Newer
Older
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# KNOWLEDGE\n",
"\n",
"The [knowledge](https://github.com/aimacode/aima-python/blob/master/knowledge.py) module covers **Chapter 19: Knowledge in Learning** from Stuart Russel's and Peter Norvig's book *Artificial Intelligence: A Modern Approach*.\n",
"\n",
"Execute the cell below to get started."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"from knowledge import *\n",
"\n",
"from notebook import pseudocode, psource"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## CONTENTS\n",
"\n",
"* Overview\n",
"* Inductive Logic Programming (FOIL)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## OVERVIEW\n",
"\n",
"Like the [learning module](https://github.com/aimacode/aima-python/blob/master/learning.ipynb), this chapter focuses on methods for generating a model/hypothesis for a domain. Unlike though the learning chapter, here we use prior knowledge to help us learn from new experiences and find a proper hypothesis.\n",
"\n",
"### First-Order Logic\n",
"\n",
"Usually knowledge in this field is represented as **first-order logic**, a type of logic that uses variables and quantifiers in logical sentences. Hypotheses are represented by logical sentences with variables, while examples are logical sentences with set values instead of variables. The goal is to assign a value to a special first-order logic predicate, called **goal predicate**, for new examples given a hypothesis. We learn this hypothesis by infering knowledge from some given examples.\n",
"\n",
"### Representation\n",
"\n",
"In this module, we use dictionaries to represent examples, with keys the attribute names and values the corresponding example values. Examples also have an extra boolean field, 'GOAL', for the goal predicate. A hypothesis is represented as a list of dictionaries. Each dictionary in that list represents a disjunction. Inside these dictionaries/disjunctions we have conjunctions.\n",
"\n",
"For example, say we want to predict if an animal (cat or dog) will take an umbrella given whether or not it rains or the animal wears a coat. The goal value is 'take an umbrella' and is denoted by the key 'GOAL'. An example:\n",
"\n",
"`{'Species': 'Cat', 'Coat': 'Yes', 'Rain': 'Yes', 'GOAL': True}`\n",
"\n",
"A hypothesis can be the following:\n",
"\n",
"`[{'Species': 'Cat'}]`\n",
"\n",
"which means an animal will take an umbrella if and only if it is a cat.\n",
"\n",
"### Consistency\n",
"\n",
"We say that an example `e` is **consistent** with an hypothesis `h` if the assignment from the hypothesis for `e` is the same as `e['GOAL']`. If the above example and hypothesis are `e` and `h` respectively, then `e` is consistent with `h` since `e['Species'] == 'Cat'`. For `e = {'Species': 'Dog', 'Coat': 'Yes', 'Rain': 'Yes', 'GOAL': True}`, the example is no longer consistent with `h`, since the value assigned to `e` is *False* while `e['GOAL']` is *True*."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Inductive Logic Programming (FOIL)\n",
"\n",
"Inductive logic programming (ILP) combines inductive methods with the power of first-order representations, concentrating in particular on the representation of hypotheses as logic programs. The general knowledge-based induction problem is to solve the entailment constrant: <br> <br>\n",
"$ Background ∧ Hypothesis ∧ Descriptions \\vDash Classifications $\n",
"\n",
"for the __unknown__ $Hypothesis$, given the $Background$ knowledge described by $Descriptions$ and $Classifications$.\n",
"\n",
"\n",
"\n",
"The first approach to ILP works by starting with a very general rule and gradually specializing\n",
"it so that it fits the data. <br> \n",
"This is essentially what happens in decision-tree learning, where a\n",
"decision tree is gradually grown until it is consistent with the observations. <br> To do ILP we\n",
"use first-order literals instead of attributes, and the $Hypothesis$ is a set of clauses (set of first order rules, where each rule is similar to a Horn clause) instead of a decision tree. <br>\n",
"\n",
"\n",
"The FOIL algorithm learns new rules, one at a time, in order to cover all given possitive and negative examples. <br>\n",
"More precicely, FOIL contains an inner and an outer while loop. <br>\n",
"- __outer loop__: <font color='blue'>(function __foil()__) </font> add rules untill all positive examples are covered. <br>\n",
" (each rule is a conjuction of literals, which are chosen inside the inner loop)\n",
" \n",
" \n",
"- __inner loop__: <font color ='blue'>(function __new_clause()__) </font> add new literals untill all negative examples are covered, and some positive examples are covered. <br>\n",
" - In each iteration, we select/add the most promising literal, according to an estimate of its utility. <font color ='blue'>(function __new_literal()__) </font> <br>\n",
" \n",
" - The evaluation function to estimate utility of adding literal $L$ to a set of rules $R$ is <font color ='blue'>(function __gain()__) </font> : \n",
" \n",
" $$ FoilGain(L,R) = t \\big( \\log_2{\\frac{p_1}{p_1+n_1}} - \\log_2{\\frac{p_0}{p_0+n_0}} \\big) $$\n",
" where: \n",
" \n",
" $p_0: \\text{is the number of possitive bindings of rule R } \\\\ n_0: \\text{is the number of negative bindings of R} \\\\ p_1: \\text{is the is the number of possitive bindings of rule R'}\\\\ n_0: \\text{is the number of negative bindings of R'}\\\\ t: \\text{is the number of possitive bindings of rule R that are still covered after adding literal L to R}$\n",
" \n",
" - Calculate the extended examples for the chosen literal <font color ='blue'>(function __extend_example()__) </font> <br>\n",
" (the set of examples created by extending example with each possible constant value for each new variable in literal)\n",
" \n",
"- Finally the algorithm returns a disjunction of first order rules (= conjuction of literals)\n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 2,
"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\">class</span> <span class=\"nc\">FOIL_container</span><span class=\"p\">(</span><span class=\"n\">FolKB</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Hold the kb and other necessary elements required by FOIL."""</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"fm\">__init__</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">clauses</span><span class=\"o\">=</span><span class=\"bp\">None</span><span class=\"p\">):</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">const_syms</span> <span class=\"o\">=</span> <span class=\"nb\">set</span><span class=\"p\">()</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">pred_syms</span> <span class=\"o\">=</span> <span class=\"nb\">set</span><span class=\"p\">()</span>\n",
" <span class=\"n\">FolKB</span><span class=\"o\">.</span><span class=\"fm\">__init__</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">clauses</span><span class=\"p\">)</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">tell</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">sentence</span><span class=\"p\">):</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">is_definite_clause</span><span class=\"p\">(</span><span class=\"n\">sentence</span><span class=\"p\">):</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">clauses</span><span class=\"o\">.</span><span class=\"n\">append</span><span class=\"p\">(</span><span class=\"n\">sentence</span><span class=\"p\">)</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">const_syms</span><span class=\"o\">.</span><span class=\"n\">update</span><span class=\"p\">(</span><span class=\"n\">constant_symbols</span><span class=\"p\">(</span><span class=\"n\">sentence</span><span class=\"p\">))</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">pred_syms</span><span class=\"o\">.</span><span class=\"n\">update</span><span class=\"p\">(</span><span class=\"n\">predicate_symbols</span><span class=\"p\">(</span><span class=\"n\">sentence</span><span class=\"p\">))</span>\n",
" <span class=\"k\">else</span><span class=\"p\">:</span>\n",
" <span class=\"k\">raise</span> <span class=\"ne\">Exception</span><span class=\"p\">(</span><span class=\"s2\">"Not a definite clause: {}"</span><span class=\"o\">.</span><span class=\"n\">format</span><span class=\"p\">(</span><span class=\"n\">sentence</span><span class=\"p\">))</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">foil</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">examples</span><span class=\"p\">,</span> <span class=\"n\">target</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Learn a list of first-order horn clauses</span>\n",
"<span class=\"sd\"> 'examples' is a tuple: (positive_examples, negative_examples).</span>\n",
"<span class=\"sd\"> positive_examples and negative_examples are both lists which contain substitutions."""</span>\n",
" <span class=\"n\">clauses</span> <span class=\"o\">=</span> <span class=\"p\">[]</span>\n",
"\n",
" <span class=\"n\">pos_examples</span> <span class=\"o\">=</span> <span class=\"n\">examples</span><span class=\"p\">[</span><span class=\"mi\">0</span><span class=\"p\">]</span>\n",
" <span class=\"n\">neg_examples</span> <span class=\"o\">=</span> <span class=\"n\">examples</span><span class=\"p\">[</span><span class=\"mi\">1</span><span class=\"p\">]</span>\n",
"\n",
" <span class=\"k\">while</span> <span class=\"n\">pos_examples</span><span class=\"p\">:</span>\n",
" <span class=\"n\">clause</span><span class=\"p\">,</span> <span class=\"n\">extended_pos_examples</span> <span class=\"o\">=</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">new_clause</span><span class=\"p\">((</span><span class=\"n\">pos_examples</span><span class=\"p\">,</span> <span class=\"n\">neg_examples</span><span class=\"p\">),</span> <span class=\"n\">target</span><span class=\"p\">)</span>\n",
" <span class=\"c1\"># remove positive examples covered by clause</span>\n",
" <span class=\"n\">pos_examples</span> <span class=\"o\">=</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">update_examples</span><span class=\"p\">(</span><span class=\"n\">target</span><span class=\"p\">,</span> <span class=\"n\">pos_examples</span><span class=\"p\">,</span> <span class=\"n\">extended_pos_examples</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\">clause</span><span class=\"p\">)</span>\n",
"\n",
" <span class=\"k\">return</span> <span class=\"n\">clauses</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">new_clause</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">examples</span><span class=\"p\">,</span> <span class=\"n\">target</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Find a horn clause which satisfies part of the positive</span>\n",
"<span class=\"sd\"> examples but none of the negative examples.</span>\n",
"<span class=\"sd\"> The horn clause is specified as [consequent, list of antecedents]</span>\n",
"<span class=\"sd\"> Return value is the tuple (horn_clause, extended_positive_examples)."""</span>\n",
" <span class=\"n\">clause</span> <span class=\"o\">=</span> <span class=\"p\">[</span><span class=\"n\">target</span><span class=\"p\">,</span> <span class=\"p\">[]]</span>\n",
" <span class=\"c1\"># [positive_examples, negative_examples]</span>\n",
" <span class=\"n\">extended_examples</span> <span class=\"o\">=</span> <span class=\"n\">examples</span>\n",
" <span class=\"k\">while</span> <span class=\"n\">extended_examples</span><span class=\"p\">[</span><span class=\"mi\">1</span><span class=\"p\">]:</span>\n",
" <span class=\"n\">l</span> <span class=\"o\">=</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">choose_literal</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">new_literals</span><span class=\"p\">(</span><span class=\"n\">clause</span><span class=\"p\">),</span> <span class=\"n\">extended_examples</span><span class=\"p\">)</span>\n",
" <span class=\"n\">clause</span><span class=\"p\">[</span><span class=\"mi\">1</span><span class=\"p\">]</span><span class=\"o\">.</span><span class=\"n\">append</span><span class=\"p\">(</span><span class=\"n\">l</span><span class=\"p\">)</span>\n",
" <span class=\"n\">extended_examples</span> <span class=\"o\">=</span> <span class=\"p\">[</span><span class=\"nb\">sum</span><span class=\"p\">([</span><span class=\"nb\">list</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">extend_example</span><span class=\"p\">(</span><span class=\"n\">example</span><span class=\"p\">,</span> <span class=\"n\">l</span><span class=\"p\">))</span> <span class=\"k\">for</span> <span class=\"n\">example</span> <span class=\"ow\">in</span>\n",
" <span class=\"n\">extended_examples</span><span class=\"p\">[</span><span class=\"n\">i</span><span class=\"p\">]],</span> <span class=\"p\">[])</span> <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=\"mi\">2</span><span class=\"p\">)]</span>\n",
"\n",
" <span class=\"k\">return</span> <span class=\"p\">(</span><span class=\"n\">clause</span><span class=\"p\">,</span> <span class=\"n\">extended_examples</span><span class=\"p\">[</span><span class=\"mi\">0</span><span class=\"p\">])</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">extend_example</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">example</span><span class=\"p\">,</span> <span class=\"n\">literal</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Generate extended examples which satisfy the literal."""</span>\n",
" <span class=\"c1\"># find all substitutions that satisfy literal</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">s</span> <span class=\"ow\">in</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">ask_generator</span><span class=\"p\">(</span><span class=\"n\">subst</span><span class=\"p\">(</span><span class=\"n\">example</span><span class=\"p\">,</span> <span class=\"n\">literal</span><span class=\"p\">)):</span>\n",
" <span class=\"n\">s</span><span class=\"o\">.</span><span class=\"n\">update</span><span class=\"p\">(</span><span class=\"n\">example</span><span class=\"p\">)</span>\n",
" <span class=\"k\">yield</span> <span class=\"n\">s</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">new_literals</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">clause</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Generate new literals based on known predicate symbols.</span>\n",
"<span class=\"sd\"> Generated literal must share atleast one variable with clause"""</span>\n",
" <span class=\"n\">share_vars</span> <span class=\"o\">=</span> <span class=\"n\">variables</span><span class=\"p\">(</span><span class=\"n\">clause</span><span class=\"p\">[</span><span class=\"mi\">0</span><span class=\"p\">])</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">l</span> <span class=\"ow\">in</span> <span class=\"n\">clause</span><span class=\"p\">[</span><span class=\"mi\">1</span><span class=\"p\">]:</span>\n",
" <span class=\"n\">share_vars</span><span class=\"o\">.</span><span class=\"n\">update</span><span class=\"p\">(</span><span class=\"n\">variables</span><span class=\"p\">(</span><span class=\"n\">l</span><span class=\"p\">))</span>\n",
" <span class=\"c1\"># creates literals with different order every time </span>\n",
" <span class=\"k\">for</span> <span class=\"n\">pred</span><span class=\"p\">,</span> <span class=\"n\">arity</span> <span class=\"ow\">in</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">pred_syms</span><span class=\"p\">:</span>\n",
" <span class=\"n\">new_vars</span> <span class=\"o\">=</span> <span class=\"p\">{</span><span class=\"n\">standardize_variables</span><span class=\"p\">(</span><span class=\"n\">expr</span><span class=\"p\">(</span><span class=\"s1\">'x'</span><span class=\"p\">))</span> <span class=\"k\">for</span> <span class=\"n\">_</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"n\">arity</span> <span class=\"o\">-</span> <span class=\"mi\">1</span><span class=\"p\">)}</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">args</span> <span class=\"ow\">in</span> <span class=\"n\">product</span><span class=\"p\">(</span><span class=\"n\">share_vars</span><span class=\"o\">.</span><span class=\"n\">union</span><span class=\"p\">(</span><span class=\"n\">new_vars</span><span class=\"p\">),</span> <span class=\"n\">repeat</span><span class=\"o\">=</span><span class=\"n\">arity</span><span class=\"p\">):</span>\n",
" <span class=\"k\">if</span> <span class=\"nb\">any</span><span class=\"p\">(</span><span class=\"n\">var</span> <span class=\"ow\">in</span> <span class=\"n\">share_vars</span> <span class=\"k\">for</span> <span class=\"n\">var</span> <span class=\"ow\">in</span> <span class=\"n\">args</span><span class=\"p\">):</span>\n",
" <span class=\"c1\"># make sure we don't return an existing rule</span>\n",
" <span class=\"k\">if</span> <span class=\"ow\">not</span> <span class=\"n\">Expr</span><span class=\"p\">(</span><span class=\"n\">pred</span><span class=\"p\">,</span> <span class=\"n\">args</span><span class=\"p\">)</span> <span class=\"ow\">in</span> <span class=\"n\">clause</span><span class=\"p\">[</span><span class=\"mi\">1</span><span class=\"p\">]:</span>\n",
" <span class=\"k\">yield</span> <span class=\"n\">Expr</span><span class=\"p\">(</span><span class=\"n\">pred</span><span class=\"p\">,</span> <span class=\"o\">*</span><span class=\"p\">[</span><span class=\"n\">var</span> <span class=\"k\">for</span> <span class=\"n\">var</span> <span class=\"ow\">in</span> <span class=\"n\">args</span><span class=\"p\">])</span>\n",
"\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">choose_literal</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">literals</span><span class=\"p\">,</span> <span class=\"n\">examples</span><span class=\"p\">):</span> \n",
" <span class=\"sd\">"""Choose the best literal based on the information gain."""</span>\n",
"\n",
" <span class=\"k\">return</span> <span class=\"nb\">max</span><span class=\"p\">(</span><span class=\"n\">literals</span><span class=\"p\">,</span> <span class=\"n\">key</span> <span class=\"o\">=</span> <span class=\"n\">partial</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">gain</span> <span class=\"p\">,</span> <span class=\"n\">examples</span> <span class=\"o\">=</span> <span class=\"n\">examples</span><span class=\"p\">))</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">gain</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">l</span> <span class=\"p\">,</span><span class=\"n\">examples</span><span class=\"p\">):</span>\n",
" <span class=\"n\">pre_pos</span><span class=\"o\">=</span> <span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"n\">examples</span><span class=\"p\">[</span><span class=\"mi\">0</span><span class=\"p\">])</span>\n",
" <span class=\"n\">pre_neg</span><span class=\"o\">=</span> <span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"n\">examples</span><span class=\"p\">[</span><span class=\"mi\">1</span><span class=\"p\">])</span>\n",
" <span class=\"n\">extended_examples</span> <span class=\"o\">=</span> <span class=\"p\">[</span><span class=\"nb\">sum</span><span class=\"p\">([</span><span class=\"nb\">list</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">extend_example</span><span class=\"p\">(</span><span class=\"n\">example</span><span class=\"p\">,</span> <span class=\"n\">l</span><span class=\"p\">))</span> <span class=\"k\">for</span> <span class=\"n\">example</span> <span class=\"ow\">in</span> <span class=\"n\">examples</span><span class=\"p\">[</span><span class=\"n\">i</span><span class=\"p\">]],</span> <span class=\"p\">[])</span> <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=\"mi\">2</span><span class=\"p\">)]</span>\n",
" <span class=\"n\">post_pos</span> <span class=\"o\">=</span> <span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"n\">extended_examples</span><span class=\"p\">[</span><span class=\"mi\">0</span><span class=\"p\">])</span> \n",
" <span class=\"n\">post_neg</span> <span class=\"o\">=</span> <span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"n\">extended_examples</span><span class=\"p\">[</span><span class=\"mi\">1</span><span class=\"p\">])</span> \n",
" <span class=\"k\">if</span> <span class=\"n\">pre_pos</span> <span class=\"o\">+</span> <span class=\"n\">pre_neg</span> <span class=\"o\">==</span><span class=\"mi\">0</span> <span class=\"ow\">or</span> <span class=\"n\">post_pos</span> <span class=\"o\">+</span> <span class=\"n\">post_neg</span><span class=\"o\">==</span><span class=\"mi\">0</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"o\">-</span><span class=\"mi\">1</span>\n",
" <span class=\"c1\"># number of positive example that are represented in extended_examples</span>\n",
" <span class=\"n\">T</span> <span class=\"o\">=</span> <span class=\"mi\">0</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">example</span> <span class=\"ow\">in</span> <span class=\"n\">examples</span><span class=\"p\">[</span><span class=\"mi\">0</span><span class=\"p\">]:</span>\n",
" <span class=\"k\">def</span> <span class=\"nf\">represents</span><span class=\"p\">(</span><span class=\"n\">d</span><span class=\"p\">):</span>\n",
" <span class=\"k\">return</span> <span class=\"nb\">all</span><span class=\"p\">(</span><span class=\"n\">d</span><span class=\"p\">[</span><span class=\"n\">x</span><span class=\"p\">]</span> <span class=\"o\">==</span> <span class=\"n\">example</span><span class=\"p\">[</span><span class=\"n\">x</span><span class=\"p\">]</span> <span class=\"k\">for</span> <span class=\"n\">x</span> <span class=\"ow\">in</span> <span class=\"n\">example</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"nb\">any</span><span class=\"p\">(</span><span class=\"n\">represents</span><span class=\"p\">(</span><span class=\"n\">l_</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">l_</span> <span class=\"ow\">in</span> <span class=\"n\">extended_examples</span><span class=\"p\">[</span><span class=\"mi\">0</span><span class=\"p\">]):</span>\n",
" <span class=\"n\">T</span> <span class=\"o\">+=</span> <span class=\"mi\">1</span>\n",
" <span class=\"n\">value</span> <span class=\"o\">=</span> <span class=\"n\">T</span> <span class=\"o\">*</span> <span class=\"p\">(</span><span class=\"n\">log</span><span class=\"p\">(</span><span class=\"n\">post_pos</span> <span class=\"o\">/</span> <span class=\"p\">(</span><span class=\"n\">post_pos</span> <span class=\"o\">+</span> <span class=\"n\">post_neg</span><span class=\"p\">)</span> <span class=\"o\">+</span> <span class=\"mf\">1e-12</span><span class=\"p\">,</span><span class=\"mi\">2</span><span class=\"p\">)</span> <span class=\"o\">-</span> <span class=\"n\">log</span><span class=\"p\">(</span><span class=\"n\">pre_pos</span> <span class=\"o\">/</span> <span class=\"p\">(</span><span class=\"n\">pre_pos</span> <span class=\"o\">+</span> <span class=\"n\">pre_neg</span><span class=\"p\">),</span><span class=\"mi\">2</span><span class=\"p\">))</span>\n",
" <span class=\"c1\">#print (l, value)</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">value</span>\n",
"\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">update_examples</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">target</span><span class=\"p\">,</span> <span class=\"n\">examples</span><span class=\"p\">,</span> <span class=\"n\">extended_examples</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Add to the kb those examples what are represented in extended_examples</span>\n",
"<span class=\"sd\"> List of omitted examples is returned."""</span>\n",
" <span class=\"n\">uncovered</span> <span class=\"o\">=</span> <span class=\"p\">[]</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">example</span> <span class=\"ow\">in</span> <span class=\"n\">examples</span><span class=\"p\">:</span>\n",
" <span class=\"k\">def</span> <span class=\"nf\">represents</span><span class=\"p\">(</span><span class=\"n\">d</span><span class=\"p\">):</span>\n",
" <span class=\"k\">return</span> <span class=\"nb\">all</span><span class=\"p\">(</span><span class=\"n\">d</span><span class=\"p\">[</span><span class=\"n\">x</span><span class=\"p\">]</span> <span class=\"o\">==</span> <span class=\"n\">example</span><span class=\"p\">[</span><span class=\"n\">x</span><span class=\"p\">]</span> <span class=\"k\">for</span> <span class=\"n\">x</span> <span class=\"ow\">in</span> <span class=\"n\">example</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"nb\">any</span><span class=\"p\">(</span><span class=\"n\">represents</span><span class=\"p\">(</span><span class=\"n\">l</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">l</span> <span class=\"ow\">in</span> <span class=\"n\">extended_examples</span><span class=\"p\">):</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">tell</span><span class=\"p\">(</span><span class=\"n\">subst</span><span class=\"p\">(</span><span class=\"n\">example</span><span class=\"p\">,</span> <span class=\"n\">target</span><span class=\"p\">))</span>\n",
" <span class=\"k\">else</span><span class=\"p\">:</span>\n",
" <span class=\"n\">uncovered</span><span class=\"o\">.</span><span class=\"n\">append</span><span class=\"p\">(</span><span class=\"n\">example</span><span class=\"p\">)</span>\n",
"\n",
" <span class=\"k\">return</span> <span class=\"n\">uncovered</span>\n",
"</pre></div>\n",
"</body>\n",
"</html>\n"
],
"text/plain": [
"<IPython.core.display.HTML object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"psource(FOIL_container)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Example Family \n",
"Suppose we have the following family relations:\n",
"<br>\n",
"\n",
"<br>\n",
"Given some positive and negative examples of the relation 'Parent(x,y)', we want to find a set of rules that satisfies all the examples. <br>\n",
"\n",
"A definition of Parent is $Parent(x,y) \\Leftrightarrow Mother(x,y) \\lor Father(x,y)$, which is the result that we expect from the algorithm. "
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"A, B, C, D, E, F, G, H, I, x, y, z = map(expr, 'ABCDEFGHIxyz')"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [],
"source": [
"small_family = FOIL_container([expr(\"Mother(Anne, Peter)\"),\n",
" expr(\"Mother(Anne, Zara)\"),\n",
" expr(\"Mother(Sarah, Beatrice)\"),\n",
" expr(\"Mother(Sarah, Eugenie)\"),\n",
" expr(\"Father(Mark, Peter)\"),\n",
" expr(\"Father(Mark, Zara)\"),\n",
" expr(\"Father(Andrew, Beatrice)\"),\n",
" expr(\"Father(Andrew, Eugenie)\"),\n",
" expr(\"Father(Philip, Anne)\"),\n",
" expr(\"Father(Philip, Andrew)\"),\n",
" expr(\"Mother(Elizabeth, Anne)\"),\n",
" expr(\"Mother(Elizabeth, Andrew)\"),\n",
" expr(\"Male(Philip)\"),\n",
" expr(\"Male(Mark)\"),\n",
" expr(\"Male(Andrew)\"),\n",
" expr(\"Male(Peter)\"),\n",
" expr(\"Female(Elizabeth)\"),\n",
" expr(\"Female(Anne)\"),\n",
" expr(\"Female(Sarah)\"),\n",
" expr(\"Female(Zara)\"),\n",
" expr(\"Female(Beatrice)\"),\n",
" expr(\"Female(Eugenie)\"),\n",
"])\n",
"\n",
"target = expr('Parent(x, y)')\n",
"\n",
"examples_pos = [{x: expr('Elizabeth'), y: expr('Anne')},\n",
" {x: expr('Elizabeth'), y: expr('Andrew')},\n",
" {x: expr('Philip'), y: expr('Anne')},\n",
" {x: expr('Philip'), y: expr('Andrew')},\n",
" {x: expr('Anne'), y: expr('Peter')},\n",
" {x: expr('Anne'), y: expr('Zara')},\n",
" {x: expr('Mark'), y: expr('Peter')},\n",
" {x: expr('Mark'), y: expr('Zara')},\n",
" {x: expr('Andrew'), y: expr('Beatrice')},\n",
" {x: expr('Andrew'), y: expr('Eugenie')},\n",
" {x: expr('Sarah'), y: expr('Beatrice')},\n",
" {x: expr('Sarah'), y: expr('Eugenie')}]\n",
"examples_neg = [{x: expr('Anne'), y: expr('Eugenie')},\n",
" {x: expr('Beatrice'), y: expr('Eugenie')},\n",
" {x: expr('Mark'), y: expr('Elizabeth')},\n",
" {x: expr('Beatrice'), y: expr('Philip')}]"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[Parent(x, y), [Mother(x, y)]], [Parent(x, y), [Father(x, y)]]]\n"
]
}
],
"source": [
"# run the FOIL algorithm \n",
"clauses = small_family.foil([examples_pos, examples_neg], target)\n",
"print (clauses)\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Indeed the algorithm returned the rule: \n",
"<br>$Parent(x,y) \\Leftrightarrow Mother(x,y) \\lor Father(x,y)$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Suppose that we have some possitive and negative results for the relation 'GrandParent(x,y)' and we want to find a set of rules that satisfies the examples. <br>\n",
"One possible set of rules for the relation $Grandparent(x,y)$ could be: <br>\n",
"\n",
"<br>\n",
"Or, if $Background$ included the sentence $Parent(x,y) \\Leftrightarrow [Mother(x,y) \\lor Father(x,y)]$ then: \n",
"\n",
"$$Grandparent(x,y) \\Leftrightarrow \\exists \\: z \\quad Parent(x,z) \\land Parent(z,y)$$\n"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[Grandparent(x, y), [Parent(x, v_5), Parent(v_5, y)]]]\n"
]
}
],
"source": [
"target = expr('Grandparent(x, y)')\n",
"\n",
"examples_pos = [{x: expr('Elizabeth'), y: expr('Peter')},\n",
" {x: expr('Elizabeth'), y: expr('Zara')},\n",
" {x: expr('Elizabeth'), y: expr('Beatrice')},\n",
" {x: expr('Elizabeth'), y: expr('Eugenie')},\n",
" {x: expr('Philip'), y: expr('Peter')},\n",
" {x: expr('Philip'), y: expr('Zara')},\n",
" {x: expr('Philip'), y: expr('Beatrice')},\n",
" {x: expr('Philip'), y: expr('Eugenie')}]\n",
"examples_neg = [{x: expr('Anne'), y: expr('Eugenie')},\n",
" {x: expr('Beatrice'), y: expr('Eugenie')},\n",
" {x: expr('Elizabeth'), y: expr('Andrew')},\n",
" {x: expr('Elizabeth'), y: expr('Anne')},\n",
" {x: expr('Elizabeth'), y: expr('Mark')},\n",
" {x: expr('Elizabeth'), y: expr('Sarah')},\n",
" {x: expr('Philip'), y: expr('Anne')},\n",
" {x: expr('Philip'), y: expr('Andrew')},\n",
" {x: expr('Anne'), y: expr('Peter')},\n",
" {x: expr('Anne'), y: expr('Zara')},\n",
" {x: expr('Mark'), y: expr('Peter')},\n",
" {x: expr('Mark'), y: expr('Zara')},\n",
" {x: expr('Andrew'), y: expr('Beatrice')},\n",
" {x: expr('Andrew'), y: expr('Eugenie')},\n",
" {x: expr('Sarah'), y: expr('Beatrice')},\n",
" {x: expr('Mark'), y: expr('Elizabeth')},\n",
" {x: expr('Beatrice'), y: expr('Philip')}, \n",
" {x: expr('Peter'), y: expr('Andrew')}, \n",
" {x: expr('Zara'), y: expr('Mark')},\n",
" {x: expr('Peter'), y: expr('Anne')},\n",
" {x: expr('Zara'), y: expr('Eugenie')}, ]\n",
"\n",
"clauses = small_family.foil([examples_pos, examples_neg], target)\n",
"\n",
"print(clauses)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Indeed the algorithm returned the rule: \n",
"<br>$Grandparent(x,y) \\Leftrightarrow \\exists \\: v \\: \\: Parent(x,v) \\land Parent(v,y)$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Example Network\n",
"\n",
"Suppose that we have the following directed graph and we want to find a rule that describes the reachability between two nodes (Reach(x,y)). <br>\n",
"Such a rule could be recursive, since y can be reached from x if and only if there is a sequence of adjacent nodes from x to y: \n",
"\n",
"$$ Reach(x,y) \\Leftrightarrow \\begin{cases} \n",
" Conn(x,y), \\: \\text{(if there is a directed edge from x to y)} \\\\\n",
" \\lor \\quad \\exists \\: z \\quad Reach(x,z) \\land Reach(z,y) \\end{cases}$$\n"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [],
"source": [
"\"\"\"\n",
"A H\n",
"|\\ /|\n",
"| \\ / |\n",
"v v v v\n",
"B D-->E-->G-->I\n",
"| / |\n",
"| / |\n",
"vv v\n",
"C F\n",
"\"\"\"\n",
"small_network = FOIL_container([expr(\"Conn(A, B)\"),\n",
" expr(\"Conn(A ,D)\"),\n",
" expr(\"Conn(B, C)\"),\n",
" expr(\"Conn(D, C)\"),\n",
" expr(\"Conn(D, E)\"),\n",
" expr(\"Conn(E ,F)\"),\n",
" expr(\"Conn(E, G)\"),\n",
" expr(\"Conn(G, I)\"),\n",
" expr(\"Conn(H, G)\"),\n",
" expr(\"Conn(H, I)\")])\n"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[Reach(x, y), [Conn(x, y)]], [Reach(x, y), [Reach(x, v_12), Reach(v_14, y), Reach(v_12, v_16), Reach(v_12, y)]], [Reach(x, y), [Reach(x, v_20), Reach(v_20, y)]]]\n"
]
}
],
"source": [
"target = expr('Reach(x, y)')\n",
"examples_pos = [{x: A, y: B},\n",
" {x: A, y: C},\n",
" {x: A, y: D},\n",
" {x: A, y: E},\n",
" {x: A, y: F},\n",
" {x: A, y: G},\n",
" {x: A, y: I},\n",
" {x: B, y: C},\n",
" {x: D, y: C},\n",
" {x: D, y: E},\n",
" {x: D, y: F},\n",
" {x: D, y: G},\n",
" {x: D, y: I},\n",
" {x: E, y: F},\n",
" {x: E, y: G},\n",
" {x: E, y: I},\n",
" {x: G, y: I},\n",
" {x: H, y: G},\n",
" {x: H, y: I}]\n",
"nodes = {A, B, C, D, E, F, G, H, I}\n",
"examples_neg = [example for example in [{x: a, y: b} for a in nodes for b in nodes]\n",
" if example not in examples_pos]\n",
"clauses = small_network.foil([examples_pos, examples_neg], target)\n",
"\n",
"print(clauses)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
MariannaSpyrakou
a validé
"The algorithm produced something close to the recursive rule: \n",
" $$ Reach(x,y) \\Leftrightarrow [Conn(x,y)] \\: \\lor \\: [\\exists \\: z \\: \\: Reach(x,z) \\, \\land \\, Reach(z,y)]$$\n",
" \n",
MariannaSpyrakou
a validé
"This happened because the size of the example is small. "
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.5.3"
}
},
"nbformat": 4,
"nbformat_minor": 2
}