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 H(↓, @). We show in detail that (↓, @) is modally natural. We begin by studying its expressivity, and provide model theoretic characterizations (via a restricted notion of Ehrenfeucht-Fraïssé game, and an enriched notion of bisimulation) and a syntactic characterization (in terms of bounded formulas). The key result to emerge is that (↓, @) corresponds to the fragment of first-order logic which is invariant for generated submodels. We then show that (↓, @) enjoys (strong) interpolation, provide counterexamples for its finite variable fragments, and show that weak interpolation holds for the sublanguage (@). Finally, we provide complexity results for (@) and other fragments and variants, and sharpen known undecidability results for (↓, @).

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We study dynamic modal operators that can change the accessibility relation of a model during the evaluation of a formula. In particular, we extend the basic modal language with modalities that are able to delete, add or swap an edge between pairs of elements of the domain. We define a generic framework to characterize this kind of operations. First, we investigate relation-changing modal logics as fragments of classical logics. Then, we use the new framework to get a suitable notion of bisimulation for the logics introduced, and we investigate their expressive power. Finally, we show that the complexity of the model checking problem for the particular operators introduced is PSpace-complete, and we study two subproblems of model checking: formula complexity and program complexity.

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We investigate dynamic modal operators that can change the model during evaluation. We define the logic SL by extending the basic modal language with the ♦ modality, which is a diamond operator that in addition has the ability to invert pairs of related elements in the domain while traversing an edge of the accessibility relation. SL is very expressive: it fails to have the finite and the tree model property. We show that SL is equivalent to a fragment of first-order logic by providing a satisfiability preserving translation. In addition, we provide an equivalence preserving translation from SL to the hybrid logic H(:, ↓). We also define a suitable notion of bisimulation for SL and investigate its expressive power, showing that it lies strictly between the basic modal logic and H(:, ↓). We finally show that its model checking problem is PSpace-complete and its satisfiability problem is undecidable.

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