Two types of universal proof systems for all variants of many-valued logics and some properties of them

Abstract

Two types of propositional proof systems are described in this paper such that proof system for each variant of propositional many-valued logic can be presented in both of the described forms. The first of introduced systems is a Gentzen-like system, the second one is based on the generalization of the notion of determinative disjunctive normal form, formerly defined by first coauthor. Some generalization of Kalmar’s proof of deducibility for two-valued tautologies in the classical propositional logic gives us a possibility to suggest the easy method of proving the completeness for first type of described systems. The completeness of the second type one is received from its construction automatically. The introduced proof systems are “weak” ones with a “simple strategist” of proof search and we have investigated the quantitative properties, related to proof complexity characteristics in them as well. In particular, for some class of many-valued tautologies simultaneously optimal bounds (asymptotically the same upper and lower bounds for each proof complexity characteristic) are obtained in the systems, considered for some versions of many-valued logic.