Finding Models Finding Models for Blockedfor Blocked 3-SAT Problems3-SAT Problems

in Linear Timein Linear Timeby Systematical Refinement of by Systematical Refinement of a a Sub-Sub-

ModelModel

Gábor Kusper

gkusper@aries.ektf.hu

Eszterházy Károly College

Hungary, Eger

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OutlineOutline

Motivation Introduction Connection to other Research My Contribution Conclusion

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MotivationMotivation

Member of the Theorema Research Group. Theorema (www.theorema.org) is a mathematical

assistant (including theorem proving), which is developed under the direction of Prof. Bruno Buchberger at Johannes Kepler University Linz.

SAT is propositional proving by refutation:Given a set of known sentences K (Knowledge base), then a sentence G (Goal) can be proven if and only if K {G} is unsatisfiable.

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IntroductionIntroduction

The Propositional Satisfiability problem is the problem of determining, for a formula of the propositional calculus, if there is an assignment of truth values to its variables for which that formula evaluates to True.

SAT is the problem of propositional satisfiability for formulae in conjunctive normal form (CNF).

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Conjunctive Normal Form (CNF)Conjunctive Normal Form (CNF)

( a c ) ( b c ) (¬a ¬b ¬c )

{ { a,c }, { b,c }, {¬a¬b, ¬c } }

+ x +

x + +

- - -

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SAT is SAT is NPNP-complete [Cook 1971]-complete [Cook 1971]

thus, there is no known polynomial time algorithm for solving it.

Because of the importance of SAT in logic, artificial intelligence, and operational research, considerable effort has been spent to determine how to cope with this problem.

Two approaches are:– develop SAT solver algorithms which usually present a

result in polynomial time;– identify special classes of SAT that can be solved or

simplified in polynomial time. My talk is concerned with the second approach.

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Connection to other ResearchConnection to other Research

blocked clause [Kullmann 1999a, 1999b]

Loveland's Davis and Putnam Procedure (DPLL) [Davis, Logemann, Loveland 1962]

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The Notion of Blocked ClauseThe Notion of Blocked Clause

A clause of the form (c C’) is blocked in a formula iff:for any other clause in the formula of the form (c B’),the resolvent (C’ B’) is a tautology.

A blocked clause can be added or deleted from a clause set without changing its satisfiability.

Example:

+ + xx - -+ x +- - xx + +

Literal c is blocked in clause C and clause set S:

Blck(c,C,S)

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My ContributionMy Contribution

Notion of Sub-Model Notion of Blocked SAT Problem Blocked SAT Solver algorithm Number of Literal Combinations (NLC) data structure

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Sub-Model

resolution-mate: rm(C, c) := C \ { c } { c }. sub-model: sm(C, c) := rm(C, c). sub-model: “negate all but the generator literal”. A sub-model is a partial assignment. A sub-model is a candidate to be a part of a model.

+ + + x

+ + - x

+ + x x - - + x

C

sm(C,c)

rm(C,c)

c

resolvent

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The Blocked SAT ProblemThe Blocked SAT Problem

A clause set is blocked iff each of its clauses are blocked. Blck(S) : Blck(c,C,S).

CS cC

- + x x

x + + x

x - - x

x x - +

--xx x x + x

x x - +

The Problem: A blocked clause set does not remain necessarily blocked after sub-model propagation.

Solution: Systematic refinement of an increasing sub-model.

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ExamplesExamples

- + x x

x + + x

x - - x

x x - +

+ + x

x + +

- - x

x - -

+ + -

+ - +

+ - -

- + +

- + -

- - +

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Significance of Blocked SATSignificance of Blocked SAT

An input clause set is very-very rarely blocked. But during the work of a general SAT solver algorithm we

may encounter a blocked clause set. Any general SAT solver uses some simplification steps:– resolution, unit-propagation, removing subsumed clauses, etc.

The fewer clauses are in a clause set the more likely that it is blocked.

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Significance of Blocked SATSignificance of Blocked SAT

SATLIB – Benchmark Problems http://www.intellektik.informatik.tu-darmstadt.de/

SATLIB/benchm.html All instances provided here are cnf formulae encoded

in DIMACS cnf format.

Test result on uf20-91 (all satisfiable, 20 variables, 91 clauses - 1000 instances) using DPLL and Moms heuristics:In average after the 90% of the unit-propagations we have a blocked clause set.

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The Blocked SAT Solver AlgorithmThe Blocked SAT Solver Algorithm

If the literal c is blocked in C, S then the sub-model sm(C, c) satisfies each clause from S which contains either c or c.

No-occurrence clauses are those not satisfied by sm(C, c).

Systematic refinement of an increasing sub-model:

- + x x

x + + x

x - - x

x x - +

- - x x

no-occurrence clause

- + - x

increasing sub-model:a blocked clause set: generator clause:

- + x x

- + + x

C1

C2

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BlockedSATSolver(S, Z)input: clause set S that is non-empty and blockedoutput: assignment Z, a model for S. A := { }; B := { }; // A: gen. clause, B: gen. literals for each clause C in S do if (A C = { }) then // C is a no-occurrence

clause A:=A C; // we refine A if (B C = { }) then Let c in C be a blocked literal in C, S; B := B { c }; // we refine B fi fi od Z := sm(A, B);

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NLC data structureNLC data structure

The Number of Literal Combinations (NLC) data structure is needed to speed up the search for blocked literals.

We create NLC by reading each clause only once. For every subset of every clause we increase the corresponding counter in NLC by one.

For example in case of {a, b, c} we increase:NLC[a], NLC[b], NLC[c],NLC[a, b], NLC[a, c], NLC[b, c],NLC[a, b, c].

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NLC data structureNLC data structure

Afterwards we read again the clause set and for every literal in every clause we calculate the number of possible resolution partners minus the number of blocking clauses.

In case of literal a in clause {a, b, c} this number is:NLC[a] – NLC[a, b] – NLC[a, c] + NLC[a, b, c]

If this number is zero then this literal is blocked.

NLC[a] is the number of possible resolution partners. NLC[a, b] + NLC[a, c] is the number of blocking

clauses, but in this case we count two times clauses which contain {a, b, c} as subset, therefore, we subtract NLC[a, b, c].

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ExamplesExamples

a b c

+ + -

+ - +

+ - -

- + +

- + -

- - +

a a b b c c

3 3 3 3 3 3

a a b b c c a 0 0 1 2 1 2 a 0 0 2 1 2 1 b 0 0 0 0 1 2

b 0 0 0 0 2 1 c 0 0 0 0 0 0

c 0 0 0 0 0 0

NLC

1D:

2D:

3D: a … 0 1 0 … 0 1 0 0 … 0 1 0 …

a … 0 1 0 … 0 0 1 0 … 0 1 0 …

b

c

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NLC data structureNLC data structure

In case of 3-SAT (the SAT instance where each clause has 3 literal at most) we need O(7m) time to create, and

O(4nm) time in the worst case to check whether the clause set is blocked is blocked.

But we need O(8n3) memory space to store NLC.

Fortunately it contains lots of zeros, therefore, using a suitable hash O(4nm) memory space is enough to store NLC.

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ConclusionConclusion

Blocked SAT Solver is linear if we have blocked literal information.

Polynomial time is needed to collect that information. In case of 3-SAT we can collect that information in

linear time by using the NLC data structure. Hence, the Blocked 3-SAT Problem is linear!

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Future WorkFuture Work

Blocked SAT Solver can solve not only blocked clause set, but also those where at least one clause is blocked from the no-occurrence clauses.How could we check this property fast?

NLC can support other techniques? How to combine NLC by lazy unit propagation?

Thank you for your attention!Thank you for your attention!

Download: http://aries.ektf.hu/~gkusper/ki06.ppt