Complexity of interpolation and related problems in positive calculi

Abstract

We consider the problem of recognizing important properties of logical calculi and find complexity bounds for some decidable properties. For a given logical system $L$, a property $P$ of logical calculi is called decidable over $L$ if there is an algorithm which for any finite set $Ax$ of new axiom schemes decides whether the calculus $L + Ax$ has the property $P$ or not. In "Complexity of some problems in modal and superintuitionistic logics," the complexity of tabularity, pre-tabularity, and interpolation problems over the intuitionistic logic Int and over modal logic $S4$ was studied, also we found the complexity of amalgamation problems in varieties of Heyting algebras and closure algebras.

In the present paper we deal with positive calculi. We prove $NP$-completeness of tabularity, $DP$- hardness of pretabularity and PSPACE-completeness of interpolation problem over $Int^+$. In addition to above-mentioned properties, we consider Beth’s definability properties. Also we improve some complexity bounds for properties of superintuitionistic calculi.