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Monodic fragments of firstorder temporal logics: 20002001 A.D.
"... The aim of this paper is to summarize and analyze some results obtained in 20002001 about decidable and undecidable fragments of various firstorder temporal logics, give some applications in the field of knowledge representation and reasoning, and attract the attention of the `temporal community&a ..."
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Cited by 49 (8 self)
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The aim of this paper is to summarize and analyze some results obtained in 20002001 about decidable and undecidable fragments of various firstorder temporal logics, give some applications in the field of knowledge representation and reasoning, and attract the attention of the `temporal community' to a number of interesting open problems.
Logics of Metric Spaces
, 2001
"... This paper investigates the expressive power and computational properties of languages designed for speaking about distances. `Distances' can be induced by difAuthors Addresses: Oliver Kutz, Frank Wolter, Institut fur Informatik, Abteilung intelligente Systeme, Universitat Leipzig, AugustusPla ..."
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Cited by 33 (24 self)
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This paper investigates the expressive power and computational properties of languages designed for speaking about distances. `Distances' can be induced by difAuthors Addresses: Oliver Kutz, Frank Wolter, Institut fur Informatik, Abteilung intelligente Systeme, Universitat Leipzig, AugustusPlatz 1011, 04109 Leipzig, Germany; Holger Sturm, Fachbereich Philosophie, Universitat Konstanz, 78457 Konstanz, Germany; NobuYuki Suzuki, Department of Mathematics, Faculty of Science, Shizuoka University, Ohya 836, Shizuoka 422 8529, Japan; Michael Zakharyaschev, Department of Computer Science, King's College, Strand, London WC2R 2LS, U.K. Emails: {kutz, wolter}@informatik.unileipzig.de, holger.sturm@unikonstanz.de, smnsuzu@ipz.shizuoka.ac.jp, and mz@dcs.kcl.ac.uk Permission to make digital/hard copy of all or part of this material without fee for personal or classroom use provided that the copies are not made or distributed for profit or commercial advantage, the ACM copyright/server notice, the title of the publication, and its date appear, and notice is given that copying is by permission of the ACM, Inc. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior specific permission and/or a fee
Complexity Results for FirstOrder TwoVariable Logic with Counting
, 2000
"... Let C 2 p denote the class of first order sentences with two variables and with additional quantifiers "there exists exactly (at most, at least) i", for i p, and let C 2 be the union of C 2 p taken over all integers p. We prove that the satisfiability problem for C 2 1 sentences is NEXPTIM ..."
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Cited by 32 (1 self)
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Let C 2 p denote the class of first order sentences with two variables and with additional quantifiers "there exists exactly (at most, at least) i", for i p, and let C 2 be the union of C 2 p taken over all integers p. We prove that the satisfiability problem for C 2 1 sentences is NEXPTIMEcomplete. This strengthens the results by E. Grädel, Ph. Kolaitis and M. Vardi [15] who showed that the satisfiability problem for the first order twovariable logic L 2 is NEXPTIMEcomplete and by E. Grädel, M. Otto and E. Rosen [16] who proved the decidability of C 2 . Our result easily implies that the satisfiability problem for C 2 is in nondeterministic, doubly exponential time. It is interesting that C 2 1 is in NEXPTIME in spite of the fact, that there are sentences whose minimal (and only) models are of doubly exponential size. It is worth noticing, that by a recent result of E. Gradel, M. Otto and E. Rosen [17], extensions of twovariables logic L 2 by a week access to car...
Monodic temporal resolution
 ACM Transactions on Computational Logic
, 2003
"... Until recently, FirstOrder Temporal Logic (FOTL) has been only partially understood. While it is well known that the full logic has no finite axiomatisation, a more detailed analysis of fragments of the logic was not previously available. However, a breakthrough by Hodkinson et al., identifying a f ..."
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Cited by 31 (15 self)
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Until recently, FirstOrder Temporal Logic (FOTL) has been only partially understood. While it is well known that the full logic has no finite axiomatisation, a more detailed analysis of fragments of the logic was not previously available. However, a breakthrough by Hodkinson et al., identifying a finitely axiomatisable fragment, termed the monodic fragment, has led to improved understanding of FOTL. Yet, in order to utilise these theoretical advances, it is important to have appropriate proof techniques for this monodic fragment. In this paper, we modify and extend the clausal temporal resolution technique, originally developed for propositional temporal logics, to enable its use in such monodic fragments. We develop a specific normal form for monodic formulae in FOTL, and provide a complete resolution calculus for formulae in this form. Not only is this clausal resolution technique useful as a practical proof technique for certain monodic classes, but the use of this approach provides us with increased understanding of the monodic fragment. In particular, we here show how several features of monodic FOTL can be established as corollaries of the completeness result for the clausal temporal resolution method. These include definitions of new decidable monodic classes, simplification of existing monodic classes by reductions, and completeness of clausal temporal resolution in the case of
Decidable Fragments of FirstOrder Modal Logics
 JOURNAL OF SYMBOLIC LOGIC
, 1999
"... The paper considers the set ML1 of firstorder polymodal formulas the modal operators in which can be applied to subformulas of at most one free variable. Using a mosaic technique, we prove a general satisfiability criterion for formulas in ML1, which reduces the modal satisfiability to the classica ..."
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Cited by 29 (8 self)
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The paper considers the set ML1 of firstorder polymodal formulas the modal operators in which can be applied to subformulas of at most one free variable. Using a mosaic technique, we prove a general satisfiability criterion for formulas in ML1, which reduces the modal satisfiability to the classical one. The criterion is then used to single out a number of new, in a sense optimal, decidable fragments of various modal predicate logics.
Towards FirstOrder Temporal Resolution
 In KI 2001, Proceedings
"... In this paper we show how to extend clausal temporal resolution to the ground eventuality fragment of monodic firstorder temporal logic, which has recently been introduced by Hodkinson, Wolter and Zakharyaschev. While a finite Hilbertlike axiomatization of complete monodic first order temporal ..."
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Cited by 27 (13 self)
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In this paper we show how to extend clausal temporal resolution to the ground eventuality fragment of monodic firstorder temporal logic, which has recently been introduced by Hodkinson, Wolter and Zakharyaschev. While a finite Hilbertlike axiomatization of complete monodic first order temporal logic was developed by Wolter and Zakharyaschev, we propose a temporal resolutionbased proof system which reduces the satisfiability problem for ground eventuality monodic firstorder temporal formulae to the satisfiability problem for formulae of classical firstorder logic.
Modal logic and the twovariable fragment
 IN ANNUAL CONF. OF THE EUROPEAN ASSOCIATION FOR COMPUTER SCIENCE LOGIC (CSL’01), LNCS
, 2001
"... We introduce a modal language L which is obtained from standard modal logic by adding the Boolean operators on accessibility relations, the identity relation, and the converse of relations. It is proved that L has the same expressive power as the twovariable fragment F O2 of firstorder logic, bu ..."
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Cited by 19 (6 self)
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We introduce a modal language L which is obtained from standard modal logic by adding the Boolean operators on accessibility relations, the identity relation, and the converse of relations. It is proved that L has the same expressive power as the twovariable fragment F O2 of firstorder logic, but speaks less succinctly about relational structures: if the number of relations is bounded, then Lsatisfiability is ExpTimecomplete but F O2 satisfiability is NExpTimecomplete. We indicate that the relation between L and F O2 provides a general framework for comparing modal and temporal languages with firstorder languages.
Equality and Monodic FirstOrder Temporal Logic
 Studia Logica
, 2002
"... It has been shown recently that monodic firstorder temporal logic without functional symbols but with equality is incomplete, i.e. the set of the valid formulae of this logic is not recursively enumerable. In this paper we show that an even simpler fragment consisting of monodic monadic twovari ..."
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Cited by 17 (7 self)
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It has been shown recently that monodic firstorder temporal logic without functional symbols but with equality is incomplete, i.e. the set of the valid formulae of this logic is not recursively enumerable. In this paper we show that an even simpler fragment consisting of monodic monadic twovariable formulae is not recursively enumerable.
Hilbert’s twentyfourth problem
 American Mathematical Monthly
, 2001
"... 1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Cong ..."
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Cited by 15 (4 self)
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1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Congress of Mathematicians (ICM) in Paris has tremendous importance for all mathematicians. Moreover, a substantial part of