Hybrid logics: Characterization, interpolation and complexity

Journal of Symbolic Logic 66 (3):977-1010 (2001)
Abstract
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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    Tero Tulenheimo (2009). Hybrid Logic Meets If Modal Logic. Journal of Logic, Language and Information 18 (4):559-591.
    Volker Weber (2009). Branching-Time Logics Repeatedly Referring to States. Journal of Logic, Language and Information 18 (4):593-624.

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