A More Efficient BDD-Based QBF Solver Oswaldo Olivo, E. Allen - - PowerPoint PPT Presentation

A More Efficient BDD-Based QBF Solver

Oswaldo Olivo, E. Allen Emerson

The University of Texas at Austin, U.S.A.

September 15, 2011

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Outline

Introduction. Preliminaries. eBDD-QBF Solver. Experiments. Conclusions and Future Work.

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Introduction

Satisfiability of quantified boolean propositional formulas (QBF). QBF Applications: Conformant planning, model checking, equivalence checking and theorem proving. Solvers classified as search (DPLL) and symbolic (BDDs, ZDDs, AIGs). Conventional wisdom : Search outperforms Symbolic.

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Introduction

New BDD-based QBF solver. BDD Constraint Propagation. Enhanced early quantification. Variable elimination order.

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Preliminaries: Quantified Boolean Formulas (QBFs)

Let formula Q1X1Q2X2...QnXnφ , where Qi ∈ {∀, ∃}, Xi set

• f propositional variables

and φ is a propositional formula over Xis. Q1X1Q2X2...QnXn is quantifier prefix and φ matrix. QBF : Is there a satisfying assignment for above formula? Assume CNF: (1) Q1X1Q2X2...QnXn(C1 ∧ C2 ∧ ... ∧ Cm). Each Cj is disjunction of literals called clause.

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Preliminaries: Binary Decision Diagrams (BDDs)

Binary Decision Diagram (BDD) is a DAG that represents all satisfying assignments. Inner node represents a variable x of formula and has two children denoted by high and low. Imposing variable ordering in paths, eliminating redundant nodes and merging of isomorphic subgraphs : ROBDDs. Apply(A, B, op) algorithm is binary operation op applied for BDDs. The RestrictBy(A, B) alg. prunes assignments of A inconsistent with B.

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Preliminaries: Binary Decision Diagrams (BDDs)

(¬x1 ∨ x2) ∧ (¬x3 ∨ x2) ∧ (x1 ∨ ¬x2 ∨ x3)

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eBDD-QBF solver

New BDD-based QBF solver. Symbolic: Transforms input CNF into set of implicitly conjoined BDDs and applies decision procedure. Includes search-inspired optimizations for BDDs : unit clause and pure literal propagation (BDD Constraint Propagation). Employs a variant of early quantification from model checking. Dynamic variable elimination heuristics.

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Enhanced Early Quantification Example.

Inner Existential Variable case : ∃x1∀x2∃x3((x1 ∨ x2) ∧ (¬x2 ∨ x3) ∧ (¬x1 ∨ ¬x3)) ≡ ∃x1∀x2((x1 ∨ x2) ∧ (∃x3(¬x2 ∨ x3) ∧ (¬x1 ∨ ¬x3))) ≡ ∃x1∀x2((x1 ∨ x2) ∧ (¬x1 ∨ ¬x2)) Inner Universal Variable case: ∃x1∀x2∀x3((x1 ∨ x2) ∧ (¬x2 ∨ x3) ∧ (¬x1 ∨ ¬x3)) ≡ ∃x1∀x2((x1 ∨ x2) ∧ (∀x3(¬x2 ∨ x3)) ∧ (∀x3(¬x1 ∨ ¬x3))) ≡ ∃x1∀x2((x1 ∨ x2) ∧ ¬x2 ∧ ¬x1)

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QBF Formulations

(1) Q1X1Q2X2...QnXn(C1 ∧ C2 ∧ ... ∧ Cm). (2) Q1X1Q2X2...Qn−1Xn−1Qn(Xn − {xi}) (Qnxi(C1 ∧ C2 ∧ ... ∧ Ck) ∧ Ck+1 ∧ ... ∧ Cm)

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Enhanced Early Quantification Algorithm.

Recall (1) and (2) from previous slide .Let var xi be innermost in (1). w.l.o.g., xi occurs. in C1, ..., Ck and not in Ck+1, ..., Cm. xi existential → rewrite (1) into (2). xi universal. → use ∀x(P(x) ∧ Q(x)) ≡ ∀x(P(x)) ∧ ∀x(Q(x)), and (1) becomes. : (3) Q1X1Q2X2...Qn−1Xn−1∀(Xn − {xi}) ((∀xi(C1) ∧ ∀xi(C2) ∧ ... ∧ ∀xi(Ck)) ∧ Ck+1 ∧ ... ∧ Cm) Motivation : quantification typically reduces diagram size, as variable is eliminated from support set. Individually quantify , conjoin all simplified BDDs, store result and apply early quantification iteratively.

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BDD Constraint Propagation (Unit Clause) Example.

Simple Case: BDDs represent clauses. Existential variable: x2 is unit clause. ∃x1∃x2(

BDD 1

• (x1 ∨ ¬x2) ∧

BDD 2

• x2 )

≡ ∃x1(

BDD 1

• (x1) )

Universal variable: x2 is unit clause. ∃x1∀x2(

BDD 1

• (x1 ∨ ¬x2) ∧

BDD 2

• x2 )

≡ UNSAT

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BDD Constraint Propagation (Unit Clause).

Unit Clause Propagation : Clause with only one literal has to be set to true in order to continue solving. Simple case:Initially every BDD represents a clause of the CNF f . Algorithm eBDD-QBF Unit Propagation:

1

Detect all BDDs of support set size 1 (called unit BDDs).

2

If any unit BDD is universal, return UNSAT.

3

Conjoin all unit BDDs into one BDD b.

4

return RestrictBy(f , b).

Limitation: Only works when BDDs represent clauses, hence useless after applying early quantification.

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BDD Constraint Propagation (Unit Clause) Example.

Complex Case: BDDs may represent non-clausal formulas. Existential variable: x1 is inner unit clause. ∃x1∀x2∃x3(

BDD 1

• (x1 ∧ ¬x3) ∨ (x1 ∧ x2) ∧(

BDD 2

x3 ∨ x1) ≡ ∃x1∀x2∃x3(

BDD 1

• (

x1

• InnerUnitClause

∧(¬x3 ∨ x2)) ∧

BDD 2

• (x3 ∨ x1))

≡ ∀x2∃x3(

BDD 1

• (¬x3 ∨ x2))

Universal variable: x1 is inner unit clause. ∀x1∀x2∃x3(

BDD 1

• (x1 ∧ ¬x3) ∨ (x1 ∧ x2) ∧(

BDD 2

x3 ∨ x1) ≡ UNSAT

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BDD Constraint Propagation (Unit Clause) (Cont.)

Complex case : Early quantification has been applied, so BDDs do not necessarily represent clauses. However, formula may contain inner unit clauses in their underlying CNF. Example (x1 ∧ x2) ∨ (x1 ∧ x3) ≡ x1 ∧ (x2 ∨ x3). Algorithm eBDD-QBF Unit Propagation :

1

For each BDD b in the formula representation f :

2

For each variable x in the support set of b

3

If RestrictBy(b, BDD(¬x)) ≡ ZERO BDD then x is inner unit clause.

4

If RestrictBy(b, BDD(x)) ≡ ZERO BDD then ¬x is inner unit clause.

5

If any inner unit clause is universal, return UNSAT.

6

Create BDD b by conjoining BDDs for every inner unit clause.

7

return RestrictBy(f , b)

Must detect selectively, otherwise just Prime Implicates.

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BDD Constraint Propagation (Pure Literals) Example.

Simple Case: BDDs represent clauses. Existential variable: x1 is positively pure. ∃x1∃x2(

BDD 1

• (x1 ∨ ¬x2) ∧

BDD 2

• x2 )

≡ ∃x2(

BDD 1

• (x2) )

Universal variable: x2 is positively pure. ∃x1∀x2∃x3(

BDD 1

• (¬x1 ∨ x2 ∨ x3) ∧

BDD 2

• (¬x3 ∨ x2 ∨ x1))

≡ ∃x1∃x3(

BDD 1

• (¬x1 ∨ x3) ∧

BDD 2

• (¬x3 ∨ x1))

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BDD Constraint Propagation (Pure Literals).

Pure Literal Propagation : Literal appears only positively (negatively) in formula. If universal literal, remove it from

• formula. Otherwise, remove clauses containing it.

Simple case: Initially every BDD represents a clause of the CNF f . Algorithm eBDD-QBF Pure Literal Propagation :

1

For each BDD b of formula f

2

For each variable x in the support set of b

3

if RestrictBy(b, BDD(x)) ≡ ONE BDD, x is positive in b.

4

if RestrictBy(b, BDD(¬x)) ≡ ONE BDD, x is negative in b.

5

For each pure literal x in f

6

if x is universal, create BDD b′ := BDD(¬x), otherwise b′ := BDD(x).

7

Conjoin all BDDs b into a BDD c.

8

return RestrictBy(f , c)

Same Limitation as for Unit Prop.

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BDD Constraint Propagation (Pure Literals) Example.

Complex Case: BDDs may represent non-clausal formulas. Existential variable: x1 is positively pure. ∃x1∀x2∃x3((

BDD 1

• (x1 ∧ ¬x3) ∨ (x1 ∧ x2)) ∧(

BDD 2

• x3 ∨ ¬x2))

Note: ∀x1(BDD 1) ≡ (¬x3 ∨ x2) ≡ BDD 1[x1 := false] ≡ ∀x2∃x3(

BDD 2

• x3 ∨ ¬x2)

Universal variable: x1 is positively pure. ∀x1∀x2∃x3((

BDD 1

• (x1 ∧ ¬x3) ∨ (x1 ∧ x2)) ∧(

BDD 2

• x3 ∨ ¬x2))

Polarity check the same as for existential case. ≡ UNSAT

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BDD Constraint Propagation (Pure Literals) (Cont.)

Complex case : Early quantification has been applied, so BDDs are not necessarily clauses. Polarity is not so easy for this case. Algorithm eBDD-QBF Unit Propagation : Modify previous algorithm by changing polarity detection procedure: x is positively pure in BDD b iff ∀x(b) ≡ RestrictBy(b, BDD(¬x)). Intuition: Quantification removes variable and setting positive to false removes variable too (analogous for negative case). Too expensive. Not used in mainstream experiments.

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Variable Ordering Heuristics.

Dynamic heuristics have been used in model checking. BDD-Based SAT and QBF solvers employ static heuristics. Reason: Bucket elimination with BDDs (a bucket being a set holding BDDs w/ the same top variable) relies on eliminating

• nly top variables at each iteration, so the ordering must be

fixed. We have implemented the following heuristics:

1

Most occurring top variable.

2

Most occurring variable.

3

Least occurring top variable.

4

Least occurring variable.

5

Smallest BDD top variable.

6

Smallest BDD variable.

7

Bounded Most occurring variable with min.

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Main algorithm

Algorithm eBDD-QBF (qp q, CNF f ):

1 Create BDD for each clause in f . 2 Apply unit propagation iteratively to all BDDs. 3 Apply pure literal propagation iteratively to all BDDs. 4 while there are still variables to be eliminated do 5

if trivially Q − SAT or Q − UNSAT return result.

6

x := least occurring variable from innermost quantifier.

7

b := Apply Enhanced Early Quantification with x.

8

Extract inner unit BDDs from b and apply unit propagation if possible.

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Experiments.

Solver implemented in C++ and BDD package CUDD. Experiments were run on Intel Dual Core 2.13GHz Unix desktop with 4 GB of RAM. Compared eBDD-QBF with other solvers that best on the verification and A.I. domains such as QuBE7.1, Semprop and DepQBF. They also dominated previous BDD-based solvers.

• Prelim. experiments to select dynamic heuristics (least
• ccurring var was best).

30 minute timeout per instance. Equivalence Checking (Ayari), Theorem Proving (Pan) and Conformant Planning (Rintanen) benchmarks.

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Experiments.

Solv.

• Inst. Solv.
• Tot. Time
• Num. Vars.
• Num. Cls’

eBDD-QBF 30(*) 246.75s 10-3290 18-19663 QuBE7.1 27 506.211s 10-2035 18-178750 Semprop 32 75.092s 10-3290 18-178750 DepQBF 23 1984.76s 10-2501 18-178750 Experimental results for Conformant Planning instances *: Solves about 3 or 4 more instances with inner pure literal propagation.

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Experiments.

Solv.

• Inst. Solv.
• Tot. Time
• Num. Vars.
• Num. Cls’

eBDD-QBF 27 3257.83s 332-5833 127-6084 QuBE7.1 19 304.214s 332-252710 127-334058 Semprop 20 361.614 s 332-57143 127-74975 DepQBF 20 61.107s 332-4558 5-73065 Experimental results for Equivalence Checking instances

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Experiments.

Solv.

• Inst. Solv.
• Tot. Time
• Num. Vars.
• Num. Cls’

eBDD-QBF 269 6283.38s 4-3004 5-131072 QuBE7.1 140 8148.41s 4-11130 5-355294 Semprop 242 10129.9s 4-5320 5-131072 DepQBF 140 12087.7s 4-4558 5-73065 Experimental results for Modal Logic instances

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Conclusions

Pure BDD-based QBF solver competitive w/ other efficient decision procedures. Generalized unit and pure literal rules for BDDs was crucial. Dynamic variable selection useful with enhanced early quantification.

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Future Work

Work still required in identifying structure of suitable formulas for BDDs, beyond empirical analysis. Combination with a top-down algorithm would allow more flexible variable selection. More robust BDD constraint propagation (a la Cadoli) is required to produce a competitive solver w.r.t. hybrid solvers such as AIGSOLVE and AQME.

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Questions?

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References

Aloul, F.A., Markov, I.L., Sakallah, K.A.: Faster SAT and Smaller BDDs via Common Function Structure. In: Technical Report #CSE-TR-445-01. University

• f Michigan (2001)

Audemard, G., Sas, L.: SAT Based BDD Solver for Quantified Boolean

• Formulas. In: 16th IEEE International Conference on Tools with Artificial

Intelligence (ICTAI), pp. 82–89. IEEE Computer Society (2004) Ayari, A. , Basin, D.: Bounded Model Construction for Monadic Second-Order

• Logics. In: 12th International Conference on Computer-Aided Verification (CAV),
• pp. 99–113. Springer-Verlag (2000)

Benedetti, M.: Evaluating QBFs via Symbolic Skolemization. In: Proc. of 11th International Conference on Logic for Programming Artificial Intelligence and Reasoning (LPAR), LNCS 3452 Springer-Verlag (2005) Biere, A., Cimatti, A., Clarke, E., Strichman, O., Zhu, Y.:Bounded model

• checking. J. Adv. Comp. Sci. 58, (2003)

Bryant, R.E.: Graph-Based Algorithms for Boolean Function Manipulation. IEEE

• Trans. on Comp. 35, 677–691 (1986)

Burch, J. R., Clarke, E.M., Mcmillan,K. L., Dill, D. L.,Hwang, L. J.:Symbolic Model Checking: 10 20 States and Beyond. In: Fifth Annual IEEE Symposium

• n Logic in Computer Science, pp. 428–439. IEEE Comput. Soc. Press (1990)

Chatalic, P., Simon, L.: Zres:The old Davis-Putnam procedure meets ZBDDs. In: 17th International Conference on Automated Deduction (CADE), pp.

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Appendix

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BDD Constraint Propagation (Formal Presentation)

Definition 1 (Unit Clause): Let f = C1 ∧ C2 ∧ .... ∧ Ci−1 ∧ Ci ∧ Ci+1 ∧ ... ∧ Ck be a CNF formula and Ci = xj, where xj is a literal . Ci a unit clause of f . . Definition 2 (Generalized Unit Clause): Let f be a boolean formula . If ∃ CNF f ′ = C1 ∧ C2 ∧ .... ∧ Ci−1 ∧ Ci ∧ Ci+1 ∧ ... ∧ Ck such that f ′ ≡ f and Ci = xj, where xj is a literal, then Ci is a generalized unit clause of f . . Theorem 1: Let Ci be a clause and Cj = xl be a unit clause (xl is an existential literal). Then RestrictBy(BDD(Ci), BDD(Cj)) = BDD(Ci[xl := true]). . Corollary 1: Let f = C1 ∧ C2 ∧ ... ∧ Ck be a CNF and Cj = xl be a unit clause (where xl is a unit literal). Then RestrictBy(BDD(f ), BDD(Cj)) = BDD(f [xl := true]). .

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BDD Constraint Propagation.

Corollary 2: Let f be a non-triviall propositional formula and Cj = xl be a generalized unit clause of f (where xl is an existential literal). Then RestrictBy(BDD(f ), BDD(Cj)) = BDD(f [xl := true]).. Theorem 2: Let f be a non-trivial propositional formula, xj a literal and f ′ an equivalent CNF representation of f obtained by applying distributivity of ∧ and ∨ and removing trivial

• clauses. Then xj is a generalized unit clause in f iff xj is a

unit clause in f ′. . Theorem 3: Let f be a non-trivial propositional formula and xj a literal. Then xj is a generalized unit clause in f iff f [xj := false] = false. . Corollary 3: Let f be a non-trivial propositional formula and xl a literal. Then Cj = xl is a generalized unit clause in f iff RestrictBy(BDD(f ), BDD(¬Cj)) = ZERO BDD. .

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BDD Constraint Propagation.

Lemma 1: Let f be a non-trivial propositional formula, and xi, xj two literals. Then RestrictBy(BDD(f ), BDD(xi) ∧ BDD(xj)) = RestrictBy(RestrictBy(BDD(f ), BDD(xi)), BDD(xj)). Theorem 4: Let f be a non-trivial propositional formula and Ci1, ..., Cim be unit clauses of existential literals. Then g(f , {Ci1, ..., Cim}) = RestrictBy(f , Ci1 ∧ .... ∧ Cim), where g(f , l) = g(RestrictBy(f , Ci1), {Ci2, ..., Cim}) if l = {Ci1, .., Cim} and f if l = ∅. Unit propagation is trivial for clause BDDs: extract BDDs with support set size one and apply unit propagation Limitation: Fails to recognize inner unit clauses for non-clause intermediate formulas. Need unit clause extraction algorithm to detect generalized unit clauses.

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BDD Constraint Propagation.

Definition 3 (Pure Literal): Let f = C1 ∧ C2 ∧ ... ∧ Ck be a CNF form. and xj a literal, where xj only appears positively (negatively) in f . We call xj a positive (negative) pure literal

• f f .

Definition 4 (Generalized Pure Literal): Let f be a boolean formula. If ∃ CNF f ′ = C1 ∧ C2 ∧ .... ∧ Ck such that f ′ ≡ f and xj occurs only positively (negatively) in f ′, where xj is a literal, then xj is a generalized positive (negative) pure literal of f . Theorem 5: Let f be a non-trivial propositional formula, xj a literal and f ′ an equivalent CNF representation of f obtained by applying distributivity of ∧ and ∨ and removing trivial

• clauses. Then xj is a generalized positive (negative) pure

literal in f iff xj is a positive (negative) pure literal in f ′. Theorem 6: Let f be a non-trivial formula and xj a literal. xj is a generalized positive (negative) pure literal in f iff f [xj := false] = ∀xj(f ) (f [xj := true] = ∀xj(f )).

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BDD Constraint Propagation.

Corollary 4: Let f be a non-trivial formula and xj a literal. xj is positively (negatively) pure in BDD(f ) iff RestrictBy(BDD(f ), BDD(¬(xj))) = ∀xj(BDD(f )) (RestrictBy(BDD(f ), BDD(xj)) = ∀xj(BDD(f )). .

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BDD Constraint Propagation.

algorithm UnitBDDExtraction(BDD b, Quantifier Prefix qp): Set of BDD S := {} for each variable v in the support set of b do if RestrictBy(b,Not(bdd(v))) = Zero BDD then if(isUniversal(v,qp)) then return {Zero BDD} else S := S ∪ {bdd(b)} else if RestrictBy(b,bdd(v)) = Zero BDD then if(isUniversal(v,qp)) then return {Zero BDD} else S := S ∪ {Not(bdd(b))} return S Algorithm for unit BDD extraction

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BDD Constraint Propagation.

algorithm PolarityVectorClauseBDDs(bdds, numVars): Vector of Int v for i = 1 to numVars do v[i] := −2 for each BDD b in bdds do for each variable v in the support set of b do if RestrictBy(b,bdd(v)) = One BDD then updatePolarity(polarity,v,1) else if RestrictBy(b,not(bdd(v)))=One BDD then updatePolarity(polarity,v,0) return polarity Algorithm for calculating the polarity vector from a set of clause BDDs

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BDD Constraint Propagation.

algorithm PureLiteralClauseBDDs(bdds, numVars, qp): boolean pure var := true while pure var do polarity := PolarityVectorClauseBDDs(bdds, numVars) BDD polarityBDD := One BDD pure var := false for each variable v in polarity do if polarity[v] = −2 ∧ polarity[v] = −1 then pure var := true if isUniversal(v,qp) then polarityBDD := polarityBDD∧ (polarity[v] = 1?not(bdd(v)) : bdd(v)) else polarityBDD := (polarity[v] = 1?bdd(v) : not(bdd(v))) for each BDD b in bdds do RestrictBy(b,polarityBDD) return bdds Algorithm for applying the pure literal rule for clause BDDs

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BDD Constraint Propagation.

algorithm PolarityVectorNonClauseBDDsPureLiteral (bdds,numVars): Vector of Int v for i = 1 to numVars do v[i] := −2 for each BDD b in bdds do for each variable v in the support set of b do if RestrictBy(b,bdd(v)) = One BDD then updatePolarity(polarity,v,1) else if RestrictBy(b,not(bdd(v))) = One BDD then updatePolarity(polarity,v,0) else if ∀v.b. ≡ RestrictBy(b,not(bdd(v))) then updatePolarity(polarity,v,1) else if ∀v.b. ≡ RestrictBy(b,bdd(v)) then updatePolarity(polarity,v,0) else updatePolarity(polarity,v,¬polarity[v]) return polarity

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Main Algorithm.

algorithm BDD-QBF(CNF F, qp, numVars, heuristic): Set of BDD bdds := createBDDForEachClause(F) bdds := ClauseBDDUnitPropagation(bdds,qp) bdds := PureLiteralClauseBDDsPureLiteral(bdds,numVars,qp) while qp = {} do if bdds = {} then return QSAT if Zero BDD ∈ bdds then return Q-UNSAT Quantifier q := innermostQuantifier(qp) qp := qp − {q} while variableSet(q) = {} do if bdds = {} then return QSAT if Zero BDD ∈ bdds then return Q-UNSAT int var := chooseVariable(bdds,variableSet(q),heuristic) variableSet(q) := variableSet(q) −{var} Set of BDD occ bdds :=chooseBDDsWithVariable(bdds,va bdds := bdds − occ bdds if isUniversal(var,qp) then

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