Bundled fragments of first-order modal logic: (un)decidability

Quantified modal logic provides a natural logical language for reasoning about modal attitudes even while retaining the richness of quantification for referring to predicates over domains. But then most fragments of the logic are undecidable, over many model classes. Over the years, only a few fragments (such as the monodic) have been shown to be decidable. In this paper, we study fragments that bundle quantifiers and modalities together, inspired by earlier work on epistemic logics of know-how/why/what. As always with quantified modal logics, it makes a significant difference whether the domain stays the same across worlds, or not. In particular, we show that the bundle forall Box is undecidable over constant domain interpretations, even with only monadic predicates, whereas exists Box bundle is decidable. On the other hand, over increasing domain interpretations, we get decidability with both forall Box and exists Box bundles with unrestricted predicates. In these cases, we also obtain tableau based procedures that run in PSPACE. We further show that the exists Box bundle cannot distinguish between constant domain and increasing domain interpretations.