A Decidable Bundled Fragment of First-Order Modal Logic Without Finite Model Property

Abstract

The satisfiability problem for First-order Modal Logic (\FOML) is undecidable even for simple fragments like having only unary predicates, two variables etc. Recently a new way to identify decidable fragments of \FOML has been introduced called the "bundled fragments", where the quantifiers and modalities are restricted to appear together. Since there are many ways to bundle the quantifiers together, some of them lead to (un)decidable fragments. In (Liu et.al, 2023) the authors prove a `trichotomy', where they show that every bundled fragment falls into one of the following three categories: (1) Those that satisfy "finite model property" (and hence decidable), (2) Those that are undecidable, and (3) Those that do not satisfy "finite model property" (whose decidability is left open). In this paper we collapse the trichotomy into a dichotomy over "increasing domain models" by proving that the one combination that falls into the last category is indeed decidable.