Genetic-fuzzy approach to the Boolean satisfiability problem

Witold Pedrycz, Giancarlo Succi, and Ofer Shai Owing to the immense size of the search space in the SAT problem, evolutionary computing arises as a viable and attractive option. The objective of this study is to formulate the SAT problem in the evolutionary setting and carry out comprehensive xperimental studies. The approach relies on the embedding principle: we generalize the Boolean problem to its continuous fuzzy (multivalued) version, find a solution to it, and convert (decode) it to the Boolean format. The concept of this transformation (embedding) was introduced initially in [10]. This study elaborates on the algorithm, presenting results of comprehensive xperimentation, and discusses necessary improvements to a generic genetic algorithm (GA) in the case of high-dimensional SAT problems. We confine discussion to the basic binary model of GA. The material is organized into six sections. First, we formulate the SAT problem in the GA environment by introducing an embedding principle that shows how a binary problem can be embedded into a continuous environment of fuzzy (multivalued) functions generated in the setting of fuzzy sets. Then we discuss details concerning the experimental environment (Section ill), including genetic optimization and a way of generating Boolean functions. In Section IV, we discuss experimental results and the efficiency of GA in solving the SAT problem, and contrast his approach with the random-search and brute-force complete enumeration methods. Moreover, we discuss an issue of scalability of the problem and experimentally identify some limits as to the number of Boolean variables. A recursive version of the genetic SAT solver is later discussed in Section V. It is shown how this recursive approach helps to handle a high-dimensional problem. Conclusions are contained in Section VI. Abstract-This study is concerned with the Boolean satisfiability ISAT) problem and its solution in setting a hybrid computational intelligence environment of genetic and fuzzy computing. In this framework, fUZ2Y sets realize an embedding principle meaning that original two-valued (Boolean) functions under investigation are extended to their continuous countel'Parts resulting in the form of fuzzy (multivalued) functions. In the sequel, the SAT problem is refonnulated for the fuzzy functions and solved using a genetic algorithm (GA). It is shown that a GA, especially its recursive version, is an efficient tool for handling multivariable SAT problems. Thorough experiments revealed that the recursive version of the GA can solve SAT problems with more than 1000 variables.