual(0, interpret('M1:x0:=x1-1;M2:HALT;'))
+ self.assertEqual(0, interpret('M1:x0:=x2-6;M2:HALT;'))
+
+ def test_assignment_number(self):
+ self.assertEqual(5, interpret('M1:x0:=5;M2:HALT;'))
+ self.assertEqual(2, interpret('M1:x0:=2;M2:HALT;'))
+ self.assertEqual(3, interpret('M1:x0:=3;M2:HALT;'))
+
+ def test_assignment_variable(self):
+ self.assertEqual(1, interpret('M1:x0:=x0+1;M2:HALT;'), 1)
+ self.assertEqual(4, interpret('M1:x0:= 5; M2:x0:=x0-1; M3:HALT;'))
+ self.assertEqual(1, interpret('M1:x0:=x1-1; M2:x0:=x0+1; M3:HALT;'))
+ self.assertEqual(3, interpret('M1:x0:=4; M2:x0:= x0 − 1; M3:HALT;'))
+
+ def test_assignment_wrong_syntax(self):
+ with self.assertRaises(SyntaxError):
+ interpret('M1:x1:=x2;M2:HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1:x1:=0+x2;M2:HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1:x5:=-1+x4;M2:HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1:x5:=-x3+x1;M2:HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1:x5:=x1-x3;M2:HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1:x2:=x1+x4;M2:HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1:x2:=x1+2;;M2:HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1:x1:=c;M2:HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1:xi:=2;M2:HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1:x0:=xj+1;M2:HALT;')
+
+ def test_goto_skip_lines(self):
+ self.assertEqual(2, interpret('M1:x0:=2;M2:GOTO M4;M3:x0:=0;M4:HALT;'))
+ self.assertEqual(3, interpret('M1:x0:=3;M2:GOTO M5;M3:GOTO M1;M4:x0:=5;M5:HALT;'))
+ self.assertEqual(0, interpret('M1:GOTO M3;M2:x0:=x2+5;M3:HALT;'))
+
+ def test_if(self):
+ self.assertEqual(10, interpret('M1:x2:=10;' +
+ 'M2:IF x2=0 THEN GOTO M6;' +
+ 'M3:x0:=x0+1;M4:x2:=x2-1;' +
+ 'M5:GOTO M2; M6:HALT;'))
+ self.assertEqual(1, interpret('M1:IF x0=5 THEN GOTO M4;M2:IF x0=2 THEN GOTO M4;M3:x0:=x0+1;M4:HALT;'))
+
+ def test_syntax_missing_operator(self):
+ with self.assertRaises(SyntaxError):
+ interpret('M1:x0:=x1 2;M2:HALT;')
+
+ def test_markers_start_with_one(self):
+ with self.assertRaises(SyntaxError):
+ interpret('M0:x0:=2;M1:HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M2:x0:=2;M3:HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M10:x0:=2;M11:HALT;')
+
+ def test_markers_increment_by_one(self):
+ with self.assertRaises(SyntaxError):
+ interpret('M1:x0:=3;M3:HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1:x0:=2;M1:HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1: GOTO M3; M3:HALT;')
+
+ def test_linebreaks(self):
+ self.assertEqual(0, interpret('''
+ M1:x1:=2;
+ M2: IF x1=2 THEN GOTO M4;
+ M3:HALT;
+ M4:GOTO M3;'''))
+
+ @mock.patch('loopinterpreter.input', side_effect=input_continue)
+ def test_break(self, custom_input):
+ self.assertEqual(2, interpret('M1:x0:=2;BREAK M2:HALT;'))
+
+ @mock.patch('loopinterpreter.input', side_effect=input_exit)
+ def test_break_exit(self, custom_input):
+ self.assertEqual(-1, interpret('M1:GOTO M1; BREAK'))
+
+ def test_assignment_missing_tokens(self):
+ with self.assertRaises(SyntaxError):
+ interpret('M1: x0')
+ with self.assertRaises(SyntaxError):
+ interpret('M1: x1 x2+2;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1: x1 := x2+2 M2')
+ with self.assertRaises(SyntaxError):
+ interpret('M1: GOTO M3; M2:x1 x2+0; M3: HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1:GOTO M3; M2:x1:=2 M3: HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1:GOTO M3; M2: x1:= M3: HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1: GOTO M3; M2: x1 := x2 M3: HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1: GOTO M3; M2: x1 := x2 - M3: HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1: GOTO M3; M2: x1:= x2 + 1 M3: HALT;')
+
+ def test_goto_missing_tokens(self):
+ with self.assertRaises(SyntaxError):
+ interpret('M1:GOTO :HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1: GOTO M2' +
+ 'M2:HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1:GOTO M3; M2: GOTO :HALT; M3: HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1: GOTO M3;' +
+ 'M2: GOTO M3' +
+ 'M3:HALT;')
+
+ def test_goto_jump_not_defined(self):
+ with self.assertRaises(SyntaxError):
+ interpret('M1: GOTO M2;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1:GOTO M3; M2: GOTO M4; M3:HALT;')
+
+ def test_halt_missing_tokens(self):
+ with self.assertRaises(SyntaxError):
+ interpret('M1:HALT M2:HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1:GOTO M3; M2: HALT M3:HALT;')
+
+ def test_if_missing_tokens(self):
+ with self.assertRaises(SyntaxError):
+ interpret('M1: IF THEN GOTO M2; M2:HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1: IF x1 THEN GOTO M2; M2:HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1: IF x2 = THEN GOTO M2; M2:HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1: IF x0 = 0 GOTO M2; M2:HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1: IF x3 = 5 THEN M2; M2:HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1:GOTO M3; M2: IF THEN GOTO M3; M3:HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1:GOTO M3; M2: IF x1 THEN GOTO M2; M3:HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1:GOTO M3; M2: IF x2 = THEN GOTO M2; M3:HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1:GOTO M3; M2: IF x0 = 0 GOTO M2; M3:HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1:GOTO M3; M2: IF x3 = 5 THEN M2; M3:HALT;')
+
+ def test_line_missing_tokens(self):
+ with self.assertRaises(SyntaxError):
+ interpret('M1 GOTO M2; M2: HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1: GOTO M3; GOTO M3; M3:HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1: GOTO M3; M2 GOTO M3; M3: HALT;')
+ with self.assertRaises(SyntaxError):
+ interpret('M1: GOTO M3; M2: M3; M3: HALT;')
+
+ def test_end_without_halt(self):
+ with self.assertRaises(SyntaxError):
+ interpret('M1: x0:=2;')
+
+ @mock.patch('whileinterpreter.input', side_effect=str_yes)
+ def test_infinite_loop(self, custom_input):
+ self.assertEqual(-1, interpret('M1: x1:=100000; M2: x1:=x1-1; M3: IF x1=0 THEN GOTO M5; M4: GOTO M1; M5:HALT;',
+ timeout=1))
+
+ def test_given_values(self):
+ self.assertEqual(5, interpret('M1: x0:=x1+0; M2: IF x2=0 THEN GOTO M6; M3: x0:=x0+1;' +
+ 'M4: x2:=x2-1; M5: GOTO M2; M6: HALT;', [2, 3]))
+ self.assertEqual(4, interpret('M1: x0:=x1+0; M2: IF x2=0 THEN GOTO M6; M3: x0:=x0+1;' +
+ 'M4: x2:=x2-1; M5: GOTO M2; M6: HALT;', [0, 4]))
+ self.assertEqual(1, interpret('M1: x0:=x0+1; M2: HALT;', [2,3,4]))
+
+ def test_wrong_given_values(self):
+ with self.assertRaises(ValueError):
+ interpret('M1: x0:=2;', [-1])
+ with self.assertRaises(ValueError):
+ interpret('M1: x0:=1;', [0.2])
+ with self.assertRaises(ValueError):
+ interpret('M1: x0:=4;', ['EINS'])
+
+ def test_homeomorphic_embedding(self):
+ with self.assertRaises(SyntaxError):
+ interpret('M1: x1:=1; M2: IF x1 = 0 THEN GOTO M5;'
+ 'M3: x1:=x1+1; M4: GOTO M2; M5: HALT;', timeout=0, debug=True)
+ with self.assertRaises(SyntaxError):
+ interpret('M1: x1:=100; M2: IF x1 = 0 THEN GOTO M5;'
+ 'M3: x1:=x1+1; M4: GOTO M2; M5: HALT;', timeout=0, debug=True)
diff --git a/info4/kapitel-8/Interpreter/test_loop_interpreter.py b/info4/kapitel-8/Interpreter/test_loop_interpreter.py
new file mode 100644
index 0000000000000000000000000000000000000000..e14b136017e16d81264b9f9daa07c811553198ee
--- /dev/null
+++ b/info4/kapitel-8/Interpreter/test_loop_interpreter.py
@@ -0,0 +1,261 @@
+from loopinterpreter import interpret
+import unittest
+from unittest import mock
+
+
+def input_exit(prompt):
+ return 'EXIT'
+
+
+def input_continue(prompt):
+ return ''
+
+
+def init_without_tokens(self, regex_to_token, program):
+ self.regex_to_token = {}
+ self.program = program
+ self.current_position = 0
+
+
+class LOOPInterpreterTest(unittest.TestCase):
+ def test_assignment_default_zero(self):
+ self.assertEqual(0, interpret('x0:=x0 + 0'))
+ self.assertEqual(0, interpret('x0:=x1 + 0'))
+ self.assertEqual(0, interpret('x0:=x2 + 0'))
+
+ def test_assignment_non_negative(self):
+ self.assertEqual(0, interpret('x0:=x0-1'))
+ self.assertEqual(0, interpret('x0:=x1-1'))
+ self.assertEqual(0, interpret('x0:=x2-6'))
+
+ def test_assignment_number(self):
+ self.assertEqual(5, interpret('x0:=5'))
+ self.assertEqual(2, interpret('x0:=2'))
+ self.assertEqual(3, interpret('x0:=3'))
+
+ def test_assignment_variable(self):
+ self.assertEqual(1, interpret('x0:=x0+1'))
+ self.assertEqual(4, interpret('x0:= 5; x0:=x0-1'))
+ self.assertEqual(1, interpret('x0:=x1-1; x0:=x0+1'))
+
+ def test_assignment_wrong_syntax(self):
+ with self.assertRaises(SyntaxError):
+ interpret('x1:=x2')
+ with self.assertRaises(SyntaxError):
+ interpret('x1:=0+x2')
+ with self.assertRaises(SyntaxError):
+ interpret('x5:=-1+x4')
+ with self.assertRaises(SyntaxError):
+ interpret('x5:=-x3+x1')
+ with self.assertRaises(SyntaxError):
+ interpret('x5:=x1-x3')
+ with self.assertRaises(SyntaxError):
+ interpret('x2:=x1+x4')
+ with self.assertRaises(SyntaxError):
+ interpret('x2:=x1+2;')
+ with self.assertRaises(SyntaxError):
+ interpret('x1:=c')
+ with self.assertRaises(SyntaxError):
+ interpret('xi:=2')
+ with self.assertRaises(SyntaxError):
+ interpret('x0:=xj+1')
+
+ def test_loop_assignment(self):
+ self.assertEqual(1, interpret('x1:=1; LOOP x1 DO x0:=1 END'))
+ self.assertEqual(4, interpret('x1:=2; LOOP x1 DO x0:=x0 + 2 END'))
+
+ def test_loop_empty_assignment(self):
+ self.assertEqual(0, interpret('LOOP x1 DO x0:=1 END'))
+ self.assertEqual(0, interpret('x2:=2;LOOP x1 DO x0:=x2+1 END'))
+
+ def test_loop_nested_assignment(self):
+ self.assertEqual(6, interpret('x1:=3; LOOP x1 DO x2:=x2+1; LOOP x2 DO x0:=x0+1 END END'))
+ self.assertEqual(3, interpret('x1:=3; x2:=3; LOOP x1 DO x2:=x2-1; LOOP x2 DO x0:=x0+1 END END'))
+
+ def test_loop_forbidden_identifier(self):
+ with self.assertRaises(SyntaxError):
+ interpret('x1:=1; LOOP x1 DO x1:=x1+1 END')
+ with self.assertRaises(SyntaxError):
+ interpret('x1:=1; LOOP x1 DO x2:=x1 + 2 END')
+
+ def test_loop_empty_forbidden_identifier(self):
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x2 DO x2:=2 END')
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x1 DO x2:=x1 - 2 END')
+
+ def test_loop_nested_forbidden_identifier(self):
+ with self.assertRaises(SyntaxError):
+ interpret('x1:=2; LOOP x1 DO LOOP x1 DO x0:=x0+1 END END')
+ with self.assertRaises(SyntaxError):
+ interpret('x1:=1; x2:=2 LOOP x1 DO LOOP x2 DO x1:=2 END END')
+ with self.assertRaises(SyntaxError):
+ interpret('x1:=1; x2:=2 LOOP x1 DO LOOP x2 DO x2:=2 END END')
+ with self.assertRaises(SyntaxError):
+ interpret('x1:=1; x2:=2 LOOP x1 DO LOOP x2 DO x0:=x2+2 END END')
+ with self.assertRaises(SyntaxError):
+ interpret('x1:=1; x2:=2 LOOP x1 DO LOOP x2 DO x0:=x1-2 END END')
+
+ def test_loop_nested_empty_forbidden_identifier(self):
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x1 DO LOOP x2 DO x2:=2 END END')
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x1 DO LOOP x2 DO x0:=x2+2 END END')
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x1 DO LOOP x2 DO x0:=x1 + 0 END END')
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x1 DO LOOP x2 DO x1:=2 END END')
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x1 DO LOOP x1 DO x2:=2 END END')
+
+ def test_loop_wrong_syntax(self):
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP 2 DO x2:=5 END')
+ with self.assertRaises(SyntaxError):
+ interpret('x1:=1; LOOP x1 DO x2:=5; END')
+ with self.assertRaises(SyntaxError):
+ interpret('x1:=1; LOOP x1 DO; x2:=5 END')
+ with self.assertRaises(SyntaxError):
+ interpret('x1:=1; LOOP x1 DO x2:=5 END;')
+
+ def test_assignment_with_loop(self):
+ self.assertEqual(2, interpret('x0:=2; LOOP x0 DO x1:=x1+1 END; x0:=x1+0'))
+ self.assertEqual(1, interpret('x1:=x1+1; LOOP x0 DO x1:=x1+1 END; x0:=x1+0'))
+
+ def test_syntax_unnecessary_semicolon(self):
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x0 DO x1:=x1+1 END;')
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x0 DO x1:=x1+1;; x1:=x1+1 END')
+ with self.assertRaises(SyntaxError):
+ interpret('x1:=x1+1;; x1:=x1+1')
+ with self.assertRaises(SyntaxError):
+ interpret(';x1:=x1+1')
+
+ def test_syntax_unnecessary_end(self):
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x0 DO x1:=x1+1 END END')
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x0 DO x1:=x1+1 END; x1:=x1+1 END')
+ with self.assertRaises(SyntaxError):
+ interpret('x1:=x1+1; END x1:=x1+1')
+ with self.assertRaises(SyntaxError):
+ interpret('END x1:=x1+1')
+
+ def test_syntax_missing_semicolon(self):
+ with self.assertRaises(SyntaxError):
+ interpret('x0:=2 LOOP x0 DO x1:=x1+1 END')
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x0 DO x1:=x1+1 x1:=x1+1 END')
+ with self.assertRaises(SyntaxError):
+ interpret('x0:=2; LOOP x0 DO x1:=x1+1 x1:=x1+1 END')
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x0 DO x1:=x1+1 END x0:=x1+0')
+ with self.assertRaises(SyntaxError):
+ interpret('x0:=2; LOOP x0 DO x1:=x1+1 END x0:=x1+0')
+
+ def test_syntax_missing_do(self):
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x1 x2:=2 END')
+ with self.assertRaises(SyntaxError):
+ interpret('x1:=2; LOOP x1 x2:=2 END')
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x0 DO LOOP x1 x2:=2 END END')
+ with self.assertRaises(SyntaxError):
+ interpret('x0:=1; LOOP x0 DO LOOP x1 x2:=2 END END')
+ with self.assertRaises(SyntaxError):
+ interpret('x0:=1; x1:=2; LOOP x0 DO LOOP x1 x2:=2 END END')
+
+ def test_syntax_missing_end(self):
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x0 DO LOOP x1 DO x2:=2')
+ with self.assertRaises(SyntaxError):
+ interpret('x0:=5; LOOP x0 DO LOOP x1 DO x2:=2')
+ with self.assertRaises(SyntaxError):
+ interpret('x0:=4; x1:=7; LOOP x0 DO LOOP x1 DO x2:=2')
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x0 DO LOOP x1 DO x2:=2 END')
+ with self.assertRaises(SyntaxError):
+ interpret('x0:=2; LOOP x0 DO LOOP x1 DO x2:=2 END')
+ with self.assertRaises(SyntaxError):
+ interpret('x0:=2; x1:=3; LOOP x0 DO LOOP x1 DO x2:=2 END')
+ with self.assertRaises(SyntaxError):
+ interpret('x0 := 2; LOOP x0 DO x1 := 1; x2 := x2 + 1')
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x0 DO x1:=2; x2:=0')
+
+ def test_syntax_missing_program(self):
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x0 DO END')
+ with self.assertRaises(SyntaxError):
+ interpret('x0:=2; LOOP x0 DO END')
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x0 DO LOOP x1 DO END END')
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x0 DO LOOP x1 DO x2:=2; END')
+
+ def test_syntax_missing_operator(self):
+ with self.assertRaises(SyntaxError):
+ interpret('x0:=x1 2')
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x2 DO x0:=x1 2 END')
+ with self.assertRaises(SyntaxError):
+ interpret('x2:=3; LOOP x2 DO x1:=x1 2 END')
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x2 DO LOOP x3 DO x1:=x1 2 END END')
+
+ def test_syntax_missing_equals(self):
+ with self.assertRaises(SyntaxError):
+ interpret('x1 2')
+ with self.assertRaises(SyntaxError):
+ interpret('x1 x2+2')
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x0 DO x1 2 END')
+ with self.assertRaises(SyntaxError):
+ interpret('x0:=2; LOOP x0 DO x1 2 END')
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x0 DO x1 x2+3 END')
+ with self.assertRaises(SyntaxError):
+ interpret('x0:=2; LOOP x0 DO x1 x2-1 END')
+
+ def test_syntax_missing_identifier(self):
+ with self.assertRaises(SyntaxError):
+ interpret('x1:=; LOOP x1 DO x2:=2 END')
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x1 DO x2:= END')
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x1 DO x2:=x0+ END')
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x0 DO LOOP DO x1:=x2+0 END END')
+
+ def test_newlines(self):
+ self.assertEqual(5, interpret('''x2:=3;
+ x0:=x2+2'''))
+ self.assertEqual(2, interpret('x1:=x1-2;\n x0:=x1+2'))
+
+ @mock.patch('loopinterpreter.input', side_effect=input_exit)
+ def test_break_exit(self, custom_input):
+ self.assertEqual(-1, interpret('x1:=2; BREAK x0:=2'))
+ self.assertEqual(-1, interpret('LOOP x1 DO BREAK x2:= 2 END'))
+
+ @mock.patch('loopinterpreter.input', side_effect=input_continue)
+ def test_break_continue(self, custom_input):
+ self.assertEqual(4, interpret('x1:=2; LOOP x1 DO x0:=x0+2 BREAK END'))
+
+ @mock.patch('lexer.Lexer.__init__', init_without_tokens)
+ def test_unknown_tokens(self):
+ with self.assertRaises(SyntaxError):
+ interpret('BLIBLABLUB')
+
+ def test_given_values(self):
+ self.assertEqual(5, interpret('x0:=x1+0; LOOP x2 DO x0:=x0+1 END', [2, 3]))
+ self.assertEqual(4, interpret('x0:=x1+0; LOOP x2 DO x0:=x0+1 END', [0, 4]))
+ self.assertEqual(1, interpret('x0:=x0+1', [2,3,4]))
+
+ def test_wrong_given_values(self):
+ with self.assertRaises(ValueError):
+ interpret('x0:=2;', [-1])
+ with self.assertRaises(ValueError):
+ interpret('x0:=1;', [0.2])
+ with self.assertRaises(ValueError):
+ interpret('x0:=4;', ['EINS'])
\ No newline at end of file
diff --git a/info4/kapitel-8/Interpreter/test_while_interpreter.py b/info4/kapitel-8/Interpreter/test_while_interpreter.py
new file mode 100644
index 0000000000000000000000000000000000000000..3fab6da7131f511c147a2913fb1bf849970437f8
--- /dev/null
+++ b/info4/kapitel-8/Interpreter/test_while_interpreter.py
@@ -0,0 +1,142 @@
+from whileinterpreter import interpret
+from test_loop_interpreter import LOOPInterpreterTest
+from unittest import mock
+
+
+def str_yes(prompt):
+ return 'J'
+
+
+class WHILEInterpreterTest(LOOPInterpreterTest):
+ def test_while_assignment(self):
+ self.assertEqual(1, interpret('x1:=1;\n WHILE x1!=0 DO\n x0:=1;\n x1:=x1-1 END'))
+ self.assertEqual(3, interpret('x1:=3; WHILE x1/=0 DO x0:=x0+1; x1:=x1-1 END'))
+
+ def test_while_empty_assignment(self):
+ self.assertEqual(0, interpret('WHILE x1 != 0 DO x0:=2; x1:=x1-1 END'))
+ self.assertEqual(0, interpret('WHILE x1 /= 0 DO x0:=1 END'))
+ self.assertEqual(0, interpret('WHILE x1!=0 DO LOOP x2 DO x0:=3 END END'))
+
+ def test_while_nested_assignment(self):
+ self.assertEqual(6, interpret('x1:=2; WHILE x1!=0 DO x2:=3;' +
+ 'WHILE x2/=0 DO x0:=x0+1; x2:=x2-1 END; x1:=x1-1 END'))
+ self.assertEqual(55, interpret('x1:=10; WHILE x1!=0 DO LOOP x1 DO x0:=x0+1 END; x1:=x1-1 END'))
+
+ def test_while_nested_empty_assignment(self):
+ self.assertEqual(0, interpret('WHILE x1!=0 DO x2:=3;' +
+ 'WHILE x2/=0 DO x0:=x0+1; x2:=x2-1 END; x1:=x1-1 END'))
+ self.assertEqual(0, interpret('WHILE x1!=0 DO LOOP x1 DO x0:=x0+1 END; x1:=x1-1 END'))
+
+ def test_while_forbidden_identifier(self):
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x1 DO WHILE x1 != 0 DO x0:=x0+1 END END ')
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x1 DO WHILE x2 != 0 DO x1:=x0+1 END END ')
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x1 DO WHILE x3 != 0 DO x0:=x1+1 END END ')
+ with self.assertRaises(SyntaxError):
+ interpret('x1:=2; LOOP x1 DO WHILE x1 != 0 DO x0:=x0+1 END END ')
+ with self.assertRaises(SyntaxError):
+ interpret('x1:=3; x2:=3; LOOP x1 DO WHILE x2 != 0 DO x1:=x0+1 END END ')
+ with self.assertRaises(SyntaxError):
+ interpret('x1:=2; x2:=4; LOOP x1 DO WHILE x2 != 0 DO x0:=x1+1 END END ')
+ with self.assertRaises(SyntaxError):
+ interpret('x1:=3; LOOP x1 DO WHILE x2 != 0 DO x1:=x0+1 END END ')
+ with self.assertRaises(SyntaxError):
+ interpret('x1:=2; LOOP x1 DO WHILE x2 != 0 DO x0:=x1+1 END END ')
+
+ def test_while_wrong_syntax(self):
+ with self.assertRaises(SyntaxError):
+ interpret('x1:= 1; WHILE x1 /= 1 DO x0:=2 END')
+ with self.assertRaises(SyntaxError):
+ interpret('x1:= 2; WHILE x1 /= 1 DO x0:=2 END')
+ with self.assertRaises(SyntaxError):
+ interpret('x1:= 1; WHILE x1 /= 0 DO WHILE x1 /= 1 DO x0:=2 END END')
+ with self.assertRaises(SyntaxError):
+ interpret('WHILE false DO x2:=2 END')
+ with self.assertRaises(SyntaxError):
+ interpret('WHILE x1>0 DO x1:=x1-1 END')
+ with self.assertRaises(SyntaxError):
+ interpret('WHILE 3>0 DO x0:=2 END')
+
+ def test_syntax_missing_semicolon_while(self):
+ with self.assertRaises(SyntaxError):
+ interpret('x0:=2 WHILE x0!=0 DO x1:=x1+1 END')
+ with self.assertRaises(SyntaxError):
+ interpret('WHILE x0/= DO x1:=x1+1 x1:=x1+1 END')
+ with self.assertRaises(SyntaxError):
+ interpret('WHILE x0/=0 DO x1:=x1+1 x1:=x1+1 END')
+ with self.assertRaises(SyntaxError):
+ interpret('x0:=2; WHILE x0/=0 DO x1:=x1+1 x1:=x1+1 END')
+ with self.assertRaises(SyntaxError):
+ interpret('WHILE x0/=0 DO x1:=x1+1 END x0:=x1+0')
+ with self.assertRaises(SyntaxError):
+ interpret('x0:=2; WHILE x0!=0 DO x1:=x1+1; x0=x0-1 END x0:=x1+0')
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x0 DO WHILE x1!=0 DO x2:=2 x1:=0 END')
+
+ def test_syntax_missing_identifier_while(self):
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x1 DO WHILE /= 0 DO x0:=x0+2 END')
+ with self.assertRaises(SyntaxError):
+ interpret('x1:=2; LOOP x1 DO WHILE /= 0 DO x0:=x0+2 END')
+
+ def test_syntax_missing_unequal_while(self):
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x1 DO WHILE x2 0 DO x0:=x0+2 END END')
+ with self.assertRaises(SyntaxError):
+ interpret('x1:=2; LOOP x1 DO WHILE x2 0 DO x0:=x0+2 END END')
+
+ def test_syntax_missing_zero_while(self):
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x1 DO WHILE x2 != DO x0:=x0+2 END END')
+ with self.assertRaises(SyntaxError):
+ interpret('x1:=2; LOOP x1 DO WHILE x2 /= DO x0:=x0+2 END END')
+
+ def test_syntax_wrong_number_while(self):
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x1 DO WHILE x2 /= 1 DO x0:=x0+2 END END')
+ with self.assertRaises(SyntaxError):
+ interpret('x1:=2; LOOP x1 DO WHILE x2 /= 3 DO x0:=x0+2 END END')
+
+ def test_syntax_missing_do_while(self):
+ with self.assertRaises(SyntaxError):
+ interpret('LOOP x1 DO WHILE x2 != 0 x0:=x0+2 END END')
+ with self.assertRaises(SyntaxError):
+ interpret('x1:=2; LOOP x1 DO WHILE x2 /= 0 x0:=x0+2 END END')
+ with self.assertRaises(SyntaxError):
+ interpret('WHILE x1 != 0 x2:=2 END')
+ with self.assertRaises(SyntaxError):
+ interpret('x1:= 2; WHILE x1 != 0 x2:=2 END')
+
+ def test_syntax_unnecessary_semicolon_while(self):
+ with self.assertRaises(SyntaxError):
+ interpret('x1:=2;; x0:=3')
+ with self.assertRaises(SyntaxError):
+ interpret('WHILE x1 != 0 DO x0:=2;;x2:=1 END')
+ with self.assertRaises(SyntaxError):
+ interpret('x0:=5;')
+
+ @mock.patch('whileinterpreter.input', side_effect=str_yes)
+ def test_infinite_loop(self, custom_input):
+ self.assertEqual(-1, interpret('x1:=100000; WHILE x1 != 0 DO x0:=1; x1:=x1-1 END', timeout=1))
+
+ def test_given_values(self):
+ self.assertEqual(5, interpret('x0:=x1+0; WHILE x2/=0 DO x0:=x0+1; x2:=x2-1 END', [2, 3]))
+ self.assertEqual(4, interpret('x0:=x1+0; WHILE x2/=0 DO x0:=x0+1; x2:=x2-1 END', [0, 4]))
+ self.assertEqual(1, interpret('x0:=x0+1', [2,3,4]))
+
+ def test_wrong_given_values(self):
+ with self.assertRaises(ValueError):
+ interpret('x0:=2', [-1])
+ with self.assertRaises(ValueError):
+ interpret('x0:=1', [0.2])
+ with self.assertRaises(ValueError):
+ interpret('x0:=4', ['EINS'])
+
+ def test_homeomorphic_embedding(self):
+ with self.assertRaises(SyntaxError):
+ interpret('x1:=1; WHILE x1 != 0 DO x1:=x1+1 END', timeout=0, debug=True)
+ with self.assertRaises(SyntaxError):
+ interpret('''x1:=100; WHILE x1 != 0 DO x1:=x1-1 END;
+ x1 = 1; WHILE x1 != 0 DO x1:=x1+1 END''', timeout=0, debug=True)
diff --git a/info4/kapitel-8/Interpreter/whileinterpreter.py b/info4/kapitel-8/Interpreter/whileinterpreter.py
new file mode 100644
index 0000000000000000000000000000000000000000..d5121ca55993c1e397af364fef0342b756d9854b
--- /dev/null
+++ b/info4/kapitel-8/Interpreter/whileinterpreter.py
@@ -0,0 +1,241 @@
+import lexer
+from loopinterpreter import LOOPInterpreter, ErrorHandler
+import re
+import signal
+from IPython.display import clear_output
+from copy import deepcopy
+
+class Timeout:
+ def __init__(self, time):
+ self.time = time
+
+ def __enter__(self):
+ if self.time > 0:
+ signal.signal(signal.SIGALRM, self.interrupt)
+ signal.alarm(self.time)
+
+ def __exit__(self, exc_type, exc_value, exc_traceback):
+ signal.alarm(0)
+
+ def interrupt(self, sig_num, stack_frame):
+ try:
+ clear_output(wait=True)
+ abort = input('''Die Funktion rechnet relativ lange.
+Vielleicht liegt eine Endlosschleife vor.
+Möchten sie abbrechen? [J,n]:''')
+ if abort.upper() in ['J', 'JA', 'Y', 'YES', '']:
+ raise KeyboardInterrupt
+ signal.alarm(self.time)
+ except EOFError:
+ pass
+
+class State:
+ def __init__(self, linenumber, values):
+ self.values = deepcopy(values)
+ self.linenumber = linenumber
+
+ def less_or_equal(self, other):
+ if self.linenumber > other.linenumber:
+ return False
+ values1 = self.values
+ values2 = other.values
+ different_keys = [k for k in values1 if values1.get(k) != values2.get(k)] +\
+ [k for k in values2 if values1.get(k) is None]
+
+ for key in different_keys:
+ if values1.get(key) > 0 and (values2.get(key) is None or values2.get(key) < values1.get(key)):
+ return False
+
+ return True
+
+
+class WHILEInterpreter(LOOPInterpreter):
+ def __init__(self, timeout=60):
+ super().__init__()
+ self.regex_to_token = [(re.compile(r'\d+'), 'NUMBER'),
+ (re.compile(r'x\d+'), 'IDENTIFIER'),
+ (re.compile(r'\+'), 'PLUS'),
+ (re.compile(r'[−-]'), 'MINUS'),
+ (re.compile(r':=|≔'), 'ALLOCATION'),
+ (re.compile(r'/=|≠|!='), 'NOTEQUALS'),
+ (re.compile(r'LOOP'), 'LOOP'),
+ (re.compile(r'WHILE'), 'WHILE'),
+ (re.compile(r'DO'), 'DO'),
+ (re.compile(r'END'), 'END'),
+ (re.compile(r';'), 'SEMICOLON'),
+ (re.compile(r'BREAK'), 'BREAK'),
+ (re.compile(r'\s+', re.MULTILINE), 'WHITESPACE'),
+ (re.compile(r'[^\n]*'), 'UNKNOWN')]
+ self.timeout = timeout
+
+ def process_program(self, forbidden_identifiers, current_token):
+ if current_token is None or current_token.k not in ['IDENTIFIER', 'LOOP', 'WHILE']:
+ self.error_handler.handle_error('Keine passende Anweisung gefunden\n' +
+ 'Erwartet: IDENTIFIER (x0, x1, ...), LOOP oder WHILE')
+ elif current_token.k == 'IDENTIFIER':
+ current_token = self.process_assignment(forbidden_identifiers, current_token)
+ elif current_token.k == 'LOOP':
+ current_token = self.process_loop(forbidden_identifiers, current_token)
+ elif current_token.k == 'WHILE':
+ current_token = self.process_while(forbidden_identifiers, current_token)
+ return current_token
+
+ def verify_program(self, forbidden_identifiers, current_token):
+ if current_token is None or current_token.k not in ['IDENTIFIER', 'LOOP', 'WHILE']:
+ self.error_handler.handle_error('Keine passende Anweisung gefunden\n' +
+ 'Erwartet: IDENTIFIER (x0, x1, ...), LOOP oder WHILE')
+ elif current_token.k == 'IDENTIFIER':
+ current_token = self.verify_assignment(forbidden_identifiers, current_token)
+ elif current_token.k == 'LOOP':
+ current_token = self.verify_loop(forbidden_identifiers, current_token)
+ elif current_token.k == 'WHILE':
+ current_token = self.verify_while(forbidden_identifiers, current_token)
+ return current_token
+
+ def process_while(self, forbidden_identifiers, current_token):
+ identifier_token = self.next_nonempty_token('WHILE', 'IDENTIFIER (x0, x1, ...)')
+ if not identifier_token.k == 'IDENTIFIER':
+ self.error_handler.handle_error('IDENTIFIER in WHILE erwartet.')
+ if identifier_token.v in forbidden_identifiers:
+ self.error_handler.handle_error('Identifier ' + identifier_token.v +
+ ' ist bereits in Loop vorhanden und darf nicht verwendet werden.')
+ if not self.next_nonempty_token('WHILE', 'UNGLEICH').k == 'NOTEQUALS':
+ self.error_handler.handle_error('UNGLEICH in WHILE erwartet.')
+
+ zero_token = self.next_nonempty_token('WHILE', '0')
+ if not zero_token.k == 'NUMBER':
+ self.error_handler.handle_error('0 in WHILE erwartet.')
+ if not int(zero_token.v) == 0:
+ self.error_handler.handle_error('0 in WHILE erwartet.')
+
+ if not self.next_nonempty_token('WHILE', 'DO').k == 'DO':
+ self.error_handler.handle_error('DO in WHILE erwartet.')
+
+ if identifier_token.v in self.values:
+ while_value = int(self.values.get(identifier_token.v))
+ else:
+ while_value = 0
+
+ saved_position = self.lex.current_position
+ saved_line = self.error_handler.line_number
+
+ if while_value == 0:
+ end_found = False
+ while not end_found:
+ token = self.verify_program(forbidden_identifiers, self.next_token())
+ if token is None or token.k not in ['SEMICOLON', 'END']:
+ self.error_handler.handle_error('SEMICOLON oder END in WHILE erwartet.')
+ elif token.k == 'SEMICOLON':
+ continue
+ elif token.k == 'END':
+ end_found = True
+
+ while not while_value == 0:
+ if self.test_homeomorphic_embedding:
+ new_state = State(self.lex.current_position, self.values)
+ for state in self.state_list:
+ if state.less_or_equal(new_state):
+ self.error_handler.handle_endless_loop()
+ self.state_list.append(new_state)
+
+ self.lex.current_position = saved_position
+ self.error_handler.line_number = saved_line
+ end_found = False
+ while not end_found:
+ token = self.process_program(forbidden_identifiers, self.next_token())
+ if token is None or token.k not in ['SEMICOLON', 'END']:
+ self.error_handler.handle_error('SEMICOLON oder END in WHILE erwartet.')
+ elif token.k == 'SEMICOLON':
+ continue
+ elif token.k == 'END':
+ end_found = True
+
+ while_value = int(self.values.get(identifier_token.v))
+
+ return self.next_token()
+
+ def verify_while(self, forbidden_identifiers, current_token):
+ identifier_token = self.next_nonempty_token('WHILE', 'IDENTIFIER')
+ if not identifier_token.k == 'IDENTIFIER':
+ self.error_handler.handle_error('IDENTIFIER in WHILE erwartet.')
+ if identifier_token.v in forbidden_identifiers:
+ self.error_handler.handle_error('Identifier ' + identifier_token.v +
+ ' ist bereits in Loop vorhanden und darf nicht verwendet werden.')
+ if not self.next_nonempty_token('WHILE', 'UNGLEICH').k == 'NOTEQUALS':
+ self.error_handler.handle_error('UNGLEICH in WHILE erwartet.')
+
+ zero_token = self.next_nonempty_token('WHILE', '0')
+ if not zero_token.k == 'NUMBER':
+ self.error_handler.handle_error('0 in WHILE erwartet.')
+ if not int(zero_token.v) == 0:
+ self.error_handler.handle_error('0 in WHILE erwartet.')
+
+ if not self.next_nonempty_token('WHILE', 'DO').k == 'DO':
+ self.error_handler.handle_error('DO in WHILE erwartet.')
+
+ end_found = False
+ while not end_found:
+ token = self.verify_program(forbidden_identifiers, self.next_token())
+ if token is None or token.k not in ['SEMICOLON', 'END']:
+ self.error_handler.handle_error('SEMICOLON oder END in WHILE erwartet.')
+ elif token.k == 'SEMICOLON':
+ continue
+ elif token.k == 'END':
+ end_found = True
+
+ return self.next_token()
+
+ def interpret(self, program, values=None, test_homeomorphic_embedding=False):
+ try:
+ with Timeout(self.timeout):
+ self.lex = lexer.Lexer(self.regex_to_token, program)
+ self.error_handler = ErrorHandler(program, self)
+ self.test_homeomorphic_embedding = test_homeomorphic_embedding
+ self.state_list = []
+ self.values = values
+ if values is None:
+ self.values = {}
+ forbidden_identifiers = []
+ current_token = self.next_token()
+ while current_token is not None:
+ current_token = self.process_program(forbidden_identifiers, current_token)
+ if current_token is not None:
+ if not current_token.k == 'SEMICOLON':
+ self.error_handler.handle_error('Semicolon erwartet')
+ current_token = self.next_token()
+ if current_token is None:
+ self.error_handler.handle_error('Semikolons werden nur zur Trennung und nicht zum ' +
+ 'Abschluss von Programmen verwendet')
+ if 'x0' in self.values:
+ return self.values.get('x0')
+ return 0
+ except KeyboardInterrupt:
+ print('Die Ausführung des Programms wurde unterbrochen.\n' +
+ 'Daher ist der Rückgabewert des Programms nicht definiert.')
+ return -1
+
+
+def interpret(program, value_list=None, timeout=60, debug=False):
+ """Funktion zum Ausführen eines WHILE-Programms.
+ :param program: WHILE-Programm als String 'x1:=10; x2:=8; x0:=x2+0; WHILE x1 /=0 DO x0:=x0+1; x1:=x1-1 END'
+ :param value_list: Array von Integern ([v1, ..., vn]).
+ Das WHILE-Programm startet mit diesen Werten in x1, ..., xn.
+ :param timeout: Zeit nach der die Ausführung eines Programms pausiert wird.
+ Gibt eine Möglichkeit zum Abbrechen bei einer Endlosschleife.
+ Ein Wert von 0 deaktiviert den Timeout.
+ :param debug: Boolean der angibt ob mit Homeomorphic Embedding auf mögliche Endlosschleifen geprüft werden soll.
+ :returns integer: Gibt bei Abbruch -1 und sonst den Wert von x0 nach dem WHILE-Programm zurück.
+ :usage interpret('x1:=10; x2:=8; x0:=x2+0; WHILE x1 /=0 DO x0:=x0+1; x1:=x1-1 END')"""
+ interpreter = WHILEInterpreter(timeout)
+ values = None
+ if value_list is not None:
+ values = {}
+ for index, value in enumerate(value_list):
+ if not isinstance(value, int) or value < 0 or not int(value) == value:
+ raise ValueError("Variablen können nur natürliche Zahlen zugewiesen bekommen.")
+ values.update({'x' + str(index+1): value})
+ return interpreter.interpret(program, values, debug)
+
+
+if __name__ == '__main__':
+ help(interpret)
diff --git a/info4/kapitel-8/LOOP-Programme.ipynb b/info4/kapitel-8/LOOP-Programme.ipynb
new file mode 100644
index 0000000000000000000000000000000000000000..3b282796b952439a3c33ecfae195789d4211bf8c
--- /dev/null
+++ b/info4/kapitel-8/LOOP-Programme.ipynb
@@ -0,0 +1,159 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# LOOP-Programme\n",
+ "\n",
+ "Im Folgenden werden wir Konzepte von prozeduralen Programmiersprachen kennenlernen. Als erste einfache Programmiersprache betrachten wir LOOP.\n",
+ "Die Syntax solcher Programme ist definiert durch:\n",
+ "* $x_i := c$, $x_i := x_j + c$ und $x_i := x_j - c$ mit $i,j,c \\in \\mathbb{N}$ sind LOOP-Programme\n",
+ "* Falls $P_1$ und $P_2$ LOOP-Programme sind ist $P_1; P_2$ ein LOOP-Programm\n",
+ "* Falls $P$ ein LOOP-Programm ist und $x_i$ nicht in $P$ vorkommt, so ist auch $LOOP\\ x_i\\ DO\\ P\\ END$ ein LOOP-Programm."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Die Semantik von LOOP-Programmen lässt sich wie folgt verstehen:\n",
+ "* Wenn eine k-stellige Funktion $f(n_1,n_2,...,n_k)$ berechnet wird sind die Startwerte $n_1,...,n_k\\in\\mathbb{N}$ in den Variablen $x_1,...,x_k$.\n",
+ "* Die restlichen Variablen starten mit dem Wert 0.\n",
+ "* Zuweisungen $x_i := c$, $x_i := x_j + c$ werden wie üblich interpretiert. Bei $x_i := x_j - c$ wird $x_i$ auf 0 gesetzt, falls $c\\geq x_j$ ist.\n",
+ "* Bei $P_1; P_2$ wird erst $P_1$ und dann $P_2$ ausgeführt.\n",
+ "* Das Programm $P$ in $LOOP\\ x_i\\ DO\\ P\\ END$ wird so oft ausgeführt, wie der Wert von $x_i$ zu Beginn des LOOPs ist.\n",
+ "* Das Programm stoppt mit dem Wert $f(n_1,n_2,...,n_k)$ in der Variablen $x_0$."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Eine Funktion heißt LOOP-berechenbar, falls es ein LOOP-Programm gibt, das mit $n_1,...,n_k$ in den Variablen $x_1,...,x_k$ gestartet wird und mit $f(n_1,n_2,...,n_k)$ in der Variablen $x_0$ endet.\n",
+ "Ein Beispiel dafür ist die Addition $f(n_1, n_2) = n_1 + n_2$:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {
+ "scrolled": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Help on function interpret in module __main__:\n",
+ "\n",
+ "interpret(program, value_list=None)\n",
+ " Funktion zum Ausführen eines LOOP-Programms.\n",
+ " :param program: LOOP-Programm als String 'x1 := 4; LOOP x1 DO x0:=x0+1 END'\n",
+ " :param value_list: Array von Integern ([v1, ..., vn]).\n",
+ " Das LOOP-Programm startet mit diesen Werten in x1, ..., xn.\n",
+ " :returns integer: Gibt bei Abbruch -1 und sonst den Wert von x0 nach dem LOOP-Programm zurück.\n",
+ " :usage interpret('x1:=10; x2:=8; x0:=x2+0; LOOP x1 DO x0:=x0+1 END')\n",
+ "\n"
+ ]
+ }
+ ],
+ "source": [
+ "%run Interpreter/loopinterpreter.py"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "9"
+ ]
+ },
+ "execution_count": 2,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "interpret('''\n",
+ "LOOP x1 DO\n",
+ " x2≔ x2 + 1\n",
+ "END;\n",
+ "x0≔ x2 + 0''', [3,6])"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Für die Visualisierung und das Verständnis haben wir die Syntax um das Keyword BREAK ereitert.\n",
+ "Wenn dieses erreicht wird, wird der Zustand ausgegeben und man wird gefragt, ob fortgefahren oder abgebrochen werden soll.\n",
+ "Dabei werden Variablen, denen noch kein Wert zugewiesen wurde ignoriert.\n",
+ "Im folgenden Fall wird $x_0$ nicht verändet, da $x_2$ nie zugewiesen wurde und die Schleife somit nicht ausgeführt wird."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "BREAK in Zeile 5\n",
+ "Aktueller Zustand:\n",
+ "Variable x1: 5\n",
+ "Variable x0: 2\n",
+ "Drücke ENTER zum Fotfahren oder schreibe EXIT zum Beenden:\n"
+ ]
+ },
+ {
+ "data": {
+ "text/plain": [
+ "2"
+ ]
+ },
+ "execution_count": 3,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "interpret('''\n",
+ "x1:=5;\n",
+ "x0:=2;\n",
+ "LOOP x2 DO\n",
+ " x0:=1\n",
+ " BREAK\n",
+ "END\n",
+ "''')"
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.8.6"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/info4/kapitel-8/WHILE-Programme.ipynb b/info4/kapitel-8/WHILE-Programme.ipynb
new file mode 100644
index 0000000000000000000000000000000000000000..ead34dc0e28e00b39260124f176f9f1023cb5628
--- /dev/null
+++ b/info4/kapitel-8/WHILE-Programme.ipynb
@@ -0,0 +1,149 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# WHILE-Programme\n",
+ "\n",
+ "Um LOOP-Programme zu erweitern führen wir eine WHILE-Schleife ein und erhalten damit die WHILE-Programme.\n",
+ "Die Syntax solcher WHILE-Programme ist definiert durch:\n",
+ "* Jedes LOOP-Programm ist ein WHILE-Programm\n",
+ "* Falls $P_1$ und $P_2$ WHILE-Programme sind ist $P_1; P_2$ ein WHILE-Programm\n",
+ "* Falls $P$ ein WHILE-Programm ist, so ist auch $WHILE\\ x_i\\neq 0\\ DO\\ P\\ END$ ein WHILE-Programm."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Die Semantik von WHILE-Programmen ist analog zu der von LOOP-Programmen zu verstehen.\n",
+ "Der einzige Unterschied sind die WHILE-Schleifen.\n",
+ "Das Programm P in einer WHILE-Schleife wird solange wiederholt, wie $x_i \\neq 0$ ist.\n",
+ "Dabei ist zu beachten, dass $x_i$ in P vorkommen sollte, da es sonst nicht verändert wird.\n",
+ "Dies ist ein Unterschied zu LOOP-Programmen, da $x_i$ in dem entsprechenden LOOP nicht verwendet werden darf."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Eine Funktion heißt WHILE-berechenbar, falls es ein WHILE-Programm gibt, das mit $n_1,...,n_k$ in den Variablen $x_1,...,x_k$ gestartet wird und mit $f(n_1,n_2,...,n_k)$ in der Variablen $x_0$ endet, falls $f(n_1,n_2,...,n_k)$ definiert ist. Andernfalls stoppt das WHILE-Programm nicht.\n",
+ "Ein Beispiel dafür ist die 2er-Potenz $f(n) = 2^n$:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Help on function interpret in module __main__:\n",
+ "\n",
+ "interpret(program, value_list=None, timeout=60)\n",
+ " Funktion zum Ausführen eines WHILE-Programms.\n",
+ " :param program: WHILE-Programm als String 'x1:=10; x2:=8; x0:=x2+0; WHILE x1 /=0 DO x0:=x0+1; x1:=x1-1 END'\n",
+ " :param value_list: Array von Integern ([v1, ..., vn]).\n",
+ " Das WHILE-Programm startet mit diesen Werten in x1, ..., xn.\n",
+ " :param timeout: Zeit nach der die Ausführung eines Programms pausiert wird.\n",
+ " Gibt eine Möglichkeit zum Abbrechen bei einer Endlosschleife.\n",
+ " Ein Wert von 0 deaktiviert den Timeout.\n",
+ " :returns integer: Gibt bei Abbruch -1 und sonst den Wert von x0 nach dem WHILE-Programm zurück.\n",
+ " :usage interpret('x1:=10; x2:=8; x0:=x2+0; WHILE x1 /=0 DO x0:=x0+1; x1:=x1-1 END')\n",
+ "\n"
+ ]
+ }
+ ],
+ "source": [
+ "%run Interpreter/whileinterpreter.py"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "1024"
+ ]
+ },
+ "execution_count": 2,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "interpret(''' \n",
+ "x0:=1;\n",
+ "WHILE x1 /= 0 DO\n",
+ " x2:=x0+0;\n",
+ " LOOP x2 DO\n",
+ " x0:=x0+1\n",
+ " END;\n",
+ " x1:=x1-1\n",
+ "END''', [10])"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Ein weiteres Beispiel ist die Funktion f(n) = n mod 2"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "0"
+ ]
+ },
+ "execution_count": 3,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "interpret('''\n",
+ "x1:=16;\n",
+ "x2:=x1-1;\n",
+ "WHILE x2!=0 DO\n",
+ " x1:=x1-2;\n",
+ " x2:=x2-2\n",
+ "END;\n",
+ "x0:=x1+0\n",
+ "''')"
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.8.6"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}