Are Bundles Good Deals for FOML?

TL;DR

This paper explores the decidability and complexity of bundled fragments in first-order modal logic, mapping their logical properties and introducing a new decidable fragment called the loosely bundled fragment.

Contribution

It provides a comprehensive analysis of bundled fragments in FOML, including new results on decidability, the finite model property, and a novel decidable fragment over increasing domain models.

Findings

Mapped the decidability landscape of bundled FOML fragments

Identified fragments lacking the finite model property

Introduced the loosely bundled fragment with decidability over increasing domains

Abstract

Bundled products are often offered as good deals to customers. When we bundle quantifiers and modalities together (as in $\exists x \Box$, $\Diamond \forall x$ etc.) in first-order modal logic (FOML), we get new logical operators whose combinations produce interesting fragments of FOML without any restriction on the arity of predicates, the number of variables, or the modal scope. It is well-known that finding decidable fragments of FOML is hard, so we may ask: do bundled fragments that exploit the distinct expressivity of FOML constitute good deals in balancing the expressivity and complexity? There are a few positive earlier results on some particular fragments. In this paper, we try to fully map the terrain of bundled fragments of FOML in (un)decidability, and in the cases without a definite answer yet, we show that they lack the finite model property. Moreover, whether the logics are interpreted over constant domains (across states/worlds) or increasing domains presents another layer of complexity. We also present the \textit{loosely bundled fragment}, which generalizes the bundles and yet retain decidability (over increasing domain models).