Satisfiability vs. Finite Satisfiability in Elementary Modal Logics

Abstract

We study elementary modal logics, i.e. modal logic considered over first-order definable classes of frames. The classical semantics of modal logic allows infinite structures, but often practical applications require to restrict our attention to finite structures. Many decidability and undecidability results for the elementary modal logics were proved separately for general satisfiability and for finite satisfiability [11, 12, 16, 17]. In this paper, we show that there is a reason why we must deal with both kinds of satisfiability separately – we prove that there is a universal first-order formula that defines an elementary modal logic with decidable (global) satisfiability problem, but undecidable finite satisfiability problem, and, the other way round, that there is a universal formula that defines an elementary modal logic with decidable finite satisfiability problem, but undecidable general satisfiability problem

Finite Satisfiability of the Two-Variable Guarded Fragment with Transitive Guards and Related Variants

“…We also point out that even though in this paper we discuss fragments for which it required more care to prove decidability of the finite satisfiability problem than to prove decidability of the satisfiability problem, this by no means is a general rule. In particular, there are logics such that Sat(L) is undecidable and FinSat(L) is decidable, or vice versa (see, e.g., Michaliszyn et al [2012] for a family of examples from the elementary modal logics).…”

Finite Model Reasoning in Expressive Fragments of First-Order Logic

“…In particular, there are logics such that Sat(L ) is undecidable and FinSat(L ) is decidable, or vice versa (see e.g. [32] for a family of examples from the elementary modal logics).…”