结合扩展规则重构的#SAT问题增量求解方法.docx

结合扩展规则重构的#SAT问题增量求解方法
Introduction
The#SATproblemreferstothetaskofcountingthenumberofsatisfyingassignmentsforaBooleanformula.Thisisafundamentalcomputationaltaskwithwide-rangingapplications,includinginartificialintelligence,probabilisticreasoning,andmodelchecking.Manyproblemsinthesefieldscanbereducedtothe#SATproblem,whichmakesitanimportanttopicofstudy.
However,solvingthe#SATproblemexactlyiscomputationallyexpensive,oftenrequiringexponentialtimeintheworstcase.Toaddressthisissue,severalapproximateandheuristicmethodshavebeendeveloped.Inthispaper,weproposeanincrementalapproachtosolvingthe#SATproblemthatleveragesthepowerofextensionrules.
Background
Tounderstandourproposedapproach,itisimportanttohavesomebackgroundknowledgeon#SATandextensionrules.#SATisaproblemofcomputingthenumberofsatisfyingsolutionsofaBooleanformula,whichcanberepresentedasaconjunctionofclauses.Aclauseisadisjunctionofliterals,whileliteralsarepropositionalvariablesortheirnegations.Forexample,considerthefollowingBooleanformula:
(A∨B)∧(¬A∨¬B∨C)
Thisformulaissatisfiablebyassigningeither(A=true,B=true,C=true)or(A=false,B=false,C=true).The#SATproblemforthisformulaistodeterminethatithastwosatisfyingassignments.
ExtensionrulesareasetofinferencerulesusedinmanySATsolverstosimplifytheformula.Theserulesexploitthestructureoftheclausestoderivenewclauses.Someexamplesofextensionrulesare:
-Unitpropagation:ifaclausecontainsonlyoneliteral,itisassignedthevaluetrueandallclausescontainingthatliteralaresimplified.
-Pureliteralelimination:ifaliteralalwaysappearswiththesamepolarity(i.e.,alwaysnegatedoralwaysnotnegated),itcanbeassignedthecorrespondingvaluetosimplifytheformula.
-Subsumptionelimination:ifaclauseisasubsetofanotherclause,itcanberemovedfromtheformula.
-Clauselearning:ifaconflictisdetected(i.e.,nosatisfyingassignmentcanbefound),anewclauseisaddedtotheformulathatpreventsthesameconflictfromoccurringagain.
Proposedapproach
Ourproposedapproachtoincremental#SATsolvinginvolvestwokeysteps:
1.Incrementalpreprocessing
Inthisstep,we