Using stochastic local search to solve quantified boolean formulae

Formal Methods in System Design, 2021

In recent years, expansion-based techniques have been shown to be very powerful in theory and practice for solving quantified Boolean formulas (QBF), the extension of propositional formulas with existential and universal quantifiers over Boolean variables. Such approaches partially expand one type of variable (either existential or universal) for obtaining a propositional abstraction of the QBF. If this formula is false, the truth value of the QBF is decided, otherwise further refinement steps are necessary. Classically, expansion-based solvers process the given formula quantifier-block wise and use one SAT solver per quantifier block. In this paper, we present a novel algorithm for expansion-based QBF solving that deals with the whole quantifier prefix at once. Hence recursive applications of the expansion principle are avoided and only two incremental SAT solvers are required. While our algorithm is naturally based on the $$\forall $$ ∀ Exp+Res calculus that is the formal foundati...

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.

Lecture Notes in Computer Science, 2007

In this paper we describe a stochastic local search (SLS) procedure for finding satisfying models of satisfiable propositional formulae. This new algorithm, gNovelty + , draws on the features of two other WalkSAT family algorithms: R+AdaptNovelty + and G 2 WSAT, while also successfully employing a dynamic local search (DLS) clause weighting heuristic to further improve performance. gNovelty + was a Gold Medal winner in the random category of the 2007 SAT competition. In this paper we present a detailed description of the algorithm and extend the SAT competition results via an empirical study of the effects of problem structure and parameter tuning on the performance of gNovelty + . The study also compares gNovelty + with two of the most representative WalkSAT-based solvers: G 2 WSAT, AdaptNovelty + , and two of the most representative DLS solvers: RSAPS and PAWS. Our new results augment the SAT competition results and show that gNovelty + is also highly competitive in the domain of solving structured satisfiability problems in comparison with other SLS techniques.

Journal of Automated Reasoning, 2000

Local search algorithms are among the standard methods for solving hard combinatorial problems from various areas of Artificial Intelligence and O perations Research. For SAT, some of the most successful and powerful algorithms are based on stochastic local search and in the past 10 years a large number of such algorithms have been proposed and investigated. In this article, we

2015

Satisfiability Modulo Theories (SMT) is essential for many practical applications, e.g., in hard- and software verification, and increasingly also in other scientific areas like computational biology. A large number of applications in these areas benefit from bit-precise reasoning over finite-domain variables. Current approaches in this area translate a formula over bit-vectors to an equisatisfiable propositional formula, which is then given to a SAT solver. In this paper, we present a novel stochastic local search (SLS) algorithm to solve SMT problems, especially those in the theory of bit-vectors, directly on the theory level. We explain how several successful techniques used in modern SLS solvers for SAT can be lifted to the SMT level. Experimental results show that our approach can compete with state-of-the-art bit-vector solvers on many practical instances and, sometimes, outperform existing solvers. This offers interesting possibilities in combining our approach with existing ...

Lecture Notes in Computer Science, 2001

This paper proposes a stochastic, and complete, backtrack search algorithm for Propositional Satisfiability (SAT). In recent years, randomization has become pervasive in SAT algorithms. Incomplete algorithms for SAT, for example the ones based on local search, often resort to randomization. Complete algorithms also resort to randomization. These include, state-of-the-art backtrack search SAT algorithms that often randomize variable selection heuristics. Moreover, it is plain that the introduction of randomization in other components of backtrack search SAT algorithms can potentially yield new competitive search strategies. As a result, we propose a stochastic backtrack search algorithm for SAT, that randomizes both the variable selection and the backtrack steps of the algorithm. In addition, we describe and compare different organizations of stochastic backtrack search. Finally, experimental results provide empirical evidence that the new search algorithm for SAT results in a very competitive approach for solving hard real-world instances.

Proc. of IJCAI-09, 2009

Lecture Notes in Computer Science, 2011

In this paper, we investigate the feasibility of applying algorithms based on the Uniform Confidence bounds applied to Trees [12] to the satisfiability of CNF formulas. We develop a new family of algorithms based on the idea of balancing exploitation (depth-first search) and exploration (breadth-first search), that can be combined with two different techniques to generate random playouts or with a heuristics-based evaluation function. We compare our algorithms with a DPLL-based algorithm and with WalkSAT, using the size of the tree and the number of flips as the performance measure. While our algorithms perform on par with DPLL on instances with little structure, they do quite well on structured instances where they can effectively reuse information gathered from one iteration on the next. We also discuss the pros and cons of our different algorithms and we conclude with a discussion of a number of avenues for future work.

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.

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.