From 8af112125d941f007320168ff6de5b3e2fbd2604 Mon Sep 17 00:00:00 2001
From: Michael Leuschel <leuschel@uni-duesseldorf.de>
Date: Tue, 25 Oct 2022 18:06:53 +0200
Subject: [PATCH] add pdf

Signed-off-by: Michael Leuschel <leuschel@uni-duesseldorf.de>
---
 logic_programming/3_PropositionalLogic.ipynb | 1048 ++++++++++++++++--
 logic_programming/3_PropositionalLogic.pdf   |  Bin 0 -> 145887 bytes
 2 files changed, 977 insertions(+), 71 deletions(-)
 create mode 100644 logic_programming/3_PropositionalLogic.pdf

diff --git a/logic_programming/3_PropositionalLogic.ipynb b/logic_programming/3_PropositionalLogic.ipynb
index f8c5253..86be5b9 100644
--- a/logic_programming/3_PropositionalLogic.ipynb
+++ b/logic_programming/3_PropositionalLogic.ipynb
@@ -6,7 +6,7 @@
    "metadata": {},
    "source": [
     "# Propositional Logic\n",
-    "Syntax and Semantics Explained using a Prolog Implementation"
+    "## Syntax and Semantics Explained using a Prolog Implementation"
    ]
   },
   {
@@ -44,7 +44,7 @@
   },
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-   "execution_count": 123,
+   "execution_count": 1,
    "id": "964ebd52",
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@@ -65,7 +65,7 @@
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@@ -89,7 +89,7 @@
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@@ -113,7 +113,7 @@
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+   "execution_count": 4,
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@@ -168,7 +168,7 @@
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@@ -191,7 +191,7 @@
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@@ -215,7 +215,7 @@
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@@ -239,7 +239,7 @@
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@@ -273,7 +273,7 @@
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@@ -297,7 +297,7 @@
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@@ -347,7 +347,7 @@
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+   "execution_count": 11,
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@@ -357,7 +357,85 @@
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+       "</g>\n",
+       "<!-- or(q,not(p))&#45;&gt;q -->\n",
+       "<g id=\"edge4\" class=\"edge\">\n",
+       "<title>or(q,not(p))&#45;&gt;q</title>\n",
+       "<path fill=\"none\" stroke=\"black\" d=\"M68,-173.8C68,-162.16 68,-146.55 68,-133.24\"/>\n",
+       "<polygon fill=\"black\" stroke=\"black\" points=\"71.5,-133.18 68,-123.18 64.5,-133.18 71.5,-133.18\"/>\n",
+       "<text text-anchor=\"middle\" x=\"71.5\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">1</text>\n",
+       "</g>\n",
+       "<!-- or(q,not(p))&#45;&gt;not(p) -->\n",
+       "<g id=\"edge5\" class=\"edge\">\n",
+       "<title>or(q,not(p))&#45;&gt;not(p)</title>\n",
+       "<path fill=\"none\" stroke=\"black\" d=\"M82.98,-173.8C93.71,-161.47 108.33,-144.68 120.33,-130.89\"/>\n",
+       "<polygon fill=\"black\" stroke=\"black\" points=\"123.12,-133.02 127.05,-123.18 117.84,-128.42 123.12,-133.02\"/>\n",
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+       "</g>\n",
+       "</g>\n",
+       "</svg>\n"
+      ],
       "text/plain": [
        "digraph {\n",
        "\"p\" [shape=\"egg\", label=\"p\"]\n",
@@ -407,7 +485,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 134,
+   "execution_count": 12,
    "id": "350b9756",
    "metadata": {
     "vscode": {
@@ -447,7 +525,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 135,
+   "execution_count": 13,
    "id": "3101448b",
    "metadata": {
     "vscode": {
@@ -471,7 +549,56 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 136,
+   "execution_count": 14,
+   "id": "79d16b97",
+   "metadata": {
+    "vscode": {
+     "languageId": "prolog"
+    }
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\u001b[1mR = [a,b],\n",
+       "R2 = [a,b,c]"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "?- append([a],[b],R), append(R,[c],R2)."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 15,
+   "id": "d989abf2",
+   "metadata": {
+    "vscode": {
+     "languageId": "prolog"
+    }
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\u001b[1mX = true"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "?- and(X,true,true)."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 16,
    "id": "922004e9",
    "metadata": {
     "vscode": {
@@ -503,7 +630,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 137,
+   "execution_count": 17,
    "id": "54b57ff8",
    "metadata": {
     "vscode": {
@@ -562,7 +689,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 138,
+   "execution_count": 18,
    "id": "8f6d804b",
    "metadata": {
     "vscode": {
@@ -574,6 +701,31 @@
      "data": {
       "text/plain": [
        "\u001b[1mInterpetation = [p/true,q/false],\n",
+       "P = true"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "?- Interpetation=[p/true, q/false],  member(p/P,Interpetation)."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 19,
+   "id": "da495155",
+   "metadata": {
+    "vscode": {
+     "languageId": "prolog"
+    }
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\u001b[1mInterpetation = [bind(p,true),bind(q,false)],\n",
        "Q = false"
       ]
      },
@@ -582,7 +734,7 @@
     }
    ],
    "source": [
-    "?- Interpetation=[p/true, q/false],  member(q/Q,Interpetation)."
+    "?- Interpetation=[bind(p,true), bind(q,false)],  member(bind(q,Q),Interpetation)."
    ]
   },
   {
@@ -597,7 +749,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 139,
+   "execution_count": 20,
    "id": "175c00d0",
    "metadata": {
     "vscode": {
@@ -609,7 +761,8 @@
     "value(X,Interpretation,Value) :- \n",
     "   atomic(X), % we have a proposition\n",
     "   member(X/Value,Interpretation).\n",
-    "value(and(A,B),I,Val) :- value(A,I,VA), value(B,I,VB),\n",
+    "value(and(A,B),I,Val) :-\n",
+    "    value(A,I,VA), value(B,I,VB),\n",
     "   and(VA,VB,Val).\n",
     "value(or(A,B),I,Val) :- value(A,I,VA), value(B,I,VB),\n",
     "   or(VA,VB,Val).\n",
@@ -630,7 +783,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 140,
+   "execution_count": 21,
    "id": "eee17774",
    "metadata": {
     "vscode": {
@@ -662,7 +815,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 141,
+   "execution_count": 22,
    "id": "575fc20f",
    "metadata": {
     "vscode": {
@@ -696,7 +849,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 142,
+   "execution_count": 23,
    "id": "fec75f65",
    "metadata": {
     "vscode": {
@@ -729,7 +882,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 143,
+   "execution_count": 24,
    "id": "78473da0",
    "metadata": {
     "vscode": {
@@ -784,7 +937,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 144,
+   "execution_count": 25,
    "id": "2ee60c15",
    "metadata": {
     "vscode": {
@@ -809,7 +962,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 145,
+   "execution_count": 26,
    "id": "2b497dad",
    "metadata": {
     "vscode": {
@@ -819,7 +972,85 @@
    "outputs": [
     {
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+       "<polygon fill=\"olive\" stroke=\"black\" points=\"70.68,-87.05 72.46,-87.15 74.22,-87.3 75.95,-87.49 77.65,-87.74 79.31,-88.03 80.92,-88.36 82.48,-88.75 83.99,-89.18 85.43,-89.65 86.8,-90.16 88.1,-90.71 89.32,-91.31 90.45,-91.94 91.51,-92.61 92.47,-93.31 93.35,-94.04 94.13,-94.8 94.81,-95.59 95.41,-96.41 95.91,-97.25 96.31,-98.11 96.62,-98.99 96.83,-99.89 96.96,-100.8 96.99,-101.72 96.93,-102.65 96.79,-103.59 96.57,-104.53 96.27,-105.47 95.89,-106.41 95.44,-107.35 94.92,-108.28 94.33,-109.2 93.69,-110.11 92.98,-111.01 92.22,-111.89 91.41,-112.75 90.56,-113.59 89.67,-114.41 88.73,-115.2 87.76,-115.96 86.76,-116.69 85.74,-117.39 84.68,-118.06 83.61,-118.69 82.52,-119.29 81.41,-119.84 80.28,-120.35 79.14,-120.82 77.99,-121.25 76.84,-121.64 75.67,-121.97 74.5,-122.26 73.33,-122.51 72.15,-122.7 70.96,-122.85 69.78,-122.95 68.59,-123 67.41,-123 66.22,-122.95 65.04,-122.85 63.85,-122.7 62.67,-122.51 61.5,-122.26 60.33,-121.97 59.16,-121.64 58.01,-121.25 56.86,-120.82 55.72,-120.35 54.59,-119.84 53.48,-119.29 52.39,-118.69 51.32,-118.06 50.26,-117.39 49.24,-116.69 48.24,-115.96 47.27,-115.2 46.33,-114.41 45.44,-113.59 44.59,-112.75 43.78,-111.89 43.02,-111.01 42.31,-110.11 41.67,-109.2 41.08,-108.28 40.56,-107.35 40.11,-106.41 39.73,-105.47 39.43,-104.53 39.21,-103.59 39.07,-102.65 39.01,-101.72 39.04,-100.8 39.17,-99.89 39.38,-98.99 39.69,-98.11 40.09,-97.25 40.59,-96.41 41.19,-95.59 41.87,-94.8 42.65,-94.04 43.53,-93.31 44.49,-92.61 45.55,-91.94 46.68,-91.31 47.9,-90.71 49.2,-90.16 50.57,-89.65 52.01,-89.18 53.52,-88.75 55.08,-88.36 56.69,-88.03 58.35,-87.74 60.05,-87.49 61.78,-87.3 63.54,-87.15 65.32,-87.05 67.1,-87 68.9,-87 70.68,-87.05\"/>\n",
+       "<text text-anchor=\"middle\" x=\"68\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">q</text>\n",
+       "</g>\n",
+       "<!-- not(p) -->\n",
+       "<g id=\"node3\" class=\"node\">\n",
+       "<title>not(p)</title>\n",
+       "<polygon fill=\"#ff8247\" stroke=\"black\" points=\"169,-123 115,-123 115,-87 169,-87 169,-123\"/>\n",
+       "<text text-anchor=\"middle\" x=\"142\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">¬</text>\n",
+       "</g>\n",
+       "<!-- not(p)&#45;&gt;p -->\n",
+       "<g id=\"edge1\" class=\"edge\">\n",
+       "<title>not(p)&#45;&gt;p</title>\n",
+       "<path fill=\"none\" stroke=\"black\" d=\"M123.79,-86.8C109.35,-73.17 89.14,-54.07 73.84,-39.62\"/>\n",
+       "<polygon fill=\"black\" stroke=\"black\" points=\"76.01,-36.86 66.34,-32.54 71.21,-41.95 76.01,-36.86\"/>\n",
+       "<text text-anchor=\"middle\" x=\"105.5\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">1</text>\n",
+       "</g>\n",
+       "<!-- and(p,or(q,not(p))) -->\n",
+       "<g id=\"node4\" class=\"node\">\n",
+       "<title>and(p,or(q,not(p)))</title>\n",
+       "<polygon fill=\"olive\" stroke=\"black\" points=\"54,-297 0,-297 0,-261 54,-261 54,-297\"/>\n",
+       "<text text-anchor=\"middle\" x=\"27\" y=\"-275.3\" font-family=\"Times,serif\" font-size=\"14.00\">∧</text>\n",
+       "</g>\n",
+       "<!-- and(p,or(q,not(p)))&#45;&gt;p -->\n",
+       "<g id=\"edge2\" class=\"edge\">\n",
+       "<title>and(p,or(q,not(p)))&#45;&gt;p</title>\n",
+       "<path fill=\"none\" stroke=\"black\" d=\"M25.59,-260.99C23.18,-227.43 19.48,-150.47 30,-87 32.35,-72.85 37.06,-57.61 41.53,-45.15\"/>\n",
+       "<polygon fill=\"black\" stroke=\"black\" points=\"44.94,-46.02 45.16,-35.42 38.38,-43.57 44.94,-46.02\"/>\n",
+       "<text text-anchor=\"middle\" x=\"28.5\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">1</text>\n",
+       "</g>\n",
+       "<!-- or(q,not(p)) -->\n",
+       "<g id=\"node5\" class=\"node\">\n",
+       "<title>or(q,not(p))</title>\n",
+       "<polygon fill=\"olive\" stroke=\"black\" points=\"95,-210 41,-210 41,-174 95,-174 95,-210\"/>\n",
+       "<text text-anchor=\"middle\" x=\"68\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">∨</text>\n",
+       "</g>\n",
+       "<!-- and(p,or(q,not(p)))&#45;&gt;or(q,not(p)) -->\n",
+       "<g id=\"edge3\" class=\"edge\">\n",
+       "<title>and(p,or(q,not(p)))&#45;&gt;or(q,not(p))</title>\n",
+       "<path fill=\"none\" stroke=\"black\" d=\"M35.3,-260.8C41.02,-248.93 48.74,-232.93 55.24,-219.45\"/>\n",
+       "<polygon fill=\"black\" stroke=\"black\" points=\"58.52,-220.7 59.72,-210.18 52.22,-217.66 58.52,-220.7\"/>\n",
+       "<text text-anchor=\"middle\" x=\"52.5\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">2</text>\n",
+       "</g>\n",
+       "<!-- or(q,not(p))&#45;&gt;q -->\n",
+       "<g id=\"edge4\" class=\"edge\">\n",
+       "<title>or(q,not(p))&#45;&gt;q</title>\n",
+       "<path fill=\"none\" stroke=\"black\" d=\"M68,-173.8C68,-162.16 68,-146.55 68,-133.24\"/>\n",
+       "<polygon fill=\"black\" stroke=\"black\" points=\"71.5,-133.18 68,-123.18 64.5,-133.18 71.5,-133.18\"/>\n",
+       "<text text-anchor=\"middle\" x=\"71.5\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">1</text>\n",
+       "</g>\n",
+       "<!-- or(q,not(p))&#45;&gt;not(p) -->\n",
+       "<g id=\"edge5\" class=\"edge\">\n",
+       "<title>or(q,not(p))&#45;&gt;not(p)</title>\n",
+       "<path fill=\"none\" stroke=\"black\" d=\"M82.98,-173.8C93.71,-161.47 108.33,-144.68 120.33,-130.89\"/>\n",
+       "<polygon fill=\"black\" stroke=\"black\" points=\"123.12,-133.02 127.05,-123.18 117.84,-128.42 123.12,-133.02\"/>\n",
+       "<text text-anchor=\"middle\" x=\"112.5\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">2</text>\n",
+       "</g>\n",
+       "</g>\n",
+       "</svg>\n"
+      ],
       "text/plain": [
        "digraph {\n",
        "\"p\" [shape=\"egg\", label=\"p\", style=\"filled\", fillcolor=\"olive\"]\n",
@@ -907,7 +1216,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 147,
+   "execution_count": 28,
    "id": "23775624",
    "metadata": {
     "vscode": {
@@ -949,7 +1258,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 148,
+   "execution_count": 29,
    "id": "7a6e29ed",
    "metadata": {
     "vscode": {
@@ -1003,7 +1312,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 149,
+   "execution_count": 30,
    "id": "6bd292f4",
    "metadata": {
     "vscode": {
@@ -1060,7 +1369,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 150,
+   "execution_count": 31,
    "id": "ab15628a",
    "metadata": {
     "vscode": {
@@ -1108,7 +1417,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 151,
+   "execution_count": 32,
    "id": "68a2a19c",
    "metadata": {
     "vscode": {
@@ -1119,7 +1428,7 @@
     {
      "data": {
       "text/markdown": [
-       "P | Q | NotPandQ | PimpliesQ | \n",
+       "P | Q | NotPorQ | PimpliesQ | \n",
        ":- | :- | :- | :- | \n",
        "true | true | true | true | \n",
        "true | false | false | false | \n",
@@ -1127,7 +1436,7 @@
        "false | false | true | true | "
       ],
       "text/plain": [
-       "P | Q | NotPandQ | PimpliesQ | \n",
+       "P | Q | NotPorQ | PimpliesQ | \n",
        ":- | :- | :- | :- | \n",
        "true | true | true | true | \n",
        "true | false | false | false | \n",
@@ -1149,7 +1458,53 @@
     }
    ],
    "source": [
-    "jupyter:print_table((user:value(or(not(p),q), [p/P, q/Q],NotPandQ),user:value(implies(p,q), [p/P, q/Q],PimpliesQ)))"
+    "jupyter:print_table((user:value(or(not(p),q), [p/P, q/Q],NotPorQ),user:value(implies(p,q), [p/P, q/Q],PimpliesQ)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 33,
+   "id": "41f43345",
+   "metadata": {
+    "vscode": {
+     "languageId": "prolog"
+    }
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/markdown": [
+       "P | Q | PQ | QP | \n",
+       ":- | :- | :- | :- | \n",
+       "true | true | true | true | \n",
+       "true | false | false | false | \n",
+       "false | true | false | false | \n",
+       "false | false | false | false | "
+      ],
+      "text/plain": [
+       "P | Q | PQ | QP | \n",
+       ":- | :- | :- | :- | \n",
+       "true | true | true | true | \n",
+       "true | false | false | false | \n",
+       "false | true | false | false | \n",
+       "false | false | false | false | "
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/plain": [
+       "\u001b[1mtrue"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "jupyter:print_table((user:value(and(p,q), [p/P, q/Q],PQ),user:value(and(q,p), [p/P, q/Q],QP)))"
    ]
   },
   {
@@ -1166,7 +1521,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 152,
+   "execution_count": 34,
    "id": "aaf46953",
    "metadata": {
     "vscode": {
@@ -1223,7 +1578,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 153,
+   "execution_count": 35,
    "id": "da0a82b5",
    "metadata": {
     "vscode": {
@@ -1245,7 +1600,55 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 154,
+   "execution_count": 36,
+   "id": "a216799f",
+   "metadata": {
+    "vscode": {
+     "languageId": "prolog"
+    }
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\u001b[1mAP = p"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "?- ap(and(p,q),AP)."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 37,
+   "id": "8a38f7cc",
+   "metadata": {
+    "vscode": {
+     "languageId": "prolog"
+    }
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\u001b[1mAP = q"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "jupyter:retry"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 38,
    "id": "82980216",
    "metadata": {
     "vscode": {
@@ -1277,7 +1680,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 155,
+   "execution_count": 39,
    "id": "23cfb806",
    "metadata": {
     "vscode": {
@@ -1297,9 +1700,49 @@
     "unsat(Formula) :- \\+ sat(Formula,_). % a formula is unsatisfiable if we can find no model, \\+ is Prolog negation"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "id": "c98b7052",
+   "metadata": {},
+   "source": [
+    "The skel predicate simply takes a list of propositions and generates a skeleton interpretation with fresh logical variables for the truth values:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 40,
+   "id": "da991b8d",
+   "metadata": {
+    "vscode": {
+     "languageId": "prolog"
+    }
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\u001b[1mR = [a/_55832,b/_55844,c/_55856]"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "?- skel([a,b,c],R)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9444e54a",
+   "metadata": {},
+   "source": [
+    "We can now use `sat/2` to compute models, without having to provide ourselves an interpretation skeleton:"
+   ]
+  },
   {
    "cell_type": "code",
-   "execution_count": 156,
+   "execution_count": 41,
    "id": "6338f6bf",
    "metadata": {
     "vscode": {
@@ -1323,7 +1766,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 157,
+   "execution_count": 42,
    "id": "266a4bd8",
    "metadata": {
     "vscode": {
@@ -1359,7 +1802,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 158,
+   "execution_count": 43,
    "id": "6957d9f3",
    "metadata": {
     "vscode": {
@@ -1376,7 +1819,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 159,
+   "execution_count": 44,
    "id": "f552a71e",
    "metadata": {
     "vscode": {
@@ -1400,7 +1843,31 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 160,
+   "execution_count": null,
+   "id": "76b8ab5a",
+   "metadata": {
+    "vscode": {
+     "languageId": "prolog"
+    }
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\u001b[1;31mfalse"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "?- prove(or(p,q),and(p,q))."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
    "id": "f8aca8b7",
    "metadata": {
     "vscode": {
@@ -1424,7 +1891,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 161,
+   "execution_count": 49,
    "id": "e7a2828d",
    "metadata": {
     "vscode": {
@@ -1435,7 +1902,7 @@
     {
      "data": {
       "text/plain": [
-       "\u001b[1;31mfalse"
+       "\u001b[1mtrue"
       ]
      },
      "metadata": {},
@@ -1443,7 +1910,7 @@
     }
    ],
    "source": [
-    "?- prove(or(p,q),and(p,q))."
+    "?- prove(and(p,not(p)),and(q,not(q)))."
    ]
   },
   {
@@ -1456,7 +1923,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 162,
+   "execution_count": 50,
    "id": "26bab55b",
    "metadata": {
     "vscode": {
@@ -1471,7 +1938,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 163,
+   "execution_count": 51,
    "id": "fe5daeb4",
    "metadata": {
     "vscode": {
@@ -1510,7 +1977,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 164,
+   "execution_count": 52,
    "id": "3187bee7",
    "metadata": {
     "vscode": {
@@ -1542,7 +2009,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 165,
+   "execution_count": 53,
    "id": "df153b0a",
    "metadata": {
     "vscode": {
@@ -1566,7 +2033,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 166,
+   "execution_count": 54,
    "id": "1712e9e0",
    "metadata": {
     "vscode": {
@@ -1576,7 +2043,144 @@
    "outputs": [
     {
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    },
    "outputs": [],
    "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "5e50f3c7",
+   "metadata": {},
+   "outputs": [],
+   "source": []
   }
  ],
  "metadata": {
diff --git a/logic_programming/3_PropositionalLogic.pdf b/logic_programming/3_PropositionalLogic.pdf
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-- 
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