The method proposed by Davis, Putnam, Logemann, and Loveland for propositional reasoning, often referred to as the Davis-Putnam method, is one of the major practical methods for the satisfiability (SAT) problem of propositional logic. We show how to implement the Davis-Putnam method efficiently using the trie data structure for propositional clauses. A new technique of indexing only the first and last literals of clauses yields a unit propagation procedure whose complexity is sublinear to the number of occurrences of the variable in the input. We also show that the Davis-Putnam method can work better when unit subsumption is not used. We illustrate the performance of our programs on some quasigroup problems. The efficiency of our programs has enabled us to solve some open quasigroup problems.

In recent years, there has been an explosion of research in AI into propositional satis ability (or Sat). There are many factors behind the increased interest in this area. One factor is the improvement in search procedures for Sat. New local search procedures like Gsat are able to solve Sat problems with thousands of variables. At the same time, implementations of complete search algorithms like Davis-Putnam have been able to solve open mathematical problems. Another factor is the identi cation of hard Sat problems at a phase transition in solubility. A third factor is the demonstration that we can often solve real world problems by encoding them into Sat. There has also seen an improved theoretical understanding of Sat, particularly in the analysis of such phase transition behaviour. This paper reviews the state of the art for research into satis ability, and discuss applications in which algorithms for satis ability have proved successful.

this paper, we will show how to produce a fair schedule with the assistance of a propositional solver, when we formulate the problem smartly

Recent work on the combinatorial search has provided experimental and theoretical evidence that randomization and restart strategies proposed by Gomes, Selman, and Kautz, can be very effective for backtrack search algorithms to solve some hard satisfiable instances of SAT. One difficulty of effectively using the restart strategy is its potential conflict with the branching heuristic. It is well-known that a completely random branching heuristic yields very poor performance. To support the restart strategy, the branching heuristic has to be random (with limitation); otherwise, every restart is a repetition of the first run. In this paper, we propose a new randomization strategy which offers the same advantage of the restart strategy but it can be used with any branching heuristics. The basic idea is to randomly jump in the search space to skip some space. Each jump corresponds to a restart in the restart strategy but there is no repetition. We ensure that no portion of the search space is visited twice during one run and the search will be going on until the allotted time is run out or the search space is exhausted. This new strategy is implemented in SATO, an efficient implementation of the Davis-Putnam-Loveland method for SAT problems. Using the new strategy, we are able to solve several previously open quasigroup problems, which could not be solved using any existing SAT systems.

Abstract. We present a scheduler with a web interface for generating fair game schedules of a tournament. The tournament can be either single or double round-robin or something in between. The search engine inside the scheduler is a SAT solver which can handle a mix of ordinary and TL (True-Literal) clauses. The latter are the formulas using the function TL which counts the number of true literals in a clause. By using TL clauses, we could solve a typical scheduling problem in a few of seconds. If we convert them into ordinary clauses, the state-of-the-art SAT solvers could not solve them in one week. We showed how to integrate TL clauses into a SAT solver and take advantages of the advanced SAT techniques. Our scheduler provides a free service to all the people who are interested in fair sports schedules. 1

Solving open quasigroup existence problems is a challenging problem to which SAT solvers have been applied successfully. However, the number of problems of this domain is relatively small and its use as a benchmark is therefore restricted. We propose a new benchmark set that generalises and extends the quasigroup existence domain by using problems generated during the construction of classification theorems for finite algebras. These problems provide a rich class of benchmarks of varying difficulty from an algebraic domain. We briefly introduce this domain, discuss the difficulties for satisfiability checking, and present first experiences of tackling the problems with SAT solvers. 1

There are many people to whom I am grateful for guidance and support during the course of this work. First and foremost I would like to thank my supervisors Rajeev Goré and John Slaney for their guidance and discussions and for teaching me many interesting things. My sincere gratitude also goes to Toby Walsh, John Lloyd, Bob Meyer, Daniel Le Berre and Matthias Fuchs for their helpful discussions and feedback on my research. I am also very grateful to the numerous members of staff and students of the Research School of Information Sciences and Engineering and the Department of Computer Science who helped me along the way. It has, undoubtedly, been an excellent experience. My gratitude also goes to the Australian National University for providing a schol-arship enabling me to perform this research. Thanks to my fellow students of reasoning with whom I have had many inter-esting discussions, Kal, Nic and Paul. I have also appreciated the company of all the other computer and coffee geeks, imbibing coffee whilst discussing theses and all things digital.