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Quantified modal logic provides a natural logical language for reasoningabout modal attitudes even while retaining the richness of quantification forreferring to predicates over domains. But then most fragments of the logic areundecidable, 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 studyfragments that bundle quantifiers and modalities together, inspired by earlierwork on epistemic logics of know-how/why/what. As always with quantified modallogics, it makes a significant difference whether the domain stays the sameacross worlds, or not. In particular, we show that the bundle $\forall \Box$ isundecidable over constant domain interpretations, even with only monadicpredicates, whereas $\exists \Box$ bundle is decidable. On the other hand, overincreasing domain interpretations, we get decidability with both $\forall \Box$and $\exists \Box$ bundles with unrestricted predicates. In these cases, wealso obtain tableau based procedures that run in \PSPACE. We further show thatthe $\exists \Box$ bundle cannot distinguish between constant domain andincreasing domain interpretations.

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