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Satisfiability Checking of Non-clausal Formulas using General Matings

Satisfiability Checking of Non-clausal Formulas using General Matings

Himanshu JainHimanshu Jain

Constantinos BartzisConstantinos Bartzis

Edmund ClarkeEdmund Clarke

Carnegie MellonCarnegie Mellon UniversityUniversity

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The first problem to be proven NP-The first problem to be proven NP-completecomplete

Applications in verificationApplications in verification– equivalence checkingequivalence checking– bounded model checkingbounded model checking– predicate abstractionpredicate abstraction– theorem provingtheorem proving– test generationtest generation

Applications in AIApplications in AI– planningplanning

Boolean SatisfiabilityBoolean Satisfiability

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Current state of SAT solvingCurrent state of SAT solving

State-of-the-art State-of-the-art completecomplete SAT solvers SAT solvers

Davis-Putnam-Logemann-Loveland (DPLL) Davis-Putnam-Logemann-Loveland (DPLL) algorithmalgorithm

Require the input formula to be in clausal form Require the input formula to be in clausal form (CNF)(CNF)– MiniSat, BerkMin, Siege, zChaff, Limmat, GRASP, MiniSat, BerkMin, Siege, zChaff, Limmat, GRASP,

SATOSATO

Conversion to CNF by adding new variablesConversion to CNF by adding new variables– Linear size but Linear size but exponential state-spaceexponential state-space– In practice does not seem to hurtIn practice does not seem to hurt

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Non-Clausal (non-cnf) SAT solvingNon-Clausal (non-cnf) SAT solving

DPLL on circuit representation of formula DPLL on circuit representation of formula – Ganai et al. (DAC 2002)Ganai et al. (DAC 2002)– Lu et al. (Lu et al. (CSATCSAT, DAC 2003), DAC 2003)– Thiffault et al. (Thiffault et al. (NoClauseNoClause, SAT 2004), SAT 2004)

This work: This work: Non-clausalNon-clausal SAT-solver based on SAT-solver based on

DPLL

General MatingsGeneral Matings

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OutlineOutline

IntroductionIntroduction

General Matings General Matings

Search space pruningSearch space pruning

LearningLearning

Non-chronological backtrackingNon-chronological backtracking

Experimental resultsExperimental results

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General Matings exampleGeneral Matings example

Formula F: (((p q) r q) (p (r s) q))

q

r

q

p

Vertical path form (vpgraph) of F

F is satisfiable iff there exists a vertical path without opposite literals

p q

r sVertical path

Each vertical path corresponds to a term in the DNF form of F

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Basic searchBasic search

a

b

-a

c

-a

-c

a

b

-c

partialassignment

Satisfiable!

vpgraph

Partial solution

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Basic IdeaBasic Idea

Given negation normal formula (NNF) FGiven negation normal formula (NNF) F

Obtain Obtain vpgraphvpgraph of F (O(k of F (O(k22), k= |F|) ), k= |F|)

Find vertical path without opposite literalsFind vertical path without opposite literals

Due to P. Andrews , W. Bibel [1981] Due to P. Andrews , W. Bibel [1981] – Focus on higher order theorem provingFocus on higher order theorem proving– Quantifier instantiation is the main problemQuantifier instantiation is the main problem

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Main problem:Main problem:

exponentially many vertical exponentially many vertical paths in the vpgraph of Fpaths in the vpgraph of F

Sample vpgraph

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Our contributionsOur contributions

Preventing enumeration of vertical pathsPreventing enumeration of vertical paths

Search space pruningSearch space pruning

LearningLearning

Non-chronological backtracking Non-chronological backtracking

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OutlineOutline

IntroductionIntroduction

General MatingsGeneral Matings

Search space pruningSearch space pruning

LearningLearning

Non-chronological backtrackingNon-chronological backtracking

Experimental resultsExperimental results

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Search space pruning on vpgraphSearch space pruning on vpgraph

a

a

b

b a b

a

b

a

a b

b

Avoids enumeration of exponentially many paths

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Conflict when pruning vpgraphConflict when pruning vpgraph

a

a

a

b

b

b

b

a

a

a

Local conflict

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Local learningLocal learning

b

b

a

a

a Locally learned clause:

(a b)

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Non-chronological backtrackingNon-chronological backtracking

a b x

Backtrack

a

b

x

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Non-chronological backtrackingNon-chronological backtracking

a b x

a

b

x

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Limitations of local learningLimitations of local learning

Learned clause is useful at a particular Learned clause is useful at a particular nodenode

Can learn same clause multiple timesCan learn same clause multiple times

Need an equivalent of learning in CNF SAT solvers

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Dual of vpgraph: hpgraphDual of vpgraph: hpgraph

Formula F: (((p q) r q) (p (r s) q))

hpgraph of F

Each horizontal path corresponds to a clause in the CNF representation of F

p q

r

q q

p

r s

horizontal path

p q

r

q q

p

r s

vpgraph of F

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Conflicts and implications in hpgraphConflicts and implications in hpgraph

r

q q

r s

pp q Conflict clause: r p

Global conflict

p q p

r

q q

r s

Unit clause:p q r s

Implied literal : r

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Conflicts and implications in hpgraphConflicts and implications in hpgraph Detecting conflict and implications Detecting conflict and implications

– Can be done in linear timeCan be done in linear time

Why use hpgraphWhy use hpgraph– GloballyGlobally learned clauses learned clauses– Obtain implications efficientlyObtain implications efficiently

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Putting vpgraph and hpgraph togetherPutting vpgraph and hpgraph together

Formula F: (((p q) r q) (p (r s) q))

p

r

p q

q q

r s

hpgraph of F

Aim to find a vertical path in vpgraph without opposite literals

p q

r

q q

p

r s

vpgraph of F

{p, r}

Global Conflict

Unit clauses(implications)Local conflict ?

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OutlineOutline

IntroductionIntroduction

General MatingsGeneral Matings

Search space pruningSearch space pruning

LearningLearning

Non-chronological backtrackingNon-chronological backtracking

Experimental resultsExperimental results

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Experimental resultsExperimental results

SatMate MiniSat BerkMin Siege zChaff

Time Solved Time Solved Time Solved Time Solved Time Solved

QG6 256 23266 235 49386 179 46625 184 46525 184 47321 180

QG6* 256 23266 235 37562 211 15975 239 30254 225 45557 186

Mboard 19 4316 12 4331 12 4947 11 4505 12 5029 11

Pigeon 19 5110 11 6114 9 5459 10 6174 9 5483 11

Benchmark

Prob-lems

Timeout of 10 minutes per problem per solver

Classification theorems for quasigroups

[Sorge et al. SAT 2005]

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Performance on individual benchmarksPerformance on individual benchmarks

2111713165958cache12

8834242315863113237q2.14

TO16412320TO721064850TOicl45

TO2629TOTO1406922683200icl39

TO1508167314412006220699181brn13

TO1238108513081558823500174dnd02

TimeTimeTimeTimeGlobal confsLocal confsTime

zChaffSiegeBerkMinMiniSatSatMate

Problem

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ConclusionConclusion

SAT solver based on SAT solver based on General MatingsGeneral Matings

Graphical representations: vpgraph, Graphical representations: vpgraph, hpgraph hpgraph

Preventing enumeration of vertical pathsPreventing enumeration of vertical paths

Experiments show promise of this Experiments show promise of this techniquetechnique

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Questions?Questions?

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Conversion to CNFConversion to CNF

Results in more complex formula. Either:Results in more complex formula. Either:– Same variables but Same variables but exponential sizeexponential size– Linear size but Linear size but exponential state-spaceexponential state-space

• Doesn’t seem to hurt in practiceDoesn’t seem to hurt in practice

Original formula:Original formula:(a (a b) b) ( (c c d) d) (e (e f)f)

CNF using expansion:CNF using expansion:(a (a c c e) e) (a (a c c f) f) (a (a d d e) e) (a (a d d f) f) (b (b c c e) e) (b (b c c f) f) (b (b d d e) e) (b (b d d f) f)

CNF using new variables:CNF using new variables:(x(x11 x x22 x x33) ) (x(x11 a a b) b) ( (xx11 a) a) ( (xx11 b) b) (x(x22 a a d) d) ( (xx22 c) c) ( (xx22 d) d) (x(x33 f f e) e) ( (xx33 f ) f ) ( (xx33 e) e)

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Using hpgraph for conflict detectionUsing hpgraph for conflict detection

p

r

p q

q q

r s

Partial truth assignment: {r=true, p=true}

r

q q

r s

pp q

Conflict clause: r p

Global conflict

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Using hpgraph to produce implicationsUsing hpgraph to produce implications

p

r

p q

q q

r s

Partial truth assignment: {p=false, q=false, s=true}

p q p

r

q q

r s

Unit clause:p q r s

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Obtaining vpgraphObtaining vpgraph

Constructed directly from a NNF formulaConstructed directly from a NNF formula

Time/Space required: O(kTime/Space required: O(k22) ) – k is the size of given formulak is the size of given formula– Recently improved to O(k)Recently improved to O(k)

Directed acyclic graphDirected acyclic graph