Adding resolution to the DPLL procedure for Boolean satisfiability

2001

The problem of proving that a propositional boolean formula is satisfiable (SAT) is one of the fundamental problems in computer science. The most popular SAT algorithms are based on the well known Davis-Putnam procedure. There, to guide search, a branching rule is applied for selecting still unassigned variables and computing an assignment to this variable. Additionally, conflict analysis methods were developed which result in non-chronological backtracking that preserves the SAT algorithm to search nonrelevant parts of the search space.

2002

ABSTRACT Many leading-edge SAT solvers are based on the Davis-Putnam procedure or the Davis-Logemann-Loveland procedure, and thus on unsatisfiable instances they can be viewed as attempting to find refutations by resolution. Therefore, exponential lower bounds on the length of resolution proofs also apply to such solvers. Empirical performance of DLL-based solvers on SAT instances from the pigeonhole and Urquhart family are consistent with this expectation.

Design Automation …, 2001

Boolean Satisfiability is probably the most studied of combinatorial optimization/search problems. Significant effort has been devoted to trying to provide practical solutions to this problem for problem instances encountered in a range of applications in Electronic Design Automation (EDA), as well as in Artificial Intelligence (AI). This study has culminated in the development of several SAT packages, both proprietary and in the public domain (e.g. GRASP, SATO) which find significant use in both research and industry. Most existing complete solvers are variants of the Davis-Putnam (DP) search algorithm. In this paper we describe the development of a new complete solver, Chaff, which achieves significant performance gains through careful engineering of all aspects of the search-especially a particularly efficient implementation of Boolean constraint propagation (BCP) and a novel low overhead decision strategy. Chaff has been able to obtain one to two orders of magnitude performance improvement on difficult SAT benchmarks in comparison with other solvers (DP or otherwise), including GRASP and SATO.

Workshop on Design and Diagnostics of Electronic Circuits and Systems, 2000

This paper proposes a new algorithm for solving the Boolean satisfiability (SAT) problem. On the basis of this algorithm a software/reconfigurable hardware SAT solver was designed, implemented and compared to a similar realization of the Davis-Putnam-like method. The satisfier suggested uses an application-specific approach, thus an instance- specific hardware compilation is completely avoided.

Artificial Intelligence, 2011

In this work, we improve on existing results on the relationship between proof systems obtained from conflict-driven clause-learning SAT solvers and general resolution. Previous contributions such as those by demonstrated that variations on conflict-driven clause-learning SAT solvers corresponded to proof systems as powerful as general resolution. However, the models used in these studies required either an extra degree of non-determinism or a preprocessing step that is not utilized by state-ofthe-art SAT solvers in practice. In this paper, we prove that conflict-driven clauselearning SAT solvers yield proof systems that indeed p-simulate general resolution without the need for any additional techniques. Moreover, we show that our result can be generalized to certain other practical variations of the solvers, which are based on different learning schemes and restart policies. $ This work extends our previous work in with generalized results (Section 6) and additional discussion about related work (Section 7).

Discrete Applied Mathematics, 2003

There are many different ways of proving formulas in proposition logic. Many of these can easily be characterized as forms of resolution (e.g. and ). Others use so-called binary decision diagrams (BDDs) . Experimental evidence suggests that BDDs and resolution based techniques are fundamentally different, in the sense that their performance can differ very much on benchmarks . In this paper we confirm these findings by mathematical proof. We provide examples that are easy for BDDs and exponentially hard for any form of resolution, and vice versa, examples that are easy for resolution and exponentially hard for BDDs. such benchmarks it can not be excluded that there exist a resolution based technique that always out-performs BDDs, provided a proper proof search strategy would be chosen. So, a mathematical comparison between the techniques is called for. This is not straightforward, as resolution and BDDs look very different. BDDs work on arbitrary formulas, whereas resolution is strictly linked to formulas in conjunctive normal form. And the resolution rule and the BDD construction algorithms appear of a totally dissimilar nature.

Siam Journal on Computing, 2002

We consider several problems related to the use of resolution-based methods for determining whether a given boolean formula in conjunctive normal form is satisfiable. First, building on work of Clegg, Edmonds and Impagliazzo, we give an algorithm for satisfiability that when given an unsatisfiable formula of F finds a resolution proof of F, and the runtime of our algorithm is nontrivial as a function of the size of the shortest resolution proof of F. Next we investigate a class of backtrack search algorithms, commonly known as Davis-Putnam procedures and provide the first average-case complexity analysis for their behavior on random formulas. In particular, for a simple algorithm in this class, called ordered DLL we prove that the running time of the algorithm on a randomly generated k-CNF formula with n variables and m clauses is 2 Θ(n(n/m) 1/(k−2) ) with probability 1 − o(1). Finally, we give new lower bounds on res(F), the size of the smallest resolution refutation of F, for a class of formulas representing the pigeonhole principle, and for randomly generated formulas. For random formulas, Chvátal and Szemerédi had shown that random 3-CNF formulas with a linear number of clauses require exponential size resolution proofs * Preliminary versions of these results appeared in the 37th IEEE Symposium on Foundations of Computer Science [BP96] and the 30th ACM Symposium of Theory of Computing [BKPS98].

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.

The past few years have seen an enormous progress in the performance of Boolean satisfiability (SAT) solvers . Despite the worst-case exponential run time of all known algorithms, satisfiability solvers are increasingly leaving their mark as a generalpurpose tool in areas as diverse as software and hardware verification , automatic test pattern generation , planning , scheduling [103], and even challenging problems from algebra . Annual SAT competitions have led to the development of dozens of clever implementations of such solvers [e.g. 13, 19, 71, 93, 109, 118, 150, 152, 161, 165, 170, 171, 173, 174, 184, 198, 211, 213, 236], an exploration of many new techniques [e.g. 15, 102, 149, 170, 174], and the creation of an extensive suite of real-world instances as well as challenging hand-crafted benchmark problems [cf. 115]. Modern SAT solvers provide a "black-box" procedure that can often solve hard structured problems with over a million variables and several million constraints.

Lecture Notes in Computer Science, 2005

We propose a new decision heuristic for DPLL-based propositional SAT solvers. Its essence is that both the initial and the conflict clauses are arranged in a list and the next decision variable is chosen from the top -most unsatisfied clause. Various methods of initially organizing the list and moving the clauses within it are studied. Our approach is an extension of one used in Berkmin, and adopted by other modern solvers, according to which only conflict clauses are organized in a list, and a literal-scoring-based secondary heuristic is used when there are no more unsatisfied conflict clauses. Our approach, implemented in the 2004 version of zChaff solver and in a generic Chaff-based SAT solver, results in a significant performance boost on hard industrial benchmarks.

2007

Local search algorithms for satisfiability testing are still the best methods for a large number of problems, despite tremendous progresses observed on complete search algorithms over the last few years. However, their intrinsic limit does not allow them to address UNSAT problems. Ten years ago, this question challenged the community without any answer: was it possible to use local search algorithm for UNSAT formulae? We propose here a first approach addressing this issue, that can beat the best resolution-based complete methods. We define the landscape of the search by approximating the number of filtered clauses by resolution proof. Furthermore, we add high-level reasoning mechanism, based on Extended Resolution and Unit Propagation Look-Ahead to make this new and challenging approach possible. Our new algorithm also tends to be the first step on two other challenging problems: obtaining short proofs for UNSAT problems and build a real local-search algorithm for QBF.

2014

In recent years, the Propositional Satisfiability (SAT) has become standard for encoding real world complex constrained problems. SAT has significant impacts on various research fields in Artificial Intelligence (AI) and Constraint Programming (CP). SAT algorithms have also been successfully used in solving many practical and industrial applications that include electronic design automation, default reasoning, diagnosis, planning, scheduling, image interpretation, circuit design, and hardware and software verification. The most common representation of a SAT formula is the Conjunctive Normal Form (CNF). A CNF formula is a conjunction of clauses where each clause is a disjunction of Boolean literals. A SAT formula is satisfiable if there is a truth assignment for each variable such that all clauses in the formula are satisfied. Solving a SAT problem is to determine a truth assignment that satisfies a CNF formula. SAT is the first problem proved to be NP-complete [20]. There are many ...

2009

Boolean satisfiability (SAT) solvers are used heavily in hardware and software verification tools for checking satisfiability of Boolean formulas. Most state-of-the-art SAT solvers are based on the Davis-Putnam-Logemann-Loveland (DPLL) algorithm and require the input formula to be in conjunctive normal form (CNF). We present a new SAT solver that operates on the negation normal form (NNF) of the given Boolean formulas/circuits. The NNF of a formula is usually more succinct than the CNF of the formula in terms of the number of variables. Our algorithm applies the DPLL algorithm to the graph-based representations of NNF formulas. We adapt the idea of the two-watched-literal scheme from CNF SAT solvers in order to efficiently carry out Boolean Constraint Propagation (BCP), a key task in the DPLL algorithm. We evaluate the new solver on a large collection of Boolean circuit benchmarks obtained from formal verification problems. The new solver outperforms the top solvers of the SAT 2007 competition and SAT-Race 2008 in terms of run time on a large majority of the benchmarks.

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.

Artificial Intelligence, 2003

The implementation of effective reasoning tools for deciding the satisfiability of Quantified Boolean Formulas (QBFs) is an important research issue in Artificial Intelligence. Many decision procedures have been proposed in the last few years, most of them based on the Davis, Logemann, Loveland procedure (DLL) for propositional satisfiability (SAT). In this paper we show how it is possible to extend the conflict-directed backjumping schema for SAT to QBF: when applicable, it allows to jump over existentially quantified literals while backtracking. We introduce solution-directed backjumping, which allows the same for universally quantified literals. Then, we show how it is possible to incorporate both conflict-directed and solution-directed backjumping in a DLL-based decision procedure for QBF satisfiability. We also implement and test the procedure: The experimental analysis shows that, because of backjumping, significant speed-ups can be obtained. While there have been several proposals for backjumping in SAT, this is the first time -as far as we know-this idea has been proposed, implemented and experimented for QBFs.