Automated Theorem Proving

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Automated Theorem Proving

• Lecture 2
• Propositional Satisfiability

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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

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Case I Boolean programs

• Boolean-valued variables and boolean operations

• ? Formula b ?? ? ? ?
• b ? SymBoolConst

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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,

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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

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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 ?

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Conversion into CNF

• Convert formula ? into normal form NF(?)
• NF(?) is polynomial in ?
• Convert ? NF(?) into equisatisfiable CNF
  formula EQCNF(?)
• EQCNF(?) is polynomial in ?

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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)
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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(?)
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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)
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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.
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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 ) ?
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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
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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
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Satisfiable example
(a ? b ? c) (b ? ?c ? ?f) (?b ? c)
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Unsatisfiable example
(a ? b) (a ? ?b) (?a ? c) (?a ? ?c)
Pick b
( a ) (?a ? c) (?a ? ?c)
Pick a
( c ) ( ?c )
Pick c
?
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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
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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(2i))
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Davis-Logemann-Loveland algorithm
Slides 42-72 of sat_course1.pdf Download from
http//research.microsoft.com/users/lintaoz/SATSol
ving/satsolving.htm
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Davis-Logemann-Loveland algorithm

• Eliminates exponential memory requirement
• Might still need exponential time

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