Undecidability of the Logic of Partial Quasiary Predicates

Abstract

We obtain an effective embedding of the classical predicate logic into the logic of partial quasiary predicates. The embedding has the property that an image of a non-theorem of the classical logic is refutable in a model of the logic of partial quasiary predicates that has the same cardinality as the classical countermodel of the non-theorem. Therefore, we also obtain an embedding of the classical predicate logic of finite models into the logic of partial quasiary predicates over finite structures. As a consequence, we prove that the logic of partial quasiary predicates is undecidable—more precisely, $\varSigma ^0_1$-complete—over arbitrary structures and not recursively enumerable—more precisely, $\varPi ^0_1$-complete—over finite structures.

Undecidability of QLTL and QCTL with two variables and one monadic predicate letter

“…The main success story of algorithmic classification research has been the classical first-order logic QCl [Börger et al, 1997]. The algorithmic classification problem for non-classical modal, temporal, and superintuitionistic logics has been much less studied: despite extensive literature [Kripke, 1962;Artemov, Dzhaparidze, 1990;Gabbay, Shehtman, 1993;Wolter, Zakharyaschev, 2001;Kontchakov et al, 2005;Rybakov, Shkatov, 2019c;Shehtman, Shkatov, 2019;Rybakov, Shkatov, 2020b;Rybakov, Shkatov, 2020c;Shehtman, Shkatov, 2020;Rybakov, Shkatov, 2021b;Rybakov, Shkatov, 2021c;Rybakov, Shkatov, 2021d;Rybakov, Shkatov, 2021e], much less is known about the algorithmic properties of fragments of non-classical logics than about the algorithmic properties of fragments of QCl.…”