On an Unsatisfiability-satisfiability Prover

Abstract

In recent decades the study of automated theorem proving has become to be accelerated with the progress of the computers. Therefore many proving methods have been implemented, for example [1], [4], [6] or theoretically investigated, for example [2], [5]. Among them most provers take their base from Robinson's resolution method. By the well known fact that the first order logic is undecidable, the most effort goes to the improvement of the efficiency of the successful test of a theorem, but not to widen the decidability of prover. In this paper we study a new unsatisfiability-satisfiability prover and its basic properties. Our main results, besides the presentation of our prover, is to show that our procedure always terminates if a given set of clauses is satisfiable in some finite domain. The unsatisfiability checking of our procedure is as strong as the existing complete proof procedures, so we examine the strength of the satisfiability checking part of our procedure in § 5 and § 6. In § 2 we define several basic concepts and show a number of propositions related to them. In §3 we give an unsatisfiability-satisfiability prover, and show its soundness that is for any unsatisfiable set of clauses the procedure says that it is unsatisfiable, and if the procedure says that a set of clause is satisfiable, then it is in fact satisfiable. In § 4 we show a number of basic properties of the procedure. In § 5, we show that for a given set of clauses assumed to be satisfiable in a simple theory of Herbrand universe, the procedure terminates saying it is satisfiable. In § 6 we show that for a given set of clauses assumed to be satisfiable in some finite domain, the procedure terminates saying it is satisfiable. In § 7 we discuss the results in this paper.