r class of SAT instances. For instance, in our study of quasigroup problems, one rule seems better than the others: choose one literal in one of the shortest positive clauses (a positive clause is a clause where all the literals are positive). On the other hand, a proved effective splitting rule is to choose a variable x such that the value f 2 (x) f 2 (:x) is maximal, where f 2 (L) is one plus the number of occurrences of literal L in binary clauses [2, 5]. We tried to combine the above two rules into one as follows: Let 0 ! a 1 and n be the number of shortest non-Horn clauses in the current set. At first, we collect all the variable names appearing in the first da ne shortest positive clauses. Then we choose x in this pool

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. 1.

Equivalency clauses (Xors or modulo 2 arithmetics) represent a common structure in the SAT-encoding of many hard real-world problems and constitute a major obstacle to DavisPutnam (DP) procedure. We propose a special look-ahead technique called equivalency reasoning to overcome the obstacle and report on the performance of an equivalency reasoning enhanced DP procedure on SAT instances containing equivalency clauses derived from problems in parity learning, cryptographic key search and model checking. Our results show that integrating equivalency reasoning renders easy many problems which were beyond DP's reach. We also compare equivalency reasoning with general CSP look-back techniques on equivalency clauses.

CNF propositional satis ability (SAT) is a special kind of the more general Constraint Satisfaction Problem (CSP). While lookback techniques appear to be of little use to solve hard random SAT problems, it is supposed that they are necessary to solve hard structured SAT problems. In this paper, we propose a very simple DPL procedure called Satz which only employs some look-ahead techniques: a variable ordering heuristic, a forward consistency checking (Unit Propagation) and a limited resolution before the search, where the heuristic is itself based on unit propagation. Satz is favorably compared on random 3-SAT problems with three DPL procedures among the best in the literature for these problems. Furthermore on a great number of problems in 4 wellknown SAT benchmarks Satz reaches or outspeeds the performance of three other DPL procedures among the best in the literature for structured SAT problems. The comparative results suggest that a suitable exploitation of look-ahead techniques, while very simple and efficient for random SAT problems, may allow to do without sophisticated look-back techniques in a DPL procedure.

The Boolean Satisfiability Problem (SAT) is a well known NP-Complete problem. While its complexity remains a source of many interesting questions for theoretical computer scientists, the problem has found many practical applications in recent years. The emergence of efficient SAT solvers which can handle large structured SAT instances has enabled the use of SAT solvers in diverse domains such as verification, planning, routing, etc. These applications continue to motivate the development of faster and more robust SAT solvers. In this paper, we describe the popular SAT solver zchaff with a focus on recent developments.

We address the parallelization and distributed execution of an algorithm from the area of symbolic computation: propositional satisfiability (SAT) checking with dynamic learning. Our parallel programming models are strict multithreading for the core SAT checking procedure, complemented by mobile agents realizing a distributed dynamic learning process. Individual threads treat dynamically created subproblems, while mobile agents collect and distribute pertinent knowledge obtained during the learning process. The parallel algorithm runs on top of our parallel system platform DOTS (Distributed Object-Oriented Threads System), which provides support for our parallel programming models in highly heterogeneous distributed systems. We present performance measurements evaluating the performance gains by our approach in different application domains with practical significance. Key words: parallel symbolic computation, parallel propositional satisfiability checking, distributed multithreading 1

The computation of prime implicants has several and significant applications in different areas, including Automated Reasoning, Non-Monotonic Reasoning, Electronic Design Automation, among others. In this paper we describe a new model and algorithm for computing minimum-size prime implicants of propositional formulas. The proposed approach is based on creating an integer linear program (ILP) formulation for computing the minimumsize prime implicant, which simplifies existing formulations. In addition, we introduce two new algorithms for solving ILPs, both of which are built on top of an algorithm for propositional satisfiability (SAT). Given the organization of the proposed SAT algorithm, the resulting ILP procedures implement powerful search pruning techniques, including a non-chronological backtracking

This thesis describes the design and implementation of a highly optimized, multithreaded algorithm for the propositional satisfiability problem. The algorithm is based on the Davis-Logemann-Loveland sequential algorithm, but includes many of the optimization techniques introduced in recent years. The document provides experimental results for the execution of the parallel algorithm on a variety of multiprocessor machines with shared memory architecture. In particular, the overwhelming e#ect of parallel execution on the performance of processor cache is studied.

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Abstract. This work presents a memory-efficient All-SAT engine which, given a propositional formula over sets of important and non-important variables, returns the set of all the assignments to the important variables, which can be extended to solutions (satisfying assignments) to the formula. The engine is built using elements of modern SAT solvers, including a scheme for learning conflict clauses and non-chronological backtracking. Re-discovering solutions that were already found is avoided by the search algorithm itself, rather than by adding blocking clauses. As a result, the space requirements of a solved instance do not increase when solutions are found. Finding the next solution is as efficient as finding the first one, making it possible to solve instances for which the number of solutions is larger than the size of the main memory. We show how to exploit our All-SAT engine for performing image computation and use it as a basic block in achieving full reachability which is purely SATbased (no BDDs involved). We implemented our All-SAT solver and reachability algorithm using the stateof-the-art SAT solver Chaff [19] as a code base. The results show that our new scheme significantly outperforms All-SAT algorithms that use blocking clauses, as measured by the execution time, the memory requirement, and the number of steps performed by the reachability analysis. 1