We generalise an interval-related interpolation theorem about abstract-time Interval Temporal Logic, which was first obtained in [GUE 01]. The generalisation is based on the abstract-time variant of a projection operator in the Duration Calculus, which was introduced in [DAN 99] and later studied extensively in [GUE 02]. We propose a way to understand interpolation in the context of formal verification. We give an example showing that, unlike abstract-time ITL, DC does not have the Craig interpolation property in general, and establish (...) a special form of Craig interpolation for abstract-time DC. Explicit definability after Beth is known to be strongly related to Craig interpolation in general. We show a limitation of a different kind to the scope of Beth definability in ITL by a counterexample too. We call the generalisation of interval-related interpolation that we present projection-related interpolation. The DC-specific restrictions apply to it too. We show that both Craig and projection-related interpolation hold about the ⌈P⌉-subset of DC without such restrictions. Our proofs of these theorems for the ⌈P⌉-subset entail algorithms for the construction of the interpolants. (shrink)

This is primarily a conceptual paper. The goal of the paper is to put into perspective the proof-theory of hybrid logic and in particular, try to give an answer to the following question: Why does the proof-theory of hybrid logic work so well compared to the proof-theory of ordinary modal logic?Roughly, there are two different kinds of proof systems for modal logic: Systems where the formulas involved in the rules are formulas of the object language, that is, ordinary modal-logical formulas, (...) and systems where the formulas involved in the rules are metalingustic formulas obtained by attaching labels representing possible worlds to ordinary modal-logical formulas. Systems of the second kind often also involve an explicit representation of the accessibility relation. From a proof-theoretic point of view, modal-logical systems of the first kind are less well-behaved than systems of the second kind.It turns out that this can be remedied by hybridization, that is, hybridization of modal logics enables the formulation of well-behaved proof systems without involving metalinguistic machinery. What has happened is that the metalinguistic machinery has been internalized in the object language. This gives an answer to the initial question, which is that the proof-theory of hybrid logic works so well because the metalinguistic semantic machinery has been internalized in the object language. (shrink)

In this paper we argue that hybrid logic is the deductive setting most natural for Kripke semantics. We do so by investigating hybrid axiomatics for a variety of systems, ranging from the basic hybrid language (a decidable system with the same complexity as orthodox propositional modal logic) to the strong Priorean language (which offers full first-order expressivity).We show that hybrid logic offers a genuinely first-order perspective on Kripke semantics: it is possible to define base logics which extend automatically to a (...) wide variety of frame classes and to prove completeness using the Henkin method. In the weaker languages, this requires the use of non-orthodox rules. We discuss these rules in detail and prove non-eliminability and eliminability results. We also show how another type of rule, which reflects the structure of the strong Priorean language, can be employed to give an even wider coverage of frame classes. We show that this deductive apparatus gets progressively simpler as we work our way up the expressivity hierarchy, and conclude the paper by showing that the approach transfers to first-order hybrid logic. (shrink)

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