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33
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' ..."
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Cited by 48 (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.
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 NEXPTIMEcomplete ..."
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Cited by 29 (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...
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 28 (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.
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, AugustusPlatz 10 ..."
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Cited by 27 (21 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
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 27 (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
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 14 (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.
The recognizability of sets of graphs is a robust property
"... Once the set of finite graphs is equipped with an algebra structure (arising from the definition of operations that generalize the concatenation of words), one can define the notion of a recognizable set of graphs in terms of finite congruences. Applications to the construction of efficient algorith ..."
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Cited by 14 (9 self)
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Once the set of finite graphs is equipped with an algebra structure (arising from the definition of operations that generalize the concatenation of words), one can define the notion of a recognizable set of graphs in terms of finite congruences. Applications to the construction of efficient algorithms and to the theory of contextfree sets of graphs follow naturally. The class of recognizable sets depends on the signature of graph operations. We consider three signatures related respectively to Hyperedge Replacement (HR) contextfree graph grammars, to Vertex Replacement (VR) contextfree graph grammars, and to modular decompositions of graphs. We compare the corresponding classes of recognizable sets. We show that they are robust in the sense that many variants of each signature (where in particular operations are defined by quantifierfree formulas, a quite flexible framework) yield the same notions of recognizability. We prove that for graphs without large complete bipartite subgraphs, HRrecognizability and VRrecognizability coincide. The same combinatorial condition equates HRcontextfree and VRcontextfree sets of graphs. Inasmuch as possible, results are formulated in the more general framework of relational structures. 1
A logic for metric and topology
 Journal of Symbolic Logic
, 2005
"... Abstract. We propose a logic for reasoning about metric spaces with the induced topologies. It combines the ‘qualitative ’ interior and closure operators with ‘quantitative’ operators ‘somewhere in the sphere of radius r, ’ including or excluding the boundary. We supply the logic with both the inten ..."
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Cited by 13 (11 self)
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Abstract. We propose a logic for reasoning about metric spaces with the induced topologies. It combines the ‘qualitative ’ interior and closure operators with ‘quantitative’ operators ‘somewhere in the sphere of radius r, ’ including or excluding the boundary. We supply the logic with both the intended metric space semantics and a natural relational semantics, and show that the latter (i) provides finite partial representations of (in general) infinite metric models and (ii) reduces the standard ‘εdefinitions ’ of closure and interior to simple constraints on relations. These features of the relational semantics suggest a finite axiomatisation of the logic and provide means to prove its EXPTIMEcompleteness (even if the rational numerical parameters are coded in binary). An extension with metric variables satisfying linear rational (in)equalities is proved to be decidable as well. Our logic can be regarded as a ‘wellbehaved ’ common denominator of logical systems constructed in temporal, spatial, and similaritybased quantitative and qualitative representation and reasoning. Interpreted on the real line (with its Euclidean metric), it is a natural fragment of decidable temporal logics for specification and verification of realtime systems. On the real plane, it is closely related to quantitative and qualitative formalisms for spatial