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Internalizing Labelled Deduction
 Journal of Logic and Computation
, 2000
"... This paper shows how to internalize the Kripke satisfaction denition using the basic hybrid language, and explores the proof theoretic consequences of doing so. As we shall see, the basic hybrid language enables us to transfer classic Gabbaystyle labelled deduction methods from the metalanguage to ..."
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Cited by 74 (20 self)
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This paper shows how to internalize the Kripke satisfaction denition using the basic hybrid language, and explores the proof theoretic consequences of doing so. As we shall see, the basic hybrid language enables us to transfer classic Gabbaystyle labelled deduction methods from the metalanguage to the object language, and to handle labelling discipline logically. This internalized approach to labelled deduction links neatly with the Gabbaystyle rules now widely used in modal Hilbertsystems, enables completeness results for a wide range of rstorder denable frame classes to be obtained automatically, and extends to many richer languages. The paper discusses related work by Jerry Seligman and Miroslava Tzakova and concludes with some reections on the status of labelling in modal logic. 1 Introduction Modern modal logic revolves around the Kripke satisfaction relation: M;w ': This says that the model M satises (or forces, or supports) the modal formula ' at the state w in M....
Hybrid languages and temporal logic
 Logic J. IGPL
, 1999
"... Hybridization is a method invented by Arthur Prior for extending the expressive power of modal languages. Although developed in interesting ways by Robert Bull, and by the So a school (notably, George Gargov, Valentin Goranko, Solomon Passy and Tinko Tinchev), the method remains little known. In our ..."
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Cited by 36 (15 self)
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Hybridization is a method invented by Arthur Prior for extending the expressive power of modal languages. Although developed in interesting ways by Robert Bull, and by the So a school (notably, George Gargov, Valentin Goranko, Solomon Passy and Tinko Tinchev), the method remains little known. In our view this has deprived temporal logic of a valuable tool. The aim of the paper is to explain why hybridization is useful in temporal logic. We make two major points, the rst technical, the second conceptual. First, we showthathybridization gives rise to wellbehaved logics that exhibit an interesting synergy between modal and classical ideas. This synergy, obvious for hybrid languages with full rstorder expressive strength, is demonstrated for a weaker local language capable of de ning the Until operator � we provide a minimal axiomatization, and show that in a wide range of temporally interesting cases extended completeness results can be obtained automatically. Second, we argue that the idea of sorted atomic symbols which underpins the hybrid enterprise can be developed further. To illustrate this, we discuss the advantages and disadvantages of a simple hybrid language which can quantify over paths. 1
Hybridizing Concept Languages
 Annals of Mathematics and Artificial Intelligence
"... This paper shows how to increase the expressivity of concept languages using a strategy called hybridization. Building on the wellknown correspondences between modal and description logics, two hybrid languages are dened. These languages are called `hybrid' because, as well as the familiar prop ..."
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Cited by 17 (8 self)
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This paper shows how to increase the expressivity of concept languages using a strategy called hybridization. Building on the wellknown correspondences between modal and description logics, two hybrid languages are dened. These languages are called `hybrid' because, as well as the familiar propositional variables and modal operators, they also contain variables across individuals and a binder that binds these variables. As is shown, combining aspects of modal and rstorder logic in this manner allows the expressivity of concept languages to be boosted in a natural way, making it possible to dene number restrictions, collections of individuals, irreexivity of roles, and TBox and ABoxstatements. Subsequent addition of the universal modality allows the notion of subsumption to internalized, and enables the representation of queries to arbitrary rstorder knowledge bases. The paper notes themes shared by the hybrid and concept language literatures, and draws attention t...
Internalization: The Case of Hybrid Logics
, 2001
"... A sequent calculus for hybrid logics is developed from a calculus for classical predicate logic by a series of transformations. We formalize the semantic theory of hybrid logic using a sequent calculus for predicate logic plus axioms. This works, but it is ugly. The unattractive features are removed ..."
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Cited by 7 (0 self)
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A sequent calculus for hybrid logics is developed from a calculus for classical predicate logic by a series of transformations. We formalize the semantic theory of hybrid logic using a sequent calculus for predicate logic plus axioms. This works, but it is ugly. The unattractive features are removed onebyone, until the final vestiges of the metalanguage can be set aside to reveal a fully internalized calculus. The techniques are quite general and can be applied to a wide range of hybrid and modal logics.