Bundled fragments of first-order modal logic: (un)decidability
This work addresses the problem of finding decidable fragments in quantified modal logic for logicians and computer scientists, but it is incremental as it builds on known fragments like the monodic one.
The paper tackled the decidability of fragments of first-order modal logic that bundle quantifiers and modalities, showing that the ∀□ bundle is undecidable over constant domains with monadic predicates, while the ∃□ bundle is decidable, and both are decidable over increasing domains with unrestricted predicates, with tableau procedures running in PSPACE.
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.