Hybrid logics: Characterization, interpolation and complexity

Journal of Symbolic Logic 66 (3):977-1010 (2001)
Hybrid languages are expansions of propositional modal languages which can refer to (or even quantify over) worlds. The use of strong hybrid languages dates back to at least [Pri67], but recent work (for example [BS98, BT98a, BT99]) has focussed on a more constrained system called $\mathscr{H}(\downarrow, @)$ . We show in detail that $\mathscr{H}(\downarrow, @)$ is modally natural. We begin by studying its expressivity, and provide model theoretic characterizations (via a restricted notion of Ehrenfeucht-Fraisse game, and an enriched notion of bisimulation) and a syntactic characterization (in terms of bounded formulas). The key result to emerge is that $\mathscr{H}(\downarrow, @)$ corresponds to the fragment of first-order logic which is invariant for generated submodels. We then show that $\mathscr{H}(\downarrow, @)$ enjoys (strong) interpolation, provide counterexamples for its finite variable fragments, and show that weak interpolation holds for the sublanguage H(@). Finally, we provide complexity results for H(@) and other fragments and variants, and sharpen known undecidability results for $\mathscr{H}(\downarrow, @)$
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DOI 10.2307/2695090
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Igor Yanovich (2015). Expressive Power of “Now” and “Then” Operators. Journal of Logic, Language and Information 24 (1):65-93.
Torben Braüner (2002). Modal Logic, Truth, and the Master Modality. Journal of Philosophical Logic 31 (4):359-386.

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