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

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....

In their simplest form, hybrid languages are propositional modal languages which can refer to states. They were introduced by Arthur Prior, the inventor of tense logic, and played an important role in his work: because they make reference to specic times possible, they remove the most serious obstacle to developing modal approaches to temporal representation and reasoning. However very little is known about the computational complexity of hybrid temporal logics. In this paper we analyze the complexity of the satisability problem of a number of hybrid temporal logics: the basic hybrid language over transitive frames; nominal tense logic over transitive frames, strict total orders, and transitive trees; nominal Until logic; and referential interval logic. We discuss the eects of including nominals, the @ operator, the somewhere modality E, and the dierence operator D. Adding nominals to tense logic leads for several frame{classes to an increase in complexity of the satisability pro...

We present the basic framework of a logic of actions and plans defined in terms of modal logic combined with a notion of dependence. The latter is used as a weak causal connection between actions and literals. In this paper we focus on the frame problem and demonstrate how it can be solved in our framework in a simple and monotonic way. We give the semantics, and associate an axiomatics and a decision procedure to it. The decision procedure is based on a sound and complete tableau method with single step rules to treat dependence. We show how it can be used to generate plans. Our solution is formally assessed by a translation of Gelfond and Lifschitz' logic A. We briefly sketch the second part of the paper, showing how we can go beyond A by some examples involving nondeterminism and ramifications.

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...

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 well-behaved logics that exhibit an interesting synergy between modal and classical ideas. This synergy, obvious for hybrid languages with full rst-order 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

1 Motivation Since Modal Logics are an extension of Propositional Logic, they provide Boolean operators for constructing complex formulae. However, most Modal Logics do not admit Boolean operators for constructing complex modal parameters to be used in the box and diamond operators. This asymmetry is not present in Boolean Modal Logics, in which box and diamond quantify over arbitrary Boolean combinations of atomic modal parameters [9]. Boolean Modal Logics have been considered in various forms and contexts: 1. "Pure " Boolean Modal Logic has been studied in [9]. Negation and intersection of modal parameters occur in some variants of Propositional Dynamic Logic, see, e.g., [7, 16, 22].

this paper, we pursue a monotonic approach to the frame problem and concentrate on the combinatorial problem and the overcommitment problem. We will propose a solution within the framework of propositional dynamic logic (PDL)---the modal logic of actions and of computer programs (see Pratt, 1976, 1980; Segerberg 1980; Harel, 1984). It is based on the idea of associating an operator [ff] with each action ff, the brackets being reminiscient of the box operator 2 of ordinary modal logic (see Hughes & Cresswell, 1984). The reading of a formula [ff]A is "after every terminating (halting) execution of ff, A is true." PDL provides a powerful language for describing compound actions such as sequential composition of actions ff and fi, written ff; fi, (non-deterministic) choice between ff and fi, written ff + fi, and (nondeterministic) iteration of ff, written ff

This paper is twofold. First, we presentes an extended system EPDL of propositional dynamic logic by allowing a proposition as a modality in order to represent and specify indirect effects of actions and causal propagation. An axiomatic deductive system is given which is sound and complete with respect to the corresponding semantics. The resultant system provides a unified treatment of direct and indirect effects of actions. Second, we reduce the EPDL into a mutlimodal logic by deleting the component of action in order to obtain an axiomatized logical system for causal propagation. A characterization theorem of the logic is given. Properties of causal reasoning with the logic are discussed.

We survey results on axiomatic completeness and complexity of formal languages for reasoning about finite quantities. These languages are interpreted on domains with and without structure. Expressions dealt with include 'at least n Xs are Ks, ' in the case of unstructured domains, and 'at leait n 'related ' Xi are Y%, ' in the structured case. The results contained in this paper are brought together from generalized quantifier theory, modal logic, and knowledge representation, while a few new results are also included. A unifying approach is offered to the wide variety of formalisms available in the literature.