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...

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 define cut-free display calculi for nominal tense logics extending the minimal nominal tense logic (MNTL) by addition of primitive axioms. To do so, we use a translation of MNTL into the minimal tense logic of inequality (MTL 6= ) which is known to be properly displayable by application of Kracht's results. The rules of the display calculus ffiMNTL for MNTL mimic those of the display calculus ffiMTL 6= for MTL 6= . Since ffiMNTL does not satisfy Belnap's condition (C8), we extend Wansing's strong normalisation theorem to get a similar theorem for any extension of ffiMNTL by addition of structural rules satisfying Belnap's conditions (C2)-(C7). Finally, we show a weak Sahlqvist-style theorem for extensions of MNTL, and by Kracht's techniques, deduce that these Sahlqvist extensions of ffiMNTL also admit cut-free display calculi. 1 Introduction Background: The addition of names (also called nominals) to modal logics has been investigated recently with different motivations; see...

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

This paper is about the good side of modal logic, the bad side of modal logic, and how hybrid logic takes the good and xes the bad. In essence, modal logic is a simple formalism for working with relational structures (or multigraphs) . But modal logic has no mechanism for referring to or reasoning about the individual nodes in such structures, and this lessens its eectiveness as a representation formalism. In their simplest form, hybrid logics are upgraded modal logics in which reference to individual nodes is possible. But hybrid logic is a rather unusual modal upgrade. It pushes one simple idea as far as it will go: represent all information as formulas. This turns out to be the key needed to draw together a surprisingly diverse range of work (for example, feature logic, description logic and labelled deduction) . Moreover, it displays a number of knowledge representation issues in a new light, notably the importance of sorting. Keywords: Labelled deduction, description logic, f...

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.

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 one-by-one, 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.

logic as dialogical logic

Automata-theoretic decision procedures for solving model-checking and satisfiability problems for temporal, dynamic, and description logics have ourished during the past decades. In the paper we define an ExpTime decision procedure based on the emptiness problem of Büchi automata on infinite trees for the very expressive information logic SIM designed for reasoning about information systems. This logic involves modal parameters satisfying certain properties to capture the relevant properties of information systems, and provides nominals at the formula level, Boolean expressions and nominals at the modal level, an implicit intersection operation for relations, and a universal modality. The original combination of known techniques allows us to solve the open question related to the ExpTime-completeness of SIM. Furthermore, we discuss how variants of SIM can be treated similarly although the decidability status of some of them is still unknown.