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,
"metadata": {},
"output_type": "execute_result"
}
@@ -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": {},
"output_type": "execute_result"
}
@@ -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": {},
"output_type": "execute_result"
}
@@ -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": {},
"output_type": "execute_result"
}
@@ -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": {},
"output_type": "execute_result"
}
@@ -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": {},
"output_type": "execute_result"
}
@@ -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": {},
"output_type": "execute_result"
}
@@ -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": {},
"output_type": "execute_result"
}
@@ -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 @@
"{[c,c,c,c,c,c],[c,c,a,a,b,c,c,c],[a,a,b,c,c,c,c,c],[c,c,c,c,a,a,b,c],[a,a,b,c,a,a,b,c,c,c],[c,c,a,a,b,c,a,a,b,c],[a,a,b,c,c,c,a,a,b,c],[a,a,b,c,a,a,b,c,a,a,b,c]}"
]
},
- "execution_count": 118,
+ "execution_count": 36,
"metadata": {},
"output_type": "execute_result"
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@@ -975,7 +975,7 @@
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{
"cell_type": "code",
- "execution_count": 119,
+ "execution_count": 37,
"metadata": {},
"outputs": [
{
@@ -987,7 +987,7 @@
"{[c,c,c,c,c,c,c,c],[c,c,c,c,a,a,b,c,c,c],[c,c,a,a,b,c,c,c,c,c],[a,a,b,c,c,c,c,c,c,c],[c,c,c,c,c,c,a,a,b,c],[a,a,b,c,a,a,b,c,c,c,c,c],[c,c,c,c,a,a,b,c,a,a,b,c],[c,c,a,a,b,c,c,c,a,a,b,c],[c,c,a,a,b,c,a,a,b,c,c,c],[a,a,b,c,c,c,c,c,a,a,b,c],[a,a,b,c,c,c,a,a,b,c,c,c],[c,c,a,a,b,c,a,a,b,c,a,a,b,c],[a,a,b,c,c,c,a,a,b,c,a,a,b,c],[a,a,b,c,a,a,b,c,c,c,a,a,b,c],[a,a,b,c,a,a,b,c,a,a,b,c,c,c],[a,a,b,c,a,a,b,c,a,a,b,c,a,a,b,c]}"
]
},
- "execution_count": 119,
+ "execution_count": 37,
"metadata": {},
"output_type": "execute_result"
}
@@ -998,7 +998,7 @@
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{
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+ "execution_count": 38,
"metadata": {},
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@@ -1007,7 +1007,7 @@
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{
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+ "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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{
"cell_type": "code",
- "execution_count": 132,
+ "execution_count": 41,
"metadata": {},
"outputs": [
{
@@ -1044,7 +1044,7 @@
"[c,b,a,a]"
]
},
- "execution_count": 132,
+ "execution_count": 41,
"metadata": {},
"output_type": "execute_result"
}
@@ -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 @@
},
{
"cell_type": "code",
- "execution_count": 138,
+ "execution_count": 47,
"metadata": {},
"outputs": [
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@@ -1206,7 +1206,7 @@
"TRUE"
]
},
- "execution_count": 138,
+ "execution_count": 47,
"metadata": {},
"output_type": "execute_result"
}
@@ -1224,7 +1224,7 @@
},
{
"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": {},
"output_type": "execute_result"
}
@@ -1247,7 +1247,7 @@
},
{
"cell_type": "code",
- "execution_count": 140,
+ "execution_count": 49,
"metadata": {},
"outputs": [
{
@@ -1259,7 +1259,7 @@
"{[],[a],[a,a],[a,b],[b],[b,a],[c],[c,b],[a,a,b],[b,a,a],[b,a,a,b],[c,b,a],[c,b,a,a],[c,b,a,a,b]}"
]
},
- "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 @@
},
{
"cell_type": "code",
- "execution_count": 143,
+ "execution_count": 52,
"metadata": {},
"outputs": [
{
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"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
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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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"<!-- Title: state Pages: 1 -->\n",
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]
},
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+ "execution_count": 52,
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}
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},
{
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- "execution_count": 144,
+ "execution_count": 53,
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"<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 @@
},
{
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- "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": {