Lintao Zhang

Chaff: engineering an efficient SAT solver

Boolean satisfiability is probably the most studied of the 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.

Efficient conflict driven learning in a boolean satisfiability solver

One of the most important features of current state-of-the-art SAT solvers is the use of conflict based backtracking and learning techniques. In this paper, we generalize various conflict driven learning strategies in terms of different partitioning schemes of the implication graph. We re-examine the learning techniques used in various SAT solvers and propose an array of new learning schemes. Extensive experiments with real world examples show that the best performing new learning scheme has at least a 2/spl times/ speedup compared with learning schemes employed in state-of-the-art SAT solvers.

Validating SAT Solvers Using an Independent Resolution-Based Checker: Practical Implementations and Other Applications

As the use of SAT solvers as core engines in EDA applications grows, it becomes increasingly important to validate their correctness. In this paper, we describe the implementation of an independent resolution-based checking procedure that can check the validity of unsatisfiable claims produced by the SAT solver zchaff. We examine the practical implementation issues of such a checker and describe two implementations with different pros and cons. Experimental results show low overhead for the checking process. Our checker can work with many other modern SAT solvers with minor modifications, and it can provide information for debugging when checking fails. Finally we describe additional results that can be obtained by the validation process and briefly discuss their applications.

The Quest for Efficient Boolean Satisfiability Solvers

The classical NP-complete problem of Boolean Satisfiability (SAT) has seen much interest in not just the theoretical computer science community, but also in areas where practical solutions to this problem enable significant practical applications. Since the first development of the basic search based algorithm proposed by Davis, Putnam, Logemann and Loveland (DPLL) about forty years ago, this area has seen active research effort with many interesting contributions that have culminated in state-of-the-art SAT solvers today being able to handle problem instances with thousands, and in same cases even millions, of variables. In this paper we examine some of the main ideas along this passage that have led to our current capabilities. Given the depth of the literature in this field, it is impossible to do this in any comprehensive way; rather we focus on techniques with consistent demonstrated efficiency in available solvers. For the most part, we focus on techniques within the basic DPLL search framework, but also briefly describe other approaches and look at some possible future research directions.

Conflict driven learning in a quantified Boolean satisfiability solver

Within the verification community, there has been a recent increase in interest in Quantified Boolean Formula evaluation (QBF) as many interesting sequential circuit verification problems can be formulated as QBF instances. A closely related research area to QBF is Boolean Satisfiability (SAT). Recent advances in SAT research have resulted in some very efficient SAT solvers. One of the critical techniques employed in these solvers is Conflict Driven Learning. In this paper, we adapt conflict driven learning for application in a QBF setting. We show that conflict driven learning can be regarded as a resolution process on the clauses. We prove that under certain conditions, tautology clauses obtained from resolution in QBF also obey the rules for implication and conflicts of regular (non-tautology) clauses; and therefore they can be treated as regular clauses and used in future search. We have implemented this idea in a new QBF solver called Quaffle and our initial experiments show that conflict driven learning can greatly speed up the solution process for most of the benchmarks we tested.

Combining strengths of circuit-based and CNF-based algorithms for a high-performance SAT solver

We propose Satisfiability Checking (SAT) techniques that lead to a consistent performance improvement of up to 3x over state-of-the-art SAT solvers like Chaff on important problem domains in VLSI CAD. We observe that in circuit oriented applications like ATPG and verification, different software engineering techniques are required for the portions of the formula corresponding to learnt clauses compared to the original formula. We demonstrate that by employing the same innovations as in advanced CNF-based SAT solvers, but in a hybrid approach where these two portions of the formula are represented differently and processed separately, it is possible to obtain the consistently highest performing SAT solver for circuit oriented problem domains. We also present controlled experiments to highlight where these gains come from. Once it is established that the hybrid approach is faster, it becomes possible to apply low overhead circuit-based heuristics that would be unavailable in the CNF domain for greater speedup.

Towards a Symmetric Treatment of Satisfaction and Conflicts in Quantified Boolean Formula Evaluation

In this paper, we describe a new framework for evaluating Quantified Boolean Formulas (QBF). The new framework is based on the Davis-Putnam (DPLL) search algorithm. In existing DPLL based QBF algorithms, the problem database is represented in Conjunctive Normal Form (CNF) as a set of clauses, implications are generated from these clauses, and backtracking in the search tree is chronological. In this work, we augment the basic DPLL algorithm with conflict driven learning as well as satisfiability directed implication and learning. In addition to the traditional clause database, we add a cube database to the data structure. We show that cubes can be used to generate satisfiability directed implications similar to conflict directed implications generated by the clauses. We show that in a QBF setting, conflicting leaves and satisfying leaves of the search tree both provide valuable information to the solver in a symmetric way. We have implemented our algorithm in the new QBF solver Quaffle. Experimental results show that for some test cases, satisfiability directed implication and learning significantly prunes the search.

Partition-based decision heuristics for image computation using SAT and BDDs

Methods based on Boolean satisfiability (SAT) typically use a conjunctive normal form (CNF) representation of the Boolean formula, and exploit the structure of the given problem through use of various decision heuristics and implication methods. We propose a new decision heuristic based on separator-set induced partitioning of the underlying CNF graph. It targets those variables whose choice generates clause partitions with disjoint variable supports. This can potentially improve performance of SAT applications by decomposing the problem dynamically within the search. In the context of a recently proposed image computation method combining SAT and BDDs, this results in simpler BDD subproblems. We provide algorithms for CNF partitioning - one based on a clause-variable dependency matrix, and another based on standard hypergraph partitioning techniques, and also for the use of partitioning information in decision heuristics for SAT. The effectiveness of the proposed partition-based heuristic is shown with practical results for reachability analysis of benchmark sequential circuits.