Propositional Reasoning
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1 Propositional Reasoning CS 440 / ECE 448 Introduction to Artificial Intelligence Instructor: Eyal Amir Grad TAs: Wen Pu, Yonatan Bisk Undergrad TAs: Sam Johnson, Nikhil Johri Spring 2010 Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
2 Outline: Propositional Logic Basics of Propositional Logic Review Semantics and Deduction Generate and Test Review Checking Satisfiability (SAT) using DPLL Syntactic Manipulation: Entailment and Refutation using Resolution Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
3 Propositional Logic Semantics Truth assignments that satisfy KB/formula Semantics VS. Deduction α β α β A deductive system or an inference algorithm is Sound If it derives only entailed sentences Complete If it can derive any sentence that is entailed Decidable If it terminates in finite steps and are equivalent if deduction is sound and complete Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
4 More Notations Interpretations - models if True Axioms - formulae that are assumed Signature - the symbols used by a KB Theory KB (a set of axioms) The complete set of sentences entailed by the axioms Sentence = Formula (in prop. logic) M[p] - value of symbol p in model M Clauses: {x 1, x 2, x 3,...} or x 1 x 2 x 3... Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
5 Outline: Propositional Logic Basics of Propositional Logic Review Semantics and Deduction Generate and Test Review Checking Satisfiability (SAT) using DPLL Syntactic Manipulation: Entailment and Refutation using Resolution Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
6 SAT Application: Hardware Verification f 5 (x 1, x 2, x 3 ) = f 3 f 4 = f 1 (f 2 x 3 ) = (x 1 x 2 ) ( x 2 x 3 ) M[x 1 ] = False, M[x 2 ] = False, M[x 3 ] = False Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
7 DPLL: Propagating a Truth-Value Model Checking KB in CNF, and we denote a = True Removing clauses with a positive from KB gives an equivalent theory. Removing negative instances of a from KB gives an equivalent theory. KB Observe a b c a b d a Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
8 DPLL: Propagating a Truth-Value Model Checking KB in CNF, and we denote a = True Removing clauses with a positive from KB gives an equivalent theory. Removing negative instances of a from KB gives an equivalent theory. KB Observe a b c a b d a Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
9 DPLL: Propagating a Truth-Value Model Checking KB in CNF, and we denote a = True Removing clauses with a positive from KB gives an equivalent theory. Removing negative instances of a from KB gives an equivalent theory. KB a b c a b d Observe a Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
10 DPLL: Search Heuristics Early termination: look only for one model Unit propagation: clauses with only one symbol Pure literal elimination (obsolete) DPLL is sound and complete Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
11 DPLL: Search Heuristics Early termination: look only for one model Unit propagation: clauses with only one symbol Pure literal elimination (obsolete) DPLL is sound and complete Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
12 DPLL Example Clauses 1.a b 2. a b 3.a c 4.c d e 5.d e 6. d f 7.f e 8. f e Steps a = F (b is pure) a = F, b = T (3 is unit) a = F, b = T, c = F a = F, b = T, c = F, d = F (4 is unit) a = F, b = T, c = F, d = F, e = T (unsatisfied, early termination, backtrace) a = F, b = T, c = F, d = T (6 is unit) a = F, b = T, c = F, d = T, f = F (e is pure) a = F, b = T, c = F, d = T, f = F, e = T (SAT!) Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
13 DPLL Example Clauses 1. a b 2. a b 3. a c 4.c d e 5.d e 6. d f 7.f e 8. f e Steps a = F (b is pure) a = F, b = T (3 is unit) a = F, b = T, c = F a = F, b = T, c = F, d = F (4 is unit) a = F, b = T, c = F, d = F, e = T (unsatisfied, early termination, backtrace) a = F, b = T, c = F, d = T (6 is unit) a = F, b = T, c = F, d = T, f = F (e is pure) a = F, b = T, c = F, d = T, f = F, e = T (SAT!) Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
14 DPLL Example Clauses 1. a b 2. a b 3. a c 4.c d e 5.d e 6. d f 7.f e 8. f e Steps a = F (b is pure) a = F, b = T (3 is unit) a = F, b = T, c = F a = F, b = T, c = F, d = F (4 is unit) a = F, b = T, c = F, d = F, e = T (unsatisfied, early termination, backtrace) a = F, b = T, c = F, d = T (6 is unit) a = F, b = T, c = F, d = T, f = F (e is pure) a = F, b = T, c = F, d = T, f = F, e = T (SAT!) Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
15 DPLL Example Clauses 1. a b 2. a b 3. a c 4. c d e 5.d e 6. d f 7.f e 8. f e Steps a = F (b is pure) a = F, b = T (3 is unit) a = F, b = T, c = F a = F, b = T, c = F, d = F (4 is unit) a = F, b = T, c = F, d = F, e = T (unsatisfied, early termination, backtrace) a = F, b = T, c = F, d = T (6 is unit) a = F, b = T, c = F, d = T, f = F (e is pure) a = F, b = T, c = F, d = T, f = F, e = T (SAT!) Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
16 DPLL Example Clauses 1. a b 2. a b 3. a c 4. c d e 5. d e 6. d f 7.f e 8. f e Steps a = F (b is pure) a = F, b = T (3 is unit) a = F, b = T, c = F a = F, b = T, c = F, d = F (4 is unit) a = F, b = T, c = F, d = F, e = T (unsatisfied, early termination, backtrace) a = F, b = T, c = F, d = T (6 is unit) a = F, b = T, c = F, d = T, f = F (e is pure) a = F, b = T, c = F, d = T, f = F, e = T (SAT!) Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
17 DPLL Example Clauses 1. a b 2. a b 3. a c 4. c d e 5. d e 6. d f 7.f e 8. f e Steps a = F (b is pure) a = F, b = T (3 is unit) a = F, b = T, c = F a = F, b = T, c = F, d = F (4 is unit) a = F, b = T, c = F, d = F, e = T (unsatisfied, early termination, backtrace) a = F, b = T, c = F, d = T (6 is unit) a = F, b = T, c = F, d = T, f = F (e is pure) a = F, b = T, c = F, d = T, f = F, e = T (SAT!) Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
18 DPLL Example Clauses 1. a b 2. a b 3. a c 4. c d e 5.d e 6. d f 7.f e 8. f e Steps a = F (b is pure) a = F, b = T (3 is unit) a = F, b = T, c = F a = F, b = T, c = F, d = F (4 is unit) a = F, b = T, c = F, d = F, e = T (unsatisfied, early termination, backtrace) a = F, b = T, c = F, d = T (6 is unit) a = F, b = T, c = F, d = T, f = F (e is pure) a = F, b = T, c = F, d = T, f = F, e = T (SAT!) Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
19 DPLL Example Clauses 1. a b 2. a b 3. a c 4. c d e 5. d e 6. d f 7.f e 8. f e Steps a = F (b is pure) a = F, b = T (3 is unit) a = F, b = T, c = F a = F, b = T, c = F, d = F (4 is unit) a = F, b = T, c = F, d = F, e = T (unsatisfied, early termination, backtrace) a = F, b = T, c = F, d = T (6 is unit) a = F, b = T, c = F, d = T, f = F (e is pure) a = F, b = T, c = F, d = T, f = F, e = T (SAT!) Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
20 DPLL Example Clauses 1. a b 2. a b 3. a c 4. c d e 5. d e 6. d f 7. f e 8. f e Steps a = F (b is pure) a = F, b = T (3 is unit) a = F, b = T, c = F a = F, b = T, c = F, d = F (4 is unit) a = F, b = T, c = F, d = F, e = T (unsatisfied, early termination, backtrace) a = F, b = T, c = F, d = T (6 is unit) a = F, b = T, c = F, d = T, f = F (e is pure) a = F, b = T, c = F, d = T, f = F, e = T (SAT!) Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
21 DPLL Example Clauses 1. a b 2. a b 3. a c 4. c d e 5. d e 6. d f 7. f e 8. f e Steps a = F (b is pure) a = F, b = T (3 is unit) a = F, b = T, c = F a = F, b = T, c = F, d = F (4 is unit) a = F, b = T, c = F, d = F, e = T (unsatisfied, early termination, backtrace) a = F, b = T, c = F, d = T (6 is unit) a = F, b = T, c = F, d = T, f = F (e is pure) a = F, b = T, c = F, d = T, f = F, e = T (SAT!) Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
22 Outline: Propositional Logic Basics of Propositional Logic Review Semantics and Deduction Generate and Test Review Checking Satisfiability (SAT) using DPLL Syntactic Manipulation: Entailment and Refutation using Resolution Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
23 Resolution Theorem Proving Resolution theorem (converse of deduction theorem): KB Q iff KB Q False Given query Q and a database of facts KB Add Q to KB Convert KB into CNF Run theorem prover, if we prove contradiction, return True, otherwise return False. Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
24 Resolution Theorem Proving Resolution theorem (converse of deduction theorem): KB Q iff KB Q False Given query Q and a database of facts KB Add Q to KB Convert KB into CNF Run theorem prover, if we prove contradiction, return True, otherwise return False. Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
25 Resolution Rule Resolution rule Intuition: x a 1 a m, x b 1 b n a 1 a m b 1 b n x α, x β α β Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
26 Resolution Rule Resolution rule Intuition: x a 1 a m, x b 1 b n a 1 a m b 1 b n x α, x β α β Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
27 Propositional Resolution Resolution Algorithm 1 While there are unresolved C 1, C 2 1 Select C 1, C 2 in KB 2 If C 1, C 2 are resolvable, resolve them into new clause C 3 and add it to KB 3 If C 3 = {} we got a contradiction 2 STOP Theorem Proving VS. Model Checking (Difference?) C 1 : p 1 C 1 C 2 : p 1 C 2 C 3 : C 1 C 2 Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
28 Propositional Resolution Resolution Algorithm 1 While there are unresolved C 1, C 2 1 Select C 1, C 2 in KB 2 If C 1, C 2 are resolvable, resolve them into new clause C 3 and add it to KB 3 If C 3 = {} we got a contradiction 2 STOP Theorem Proving VS. Model Checking (Difference?) C 1 : p 1 C 1 C 2 : p 1 C 2 C 3 : C 1 C 2 Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
29 Resolution Example: Constructive Dilemma Problem: Constructive Dilemma KB: p q, r s, p r Theorem to prove: q s (p q) (r s) (p r) q s Convert to CNF KB: p q, r s, p r Negated query: q, s Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
30 Resolution Example: Constructive Dilemma Problem: Constructive Dilemma KB: p q, r s, p r Theorem to prove: q s (p q) (r s) (p r) q s Convert to CNF KB: p q, r s, p r Negated query: q, s Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
31 Resolution Example (cont.) Formulas: 1. p q 2. r s 3. p r 4. q 5. s Steps 6. p (1, 4) 7. r (3, 6) 8. s (2, 7) 9. (empty) (5, 8) Contradiction! Conclude: (p q) (r s) (p r) resolution q s Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
32 Resolution Example (cont.) Formulas: 1. p q 2. r s 3. p r 4. q 5. s Steps 6. p (1, 4) 7. r (3, 6) 8. s (2, 7) 9. (empty) (5, 8) Contradiction! Conclude: (p q) (r s) (p r) resolution q s Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
33 Resolution Example (cont.) Formulas: 1. p q 2. r s 3. p r 4. q 5. s Steps 6. p (1, 4) 7. r (3, 6) 8. s (2, 7) 9. (empty) (5, 8) Contradiction! Conclude: (p q) (r s) (p r) resolution q s Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
34 Resolution Example (cont.) Formulas: 1. p q 2. r s 3. p r 4. q 5. s Steps 6. p (1, 4) 7. r (3, 6) 8. s (2, 7) 9. (empty) (5, 8) Contradiction! Conclude: (p q) (r s) (p r) resolution q s Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
35 Resolution Example (cont.) Formulas: 1. p q 2. r s 3. p r 4. q 5. s Steps 6. p (1, 4) 7. r (3, 6) 8. s (2, 7) 9. (empty) (5, 8) Contradiction! Conclude: (p q) (r s) (p r) resolution q s Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
36 Resolution Example (cont.) Formulas: 1. p q 2. r s 3. p r 4. q 5. s Steps 6. p (1, 4) 7. r (3, 6) 8. s (2, 7) 9. (empty) (5, 8) Contradiction! Conclude: (p q) (r s) (p r) resolution q s Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
37 Properties of Resolution Theorem: Resolution is sound Resolving clauses in KB generates valid consequences of KB Theorem: Resolution is refutation complete Resolution of KB with Q yields the empty clause iff KB Q (p q) (r s) (p r) q s Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
38 Properties of Resolution Theorem: Resolution is sound Resolving clauses in KB generates valid consequences of KB Theorem: Resolution is refutation complete Resolution of KB with Q yields the empty clause iff KB Q (p q) (r s) (p r) q s Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
39 Properties of Resolution Will resolution only in KB generate Q? Resolution does not always generate Q KB = {{a, b}, { a, b}, {b, c}} Q = b c = {b, c} Theorem: Resolution always generates a clause that subsumes Q iff KB Q. Example: Resolving KB generates b Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
40 Properties of Resolution Will resolution only in KB generate Q? Resolution does not always generate Q KB = {{a, b}, { a, b}, {b, c}} Q = b c = {b, c} Theorem: Resolution always generates a clause that subsumes Q iff KB Q. Example: Resolving KB generates b Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
41 Exercise Use resolution refutation prove KB: P Q, P R Query: Q R Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
42 Solution Convert to CNF KB: P Q, P R, Negated query: Q, R Formulas: 1. P Q 2. P R 3. Q 4. R 5. P Q Steps: 5. P (1,3) 6. R (2, 5) 7. Q R (1, 2) Terminate without producing empty clause! KB Q R Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
43 Solution Convert to CNF KB: P Q, P R, Negated query: Q, R Formulas: 1. P Q 2. P R 3. Q 4. R 5. P Q Steps: 5. P (1,3) 6. R (2, 5) 7. Q R (1, 2) Terminate without producing empty clause! KB Q R Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
44 Solution Convert to CNF KB: P Q, P R, Negated query: Q, R Formulas: 1. P Q 2. P R 3. Q 4. R 5. P Q Steps: 5. P (1,3) 6. R (2, 5) 7. Q R (1, 2) Terminate without producing empty clause! KB Q R Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
45 Solution Convert to CNF KB: P Q, P R, Negated query: Q, R Formulas: 1. P Q 2. P R 3. Q 4. R 5. P Q Steps: 5. P (1,3) 6. R (2, 5) 7. Q R (1, 2) Terminate without producing empty clause! KB Q R Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
46 Solution Convert to CNF KB: P Q, P R, Negated query: Q, R Formulas: 1. P Q 2. P R 3. Q 4. R 5. P Q Steps: 5. P (1,3) 6. R (2, 5) 7. Q R (1, 2) Terminate without producing empty clause! KB Q R Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
47 Resolution Strategies Unit Preference: resolve unit clauses (one literal clause) first Generate shorter clause (our target is a zero length clause, i.e. contradiction) Set of Support: Choose a resolution involving the negated goal or any clause derived from the negated goal Target is to produce contradiction from negated query, so these are more relevant If contradiction exists, we can find one using this strategy Other improvements Remove subsumed clauses {p} subsumes {p, q} {p, q} subsumes {p, q, r} { p} does not subsume {p, q} Contract same literals {p, p, q} becomes {p, q} Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
48 Resolution Strategies Unit Preference: resolve unit clauses (one literal clause) first Generate shorter clause (our target is a zero length clause, i.e. contradiction) Set of Support: Choose a resolution involving the negated goal or any clause derived from the negated goal Target is to produce contradiction from negated query, so these are more relevant If contradiction exists, we can find one using this strategy Other improvements Remove subsumed clauses {p} subsumes {p, q} {p, q} subsumes {p, q, r} { p} does not subsume {p, q} Contract same literals {p, p, q} becomes {p, q} Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
49 Resolution Strategies Unit Preference: resolve unit clauses (one literal clause) first Generate shorter clause (our target is a zero length clause, i.e. contradiction) Set of Support: Choose a resolution involving the negated goal or any clause derived from the negated goal Target is to produce contradiction from negated query, so these are more relevant If contradiction exists, we can find one using this strategy Other improvements Remove subsumed clauses {p} subsumes {p, q} {p, q} subsumes {p, q, r} { p} does not subsume {p, q} Contract same literals {p, p, q} becomes {p, q} Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
50 Resolution vs. SAT Theorem proving vs Model checking SAT solvers can find models Resolution sometimes better at finding contradictions With resolution it is easier to explain and provide a proof Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
51 Resolution in Prop. Logc and FOL Similarities Both use clausal form Differences Unification is needed in FOL Always terminates in prop. logic, but does not guarantee to terminate in FOL Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
52 Summary SAT checking using DPLL (instantiate, propagate, backtrack) Entailment/SAT checking using resolution (create more and more clauses until KB is saturated) Formal verification uses mainly SAT checking such as DPLL, but also sometimes resolution. Intro to AI (CS 440 / ECE 448) Propositional Reasoning Spring / 23
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