Backjumping for Quantified Boolean Logic satisfiability

Lecture Notes in Computer Science, 2005

Solving Quantified Boolean Formulas (QBF) has become an important and attractive research area, since several problem classes might be formulated efficiently as QBF instances (e.g. planning, non monotonic reasoning, twoplayer games, model checking, etc). Many QBF solvers has been proposed, most of them perform decision tree search using the DPLL-like techniques. To set free the variable ordering heuristics that are traditionally constrained by the static order of the QBF quantifiers, a new symbolic search based approach (QBDD(SAT)) is proposed. It makes an original use of binary decision diagram to represent the set of models (or prime implicants) of the boolean formula found using searchbased satisfiability solver. Our approach is enhanced with two interesting extensions. First, powerful reduction operators are introduced in order to dynamically reduce the BDD size and to answer the validity of the QBF. Second, useful cuts are achieved on the search tree thanks to the nogoods generated from the BDD representation. Using DPLL-likes (resp. local search) techniques, our approach gives rise to a complete QBDD(DPLL) (resp. incomplete QBDD(LS)) solver. Our preliminary experimental results show that on some classes of instances from the QBF evaluation, QBDD(DPLL) and QBDD(LS) are competitive with stateof-the-art QBF solvers.

2008

In this paper we introduce QuBIS an (in)complete solver for quantified Boolean formulas (QBFs). The particularity of QuBIS is that it is not inherently incomplete, but it has the ability to surrender upon realizing that its deduction mechanism is becoming ineffective. Whenever this happens, QuBIS outputs a partial result which can be fed to a complete QBF solver for further processing. As our experiments show, not only QuBIS is competitive as an incomplete solver, but providing the output of QuBIS as an input to complete solvers can boost their performances on several instances.

2004

Solving Quantified Boolean Formulas (QBF) has become an attractive research area in Artificial intelligence. Many important artificial intelligence problems (planning, non monotonic reasoning, formal verification, etc.) can be reduced to QBFs. In this paper, a new DLL-based method is proposed that integrates binary decision diagram (BDD) to set free the variable ordering heuristics that are traditionally constrained by the static order of the QBF quantifiers. BDD is used to represent in a compact form the set of models of the boolean formula. Interesting reduction operators are proposed in order to dynamically reduce the BDD size and to answer the validity of the QBF. Experimental results on instances from the QBF'03 evaluation show that our approach can efficiently solve instances that are very hard for current QBF solvers.

2009

In this paper we approach the problem of reasoning with quantified Boolean formulas (QBFs) by combining search and resolution, and by switching between them according to structural properties of QBFs. We provide empirical evidence that QBFs which cannot be solved by search or resolution alone, can be solved by combining them, and that our approach makes a proof-of-concept implementation competitive with current QBF solvers.

2005

Quantified Boolean formulas (QBFs) play an important role in theoretical computer science. QBF extends propositional logic in such a way that many advanced forms of reasoning can be easily formulated and evaluated. In this dissertation we present our ZQSAT, which is an algorithm for evaluating quantified Boolean formulas. ZQSAT is based on ZBDD: Zero-Suppressed Binary Decision Diagram , which is a variant of BDD, and an adopted version of the DPLL algorithm. It has been implemented in C using the CUDD: Colorado University Decision Diagram package. The capability of ZBDDs in storing sets of subsets efficiently enabled us to store the clauses of a QBF very compactly and let us to embed the notion of memoization to the DPLL algorithm. These points led us to implement the search algorithm in such a way that we could store and reuse the results of all previously solved subformulas with a little overheads. ZQSAT can solve some sets of standard QBF benchmark problems (known to be hard for ...

2019 IEEE 31st International Conference on Tools with Artificial Intelligence (ICTAI), 2019

The decision problem of quantified Boolean formulas (QBFs) is the archetypical problem for the complexity class PSPACE that contains many reasoning problems of practical relevance. Because of the availability of a rich solving infrastructure aspects QBFs provide an attractive framework for encoding and solving such reasoning problems ranging from symbolic reasoning in artificial intelligence to the formal verification and synthesis of computing systems. In this paper, we survey the different application areas that exploit QBF technology for solving their specific problems.

2002

The availability of decision procedures for combinations of boolean and linear mathematical propositions opens the ability to solve problems arising from real-world domains such as verification of timed systems and planning with resources. In this paper we present a general and efficient approach to the problem, based on two main ingredients. The first is a DPLL-based SAT procedure, for dealing efficiently with the propositional component of the problem. The second is a tight integration, within the DPLL architecture, of a set of mathematical deciders for theories of increasing expressive power. A preliminary experimental evaluation shows the potential of the approach.

2002

We present algorithms for solving quantified Boolean formulas (QBF, or sometimes QSAT) with worst case runtime asymptotically less than O(2 n ) when the clause-to-variable ratio is smaller or larger than some constant. We solve QBFs in conjunctive normal form (CNF) in O(1.709 m ) time and space, where m is the number of clauses. Extending the technique to a quantified version of constraint satisfaction problems (QCSP), we solve QCSP with domain size d = 3 in O(1.953 m ) time, and QCSPs with d ≥ 4 in O(d m/2+ ) time and space for > 0, where m is the number of constraints. For 3-CNF QBF, we describe an polynomial space algorithm with time complexity O(1. when the number of 3-CNF clauses is equal to n; the bound approaches 2 n as the clause-to-variable ratio approaches 2. For 3-CNF Π 2 -SAT (3-CNF QBFs of the form ∀u 1 · · · u j ∃x j+1 · · · x n F ), an improved polyspace algorithm has runtime varying from O(1.840 m ) to O(1.415 m ), as a particular clause-to-variable ratio increases from 1.

Journal of Artificial Intelligence Research

This is the third of three papers describing ZAP, a satisfiability engine that substantially generalizes existing tools while retaining the performance characteristics of modern high-performance solvers. The fundamental idea underlying ZAP is that many problems passed to such engines contain rich internal structure that is obscured by the Boolean representation used; our goal has been to define a representation in which this structure is apparent and can be exploited to improve computational performance. The first paper surveyed existing work that (knowingly or not) exploited problem structure to improve the performance of satisfiability engines, and the second paper showed that this structure could be understood in terms of groups of permutations acting on individual clauses in any particular Boolean theory. We conclude the series by discussing the techniques needed to implement our ideas, and by reporting on their performance on a variety of problem instances.

Lecture Notes in Computer Science, 2003

In recent years backtrack search algorithms for Propositional Satisfiability (SAT) have been the subject of dramatic improvements. These improvements allowed SAT solvers to successfully solve instances with thousands of variables and hundreds of thousands of clauses, and also motivated the development of many new challenging problem instances, many of which still too hard for the current generation of SAT solvers. As a result, further improvements to SAT technology are expected to have key consequences in solving hard real-world instances. The objective of this paper is to propose heuristic approaches to the backtrack step of backtrack search SAT solvers, with the goal of increasing the ability of a SAT solver to search different parts of the search space. The proposed heuristics are inspired by the heuristics proposed in recent years for the branching step of SAT solvers, namely VSIDS and some of its improvements. Moreover, the completeness of the new algorithm is guaranteed. The preliminary experimental results are promising, and motivate the integration of heuristic backtracking in state-of-the-art SAT solvers.