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Hybrid logics: characterization, interpolation and complexity

Published online by Cambridge University Press:  12 March 2014

Carlos Areces
Affiliation:
ILLC, University of Amsterdam, Plantage Muidergracht 24, 1018TV, Amsterdam, The Netherlands, E-mail: carlos@science.uva.nl
Patrick Blackburn
Affiliation:
INRIA, Lorraine, 615, Rue du Jardin Botanique, 54602 Villers lès Nancy Cedex, France, E-mail: patrick@aplog.org
Maarten Marx
Affiliation:
Department of Sociology and Anthropology, University of Amsterdam, Plantage Muidergracht 24, 1018TV, Amsterdam, The Netherlands, E-mail: marx@science.uva.nl

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 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 (↓, @).

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2001

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