Proofs from SAT Solvers

Yeting Ge

ACSys NYU

Nov 20 2007

2

SAT solvers and proofs SAT problem and solvers

Given a propositional logic formula, a SAT solver outputs sat or unsat

Proofs from SAT solvers are needed A certificate to show the solver is correct Required by applications

Planning Calculation of interpolants ……

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A bunch of SAT solvers Complete / incomplete Internal representation

BDD, circuit, CNF,…… Search method

Depth first, breadth first Most modern SAT solvers

Complete, CNF, breadth first DPLL based

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4

Naïve SAT algorithm To solve:

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)()()()( 21212121 xxxxxxxx

Τx 1 Fx 1

Τx 2 Fx 2 Τx 2 Fx 2

UNSAT UNSAT

UNSAT

UNSATUNSAT

UNSAT

UNSAT

22 xx 22 xx

5

DPLL algorithmdpll(Clauses C){ C = simplify(C); if ( C contains contradictions ) return UNSAT ; if ( no more free variables in C) return SAT ; choose a free variable v in C ; C1 = substitute(C, v, T ) ; if (SAT == dpll(Clauses C1)) return SAT ; else { C2 = substitute(C, v, F ) ;

return dpll(Clauses C2) ; } }

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DPLL algorithm Two rules to simplify the CNF

clauses Unit propagation rule

If there is a clause contains only one literal, the literal is forced to be true. Propagate this new assignment immediately.

Given { , }, deduce

and { }, then One literal rule

Not used in most modern SAT solvers 11/20/2007

31 xx 1x Fx 1

Fx 33x

7

Modern SAT solvers Efficient unit propagation

BCP(Boolean Constraints Propagation) Efficient back-tracking

Iterative algorithm Almost constant cost back-tracking

Better heuristics on what to do next Conflict analysis Look ahead heuristics

Fine tuning Restart, preprocessing,…

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Resolution and SAT problem Resolution

Given two clauses and , derive From and , derive empty clause

Theorem A set of CNF clauses is unsatisfiable if and

only if there is a resolution derivation of empty clause

Proof: Based on induction.

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klll ...21

kljj ...21 ...... 2121 jjllx x

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From DPLL search tree to resolution proof

Observation: A contraction a resolution

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)()()()( 21212121 xxxxxxxx

Τx 1 Fx 1

UNSATUNSAT22 xx

21

21

xx

xx

11 xx 11 xx

1 1 , xx UNSAT

10

Naïve proof generation Record the entire proof tree and

reconstruct the resolution proof Dump search trace

Modern SAT solvers employ unit propagation

Modern SAT solvers employ some learning techniques New clauses are learned and added into the

CNF clause set. New clause could be used later

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Unit propagation (BCP) and resolution Given a unit clause , only if there

is a clause , we can generate a new clause A special case of resolution New unit clauses will be linked to

their source clauses (implication graph)

Modern SAT solvers spent most of the time on BCP

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x...21 yyx

...21 yy

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Learning and resolution:Implication graph

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Implication graph

)( 64 xx The contraction is due to: or 64 xx

We can learn a clause , which is the result of resolution of clause 4) and 5) contraction resolution

64 xx

1411/20/2007

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Implication graph:more learned clause

Another clause )( 632 xxx

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Implication graph:more learned clause

Yet another clause 61 xx

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Implication graph:more learned clause

1) 8) 9)

2) 7) 8)

3) 6) 7)

5) 4) 6)

61

621

632

64

xx

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Resolution and learned clauses Conflict clause

The learned clause that are fed back into SAT solver

Conflict clauses can be seen as the result of some resolution

Conflict clauses are redundant Could be deleted later

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Proof generation Whenever a learned clause is

generated, record the clause and dump the reason for that clause

Recode all variables assigned at root level and the reasons

Re-construct the resolution proof from the last contradiction by searching the dumped trace and recorded information Learned clauses are constructed if

necessary11/20/2007

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Application of proofs:Small unsat core Some applications require small unsat

core Given a set of unsatisfiable CNF clauses

C, if and , then S is

a minimal unsat core of C If s is the smallest among all minimal

unsat core, then s is the minimum unsat core

It is difficult to obtain a minimum unsat core

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CS esatisfiabl is . tStt

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Small unsat core from proofs By Zhang et al. Given a unsatisfiable set of clauses C Collect all clauses appears in the unsat

proof, say C1, which must be unsatifiable

Run SAT solver on C1 and collect all clauses appears in the unsat proof of C1, say C2

Repeat until reach a fixpoint There is no guarantee that the fixpoint

is minimal11/20/2007

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Another kind of proof Given a set of unsatisfiable CNF clauses F and

all conflict clauses C found by the SAT solver, a conflict pair can be derived by performing BCP on

F C only If a set of CNF clauses is unsatisfiable, a conflict pair

can be derive by resolution All the result of non-BCP resolution are in C

The proof checking is time consuming The checking procedure could give a small

unsat core as a by-product

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},{ xx

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Some future problems A standard proof format

Dump of the resolution proof Dump of conflict clauses

Better proof generation and checking Big proofs Proofs from SMT solvers

More difficult There is no general standard format

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Summary Proofs from SAT solvers are useful Modern CNF based SAT solvers can

generate proofs with little overhead

Construct the resolution proof might be a problem for large cases

Small unsat core could be obtained from proofs

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