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UIUC CS 497: Section EA Lecture #2

UIUC CS 497: Section EA Lecture #2. Reasoning in Artificial Intelligence Professor: Eyal Amir Spring Semester 2004. Last Time. Propositional logic as a language for representing knowledge Did not touch on reasoning procedures Defined language, signature, models

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UIUC CS 497: Section EA Lecture #2

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  1. UIUC CS 497: Section EALecture #2 Reasoning in Artificial Intelligence Professor: Eyal Amir Spring Semester 2004

  2. Last Time • Propositional logic as a language for representing knowledge • Did not touch on reasoning procedures • Defined language, signature, models • From homeworks: you should know • Soundness; Completeness theorem • Deduction theorem • De Morgan Laws

  3. Today • Reasoning procedures for propositional logic • Checking Satisfiability (SAT) using DPLL • Proving entailment using Resolution • Application du jour: Formal Verification • Applications we will nottouch • AI planning, graph algorithms, cryptography, …

  4. SAT via Generate and Test • If we have a truth table of KB, then we can check that KB satisfiable by looking at it. • Problem: n propositional symbols  2n rows in truth table • Checking interpretation I takes time O(|KB|) • Generating table is expensive: O(2n |KB|) time • Observation: SAT requires us to look only for one model

  5. º Ú Ø Ù Ú Ù Ú Ø Ù Ú ( a a ) ( a b ) ( b a ) ( b b ) Clausal Form • Every formula can be reformulated into an equivalent CNF formula (conjunction of clauses). • Examples (using De Morgan Laws):

  6. Clausal Form • Every formula can be reformulated into an equivalent CNF formula (conjunction of clauses). • Examples:

  7. Clausal Form • Every formula can be reformulated into an equivalent CNF formula (conjunction of clauses). • Examples:

  8. Observe Propagating a Truth-Value • KB in CNF, and we observe p=TRUE • Then, removing clauses with p positive from KB gives an equivalent theory. • Example: KB

  9. Propagating a Truth-Value • KB in CNF, and we observe p=TRUE • Then, removing negative instances of p from KB gives an equivalent theory. • Example: KB Observe

  10. DPLL Search Procedure for CNF • If no clauses in KB, return T • If a clause in KB is empty (FALSE), return F • If KB has a unit clause C with prop. p, then return DPLL(KB,p←polarity(p,C)) • Choose an uninstantiated variable p • If DPLL(KB, p←TRUE) returns T, return T • If DPLL(KB, p←FALSE) returns T, return T • Return F

  11. DPLL in Action On board

  12. DPLL in Action On board Note: we could know without thorough checking that this KB is satisfiable

  13. DPLL in Action On board

  14. Related: SAT Solving • Order of selection of variables (lecture #5) • Stochastic local search (paper #1) • Binary Decision Diagrams (paper #2) • Strategies other than unit (paper #23) • 2-SAT is solvable in linear time • Smart backtracking (paper #21) • Clauses/Vars in Random SAT (paper #22) • SAT via probabilistic models (paper #15)

  15. Take a Breath • Until now: SAT solving • Search in the space of models • From now: Resolution theorem proving • Search in the space of proofs • Later: Formal Verification

  16. Resolution Theorem Proving • Given: • KB – a set of propositional sentences • Query Q – a logical sentence • Calling procedure: • Add Q to KB • Convert KB into clausal form • Run theorem prover. If we prove contradiction, return T. Otherwise, return F.

  17. Resolution Theorem Proving • Add Q to KB • Convert KB into clausal form • Run theorem prover. If we prove contradiction, return T. Otherwise, return F. Deduction theorem: KB Q iff KB  Q FALSE ╨ ╨

  18. Resolution Theorem Proving • Add Q to KB • Convert KB into clausal form • Run theorem prover. If we prove contradiction, return T. Otherwise, return F. Deduction theorem: KB Q iff KB  Q FALSE ╨ ╨

  19. Propositional Resolution • Resolution inference rule: C1: p1  C1’ C2: p1  C2’ -------------------- C3: C1’  C2’ C1  p1  C1’ C2  p1  C2’

  20. Propositional Resolution • Resolution algorithm (saturation): • While there are unresolved C1,C2: • Select C1, C2 in KB • If C1, C2 are resolvable, resolve them into a new clause C3 • Add C3 to KB • If C3={ } (empty clause), we got a contradiction. • STOP C1: p1  C1’ C2: p1  C2’ -------------------- C3: C1’  C2’

  21. Resolution in Action On board C1: p1  C1’ C2: p1  C2’ -------------------- C3: C1’  C2’ KB Negated Query

  22. Resolution in Action On board C1: p1  C1’ C2: p1  C2’ -------------------- C3: C1’  C2’ KB Negated Query

  23. Properties of Resolution • Running time for n variables, m clauses: • Resolving two clauses: O(n) • Finding two resolvable clauses: O(1) • Overall algorithm: O(3nn)

  24. 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 ╨

  25. Properties of Resolution • 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 ╨

  26. Simple Enhancements • 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 } • Unit resolution: resolve unit clauses first

  27. Related to Prop. Resolution • Clause selection for resolution (lecture #5, paper #19) • Consequence finding (paper #3) • Prime implicates/implicants (paper #4)

  28. Resolution vs SAT • SAT solvers can find models • Resolution sometimes better at finding contradictions • With resolution it is easier to explain and provide a proof

  29. Summary So Far • Finding models using DPLL • Resolution theorem proving allows us to find contradictions and explanation. • The deduction theorem tells us how to ask queries from either SAT solvers or Resolution

  30. Application: Hardware Verification f3 x1 f1 not AND x2 f5 AND not f2 OR x3 f4 Question: Can we set this boolean cirtuit to TRUE? f5(x1,x2,x3) = a function of the input signal

  31. Application: Hardware Verification f3 x1 f1 not AND x2 f5 AND not f2 OR SAT(f5) ? x3 f4 Question: Can we set this boolean cirtuit to TRUE? f5(x1,x2,x3) = f3 f4 = f1  (f2  x3) = (x1  x2)  (x2  x3) M[x1]=FALSE M[x2]=FALSE M[x3]=FALSE

  32. Hardware Verification • Questions in logical circuit verification • Equivalence of circuits • Arrival of the circuit to a state (required a temporal model, which gets propositionalized) • Achieving an output from the circuit

  33. 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

  34. Next Time • FOL Resolution • Homework: • Read readings (incl. application reading) • Make sure you know: • Deduction theorem for FOL • Language of FOL • Soundness, completeness, and incompleteness theorems • Models of first-order logic

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