On Translation of Finitely-Valued Logics to Classical First-Order Logic

The main goal of this paper is to give a better understanding of existing many-valued resolution procedures, and thus help to improve the efficiency of automated theorem proving in finitely-valued logics. First, we briefly present a method for translation to clause form in finitely-valued logics based on distributive lattices with operators, which uses the Priestley dual of the algebra of truth values. We show that the unsatisfiability of the set of signed clauses thus obtained can be checked by a version of signed negative hyperresolution. This extends the results established by H\"ahnle in the case of regular logics, where the set of truth values is linearly ordered. As in the case of regular hyperresolution, also our version of signed hyperresolution is surprisingly similar to the classical version. In the second part of the paper we explain why regular logics and the logics we consider are so well-behaved. We show that in both cases the translation to clause form is actually a translation to classical logic, and that soundness and completeness of various refinements of the (signed) resolution procedure follow as a consequence of results from first-order logic. Decidability and complexity results for signed clauses follow as well, by using results from first-order logic. This explains and extends earlier results on theorem proving in finitely-valued logics