Are bundles good deals for first-order modal logic?

Abstract

Bundled products are often offered as good deals to customers. In first-order modal logic (FOML), when we bundle quantifiers and modalities together (as in ∃x□, ◇∀x, etc.), 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 towards identifying the decidable ones. In this paper, we 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 class of models considered has constant domains (across states/worlds) or increasing domains presents another layer of complexity. We also present the loosely bundled fragment, which generalizes the bundles and yet retains decidability (over increasing domain models).