Newer
Older
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2031
2032
2033
2034
2035
2036
2037
2038
2039
2040
2041
2042
2043
2044
2045
2046
2047
2048
2049
2050
2051
2052
2053
2054
2055
2056
2057
2058
2059
2060
2061
2062
2063
2064
2065
2066
2067
2068
2069
2070
2071
2072
2073
2074
2075
2076
2077
2078
2079
2080
2081
2082
2083
2084
2085
2086
2087
2088
2089
2090
2091
2092
2093
2094
2095
2096
2097
2098
2099
2100
2101
2102
2103
2104
2105
2106
2107
2108
2109
2110
2111
2112
2113
2114
2115
2116
2117
2118
2119
2120
2121
2122
2123
2124
2125
2126
2127
2128
2129
2130
2131
2132
2133
2134
2135
2136
2137
2138
2139
2140
2141
2142
2143
2144
2145
2146
2147
2148
2149
2150
2151
2152
2153
2154
2155
2156
2157
2158
2159
2160
2161
2162
2163
2164
2165
2166
2167
2168
2169
2170
2171
2172
2173
2174
2175
2176
2177
2178
2179
2180
2181
2182
2183
2184
2185
2186
2187
2188
2189
2190
2191
2192
2193
2194
2195
2196
2197
2198
2199
2200
2201
2202
2203
2204
2205
2206
2207
2208
2209
2210
2211
2212
2213
2214
2215
2216
2217
2218
2219
2220
2221
2222
2223
2224
2225
2226
2227
2228
2229
2230
2231
2232
2233
2234
2235
2236
2237
2238
2239
2240
2241
2242
2243
2244
2245
2246
2247
2248
2249
2250
2251
2252
2253
2254
2255
2256
2257
2258
2259
2260
2261
2262
2263
2264
2265
2266
2267
2268
2269
2270
2271
2272
2273
2274
2275
2276
2277
2278
2279
2280
2281
2282
2283
2284
2285
2286
2287
2288
2289
2290
2291
2292
2293
2294
2295
2296
2297
2298
2299
2300
2301
2302
2303
2304
2305
2306
2307
2308
2309
2310
2311
2312
2313
2314
2315
2316
2317
2318
2319
2320
2321
2322
2323
2324
2325
2326
2327
2328
2329
2330
2331
2332
2333
2334
2335
2336
2337
2338
"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\">WalkSAT</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">,</span> <span class=\"n\">p</span><span class=\"o\">=</span><span class=\"mf\">0.5</span><span class=\"p\">,</span> <span class=\"n\">max_flips</span><span class=\"o\">=</span><span class=\"mi\">10000</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Checks for satisfiability of all clauses by randomly flipping values of variables</span>\n",
"<span class=\"sd\"> """</span>\n",
" <span class=\"c1\"># Set of all symbols in all clauses</span>\n",
" <span class=\"n\">symbols</span> <span class=\"o\">=</span> <span class=\"p\">{</span><span class=\"n\">sym</span> <span class=\"k\">for</span> <span class=\"n\">clause</span> <span class=\"ow\">in</span> <span class=\"n\">clauses</span> <span class=\"k\">for</span> <span class=\"n\">sym</span> <span class=\"ow\">in</span> <span class=\"n\">prop_symbols</span><span class=\"p\">(</span><span class=\"n\">clause</span><span class=\"p\">)}</span>\n",
" <span class=\"c1\"># model is a random assignment of true/false to the symbols in clauses</span>\n",
" <span class=\"n\">model</span> <span class=\"o\">=</span> <span class=\"p\">{</span><span class=\"n\">s</span><span class=\"p\">:</span> <span class=\"n\">random</span><span class=\"o\">.</span><span class=\"n\">choice</span><span class=\"p\">([</span><span class=\"bp\">True</span><span class=\"p\">,</span> <span class=\"bp\">False</span><span class=\"p\">])</span> <span class=\"k\">for</span> <span class=\"n\">s</span> <span class=\"ow\">in</span> <span class=\"n\">symbols</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\">max_flips</span><span class=\"p\">):</span>\n",
" <span class=\"n\">satisfied</span><span class=\"p\">,</span> <span class=\"n\">unsatisfied</span> <span class=\"o\">=</span> <span class=\"p\">[],</span> <span class=\"p\">[]</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">clause</span> <span class=\"ow\">in</span> <span class=\"n\">clauses</span><span class=\"p\">:</span>\n",
" <span class=\"p\">(</span><span class=\"n\">satisfied</span> <span class=\"k\">if</span> <span class=\"n\">pl_true</span><span class=\"p\">(</span><span class=\"n\">clause</span><span class=\"p\">,</span> <span class=\"n\">model</span><span class=\"p\">)</span> <span class=\"k\">else</span> <span class=\"n\">unsatisfied</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">append</span><span class=\"p\">(</span><span class=\"n\">clause</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"ow\">not</span> <span class=\"n\">unsatisfied</span><span class=\"p\">:</span> <span class=\"c1\"># if model satisfies all the clauses</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">model</span>\n",
" <span class=\"n\">clause</span> <span class=\"o\">=</span> <span class=\"n\">random</span><span class=\"o\">.</span><span class=\"n\">choice</span><span class=\"p\">(</span><span class=\"n\">unsatisfied</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">probability</span><span class=\"p\">(</span><span class=\"n\">p</span><span class=\"p\">):</span>\n",
" <span class=\"n\">sym</span> <span class=\"o\">=</span> <span class=\"n\">random</span><span class=\"o\">.</span><span class=\"n\">choice</span><span class=\"p\">(</span><span class=\"nb\">list</span><span class=\"p\">(</span><span class=\"n\">prop_symbols</span><span class=\"p\">(</span><span class=\"n\">clause</span><span class=\"p\">)))</span>\n",
" <span class=\"k\">else</span><span class=\"p\">:</span>\n",
" <span class=\"c1\"># Flip the symbol in clause that maximizes number of sat. clauses</span>\n",
" <span class=\"k\">def</span> <span class=\"nf\">sat_count</span><span class=\"p\">(</span><span class=\"n\">sym</span><span class=\"p\">):</span>\n",
" <span class=\"c1\"># Return the the number of clauses satisfied after flipping the symbol.</span>\n",
" <span class=\"n\">model</span><span class=\"p\">[</span><span class=\"n\">sym</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"ow\">not</span> <span class=\"n\">model</span><span class=\"p\">[</span><span class=\"n\">sym</span><span class=\"p\">]</span>\n",
" <span class=\"n\">count</span> <span class=\"o\">=</span> <span class=\"nb\">len</span><span class=\"p\">([</span><span class=\"n\">clause</span> <span class=\"k\">for</span> <span class=\"n\">clause</span> <span class=\"ow\">in</span> <span class=\"n\">clauses</span> <span class=\"k\">if</span> <span class=\"n\">pl_true</span><span class=\"p\">(</span><span class=\"n\">clause</span><span class=\"p\">,</span> <span class=\"n\">model</span><span class=\"p\">)])</span>\n",
" <span class=\"n\">model</span><span class=\"p\">[</span><span class=\"n\">sym</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"ow\">not</span> <span class=\"n\">model</span><span class=\"p\">[</span><span class=\"n\">sym</span><span class=\"p\">]</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">count</span>\n",
" <span class=\"n\">sym</span> <span class=\"o\">=</span> <span class=\"n\">argmax</span><span class=\"p\">(</span><span class=\"n\">prop_symbols</span><span class=\"p\">(</span><span class=\"n\">clause</span><span class=\"p\">),</span> <span class=\"n\">key</span><span class=\"o\">=</span><span class=\"n\">sat_count</span><span class=\"p\">)</span>\n",
" <span class=\"n\">model</span><span class=\"p\">[</span><span class=\"n\">sym</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"ow\">not</span> <span class=\"n\">model</span><span class=\"p\">[</span><span class=\"n\">sym</span><span class=\"p\">]</span>\n",
" <span class=\"c1\"># If no solution is found within the flip limit, we return failure</span>\n",
" <span class=\"k\">return</span> <span class=\"bp\">None</span>\n",
"</pre></div>\n",
"</body>\n",
"</html>\n"
],
"text/plain": [
"<IPython.core.display.HTML object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"psource(WalkSAT)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The function takes three arguments:\n",
"<br>\n",
"1. The `clauses` we want to satisfy.\n",
"<br>\n",
"2. The probability `p` of randomly changing a symbol.\n",
"<br>\n",
"3. The maximum number of flips (`max_flips`) the algorithm will run for. If the clauses are still unsatisfied, the algorithm returns `None` to denote failure.\n",
"<br>\n",
"The algorithm is identical in concept to Hill climbing and the code isn't difficult to understand.\n",
"<br>\n",
"<br>\n",
"Let's see a few examples of usage."
]
},
{
"cell_type": "code",
"execution_count": 57,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"A, B, C, D = expr('A, B, C, D')"
]
},
{
"cell_type": "code",
"execution_count": 58,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{C: False, A: True, D: True, B: True}"
]
},
"execution_count": 58,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"WalkSAT([A, B, ~C, D], 0.5, 100)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This is a simple case to show that the algorithm converges."
]
},
{
"cell_type": "code",
"execution_count": 59,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{C: True, A: True, B: True}"
]
},
"execution_count": 59,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"WalkSAT([A & B, A & C], 0.5, 100)"
]
},
{
"cell_type": "code",
"execution_count": 60,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{C: True, A: True, D: True, B: True}"
]
},
"execution_count": 60,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"WalkSAT([A & B, C & D, C & B], 0.5, 100)"
]
},
{
"cell_type": "code",
"execution_count": 61,
"metadata": {},
"outputs": [],
"source": [
"WalkSAT([A & B, C | D, ~(D | B)], 0.5, 1000)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This one doesn't give any output because WalkSAT did not find any model where these clauses hold. We can solve these clauses to see that they together form a contradiction and hence, it isn't supposed to have a solution."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"One point of difference between this algorithm and the `dpll_satisfiable` algorithms is that both these algorithms take inputs differently. \n",
"For WalkSAT to take complete sentences as input, \n",
"we can write a helper function that converts the input sentence into conjunctive normal form and then calls WalkSAT with the list of conjuncts of the CNF form of the sentence."
]
},
{
"cell_type": "code",
"execution_count": 62,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def WalkSAT_CNF(sentence, p=0.5, max_flips=10000):\n",
" return WalkSAT(conjuncts(to_cnf(sentence)), 0, max_flips)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now we can call `WalkSAT_CNF` and `DPLL_Satisfiable` with the same arguments."
]
},
{
"cell_type": "code",
"execution_count": 63,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{A: False, D: False, C: True, B: False}"
]
},
"execution_count": 63,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"WalkSAT_CNF((A & B) | (C & ~A) | (B & ~D), 0.5, 1000)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"It works!\n",
"<br>\n",
"Notice that the solution generated by WalkSAT doesn't omit variables that the sentence doesn't depend upon. \n",
"If the sentence is independent of a particular variable, the solution contains a random value for that variable because of the stochastic nature of the algorithm.\n",
"<br>\n",
"<br>\n",
"Let's compare the runtime of WalkSAT and DPLL for a few cases. We will use the `%%timeit` magic to do this."
]
},
{
"cell_type": "code",
"execution_count": 64,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"sentence_1 = A |'<=>'| B\n",
"sentence_2 = (A & B) | (C & ~A) | (B & ~D)\n",
"sentence_3 = (A | (B & C)) |'<=>'| ((A | B) & (A | C))"
]
},
{
"cell_type": "code",
"execution_count": 65,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"100 loops, best of 3: 2.46 ms per loop\n"
]
}
],
"source": [
"%%timeit\n",
"dpll_satisfiable(sentence_1)\n",
"dpll_satisfiable(sentence_2)\n",
"dpll_satisfiable(sentence_3)"
]
},
{
"cell_type": "code",
"execution_count": 66,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"100 loops, best of 3: 1.91 ms per loop\n"
]
}
],
"source": [
"%%timeit\n",
"WalkSAT_CNF(sentence_1)\n",
"WalkSAT_CNF(sentence_2)\n",
"WalkSAT_CNF(sentence_3)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"On an average, for solvable cases, `WalkSAT` is quite faster than `dpll` because, for a small number of variables, \n",
"`WalkSAT` can reduce the search space significantly. \n",
"Results can be different for sentences with more symbols though.\n",
"Feel free to play around with this to understand the trade-offs of these algorithms better."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## First-Order Logic Knowledge Bases: `FolKB`\n",
"\n",
"The class `FolKB` can be used to represent a knowledge base of First-order logic sentences. You would initialize and use it the same way as you would for `PropKB` except that the clauses are first-order definite clauses. We will see how to write such clauses to create a database and query them in the following sections."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Criminal KB\n",
"In this section we create a `FolKB` based on the following paragraph.<br/>\n",
"<em>The law says that it is a crime for an American to sell weapons to hostile nations. The country Nono, an enemy of America, has some missiles, and all of its missiles were sold to it by Colonel West, who is American.</em><br/>\n",
"The first step is to extract the facts and convert them into first-order definite clauses. Extracting the facts from data alone is a challenging task. Fortunately, we have a small paragraph and can do extraction and conversion manually. We'll store the clauses in list aptly named `clauses`."
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"clauses = []"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<em>“... it is a crime for an American to sell weapons to hostile nations”</em><br/>\n",
"The keywords to look for here are 'crime', 'American', 'sell', 'weapon' and 'hostile'. We use predicate symbols to make meaning of them.\n",
"\n",
"* `Criminal(x)`: `x` is a criminal\n",
"* `American(x)`: `x` is an American\n",
"* `Sells(x ,y, z)`: `x` sells `y` to `z`\n",
"* `Weapon(x)`: `x` is a weapon\n",
"* `Hostile(x)`: `x` is a hostile nation\n",
"\n",
"Let us now combine them with appropriate variable naming to depict the meaning of the sentence. The criminal `x` is also the American `x` who sells weapon `y` to `z`, which is a hostile nation.\n",
2383
2384
2385
2386
2387
2388
2389
2390
2391
2392
2393
2394
2395
2396
2397
2398
2399
2400
2401
2402
2403
2404
2405
2406
2407
2408
2409
2410
2411
2412
2413
2414
2415
2416
2417
2418
2419
2420
2421
2422
2423
"\n",
"$\\text{American}(x) \\land \\text{Weapon}(y) \\land \\text{Sells}(x, y, z) \\land \\text{Hostile}(z) \\implies \\text{Criminal} (x)$"
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"clauses.append(expr(\"(American(x) & Weapon(y) & Sells(x, y, z) & Hostile(z)) ==> Criminal(x)\"))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<em>\"The country Nono, an enemy of America\"</em><br/>\n",
"We now know that Nono is an enemy of America. We represent these nations using the constant symbols `Nono` and `America`. the enemy relation is show using the predicate symbol `Enemy`.\n",
"\n",
"$\\text{Enemy}(\\text{Nono}, \\text{America})$"
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"clauses.append(expr(\"Enemy(Nono, America)\"))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<em>\"Nono ... has some missiles\"</em><br/>\n",
"This states the existence of some missile which is owned by Nono. $\\exists x \\text{Owns}(\\text{Nono}, x) \\land \\text{Missile}(x)$. We invoke existential instantiation to introduce a new constant `M1` which is the missile owned by Nono.\n",
2425
2426
2427
2428
2429
2430
2431
2432
2433
2434
2435
2436
2437
2438
2439
2440
2441
2442
2443
2444
2445
2446
2447
2448
2449
2450
2451
2452
2453
2454
2455
2456
2457
2458
2459
2460
2461
2462
2463
2464
2465
2466
2467
2468
2469
2470
2471
2472
2473
2474
2475
2476
2477
2478
2479
2480
2481
2482
2483
2484
2485
2486
2487
2488
2489
2490
2491
2492
2493
2494
2495
2496
2497
2498
2499
2500
2501
2502
2503
2504
2505
2506
2507
2508
2509
2510
2511
2512
2513
2514
2515
2516
2517
2518
2519
2520
2521
2522
2523
2524
2525
2526
2527
2528
"\n",
"$\\text{Owns}(\\text{Nono}, \\text{M1}), \\text{Missile}(\\text{M1})$"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"clauses.append(expr(\"Owns(Nono, M1)\"))\n",
"clauses.append(expr(\"Missile(M1)\"))"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": true
},
"source": [
"<em>\"All of its missiles were sold to it by Colonel West\"</em><br/>\n",
"If Nono owns something and it classifies as a missile, then it was sold to Nono by West.\n",
"\n",
"$\\text{Missile}(x) \\land \\text{Owns}(\\text{Nono}, x) \\implies \\text{Sells}(\\text{West}, x, \\text{Nono})$"
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"clauses.append(expr(\"(Missile(x) & Owns(Nono, x)) ==> Sells(West, x, Nono)\"))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<em>\"West, who is American\"</em><br/>\n",
"West is an American.\n",
"\n",
"$\\text{American}(\\text{West})$"
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"clauses.append(expr(\"American(West)\"))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We also know, from our understanding of language, that missiles are weapons and that an enemy of America counts as “hostile”.\n",
"\n",
"$\\text{Missile}(x) \\implies \\text{Weapon}(x), \\text{Enemy}(x, \\text{America}) \\implies \\text{Hostile}(x)$"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"clauses.append(expr(\"Missile(x) ==> Weapon(x)\"))\n",
"clauses.append(expr(\"Enemy(x, America) ==> Hostile(x)\"))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now that we have converted the information into first-order definite clauses we can create our first-order logic knowledge base."
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"crime_kb = FolKB(clauses)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Inference in First-Order Logic\n",
"In this section we look at a forward chaining and a backward chaining algorithm for `FolKB`. Both aforementioned algorithms rely on a process called <strong>unification</strong>, a key component of all first-order inference algorithms."
2530
2531
2532
2533
2534
2535
2536
2537
2538
2539
2540
2541
2542
2543
2544
2545
2546
2547
2548
2549
2550
2551
2552
2553
2554
2555
2556
2557
2558
2559
2560
2561
2562
2563
2564
2565
2566
2567
2568
2569
2570
2571
2572
2573
2574
2575
2576
2577
2578
2579
2580
2581
2582
2583
2584
2585
2586
2587
2588
2589
2590
2591
2592
2593
2594
2595
2596
2597
2598
2599
2600
2601
2602
2603
2604
2605
2606
2607
2608
2609
2610
2611
2612
2613
2614
2615
2616
2617
2618
2619
2620
2621
2622
2623
2624
2625
2626
2627
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Unification\n",
"We sometimes require finding substitutions that make different logical expressions look identical. This process, called unification, is done by the `unify` algorithm. It takes as input two sentences and returns a <em>unifier</em> for them if one exists. A unifier is a dictionary which stores the substitutions required to make the two sentences identical. It does so by recursively unifying the components of a sentence, where the unification of a variable symbol `var` with a constant symbol `Const` is the mapping `{var: Const}`. Let's look at a few examples."
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{x: 3}"
]
},
"execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"unify(expr('x'), 3)"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{x: B}"
]
},
"execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"unify(expr('A(x)'), expr('A(B)'))"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{x: Bella, y: Dobby}"
]
},
"execution_count": 37,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"unify(expr('Cat(x) & Dog(Dobby)'), expr('Cat(Bella) & Dog(y)'))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In cases where there is no possible substitution that unifies the two sentences the function return `None`."
]
},
{
"cell_type": "code",
"execution_count": 38,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"None\n"
]
}
],
"source": [
"print(unify(expr('Cat(x)'), expr('Dog(Dobby)')))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We also need to take care we do not unintentionally use the same variable name. Unify treats them as a single variable which prevents it from taking multiple value."
2629
2630
2631
2632
2633
2634
2635
2636
2637
2638
2639
2640
2641
2642
2643
2644
2645
2646
2647
2648
2649
2650
2651
2652
]
},
{
"cell_type": "code",
"execution_count": 39,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"None\n"
]
}
],
"source": [
"print(unify(expr('Cat(x) & Dog(Dobby)'), expr('Cat(Bella) & Dog(x)')))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Forward Chaining Algorithm\n",
"We consider the simple forward-chaining algorithm presented in <em>Figure 9.3</em>. We look at each rule in the knoweldge base and see if the premises can be satisfied. This is done by finding a substitution which unifies each of the premise with a clause in the `KB`. If we are able to unify the premises, the conclusion (with the corresponding substitution) is added to the `KB`. This inferencing process is repeated until either the query can be answered or till no new sentences can be added. We test if the newly added clause unifies with the query in which case the substitution yielded by `unify` is an answer to the query. If we run out of sentences to infer, this means the query was a failure.\n",
2655
2656
2657
2658
2659
2660
2661
2662
2663
2664
2665
2666
2667
2668
2669
2670
2671
2672
2673
2674
2675
2676
2677
2678
2679
2680
2681
2682
2683
2684
2685
2686
2687
2688
2689
2690
2691
2692
2693
2694
2695
2696
2697
2698
2699
2700
2701
2702
2703
2704
2705
2706
2707
2708
2709
2710
2711
2712
2713
2714
2715
2716
2717
2718
2719
2720
2721
2722
2723
2724
2725
2726
2727
2728
2729
2730
2731
2732
2733
2734
2735
2736
2737
2738
2739
2740
2741
2742
2743
2744
2745
2746
2747
2748
2749
2750
2751
2752
2753
2754
2755
2756
2757
2758
2759
2760
2761
2762
2763
2764
2765
2766
2767
2768
2769
2770
2771
2772
2773
2774
2775
2776
2777
2778
2779
2780
2781
2782
2783
2784
2785
2786
2787
2788
2789
2790
2791
2792
2793
2794
2795
2796
2797
2798
2799
2800
2801
2802
2803
2804
2805
2806
2807
2808
2809
2810
2811
2812
2813
2814
2815
2816
"The function `fol_fc_ask` is a generator which yields all substitutions which validate the query."
]
},
{
"cell_type": "code",
"execution_count": 40,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"%psource fol_fc_ask"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's find out all the hostile nations. Note that we only told the `KB` that Nono was an enemy of America, not that it was hostile."
]
},
{
"cell_type": "code",
"execution_count": 41,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[{x: Nono}]\n"
]
}
],
"source": [
"answer = fol_fc_ask(crime_kb, expr('Hostile(x)'))\n",
"print(list(answer))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The generator returned a single substitution which says that Nono is a hostile nation. See how after adding another enemy nation the generator returns two substitutions."
]
},
{
"cell_type": "code",
"execution_count": 42,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[{x: Nono}, {x: JaJa}]\n"
]
}
],
"source": [
"crime_kb.tell(expr('Enemy(JaJa, America)'))\n",
"answer = fol_fc_ask(crime_kb, expr('Hostile(x)'))\n",
"print(list(answer))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<strong><em>Note</em>:</strong> `fol_fc_ask` makes changes to the `KB` by adding sentences to it."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Backward Chaining Algorithm\n",
"This algorithm works backward from the goal, chaining through rules to find known facts that support the proof. Suppose `goal` is the query we want to find the substitution for. We find rules of the form $\\text{lhs} \\implies \\text{goal}$ in the `KB` and try to prove `lhs`. There may be multiple clauses in the `KB` which give multiple `lhs`. It is sufficient to prove only one of these. But to prove a `lhs` all the conjuncts in the `lhs` of the clause must be proved. This makes it similar to <em>And/Or</em> search."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### OR\n",
"The <em>OR</em> part of the algorithm comes from our choice to select any clause of the form $\\text{lhs} \\implies \\text{goal}$. Looking at all rules's `lhs` whose `rhs` unify with the `goal`, we yield a substitution which proves all the conjuncts in the `lhs`. We use `parse_definite_clause` to attain `lhs` and `rhs` from a clause of the form $\\text{lhs} \\implies \\text{rhs}$. For atomic facts the `lhs` is an empty list."
]
},
{
"cell_type": "code",
"execution_count": 43,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"%psource fol_bc_or"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### AND\n",
"The <em>AND</em> corresponds to proving all the conjuncts in the `lhs`. We need to find a substitution which proves each <em>and</em> every clause in the list of conjuncts."
]
},
{
"cell_type": "code",
"execution_count": 44,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"%psource fol_bc_and"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now the main function `fl_bc_ask` calls `fol_bc_or` with substitution initialized as empty. The `ask` method of `FolKB` uses `fol_bc_ask` and fetches the first substitution returned by the generator to answer query. Let's query the knowledge base we created from `clauses` to find hostile nations."
]
},
{
"cell_type": "code",
"execution_count": 45,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"# Rebuild KB because running fol_fc_ask would add new facts to the KB\n",
"crime_kb = FolKB(clauses)"
]
},
{
"cell_type": "code",
"execution_count": 46,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{v_5: x, x: Nono}"
]
},
"execution_count": 46,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"crime_kb.ask(expr('Hostile(x)'))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"You may notice some new variables in the substitution. They are introduced to standardize the variable names to prevent naming problems as discussed in the [Unification section](#Unification)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Appendix: The Implementation of `|'==>'|`\n",
"Consider the `Expr` formed by this syntax:"
"outputs": [
{
"data": {
"text/plain": [
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"What is the funny `|'==>'|` syntax? The trick is that \"`|`\" is just the regular Python or-operator, and so is exactly equivalent to this: "
"outputs": [
{
"data": {
"text/plain": [
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In other words, there are two applications of or-operators. Here's the first one:"
"outputs": [
{
"data": {
"text/plain": [
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"What is going on here is that the `__or__` method of `Expr` serves a dual purpose. If the right-hand-side is another `Expr` (or a number), then the result is an `Expr`, as in `(P | Q)`. But if the right-hand-side is a string, then the string is taken to be an operator, and we create a node in the abstract syntax tree corresponding to a partially-filled `Expr`, one where we know the left-hand-side is `P` and the operator is `==>`, but we don't yet know the right-hand-side.\n",
"The `PartialExpr` class has an `__or__` method that says to create an `Expr` node with the right-hand-side filled in. Here we can see the combination of the `PartialExpr` with `Q` to create a complete `Expr`:"
"outputs": [
{
"data": {
"text/plain": [
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"partial = PartialExpr('==>', P) \n",
"partial | ~Q"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This [trick](http://code.activestate.com/recipes/384122-infix-operators/) is due to [Ferdinand Jamitzky](http://code.activestate.com/recipes/users/98863/), with a modification by [C. G. Vedant](https://github.com/Chipe1),\n",
"who suggested using a string inside the or-bars.\n",
"\n",
"## Appendix: The Implementation of `expr`\n",
"\n",
"How does `expr` parse a string into an `Expr`? It turns out there are two tricks (besides the Jamitzky/Vedant trick):\n",
"\n",
"1. We do a string substitution, replacing \"`==>`\" with \"`|'==>'|`\" (and likewise for other operators).\n",
"2. We `eval` the resulting string in an environment in which every identifier\n",
"is bound to a symbol with that identifier as the `op`.\n",
"\n",
"In other words,"
{
"cell_type": "code",
"outputs": [
{
"data": {
"text/plain": [
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"outputs": [
{
"data": {
"text/plain": [
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"P, Q = symbols('P, Q')\n",
"~(P & Q) |'==>'| (~P | ~Q)"
]
},
{
"cell_type": "markdown",
"metadata": {},