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Automated Theorem Proving

Automated Theorem Proving. Lecture 2 Propositional Satisfiability . Decision procedures. Boolean programs Propositional satisfiability Arithmetic programs Propositional satisfiability modulo theory of linear arithmetic Memory programs

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Automated Theorem Proving

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  1. Automated Theorem Proving Lecture 2 Propositional Satisfiability

  2. Decision procedures • Boolean programs • Propositional satisfiability • Arithmetic programs • Propositional satisfiability modulo theory of linear arithmetic • Memory programs • Propositional satisfiability modulo theory of linear arithmetic + arrays

  3. Case I: Boolean programs • Boolean-valued variables and boolean operations •  Formula := b |  |    b  SymBoolConst

  4. SAT • First NP-complete problem (Cook 1972) • Davis-Putnam algorithm (1960) • resolution-based • may use exponential memory • Davis-Logemann-Loveland algorithm (1962) • search-based • basis for all successful modern solvers • Conflict-driven learning and non-chronological backtracking (1996) • resolution strikes back! • Amazing progress • GRASP, SATO, Chaff, ZChaff, BerkMin, …

  5. Conjunctive Normal Form •  CNF Formula ::= c1  c2  … cm • c  Clause ::= l1  l2  … ln • l  Literal ::= b | b • b  SymBoolConst • Unit clause ( l ) • a clause containing a single literal • Empty clause ( ) • a clause containing no literal • equivalent to false

  6. Conversion into CNF • In general, converting  into an equivalent CNF formula may result in an exponential blow-up • We are only interested in satisfiability of  • Convert into an equi-satisfiable CNF formula EQCNF() •  is satisfiable iff EQCNF() is satisfiable • size of EQCNF() is polynomial in size of 

  7. Conversion into CNF • Convert formula  into normal form NF() • NF() is polynomial in  • Convert  = NF() into equisatisfiable CNF formula EQCNF() • EQCNF() is polynomial in 

  8. Normal Form Normal form: NF()   Negated normal form: NNF()   NF(b) = b NNF(b) = b NF() = NNF() NNF() = NF() NF(1  2) = NF(1)  NF(1) NNF(1  2) = NNF(1)  NNF(2)

  9. Equi-satisfiable CNF Let  be a formula in normal form. For each subformula  of : - create a fresh symbol v in SymBoolConst Identify vb with b and vb with b Cl(b) = Cl(b) = true Cl() = Cl()  Cl()  (v v v)  (v  v)  (v  v) Cl() = Cl()  Cl()  (v  v v)  (v  v)  (v  v) EQCNF() = v  Cl()

  10. Resolution c1, c2 independent of b clauses (c1 b) (c2  b) (c1  c2) resolvent resolvent(b, c1 b, c2  b) = c1 c2 = b. (c1 b)  (c2  b)

  11. Theorem   (c1 b)  (c2  b) iff   (c1 b)  (c2  b)  (c1 c2) Adding the resolvent to the set of clauses does not affect the satisfiability of the clause set.

  12. Unit resolution One of the clauses being resolved is a unit clause ( b ) (c2  b) ( c2 ) ( b ) (c2  b) ( c2 ) Derivation of the empty clause (denoted by ) ( b ) ( b ) 

  13. Davis-Putnam algorithm (I) Given clause set C: Rule 1: If a clause (c  l  l) C, replace it with (c  l) Rule 2: If a clause (c  b  b) C, remove it from C Rule 3a: If b does not occur in any clause in C, remove every clause containing b from C Rule 3b: If b does not occur in any clause in C, remove every clause containing b from C

  14. Davis-Putnam algorithm (II) Saturate C w.r.t Rules 1, 2, 3a, and 3b while (C is nonempty) { Pick a variable b appearing in some clause in C C’ = { resolvent(b,c1,c2) | c1,c2 C } Saturate C’ w.r.t. Rules 1, 2, 3a, and 3b if (  C’) return unsatisfiable C = C’ } return satisfiable

  15. Rule 3a (b  c  f) (b  c) Resolve on b (c  c  f) Rule 2 Clause set is empty Satisfiable example (a  b  c) (b  c  f) (b  c)

  16. Unsatisfiable example (a  b) (a b) (a  c) (a  c) Pick b ( a ) (a  c) (a  c) Pick a ( c ) ( c ) Pick c 

  17. Correctness Saturate C w.r.t Rules 1, 2, 3a, and 3b while (C is nonempty) { Pick a variable b appearing in some clause in C C’ = { resolvent(b,c1,c2) | c1,c2 C } Saturate C’ w.r.t. Rules 1, 2, 3a, and 3b if (  C’) return unsatisfiable C = C’ } return satisfiable Two observations: - Each of the rules 1, 2, 3a, and 3b preserve satisfiability - C’ = b. C

  18. Memory explosion Saturate C w.r.t Rules 1, 2, 3a, and 3b while (C is nonempty) { Pick a variable b appearing in some clause in C C’ = { resolvent(b,c1,c2) | c1,c2 C } Saturate C’ w.r.t. Rules 1, 2, 3a, and 3b if (  C’) return unsatisfiable C = C’ } return satisfiable Let n be the number of clauses in the input clause set Number of clauses after i-th iteration of loop: O(n^(2^i))

  19. Davis-Logemann-Loveland algorithm Slides 42-72 of sat_course1.pdf Download from: http://research.microsoft.com/users/lintaoz/SATSolving/satsolving.htm

  20. Davis-Logemann-Loveland algorithm • Eliminates exponential memory requirement • Might still need exponential time

  21. Conflict-driven learning and non-chronological backtracking Slides 2-20 of sat_course2.pdf Download from: http://research.microsoft.com/users/lintaoz/SATSolving/satsolving.htm

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