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 Gabbay-style 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 Gabbay-style rules now widely used in modal Hilbert-systems, enables completeness results for a wide range of rst-order 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....

Many of the formalisms used in Attribute Value grammar are notational variants of languages of propositional modal logic, and testing whether two Attribute Value descriptions unify amounts to testing for modal satisfiablity. In this paper we put this observation to work. We study the complexity of the satisfiability problem for nine modal languages which mirror different aspects of AVS description formalisms, including the ability to express re-entrancy, the ability to express generalisations, and the ability to express recursive constraints. Two main techniques are used: either Kripke models with desirable properties are constructed, or modalities are used to simulate fragments of Propositional Dynamic Logic. Further possibilities for the application of modal logic in computational linguistics are noted. Attribute Value Structures (AVSs) are probably the most widely used means of representing linguistic structure in current computational linguistics, and the process of unifying...

This chapter provides a modern overview of the field of hybrid logic. Hybrid logics are extensions of standard modal logics, involving symbols that name individual states in models. The first results that are nowadays considered as part of the field date back to the early work of Arthur

We analyze several classes of path constraints for semistructured data and prove a umber of decidability and complexity results for such constraints. While some of our decidability results were known before, we believe that our improved complexity bounds are new. Our proofs are based on techniques from modal logic and automata theory. We believe that our modal logic perspective sheds additional light on the reasons for previously known decidability and complexity results.

This paper discusses an approach to feature logic called the structural approach. The method consists in first viewing feature structures as models (or relational structures) in the standard model theoretic sense, and then going on to consider various languages that can be interpeted on these structures and the way these languages are interrelated. It will be shown that the structural approach offers...

this paper, Jerry Seligman takes us on an interesting journey. The satisfaction denition of most modal operators is specied in terms of rst-order conditions. Hence we can always obtain a complete calculus for the basic logic characterizing any collection of such operators by appealing to a calculus which is complete for the full rst-order language. Seligman shows here that by making use of the expressiveness provided by the hybrid apparatus, we can, step by step, transform a rst-order sequent calculus into an internalized sequent calculus specically tailored for a particular hybrid fragment

We analyze several classes of path constraints for semistructured data in a uni- ed framework and prove some decidability and complexity results for these constraints by embedding them in Propositional Dynamic Logic. While some of our decidability results were known before, we believe that our improved complexity bounds are new. Our proofs, based on techniques from modal logic, shed additional light on the reasons for previously known decidability and complexity results.

this paper tries to takes it seriously. The sorted systems considered here were not developed for use in AI; they are parts of richer languages designed with the needs of natural language semantics in mind (see Blackburn [5, 8]). Only subsequently was it observed that these systems oered an interesting perspective on temporal knowledge representation.

session examined general issues in hybrid logic and their relation to other frameworks. Three talks were given, and all adopted a largely model-theoretic perspective: Hybrid Languages Patrick Blackburn Computerlinguistik, Universitat des Saarlandes, Saarbrucken, Germany. Patrick gave an overview of hybrid languages that introduced various key concepts such as nominals, the @ operator, the various hybrid binders now commonly used (notably and #), and other tools (such as the universal modality, and the di#erence operator D) encountered in the literature on hybrid languages. As well as outlining the `mainstream' history of hybrid languages, Patrick drew attention to many other formalisms which can be viewed as variants or subsystems. Hybrid Logic and Description Logic Carlos Areces ILLC, University of Amsterdam, The Netherlands. While the basic link between hybrid languages and description logic has long been clear (the one-of operator O corresponds to disjunctions of nomi

The evolution of Description Logics (DLs) and Propositional Dynamic Logics produced a hierarchy of decidable logics with multiple maximal elements. It would be desirable to combine different maximal logics into one super-logic, but then inference may turn out to be undecidable. Then it is important to characterize the decidability threshold for these logics. In this perspective, an interesting open question pointed out by Sattler and Vardi [20] is whether inference in a hybrid µ-calculus with restricted forms of graded modalities is decidable, and which complexity class it belongs to. In this paper we improve a previous result [5] and prove that this calculus and the corresponding DL µALCIOfa are undecidable. We show also that nested fixpoints are not necessary for undecidability.