Algorithmic properties of first-order superintuitionistic logics of finite Kripke frames in restricted languages

Abstract

Abstract We consider the effect of restricting the number of individual variables, as well as the number and arity of predicate letters, in languages of first-order predicate superintuitionistic logics of finite Kripke frames on the logics’ algorithmic properties. By a finite frame we mean a frame with a finite set of possible worlds. The languages we consider have no constants, function symbols or the equality symbol. We show that positive fragments of many predicate superintuitionistic logics of natural classes of finite Kripke frames are not recursively enumerable—more precisely, $\varPi ^0_1$-hard—in languages with three individual variables and a single monadic predicate letter; this applies to the logics of finite frames of the predicate counterparts of propositional logics lying between the intuitionistic logic and the logic of the weak law of the excluded middle.