This document is a complete draft of a chapter by Rajeev Gor'e on "Tableau Methods for Modal and Temporal Logics" which is part of the "Handbook of Tableau Methods", edited by M. D'Agostino, D. Gabbay, R. Hahnle and J. Posegga, to be published in 1998 by Kluwer, Dordrecht. Any comments and corrections are highly welcome. Please email me at rpg@arp.anu.edu.au The latest version of this document can be obtained via my WWW home page: http://arp.anu.edu.au/ Tableau Methods for Modal and Temporal Logics Rajeev Gor'e Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1 Syntax and Notational Conventions . . . . . . . . . . . . 3 2.2 Axiomatics of Modal Logics . . . . . . . . . . . . . . . . 4 2.3 Kripke Semantics For Modal Logics . . . . . . . . . . . . 5 2.4 Known Correspondence and Completeness Results . . . . 6 2.5 Logical Consequence . . . . . . . . . . . . . . . . . . . . 8 2....

Substructural logics are traditionally obtained by dropping some or all of the structural rules from Gentzen's sequent calculi LK or LJ. It is well known that the usual logical connectives then split into more than one connective. Alternatively, one can start with the (intuitionistic) Lambek calculus, which contains these multiple connectives, and obtain numerous logics like: exponential-free linear logic, relevant logic, BCK logic, and intuitionistic logic, in an incremental way. Each of these logics also has a classical counterpart, and some also have a "cyclic" counterpart. These logics have been studied extensively and are quite well understood. Generalising further, one can start with intuitionistic Bi-Lambek logic, which contains the dual of every connective from the Lambek calculus. The addition of the structural rules then gives Bi-linear, Bi-relevant, Bi-BCK and Bi-intuitionistic logic, again in an incremental way. Each of these logics also has a classical counterpart, and som...

Natural language generation (NLG) is first and foremost a reasoning task. In this reasoning, a system plans a communicative act that will signal key facts about the domain to the hearer. In generating action descriptions, this reasoning draws on characterizations both of the causal properties of the domain and the states of knowledge of the participants in the conversation. This dissertation shows how such characterizations can be specified declaratively and accessed efficiently in NLG. The heart of this dissertation is a study of logical statements about knowledge and action in modal logic. By investigating the proof-theory of modal logic from a logic programming point of view, I show how many kinds of modal statements can be seen as straightforward instructions for computationally manageable search, just as Prolog clauses can. These modal statements provide sufficient expressive resources for an NLG system to represent the effects of actions in the world or to model an addressee whose knowledge in some respects exceeds and in other respects falls short of its own. To illustrate the use of such statements, I describe how the SPUD sentence planner exploits a modal knowledge base to

. 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 studies the relative proof complexity of variations of a tableau method for Boolean circuit satisfiability checking obtained by restricting the use of the cut rule in several natural ways. The results show that the unrestricted cut rule can be exponentially more effective than any of the considered restrictions. Moreover, there are exponential differences between the restricted versions, too. The results also apply to the Davis-Putnam procedure for conjunctive normal form formulae obtained from Boolean circuits with a standard linear size translation.

. We define sequent-style calculi for nominal tense logics characterized by classes of modal frames that are first-order definable by certain \Pi 0 1 -formulae and \Pi 0 2 -formulae. The calculi are based on d'Agostino and Mondadori's calculus KE and therefore they admit a restricted cutrule that is not eliminable. A nice computational property of the restriction is, for instance, that at any stage of the proof, only a finite number of potential cut-formulae needs to be taken under consideration. Although restrictions on the proof search (preserving completeness) are given in the paper and most of them are theoretically appealing, the use of those calculi for mechanization is however doubtful. Indeed, we present sequent calculi for fragments of classical logic that are syntactic variants of the sequent calculi for the nominal tense logics. 1 Introduction Background. The nominal tense logics are extensions of Prior tense logics (see e.g. [Pri57, RU71]) by adding nomina...

In this paper we present a new fast tableau-based decision procedure for the ground set-theoretic fragment Multi-Level Syllogistic with Singleton (in short MLSS) which avoids the interleaving of model checking steps. The underlying

. This paper is concerned with various schemes for enhancing the performance of modal tableau procedures. It discusses techniques and strategies for dealing with the nondeterminism in tableau calculi, as well as simplification and backjumping. Benchmark results obtained with randomly generated modal formulae show the e#ect of combinations of di#erent schemes. 1 Introduction Usually the literature on theorem provers for modal logic confines itself to a description of the underlying calculus and methodology. Sometimes the description is accompanied with a consideration of the worst-case complexity of an algorithm based on the presented calculus or a small collection of benchmark results. Problems arising when implementing modal theorem provers and also considerations concerning optimisations towards increased e#ciency have received much less attention, which, of course, is typical in a field under development. Sometimes the description of the theorem prover mentions some simplification ...

This chapter presents a logic for the retrieval of multimedia information, whose ultimate goal is to model retrieval as an uncertain logical inference, in accordance to the logic-based view of retrieval. The logic being presented is the product of a number of extensions to a Description Logic, which constitutes the kernel of our model. Each extension is meant to capture an important aspect of the retrieval endeavour that is not adequately dealt with by the kernel logic. The extensions are: a 4-valued semantics, aiming at capturing relevance in retrieval; closure assertions, aiming at allowing closed-world reading of selectively specified parts of a document base; fuzzy assertions, aiming at handling the uncertainty intrinsic in retrieval. The resulting logic is to be understood as a modelling retrieval tool, which can be used for the specification and the rapid prototyping of applications.

We present the first tableau-based decision procedure for basic hybrid logic with the difference modality. The decision procedure is gracefully degrading in that the less expressive constructs don’t pay for the computationally expensive difference modality. The procedure can be specialized to reflexive and transitive frames. Key features of our approach are nominal elimination, pattern-based blocking, and expansion control.