Automated Theorem Proving

Lecture 2Propositional Satisfiability

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

Case I: Boolean programs

• Boolean-valued variables and boolean operations

Formula := b | | b SymBoolConst

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, …

Conjunctive Normal Form

CNF Formula ::= c1 c2 … cm

c Clause ::= l1 l2 … lnl Literal ::= b | bb SymBoolConst

Unit clause ( l )-a clause containing a single literal

Empty clause ( )- a clause containing no literal - equivalent to false

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

Conversion into CNF

• Convert formula into normal form NF()– NF() is polynomial in

• Convert = NF() into equisatisfiable CNF formula EQCNF()– EQCNF() is polynomial in

Normal form: NF() Negated normal form: NNF()

Normal Form

NF(b) = bNNF(b) = b

NF() = NNF()NNF() = NF()

NF(1 2) = NF(1) NF(1)NNF(1 2) = NNF(1) NNF(2)

Equi-satisfiable CNF

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)

Let be a formula in normal form.For each subformula of : - create a fresh symbol v in SymBoolConstIdentify vb with b and vb with b

EQCNF() = v Cl()

Resolution

(c1 b) (c2 b)

(c1 c2)

clauses

resolvent

resolvent(b, c1 b, c2 b) = c1 c2 = b. (c1 b) (c2 b)

c1, c2 independent of b

(c1 b) (c2 b)iff

(c1 b) (c2 b) (c1 c2)

Theorem

Adding the resolvent to the set of clauses does not affect the satisfiability of the clause set.

Unit resolution

( b ) (c2 b)

( c2 )

One of the clauses being resolved is a unit clause

Derivation of the empty clause (denoted by )

( b ) ( b )

( b ) (c2 b)

( c2 )

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

Davis-Putnam algorithm (II)

Saturate C w.r.t Rules 1, 2, 3a, and 3bwhile (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

(a b c) (b c f) (b c)

Satisfiable example

(b c f) (b c)

Rule 3a

(c c f)

Resolve on b

Rule 2

Clause set is empty

(a b) (a b) (a c) (a c)

( a ) (a c) (a c)

( c ) ( c )

Unsatisfiable example

Pick b

Pick a

Pick c

Correctness

Saturate C w.r.t Rules 1, 2, 3a, and 3bwhile (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

Memory explosion

Saturate C w.r.t Rules 1, 2, 3a, and 3bwhile (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))

Davis-Logemann-Loveland algorithm

Slides 42-72 of sat_course1.pdfDownload from:http://research.microsoft.com/users/lintaoz/SATSolving/satsolving.htm

Davis-Logemann-Loveland algorithm

• Eliminates exponential memory requirement

• Might still need exponential time

Conflict-driven learning and non-chronological backtracking

Slides 2-20 of sat_course2.pdfDownload from:

http://research.microsoft.com/users/lintaoz/SATSolving/satsolving.htm