Newer
Older
"\n",
"* There are no new clauses that can be added, in which case $\\text{KB} \\nvDash \\alpha$.\n",
"* Two clauses resolve to yield the <em>empty clause</em>, in which case $\\text{KB} \\vDash \\alpha$.\n",
"\n",
"The <em>empty clause</em> is equivalent to <em>False</em> because it arises only from resolving two complementary\n",
"unit clauses such as $P$ and $\\neg P$ which is a contradiction as both $P$ and $\\neg P$ can't be <em>True</em> at the same time."
]
},
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
{
"cell_type": "markdown",
"metadata": {},
"source": [
"There is one catch however, the algorithm that implements proof by resolution cannot handle complex sentences. \n",
"Implications and bi-implications have to be simplified into simpler clauses. \n",
"We already know that *every sentence of a propositional logic is logically equivalent to a conjunction of clauses*.\n",
"We will use this fact to our advantage and simplify the input sentence into the **conjunctive normal form** (CNF) which is a conjunction of disjunctions of literals.\n",
"For eg:\n",
"<br>\n",
"$$(A\\lor B)\\land (\\neg B\\lor C\\lor\\neg D)\\land (D\\lor\\neg E)$$\n",
"This is equivalent to the POS (Product of sums) form in digital electronics.\n",
"<br>\n",
"Here's an outline of how the conversion is done:\n",
"1. Convert bi-implications to implications\n",
"<br>\n",
"$\\alpha\\iff\\beta$ can be written as $(\\alpha\\implies\\beta)\\land(\\beta\\implies\\alpha)$\n",
"<br>\n",
"This also applies to compound sentences\n",
"<br>\n",
"$\\alpha\\iff(\\beta\\lor\\gamma)$ can be written as $(\\alpha\\implies(\\beta\\lor\\gamma))\\land((\\beta\\lor\\gamma)\\implies\\alpha)$\n",
"<br>\n",
"2. Convert implications to their logical equivalents\n",
"<br>\n",
"$\\alpha\\implies\\beta$ can be written as $\\neg\\alpha\\lor\\beta$\n",
"<br>\n",
"3. Move negation inwards\n",
"<br>\n",
"CNF requires atomic literals. Hence, negation cannot appear on a compound statement.\n",
"De Morgan's laws will be helpful here.\n",
"<br>\n",
"$\\neg(\\alpha\\land\\beta)\\equiv(\\neg\\alpha\\lor\\neg\\beta)$\n",
"<br>\n",
"$\\neg(\\alpha\\lor\\beta)\\equiv(\\neg\\alpha\\land\\neg\\beta)$\n",
"<br>\n",
"4. Distribute disjunction over conjunction\n",
"<br>\n",
"Disjunction and conjunction are distributive over each other.\n",
"Now that we only have conjunctions, disjunctions and negations in our expression, \n",
"we will distribute disjunctions over conjunctions wherever possible as this will give us a sentence which is a conjunction of simpler clauses, \n",
"which is what we wanted in the first place.\n",
"<br>\n",
"We need a term of the form\n",
"<br>\n",
"$(\\alpha_{1}\\lor\\alpha_{2}\\lor\\alpha_{3}...)\\land(\\beta_{1}\\lor\\beta_{2}\\lor\\beta_{3}...)\\land(\\gamma_{1}\\lor\\gamma_{2}\\lor\\gamma_{3}...)\\land...$\n",
"<br>\n",
"<br>\n",
"The `to_cnf` function executes this conversion using helper subroutines."
]
},
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
"execution_count": 29,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<!DOCTYPE html PUBLIC \"-//W3C//DTD HTML 4.01//EN\"\n",
" \"http://www.w3.org/TR/html4/strict.dtd\">\n",
"\n",
"<html>\n",
"<head>\n",
" <title></title>\n",
" <meta http-equiv=\"content-type\" content=\"text/html; charset=None\">\n",
" <style type=\"text/css\">\n",
"td.linenos { background-color: #f0f0f0; padding-right: 10px; }\n",
"span.lineno { background-color: #f0f0f0; padding: 0 5px 0 5px; }\n",
"pre { line-height: 125%; }\n",
"body .hll { background-color: #ffffcc }\n",
"body { background: #f8f8f8; }\n",
"body .c { color: #408080; font-style: italic } /* Comment */\n",
"body .err { border: 1px solid #FF0000 } /* Error */\n",
"body .k { color: #008000; font-weight: bold } /* Keyword */\n",
"body .o { color: #666666 } /* Operator */\n",
"body .ch { color: #408080; font-style: italic } /* Comment.Hashbang */\n",
"body .cm { color: #408080; font-style: italic } /* Comment.Multiline */\n",
"body .cp { color: #BC7A00 } /* Comment.Preproc */\n",
"body .cpf { color: #408080; font-style: italic } /* Comment.PreprocFile */\n",
"body .c1 { color: #408080; font-style: italic } /* Comment.Single */\n",
"body .cs { color: #408080; font-style: italic } /* Comment.Special */\n",
"body .gd { color: #A00000 } /* Generic.Deleted */\n",
"body .ge { font-style: italic } /* Generic.Emph */\n",
"body .gr { color: #FF0000 } /* Generic.Error */\n",
"body .gh { color: #000080; font-weight: bold } /* Generic.Heading */\n",
"body .gi { color: #00A000 } /* Generic.Inserted */\n",
"body .go { color: #888888 } /* Generic.Output */\n",
"body .gp { color: #000080; font-weight: bold } /* Generic.Prompt */\n",
"body .gs { font-weight: bold } /* Generic.Strong */\n",
"body .gu { color: #800080; font-weight: bold } /* Generic.Subheading */\n",
"body .gt { color: #0044DD } /* Generic.Traceback */\n",
"body .kc { color: #008000; font-weight: bold } /* Keyword.Constant */\n",
"body .kd { color: #008000; font-weight: bold } /* Keyword.Declaration */\n",
"body .kn { color: #008000; font-weight: bold } /* Keyword.Namespace */\n",
"body .kp { color: #008000 } /* Keyword.Pseudo */\n",
"body .kr { color: #008000; font-weight: bold } /* Keyword.Reserved */\n",
"body .kt { color: #B00040 } /* Keyword.Type */\n",
"body .m { color: #666666 } /* Literal.Number */\n",
"body .s { color: #BA2121 } /* Literal.String */\n",
"body .na { color: #7D9029 } /* Name.Attribute */\n",
"body .nb { color: #008000 } /* Name.Builtin */\n",
"body .nc { color: #0000FF; font-weight: bold } /* Name.Class */\n",
"body .no { color: #880000 } /* Name.Constant */\n",
"body .nd { color: #AA22FF } /* Name.Decorator */\n",
"body .ni { color: #999999; font-weight: bold } /* Name.Entity */\n",
"body .ne { color: #D2413A; font-weight: bold } /* Name.Exception */\n",
"body .nf { color: #0000FF } /* Name.Function */\n",
"body .nl { color: #A0A000 } /* Name.Label */\n",
"body .nn { color: #0000FF; font-weight: bold } /* Name.Namespace */\n",
"body .nt { color: #008000; font-weight: bold } /* Name.Tag */\n",
"body .nv { color: #19177C } /* Name.Variable */\n",
"body .ow { color: #AA22FF; font-weight: bold } /* Operator.Word */\n",
"body .w { color: #bbbbbb } /* Text.Whitespace */\n",
"body .mb { color: #666666 } /* Literal.Number.Bin */\n",
"body .mf { color: #666666 } /* Literal.Number.Float */\n",
"body .mh { color: #666666 } /* Literal.Number.Hex */\n",
"body .mi { color: #666666 } /* Literal.Number.Integer */\n",
"body .mo { color: #666666 } /* Literal.Number.Oct */\n",
"body .sa { color: #BA2121 } /* Literal.String.Affix */\n",
"body .sb { color: #BA2121 } /* Literal.String.Backtick */\n",
"body .sc { color: #BA2121 } /* Literal.String.Char */\n",
"body .dl { color: #BA2121 } /* Literal.String.Delimiter */\n",
"body .sd { color: #BA2121; font-style: italic } /* Literal.String.Doc */\n",
"body .s2 { color: #BA2121 } /* Literal.String.Double */\n",
"body .se { color: #BB6622; font-weight: bold } /* Literal.String.Escape */\n",
"body .sh { color: #BA2121 } /* Literal.String.Heredoc */\n",
"body .si { color: #BB6688; font-weight: bold } /* Literal.String.Interpol */\n",
"body .sx { color: #008000 } /* Literal.String.Other */\n",
"body .sr { color: #BB6688 } /* Literal.String.Regex */\n",
"body .s1 { color: #BA2121 } /* Literal.String.Single */\n",
"body .ss { color: #19177C } /* Literal.String.Symbol */\n",
"body .bp { color: #008000 } /* Name.Builtin.Pseudo */\n",
"body .fm { color: #0000FF } /* Name.Function.Magic */\n",
"body .vc { color: #19177C } /* Name.Variable.Class */\n",
"body .vg { color: #19177C } /* Name.Variable.Global */\n",
"body .vi { color: #19177C } /* Name.Variable.Instance */\n",
"body .vm { color: #19177C } /* Name.Variable.Magic */\n",
"body .il { color: #666666 } /* Literal.Number.Integer.Long */\n",
"\n",
" </style>\n",
"</head>\n",
"<body>\n",
"<h2></h2>\n",
"\n",
"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">to_cnf</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Convert a propositional logical sentence to conjunctive normal form.</span>\n",
"<span class=\"sd\"> That is, to the form ((A | ~B | ...) & (B | C | ...) & ...) [p. 253]</span>\n",
"<span class=\"sd\"> >>> to_cnf('~(B | C)')</span>\n",
"<span class=\"sd\"> (~B & ~C)</span>\n",
"<span class=\"sd\"> """</span>\n",
" <span class=\"n\">s</span> <span class=\"o\">=</span> <span class=\"n\">expr</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"nb\">isinstance</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"nb\">str</span><span class=\"p\">):</span>\n",
" <span class=\"n\">s</span> <span class=\"o\">=</span> <span class=\"n\">expr</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span>\n",
" <span class=\"n\">s</span> <span class=\"o\">=</span> <span class=\"n\">eliminate_implications</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span> <span class=\"c1\"># Steps 1, 2 from p. 253</span>\n",
" <span class=\"n\">s</span> <span class=\"o\">=</span> <span class=\"n\">move_not_inwards</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span> <span class=\"c1\"># Step 3</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">distribute_and_over_or</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span> <span class=\"c1\"># Step 4</span>\n",
"</pre></div>\n",
"</body>\n",
"</html>\n"
],
"text/plain": [
"<IPython.core.display.HTML object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"psource(to_cnf)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"`to_cnf` calls three subroutines.\n",
"<br>\n",
"`eliminate_implications` converts bi-implications and implications to their logical equivalents.\n",
"<br>\n",
"`move_not_inwards` removes negations from compound statements and moves them inwards using De Morgan's laws.\n",
"<br>\n",
"`distribute_and_over_or` distributes disjunctions over conjunctions.\n",
"<br>\n",
"Run the cells below for implementation details.\n"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"%psource eliminate_implications"
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
1444
1445
1446
1447
1448
1449
"%psource move_not_inwards"
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"%psource distribute_and_over_or"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's convert some sentences to see how it works\n"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"((A | ~B) & (B | ~A))"
]
},
"execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A, B, C, D = expr('A, B, C, D')\n",
"to_cnf(A |'<=>'| B)"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"((A | ~B | ~C) & (B | ~A) & (C | ~A))"
]
},
"execution_count": 34,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"to_cnf(A |'<=>'| (B & C))"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(A & (C | B) & (D | B))"
]
},
"execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"to_cnf(A & (B | (C & D)))"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"((B | ~A | C | ~D) & (A | ~A | C | ~D) & (B | ~B | C | ~D) & (A | ~B | C | ~D))"
]
},
"execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"to_cnf((A |'<=>'| ~B) |'==>'| (C | ~D))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Coming back to our resolution problem, we can see how the `to_cnf` function is utilized here"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<!DOCTYPE html PUBLIC \"-//W3C//DTD HTML 4.01//EN\"\n",
" \"http://www.w3.org/TR/html4/strict.dtd\">\n",
"\n",
"<html>\n",
"<head>\n",
" <title></title>\n",
" <meta http-equiv=\"content-type\" content=\"text/html; charset=None\">\n",
" <style type=\"text/css\">\n",
"td.linenos { background-color: #f0f0f0; padding-right: 10px; }\n",
"span.lineno { background-color: #f0f0f0; padding: 0 5px 0 5px; }\n",
"pre { line-height: 125%; }\n",
"body .hll { background-color: #ffffcc }\n",
"body { background: #f8f8f8; }\n",
"body .c { color: #408080; font-style: italic } /* Comment */\n",
"body .err { border: 1px solid #FF0000 } /* Error */\n",
"body .k { color: #008000; font-weight: bold } /* Keyword */\n",
"body .o { color: #666666 } /* Operator */\n",
"body .ch { color: #408080; font-style: italic } /* Comment.Hashbang */\n",
"body .cm { color: #408080; font-style: italic } /* Comment.Multiline */\n",
"body .cp { color: #BC7A00 } /* Comment.Preproc */\n",
"body .cpf { color: #408080; font-style: italic } /* Comment.PreprocFile */\n",
"body .c1 { color: #408080; font-style: italic } /* Comment.Single */\n",
"body .cs { color: #408080; font-style: italic } /* Comment.Special */\n",
"body .gd { color: #A00000 } /* Generic.Deleted */\n",
"body .ge { font-style: italic } /* Generic.Emph */\n",
"body .gr { color: #FF0000 } /* Generic.Error */\n",
"body .gh { color: #000080; font-weight: bold } /* Generic.Heading */\n",
"body .gi { color: #00A000 } /* Generic.Inserted */\n",
"body .go { color: #888888 } /* Generic.Output */\n",
"body .gp { color: #000080; font-weight: bold } /* Generic.Prompt */\n",
"body .gs { font-weight: bold } /* Generic.Strong */\n",
"body .gu { color: #800080; font-weight: bold } /* Generic.Subheading */\n",
"body .gt { color: #0044DD } /* Generic.Traceback */\n",
"body .kc { color: #008000; font-weight: bold } /* Keyword.Constant */\n",
"body .kd { color: #008000; font-weight: bold } /* Keyword.Declaration */\n",
"body .kn { color: #008000; font-weight: bold } /* Keyword.Namespace */\n",
"body .kp { color: #008000 } /* Keyword.Pseudo */\n",
"body .kr { color: #008000; font-weight: bold } /* Keyword.Reserved */\n",
"body .kt { color: #B00040 } /* Keyword.Type */\n",
"body .m { color: #666666 } /* Literal.Number */\n",
"body .s { color: #BA2121 } /* Literal.String */\n",
"body .na { color: #7D9029 } /* Name.Attribute */\n",
"body .nb { color: #008000 } /* Name.Builtin */\n",
"body .nc { color: #0000FF; font-weight: bold } /* Name.Class */\n",
"body .no { color: #880000 } /* Name.Constant */\n",
"body .nd { color: #AA22FF } /* Name.Decorator */\n",
"body .ni { color: #999999; font-weight: bold } /* Name.Entity */\n",
"body .ne { color: #D2413A; font-weight: bold } /* Name.Exception */\n",
"body .nf { color: #0000FF } /* Name.Function */\n",
"body .nl { color: #A0A000 } /* Name.Label */\n",
"body .nn { color: #0000FF; font-weight: bold } /* Name.Namespace */\n",
"body .nt { color: #008000; font-weight: bold } /* Name.Tag */\n",
"body .nv { color: #19177C } /* Name.Variable */\n",
"body .ow { color: #AA22FF; font-weight: bold } /* Operator.Word */\n",
"body .w { color: #bbbbbb } /* Text.Whitespace */\n",
"body .mb { color: #666666 } /* Literal.Number.Bin */\n",
"body .mf { color: #666666 } /* Literal.Number.Float */\n",
"body .mh { color: #666666 } /* Literal.Number.Hex */\n",
"body .mi { color: #666666 } /* Literal.Number.Integer */\n",
"body .mo { color: #666666 } /* Literal.Number.Oct */\n",
"body .sa { color: #BA2121 } /* Literal.String.Affix */\n",
"body .sb { color: #BA2121 } /* Literal.String.Backtick */\n",
"body .sc { color: #BA2121 } /* Literal.String.Char */\n",
"body .dl { color: #BA2121 } /* Literal.String.Delimiter */\n",
"body .sd { color: #BA2121; font-style: italic } /* Literal.String.Doc */\n",
"body .s2 { color: #BA2121 } /* Literal.String.Double */\n",
"body .se { color: #BB6622; font-weight: bold } /* Literal.String.Escape */\n",
"body .sh { color: #BA2121 } /* Literal.String.Heredoc */\n",
"body .si { color: #BB6688; font-weight: bold } /* Literal.String.Interpol */\n",
"body .sx { color: #008000 } /* Literal.String.Other */\n",
"body .sr { color: #BB6688 } /* Literal.String.Regex */\n",
"body .s1 { color: #BA2121 } /* Literal.String.Single */\n",
"body .ss { color: #19177C } /* Literal.String.Symbol */\n",
"body .bp { color: #008000 } /* Name.Builtin.Pseudo */\n",
"body .fm { color: #0000FF } /* Name.Function.Magic */\n",
"body .vc { color: #19177C } /* Name.Variable.Class */\n",
"body .vg { color: #19177C } /* Name.Variable.Global */\n",
"body .vi { color: #19177C } /* Name.Variable.Instance */\n",
"body .vm { color: #19177C } /* Name.Variable.Magic */\n",
"body .il { color: #666666 } /* Literal.Number.Integer.Long */\n",
"\n",
" </style>\n",
"</head>\n",
"<body>\n",
"<h2></h2>\n",
"\n",
"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">pl_resolution</span><span class=\"p\">(</span><span class=\"n\">KB</span><span class=\"p\">,</span> <span class=\"n\">alpha</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Propositional-logic resolution: say if alpha follows from KB. [Figure 7.12]"""</span>\n",
" <span class=\"n\">clauses</span> <span class=\"o\">=</span> <span class=\"n\">KB</span><span class=\"o\">.</span><span class=\"n\">clauses</span> <span class=\"o\">+</span> <span class=\"n\">conjuncts</span><span class=\"p\">(</span><span class=\"n\">to_cnf</span><span class=\"p\">(</span><span class=\"o\">~</span><span class=\"n\">alpha</span><span class=\"p\">))</span>\n",
" <span class=\"n\">new</span> <span class=\"o\">=</span> <span class=\"nb\">set</span><span class=\"p\">()</span>\n",
" <span class=\"k\">while</span> <span class=\"bp\">True</span><span class=\"p\">:</span>\n",
" <span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">)</span>\n",
" <span class=\"n\">pairs</span> <span class=\"o\">=</span> <span class=\"p\">[(</span><span class=\"n\">clauses</span><span class=\"p\">[</span><span class=\"n\">i</span><span class=\"p\">],</span> <span class=\"n\">clauses</span><span class=\"p\">[</span><span class=\"n\">j</span><span class=\"p\">])</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">j</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"n\">i</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">)]</span>\n",
" <span class=\"k\">for</span> <span class=\"p\">(</span><span class=\"n\">ci</span><span class=\"p\">,</span> <span class=\"n\">cj</span><span class=\"p\">)</span> <span class=\"ow\">in</span> <span class=\"n\">pairs</span><span class=\"p\">:</span>\n",
" <span class=\"n\">resolvents</span> <span class=\"o\">=</span> <span class=\"n\">pl_resolve</span><span class=\"p\">(</span><span class=\"n\">ci</span><span class=\"p\">,</span> <span class=\"n\">cj</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"bp\">False</span> <span class=\"ow\">in</span> <span class=\"n\">resolvents</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"bp\">True</span>\n",
" <span class=\"n\">new</span> <span class=\"o\">=</span> <span class=\"n\">new</span><span class=\"o\">.</span><span class=\"n\">union</span><span class=\"p\">(</span><span class=\"nb\">set</span><span class=\"p\">(</span><span class=\"n\">resolvents</span><span class=\"p\">))</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">new</span><span class=\"o\">.</span><span class=\"n\">issubset</span><span class=\"p\">(</span><span class=\"nb\">set</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">)):</span>\n",
" <span class=\"k\">return</span> <span class=\"bp\">False</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">c</span> <span class=\"ow\">in</span> <span class=\"n\">new</span><span class=\"p\">:</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">c</span> <span class=\"ow\">not</span> <span class=\"ow\">in</span> <span class=\"n\">clauses</span><span class=\"p\">:</span>\n",
" <span class=\"n\">clauses</span><span class=\"o\">.</span><span class=\"n\">append</span><span class=\"p\">(</span><span class=\"n\">c</span><span class=\"p\">)</span>\n",
"</pre></div>\n",
"</body>\n",
"</html>\n"
],
"text/plain": [
"<IPython.core.display.HTML object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"psource(pl_resolution)"
1450
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
1485
1486
1487
1488
1489
1490
1491
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(True, False)"
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"pl_resolution(wumpus_kb, ~P11), pl_resolution(wumpus_kb, P11)"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(False, False)"
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"pl_resolution(wumpus_kb, ~P22), pl_resolution(wumpus_kb, P22)"
]
},
1492
1493
1494
1495
1496
1497
1498
1499
1500
1501
1502
1503
1504
1505
1506
1507
1508
1509
1510
1511
1512
1513
1514
1515
1516
1517
1518
1519
1520
1521
1522
1523
1524
1525
1526
1527
1528
1529
1530
1531
1532
1533
1534
1535
1536
1537
1538
1539
1540
1541
1542
1543
1544
1545
1546
1547
1548
1549
1550
1551
1552
1553
1554
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
1565
1566
1567
1568
1569
1570
1571
1572
1573
1574
1575
1576
1577
1578
1579
1580
1581
1582
1583
1584
1585
1586
1587
1588
1589
1590
1591
1592
1593
1594
1595
1596
1597
1598
1599
1600
1601
1602
1603
1604
1605
1606
1607
1608
1609
1610
1611
1612
1613
1614
1615
1616
1617
1618
1619
1620
1621
1622
1623
1624
1625
1626
1627
1628
1629
1630
1631
1632
1633
1634
1635
1636
1637
1638
1639
1640
1641
1642
1643
1644
1645
1646
1647
1648
1649
1650
1651
1652
1653
1654
1655
1656
1657
1658
1659
1660
1661
1662
1663
1664
1665
1666
1667
1668
1669
1670
1671
1672
1673
1674
1675
1676
1677
1678
1679
1680
1681
1682
1683
1684
1685
1686
1687
1688
1689
1690
1691
1692
1693
1694
1695
1696
1697
1698
1699
1700
1701
1702
1703
1704
1705
1706
1707
1708
1709
1710
1711
1712
1713
1714
1715
1716
1717
1718
1719
1720
1721
1722
1723
1724
1725
1726
1727
1728
1729
1730
1731
1732
1733
1734
1735
1736
1737
1738
1739
1740
1741
1742
1743
1744
1745
1746
1747
1748
1749
1750
1751
1752
1753
1754
1755
1756
1757
1758
1759
1760
1761
1762
1763
1764
1765
1766
1767
1768
1769
1770
1771
1772
1773
1774
1775
1776
1777
1778
1779
1780
1781
1782
1783
1784
1785
1786
1787
1788
1789
1790
1791
1792
1793
1794
1795
1796
1797
1798
1799
1800
1801
1802
1803
1804
1805
1806
1807
1808
1809
1810
1811
1812
1813
1814
1815
1816
1817
1818
1819
1820
1821
1822
1823
1824
1825
1826
1827
1828
1829
1830
1831
1832
1833
1834
1835
1836
1837
1838
1839
1840
1841
1842
1843
1844
1845
1846
1847
1848
1849
1850
1851
1852
1853
1854
1855
1856
1857
1858
1859
1860
1861
1862
1863
1864
1865
1866
1867
1868
1869
1870
1871
1872
1873
1874
1875
1876
1877
1878
1879
1880
1881
1882
1883
1884
1885
1886
1887
1888
1889
1890
1891
1892
1893
1894
1895
1896
1897
1898
1899
1900
1901
1902
1903
1904
1905
1906
1907
1908
1909
1910
1911
1912
1913
1914
1915
1916
1917
1918
1919
1920
1921
1922
1923
1924
1925
1926
1927
1928
1929
1930
1931
1932
1933
1934
1935
1936
1937
1938
1939
1940
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Effective Propositional Model Checking\n",
"\n",
"The previous segments elucidate the algorithmic procedure for model checking. \n",
"In this segment, we look at ways of making them computationally efficient.\n",
"<br>\n",
"The problem we are trying to solve is conventionally called the _propositional satisfiability problem_, abbreviated as the _SAT_ problem.\n",
"In layman terms, if there exists a model that satisfies a given Boolean formula, the formula is called satisfiable.\n",
"<br>\n",
"The SAT problem was the first problem to be proven _NP-complete_.\n",
"The main characteristics of an NP-complete problem are:\n",
"- Given a solution to such a problem, it is easy to verify if the solution solves the problem.\n",
"- The time required to actually solve the problem using any known algorithm increases exponentially with respect to the size of the problem.\n",
"<br>\n",
"<br>\n",
"Due to these properties, heuristic and approximational methods are often applied to find solutions to these problems.\n",
"<br>\n",
"It is extremely important to be able to solve large scale SAT problems efficiently because \n",
"many combinatorial problems in computer science can be conveniently reduced to checking the satisfiability of a propositional sentence under some constraints.\n",
"<br>\n",
"We will introduce two new algorithms that perform propositional model checking in a computationally effective way.\n",
"<br>\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 1. DPLL (Davis-Putnam-Logeman-Loveland) algorithm\n",
"This algorithm is very similar to Backtracking-Search.\n",
"It recursively enumerates possible models in a depth-first fashion with the following improvements over algorithms like `tt_entails`:\n",
"1. Early termination:\n",
"<br>\n",
"In certain cases, the algorithm can detect the truth value of a statement using just a partially completed model.\n",
"For example, $(P\\lor Q)\\land(P\\lor R)$ is true if P is true, regardless of other variables.\n",
"This reduces the search space significantly.\n",
"2. Pure symbol heuristic:\n",
"<br>\n",
"A symbol that has the same sign (positive or negative) in all clauses is called a _pure symbol_.\n",
"It isn't difficult to see that any satisfiable model will have the pure symbols assigned such that its parent clause becomes _true_.\n",
"For example, $(P\\lor\\neg Q)\\land(\\neg Q\\lor\\neg R)\\land(R\\lor P)$ has P and Q as pure symbols\n",
"and for the sentence to be true, P _has_ to be true and Q _has_ to be false.\n",
"The pure symbol heuristic thus simplifies the problem a bit.\n",
"3. Unit clause heuristic:\n",
"<br>\n",
"In the context of DPLL, clauses with just one literal and clauses with all but one _false_ literals are called unit clauses.\n",
"If a clause is a unit clause, it can only be satisfied by assigning the necessary value to make the last literal true.\n",
"We have no other choice.\n",
"<br>\n",
"Assigning one unit clause can create another unit clause.\n",
"For example, when P is false, $(P\\lor Q)$ becomes a unit clause, causing _true_ to be assigned to Q.\n",
"A series of forced assignments derived from previous unit clauses is called _unit propagation_.\n",
"In this way, this heuristic simplifies the problem further.\n",
"<br>\n",
"The algorithm often employs other tricks to scale up to large problems.\n",
"However, these tricks are currently out of the scope of this notebook. Refer to section 7.6 of the book for more details.\n",
"<br>\n",
"<br>\n",
"Let's have a look at the algorithm."
]
},
{
"cell_type": "code",
"execution_count": 48,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<!DOCTYPE html PUBLIC \"-//W3C//DTD HTML 4.01//EN\"\n",
" \"http://www.w3.org/TR/html4/strict.dtd\">\n",
"\n",
"<html>\n",
"<head>\n",
" <title></title>\n",
" <meta http-equiv=\"content-type\" content=\"text/html; charset=None\">\n",
" <style type=\"text/css\">\n",
"td.linenos { background-color: #f0f0f0; padding-right: 10px; }\n",
"span.lineno { background-color: #f0f0f0; padding: 0 5px 0 5px; }\n",
"pre { line-height: 125%; }\n",
"body .hll { background-color: #ffffcc }\n",
"body { background: #f8f8f8; }\n",
"body .c { color: #408080; font-style: italic } /* Comment */\n",
"body .err { border: 1px solid #FF0000 } /* Error */\n",
"body .k { color: #008000; font-weight: bold } /* Keyword */\n",
"body .o { color: #666666 } /* Operator */\n",
"body .ch { color: #408080; font-style: italic } /* Comment.Hashbang */\n",
"body .cm { color: #408080; font-style: italic } /* Comment.Multiline */\n",
"body .cp { color: #BC7A00 } /* Comment.Preproc */\n",
"body .cpf { color: #408080; font-style: italic } /* Comment.PreprocFile */\n",
"body .c1 { color: #408080; font-style: italic } /* Comment.Single */\n",
"body .cs { color: #408080; font-style: italic } /* Comment.Special */\n",
"body .gd { color: #A00000 } /* Generic.Deleted */\n",
"body .ge { font-style: italic } /* Generic.Emph */\n",
"body .gr { color: #FF0000 } /* Generic.Error */\n",
"body .gh { color: #000080; font-weight: bold } /* Generic.Heading */\n",
"body .gi { color: #00A000 } /* Generic.Inserted */\n",
"body .go { color: #888888 } /* Generic.Output */\n",
"body .gp { color: #000080; font-weight: bold } /* Generic.Prompt */\n",
"body .gs { font-weight: bold } /* Generic.Strong */\n",
"body .gu { color: #800080; font-weight: bold } /* Generic.Subheading */\n",
"body .gt { color: #0044DD } /* Generic.Traceback */\n",
"body .kc { color: #008000; font-weight: bold } /* Keyword.Constant */\n",
"body .kd { color: #008000; font-weight: bold } /* Keyword.Declaration */\n",
"body .kn { color: #008000; font-weight: bold } /* Keyword.Namespace */\n",
"body .kp { color: #008000 } /* Keyword.Pseudo */\n",
"body .kr { color: #008000; font-weight: bold } /* Keyword.Reserved */\n",
"body .kt { color: #B00040 } /* Keyword.Type */\n",
"body .m { color: #666666 } /* Literal.Number */\n",
"body .s { color: #BA2121 } /* Literal.String */\n",
"body .na { color: #7D9029 } /* Name.Attribute */\n",
"body .nb { color: #008000 } /* Name.Builtin */\n",
"body .nc { color: #0000FF; font-weight: bold } /* Name.Class */\n",
"body .no { color: #880000 } /* Name.Constant */\n",
"body .nd { color: #AA22FF } /* Name.Decorator */\n",
"body .ni { color: #999999; font-weight: bold } /* Name.Entity */\n",
"body .ne { color: #D2413A; font-weight: bold } /* Name.Exception */\n",
"body .nf { color: #0000FF } /* Name.Function */\n",
"body .nl { color: #A0A000 } /* Name.Label */\n",
"body .nn { color: #0000FF; font-weight: bold } /* Name.Namespace */\n",
"body .nt { color: #008000; font-weight: bold } /* Name.Tag */\n",
"body .nv { color: #19177C } /* Name.Variable */\n",
"body .ow { color: #AA22FF; font-weight: bold } /* Operator.Word */\n",
"body .w { color: #bbbbbb } /* Text.Whitespace */\n",
"body .mb { color: #666666 } /* Literal.Number.Bin */\n",
"body .mf { color: #666666 } /* Literal.Number.Float */\n",
"body .mh { color: #666666 } /* Literal.Number.Hex */\n",
"body .mi { color: #666666 } /* Literal.Number.Integer */\n",
"body .mo { color: #666666 } /* Literal.Number.Oct */\n",
"body .sa { color: #BA2121 } /* Literal.String.Affix */\n",
"body .sb { color: #BA2121 } /* Literal.String.Backtick */\n",
"body .sc { color: #BA2121 } /* Literal.String.Char */\n",
"body .dl { color: #BA2121 } /* Literal.String.Delimiter */\n",
"body .sd { color: #BA2121; font-style: italic } /* Literal.String.Doc */\n",
"body .s2 { color: #BA2121 } /* Literal.String.Double */\n",
"body .se { color: #BB6622; font-weight: bold } /* Literal.String.Escape */\n",
"body .sh { color: #BA2121 } /* Literal.String.Heredoc */\n",
"body .si { color: #BB6688; font-weight: bold } /* Literal.String.Interpol */\n",
"body .sx { color: #008000 } /* Literal.String.Other */\n",
"body .sr { color: #BB6688 } /* Literal.String.Regex */\n",
"body .s1 { color: #BA2121 } /* Literal.String.Single */\n",
"body .ss { color: #19177C } /* Literal.String.Symbol */\n",
"body .bp { color: #008000 } /* Name.Builtin.Pseudo */\n",
"body .fm { color: #0000FF } /* Name.Function.Magic */\n",
"body .vc { color: #19177C } /* Name.Variable.Class */\n",
"body .vg { color: #19177C } /* Name.Variable.Global */\n",
"body .vi { color: #19177C } /* Name.Variable.Instance */\n",
"body .vm { color: #19177C } /* Name.Variable.Magic */\n",
"body .il { color: #666666 } /* Literal.Number.Integer.Long */\n",
"\n",
" </style>\n",
"</head>\n",
"<body>\n",
"<h2></h2>\n",
"\n",
"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">dpll</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">,</span> <span class=\"n\">symbols</span><span class=\"p\">,</span> <span class=\"n\">model</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""See if the clauses are true in a partial model."""</span>\n",
" <span class=\"n\">unknown_clauses</span> <span class=\"o\">=</span> <span class=\"p\">[]</span> <span class=\"c1\"># clauses with an unknown truth value</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">c</span> <span class=\"ow\">in</span> <span class=\"n\">clauses</span><span class=\"p\">:</span>\n",
" <span class=\"n\">val</span> <span class=\"o\">=</span> <span class=\"n\">pl_true</span><span class=\"p\">(</span><span class=\"n\">c</span><span class=\"p\">,</span> <span class=\"n\">model</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">val</span> <span class=\"ow\">is</span> <span class=\"bp\">False</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"bp\">False</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">val</span> <span class=\"ow\">is</span> <span class=\"ow\">not</span> <span class=\"bp\">True</span><span class=\"p\">:</span>\n",
" <span class=\"n\">unknown_clauses</span><span class=\"o\">.</span><span class=\"n\">append</span><span class=\"p\">(</span><span class=\"n\">c</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"ow\">not</span> <span class=\"n\">unknown_clauses</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">model</span>\n",
" <span class=\"n\">P</span><span class=\"p\">,</span> <span class=\"n\">value</span> <span class=\"o\">=</span> <span class=\"n\">find_pure_symbol</span><span class=\"p\">(</span><span class=\"n\">symbols</span><span class=\"p\">,</span> <span class=\"n\">unknown_clauses</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">P</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">dpll</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">,</span> <span class=\"n\">removeall</span><span class=\"p\">(</span><span class=\"n\">P</span><span class=\"p\">,</span> <span class=\"n\">symbols</span><span class=\"p\">),</span> <span class=\"n\">extend</span><span class=\"p\">(</span><span class=\"n\">model</span><span class=\"p\">,</span> <span class=\"n\">P</span><span class=\"p\">,</span> <span class=\"n\">value</span><span class=\"p\">))</span>\n",
" <span class=\"n\">P</span><span class=\"p\">,</span> <span class=\"n\">value</span> <span class=\"o\">=</span> <span class=\"n\">find_unit_clause</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">,</span> <span class=\"n\">model</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">P</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">dpll</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">,</span> <span class=\"n\">removeall</span><span class=\"p\">(</span><span class=\"n\">P</span><span class=\"p\">,</span> <span class=\"n\">symbols</span><span class=\"p\">),</span> <span class=\"n\">extend</span><span class=\"p\">(</span><span class=\"n\">model</span><span class=\"p\">,</span> <span class=\"n\">P</span><span class=\"p\">,</span> <span class=\"n\">value</span><span class=\"p\">))</span>\n",
" <span class=\"k\">if</span> <span class=\"ow\">not</span> <span class=\"n\">symbols</span><span class=\"p\">:</span>\n",
" <span class=\"k\">raise</span> <span class=\"ne\">TypeError</span><span class=\"p\">(</span><span class=\"s2\">"Argument should be of the type Expr."</span><span class=\"p\">)</span>\n",
" <span class=\"n\">P</span><span class=\"p\">,</span> <span class=\"n\">symbols</span> <span class=\"o\">=</span> <span class=\"n\">symbols</span><span class=\"p\">[</span><span class=\"mi\">0</span><span class=\"p\">],</span> <span class=\"n\">symbols</span><span class=\"p\">[</span><span class=\"mi\">1</span><span class=\"p\">:]</span>\n",
" <span class=\"k\">return</span> <span class=\"p\">(</span><span class=\"n\">dpll</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">,</span> <span class=\"n\">symbols</span><span class=\"p\">,</span> <span class=\"n\">extend</span><span class=\"p\">(</span><span class=\"n\">model</span><span class=\"p\">,</span> <span class=\"n\">P</span><span class=\"p\">,</span> <span class=\"bp\">True</span><span class=\"p\">))</span> <span class=\"ow\">or</span>\n",
" <span class=\"n\">dpll</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">,</span> <span class=\"n\">symbols</span><span class=\"p\">,</span> <span class=\"n\">extend</span><span class=\"p\">(</span><span class=\"n\">model</span><span class=\"p\">,</span> <span class=\"n\">P</span><span class=\"p\">,</span> <span class=\"bp\">False</span><span class=\"p\">)))</span>\n",
"</pre></div>\n",
"</body>\n",
"</html>\n"
],
"text/plain": [
"<IPython.core.display.HTML object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"psource(dpll)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The algorithm uses the ideas described above to check satisfiability of a sentence in propositional logic.\n",
"It recursively calls itself, simplifying the problem at each step. It also uses helper functions `find_pure_symbol` and `find_unit_clause` to carry out steps 2 and 3 above.\n",
"<br>\n",
"The `dpll_satisfiable` helper function converts the input clauses to _conjunctive normal form_ and calls the `dpll` function with the correct parameters."
]
},
{
"cell_type": "code",
"execution_count": 49,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<!DOCTYPE html PUBLIC \"-//W3C//DTD HTML 4.01//EN\"\n",
" \"http://www.w3.org/TR/html4/strict.dtd\">\n",
"\n",
"<html>\n",
"<head>\n",
" <title></title>\n",
" <meta http-equiv=\"content-type\" content=\"text/html; charset=None\">\n",
" <style type=\"text/css\">\n",
"td.linenos { background-color: #f0f0f0; padding-right: 10px; }\n",
"span.lineno { background-color: #f0f0f0; padding: 0 5px 0 5px; }\n",
"pre { line-height: 125%; }\n",
"body .hll { background-color: #ffffcc }\n",
"body { background: #f8f8f8; }\n",
"body .c { color: #408080; font-style: italic } /* Comment */\n",
"body .err { border: 1px solid #FF0000 } /* Error */\n",
"body .k { color: #008000; font-weight: bold } /* Keyword */\n",
"body .o { color: #666666 } /* Operator */\n",
"body .ch { color: #408080; font-style: italic } /* Comment.Hashbang */\n",
"body .cm { color: #408080; font-style: italic } /* Comment.Multiline */\n",
"body .cp { color: #BC7A00 } /* Comment.Preproc */\n",
"body .cpf { color: #408080; font-style: italic } /* Comment.PreprocFile */\n",
"body .c1 { color: #408080; font-style: italic } /* Comment.Single */\n",
"body .cs { color: #408080; font-style: italic } /* Comment.Special */\n",
"body .gd { color: #A00000 } /* Generic.Deleted */\n",
"body .ge { font-style: italic } /* Generic.Emph */\n",
"body .gr { color: #FF0000 } /* Generic.Error */\n",
"body .gh { color: #000080; font-weight: bold } /* Generic.Heading */\n",
"body .gi { color: #00A000 } /* Generic.Inserted */\n",
"body .go { color: #888888 } /* Generic.Output */\n",
"body .gp { color: #000080; font-weight: bold } /* Generic.Prompt */\n",
"body .gs { font-weight: bold } /* Generic.Strong */\n",
"body .gu { color: #800080; font-weight: bold } /* Generic.Subheading */\n",
"body .gt { color: #0044DD } /* Generic.Traceback */\n",
"body .kc { color: #008000; font-weight: bold } /* Keyword.Constant */\n",
"body .kd { color: #008000; font-weight: bold } /* Keyword.Declaration */\n",
"body .kn { color: #008000; font-weight: bold } /* Keyword.Namespace */\n",
"body .kp { color: #008000 } /* Keyword.Pseudo */\n",
"body .kr { color: #008000; font-weight: bold } /* Keyword.Reserved */\n",
"body .kt { color: #B00040 } /* Keyword.Type */\n",
"body .m { color: #666666 } /* Literal.Number */\n",
"body .s { color: #BA2121 } /* Literal.String */\n",
"body .na { color: #7D9029 } /* Name.Attribute */\n",
"body .nb { color: #008000 } /* Name.Builtin */\n",
"body .nc { color: #0000FF; font-weight: bold } /* Name.Class */\n",
"body .no { color: #880000 } /* Name.Constant */\n",
"body .nd { color: #AA22FF } /* Name.Decorator */\n",
"body .ni { color: #999999; font-weight: bold } /* Name.Entity */\n",
"body .ne { color: #D2413A; font-weight: bold } /* Name.Exception */\n",
"body .nf { color: #0000FF } /* Name.Function */\n",
"body .nl { color: #A0A000 } /* Name.Label */\n",
"body .nn { color: #0000FF; font-weight: bold } /* Name.Namespace */\n",
"body .nt { color: #008000; font-weight: bold } /* Name.Tag */\n",
"body .nv { color: #19177C } /* Name.Variable */\n",
"body .ow { color: #AA22FF; font-weight: bold } /* Operator.Word */\n",
"body .w { color: #bbbbbb } /* Text.Whitespace */\n",
"body .mb { color: #666666 } /* Literal.Number.Bin */\n",
"body .mf { color: #666666 } /* Literal.Number.Float */\n",
"body .mh { color: #666666 } /* Literal.Number.Hex */\n",
"body .mi { color: #666666 } /* Literal.Number.Integer */\n",
"body .mo { color: #666666 } /* Literal.Number.Oct */\n",
"body .sa { color: #BA2121 } /* Literal.String.Affix */\n",
"body .sb { color: #BA2121 } /* Literal.String.Backtick */\n",
"body .sc { color: #BA2121 } /* Literal.String.Char */\n",
"body .dl { color: #BA2121 } /* Literal.String.Delimiter */\n",
"body .sd { color: #BA2121; font-style: italic } /* Literal.String.Doc */\n",
"body .s2 { color: #BA2121 } /* Literal.String.Double */\n",
"body .se { color: #BB6622; font-weight: bold } /* Literal.String.Escape */\n",
"body .sh { color: #BA2121 } /* Literal.String.Heredoc */\n",
"body .si { color: #BB6688; font-weight: bold } /* Literal.String.Interpol */\n",
"body .sx { color: #008000 } /* Literal.String.Other */\n",
"body .sr { color: #BB6688 } /* Literal.String.Regex */\n",
"body .s1 { color: #BA2121 } /* Literal.String.Single */\n",
"body .ss { color: #19177C } /* Literal.String.Symbol */\n",
"body .bp { color: #008000 } /* Name.Builtin.Pseudo */\n",
"body .fm { color: #0000FF } /* Name.Function.Magic */\n",
"body .vc { color: #19177C } /* Name.Variable.Class */\n",
"body .vg { color: #19177C } /* Name.Variable.Global */\n",
"body .vi { color: #19177C } /* Name.Variable.Instance */\n",
"body .vm { color: #19177C } /* Name.Variable.Magic */\n",
"body .il { color: #666666 } /* Literal.Number.Integer.Long */\n",
"\n",
" </style>\n",
"</head>\n",
"<body>\n",
"<h2></h2>\n",
"\n",
"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">dpll_satisfiable</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Check satisfiability of a propositional sentence.</span>\n",
"<span class=\"sd\"> This differs from the book code in two ways: (1) it returns a model</span>\n",
"<span class=\"sd\"> rather than True when it succeeds; this is more useful. (2) The</span>\n",
"<span class=\"sd\"> function find_pure_symbol is passed a list of unknown clauses, rather</span>\n",
"<span class=\"sd\"> than a list of all clauses and the model; this is more efficient."""</span>\n",
" <span class=\"n\">clauses</span> <span class=\"o\">=</span> <span class=\"n\">conjuncts</span><span class=\"p\">(</span><span class=\"n\">to_cnf</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">))</span>\n",
" <span class=\"n\">symbols</span> <span class=\"o\">=</span> <span class=\"nb\">list</span><span class=\"p\">(</span><span class=\"n\">prop_symbols</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">))</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">dpll</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">,</span> <span class=\"n\">symbols</span><span class=\"p\">,</span> <span class=\"p\">{})</span>\n",
"</pre></div>\n",
"</body>\n",
"</html>\n"
],
"text/plain": [
"<IPython.core.display.HTML object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"psource(dpll_satisfiable)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's see a few examples of usage."
]
},
{
"cell_type": "code",
"execution_count": 50,
"metadata": {},
"outputs": [],
"source": [
"A, B, C, D = expr('A, B, C, D')"
]
},
{
"cell_type": "code",
"execution_count": 51,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{C: False, A: True, D: True, B: True}"
]
},
"execution_count": 51,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dpll_satisfiable(A & B & ~C & D)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This is a simple case to highlight that the algorithm actually works."
]
},
{
"cell_type": "code",
"execution_count": 52,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{C: True, D: False, B: True}"
]
},
"execution_count": 52,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dpll_satisfiable((A & B) | (C & ~A) | (B & ~D))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"If a particular symbol isn't present in the solution, \n",
"it means that the solution is independent of the value of that symbol.\n",
"In this case, the solution is independent of A."
]
},
{
"cell_type": "code",
"execution_count": 53,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{A: True, B: True}"
]
},
"execution_count": 53,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dpll_satisfiable(A |'<=>'| B)"
]
},
{
"cell_type": "code",
"execution_count": 54,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{C: True, A: True, B: False}"
]
},
"execution_count": 54,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dpll_satisfiable((A |'<=>'| B) |'==>'| (C & ~A))"
]
},
{
"cell_type": "code",
"execution_count": 55,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{C: True, A: True}"
]
},
"execution_count": 55,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dpll_satisfiable((A | (B & C)) |'<=>'| ((A | B) & (A | C)))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 2. WalkSAT algorithm\n",
"This algorithm is very similar to Hill climbing.\n",
"On every iteration, the algorithm picks an unsatisfied clause and flips a symbol in the clause.\n",
"This is similar to finding a neighboring state in the `hill_climbing` algorithm.\n",
"<br>\n",
"The symbol to be flipped is decided by an evaluation function that counts the number of unsatisfied clauses.\n",
"Sometimes, symbols are also flipped randomly, to avoid local optima. A subtle balance between greediness and randomness is required. Alternatively, some versions of the algorithm restart with a completely new random assignment if no solution has been found for too long, as a way of getting out of local minima of numbers of unsatisfied clauses.\n",
"<br>\n",
"<br>\n",
"Let's have a look at the algorithm."
]
},
{
"cell_type": "code",
"execution_count": 56,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<!DOCTYPE html PUBLIC \"-//W3C//DTD HTML 4.01//EN\"\n",
" \"http://www.w3.org/TR/html4/strict.dtd\">\n",
"\n",
"<html>\n",
"<head>\n",
" <title></title>\n",
" <meta http-equiv=\"content-type\" content=\"text/html; charset=None\">\n",
" <style type=\"text/css\">\n",
"td.linenos { background-color: #f0f0f0; padding-right: 10px; }\n",
"span.lineno { background-color: #f0f0f0; padding: 0 5px 0 5px; }\n",
"pre { line-height: 125%; }\n",
"body .hll { background-color: #ffffcc }\n",
"body { background: #f8f8f8; }\n",
"body .c { color: #408080; font-style: italic } /* Comment */\n",
"body .err { border: 1px solid #FF0000 } /* Error */\n",
"body .k { color: #008000; font-weight: bold } /* Keyword */\n",
"body .o { color: #666666 } /* Operator */\n",
"body .ch { color: #408080; font-style: italic } /* Comment.Hashbang */\n",
"body .cm { color: #408080; font-style: italic } /* Comment.Multiline */\n",
"body .cp { color: #BC7A00 } /* Comment.Preproc */\n",
"body .cpf { color: #408080; font-style: italic } /* Comment.PreprocFile */\n",
"body .c1 { color: #408080; font-style: italic } /* Comment.Single */\n",
"body .cs { color: #408080; font-style: italic } /* Comment.Special */\n",
"body .gd { color: #A00000 } /* Generic.Deleted */\n",
"body .ge { font-style: italic } /* Generic.Emph */\n",
"body .gr { color: #FF0000 } /* Generic.Error */\n",
"body .gh { color: #000080; font-weight: bold } /* Generic.Heading */\n",
"body .gi { color: #00A000 } /* Generic.Inserted */\n",