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.

Abstract. RACER implements a TBox and ABox reasoner for the logic SHIQ. RACER was the first full-fledged ABox description logic system for a very expressive logic and is based on optimized sound and complete algorithms. RACER also implements a decision procedure for modal logic satisfiability problems (possibly with global axioms). 1

AbstractÐThis paper introduces GRASP (Generic seaRch Algorithm for the Satisfiability Problem), a new search algorithm for Propositional Satisfiability (SAT). GRASP incorporates several search-pruning techniques that proved to be quite powerful on a wide variety of SAT problems. Some of these techniques are specific to SAT, whereas others are similar in spirit to approaches in other fields of Artificial Intelligence. GRASP is premised on the inevitability of conflicts during the search and its most distinguishing feature is the augmentation of basic backtracking search with a powerful conflict analysis procedure. Analyzing conflicts to determine their causes enables GRASP to backtrack nonchronologically to earlier levels in the search tree, potentially pruning large portions of the search space. In addition, by ªrecordingº the causes of conflicts, GRASP can recognize and preempt the occurrence of similar conflicts later on in the search. Finally, straightforward bookkeeping of the causality chains leading up to conflicts allows GRASP to identify assignments that are necessary for a solution to be found. Experimental results obtained from a large number of benchmarks indicate that application of the proposed conflict analysis techniques to SAT algorithms can be extremely effective for a large number of representative classes of SAT instances. Index TermsÐSatisfiability, search algorithms, conflict diagnosis, conflict-directed nonchronological backtracking, conflict-based equivalence, failure-driven assertions, unique implication points. 1

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 2X speedup compared with learning schemes employed in state-of-the-art SAT solvers.

We describe a SAT-solver, BerkMin, that inherits such features of GRASP, SATO, and Chaff as clause recording, fast BCP, restarts, and conflict clause “aging”. At the same time BerkMin introduces a new decision making procedure and a new method of clause database management. We experimentally compare BerkMin with Chaff, the leader among SAT-solvers used in the EDA domain. Experiments show that our solver is more robust than Chaff. BerkMin solved all the instances we used in experiments including very large CNFs from a microprocessor verification benchmark suite. On the other hand, Chaff was not able to complete some instances even with the timeout limit of 16 hours. 1.

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

Description Logics (DLs) are a family of knowledge representation formalisms mainly characterised by constructors to build complex concepts and roles from atomic ones. Expressive role constructors are important in many applications, but can be computationally problematical. We present an algorithm that decides satisfiability of the DL ALC extended with transitive and inverse roles and functional restrictions with respect to general concept inclusion axioms and role hierarchies; early experiments indicate that this algorithm is well-suited for implementation. Additionally, we show that ALC extended with just transitive and inverse roles is still in PSpace. We investigate the limits of decidability for this family of DLs, showing that relaxing the constraints placed on the kinds of roles used in number restrictions leads to the undecidability of all inference problems. Finally, we describe a number of optimisation techniques that are crucial in obtaining implementations of the decision procedures, which, despite the hight worst-case complexity of the problem, exhibit good performance with real-life problems. 1

Abstract. FaCT is a Description Logic classifier which has been implemented as a test-bed for a highly optimised tableaux satisfiability (subsumption) testing algorithm. The correspondence between modal and description logics also allows FaCT to be used as a theorem prover for the propositional modal logics K, KT, K4 and S4. Empirical tests have demonstrated the effectiveness of the optimised implementation and, in particular, of the dependency directed backtracking optimisation. 1

The research on conditional planning rejects the assumptions that there is no uncertainty or incompleteness of knowledge with respect to the state and changes of the system the plans operate on. Without these assumptions the sequences of operations that achieve the goals depend on the initial state and the outcomes of nondeterministic changes in the system. This setting raises the questions of how to represent the plans and how to perform plan search. The answers are quite different from those in the simpler classical framework. In this paper, we approach conditional planning from a new viewpoint that is motivated by the use of satisfiability algorithms in classical planning. Translating conditional planning to formulae in the propositional logic is not feasible because of inherent computational limitations. Instead, we translate conditional planning to quantified Boolean formulae. We discuss three formalizations of conditional planning as quantified Boolean formulae, and pr...

Determining whether a propositional theory is satisfiable is a prototypical example of an NP-complete problem. Further, a large number of problems that occur in knowledge-representation, learning, planning, and other ares of AI are essentially satisfiability problems. This paper reports on the most extensive set of experiments to date on the location and nature of the cross-over point in satisfiability problems. These experiments generally confirm previous results with two notable exceptions. First, we have found that neither of the functions previously proposed accurately models the location of the cross-over point. Second, we have found no evidence of any hard problems in the underconstrained region. In fact the hardest problems found in the underconstrained region were many times easier than the easiest unsatisfiable problems found in the neighborhood of the cross-over point. We offer explanations for these apparent contradictions of previous results. This work has been supported ...