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23
On the Decision Problem for TwoVariable FirstOrder Logic
, 1997
"... We identify the computational complexity of the satisfiability problem for FO², the fragment of firstorder logic consisting of all relational firstorder sentences with at most two distinct variables. Although this fragment was shown to be decidable a long time ago, the computational complexity ..."
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Cited by 48 (1 self)
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We identify the computational complexity of the satisfiability problem for FO², the fragment of firstorder logic consisting of all relational firstorder sentences with at most two distinct variables. Although this fragment was shown to be decidable a long time ago, the computational complexity of its decision problem has not been pinpointed so far. In 1975 Mortimer proved that FO² has the finitemodel property, which means that if an FO²sentence is satisfiable, then it has a finite model. Moreover, Mortimer showed that every satisfiable FO²sentence has a model whose size is at most doubly exponential in the size of the sentence. In this paper, we improve Mortimer's bound by one exponential and show that every satisfiable FO²sentence has a model whose size is at most exponential in the size of the sentence. As a consequence, we establish that the satisfiability problem for FO² is NEXPTIMEcomplete.
On Logics with Two Variables
 Theoretical Computer Science
, 1999
"... This paper is a survey and systematic presentation of decidability and complexity issues for modal and nonmodal twovariable logics. A classical result due to Mortimer says that the twovariable fragment of firstorder logic, denoted FO 2 , has the finite model property and is therefore decidable ..."
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Cited by 41 (8 self)
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This paper is a survey and systematic presentation of decidability and complexity issues for modal and nonmodal twovariable logics. A classical result due to Mortimer says that the twovariable fragment of firstorder logic, denoted FO 2 , has the finite model property and is therefore decidable for satisfiability. One of the reasons for the significance of this result is that many propositional modal logics can be embedded into FO 2 . Logics that are of interest for knowledge representation, for the specification and verification of concurrent systems and for other areas of computer science are often defined (or can be viewed) as extensions of modal logics by features like counting constructs, path quantifiers, transitive closure operators, least and greatest fixed points etc. Examples of such logics are computation tree logic CTL, the modal ¯calculus L¯ , or popular description logics used in artificial intelligence. Although the additional features are usually not firstorder...
The TwoVariable Guarded Fragment with Transitive Relations
 In Proc. LICS'99
, 1999
"... We consider the restriction of the guarded fragment to the twovariable case where, in addition, binary relations may be specified as transitive. We show that (i) this very restricted form of the guarded fragment without equality is undecidable and that (ii) when allowing nonunary relations to occu ..."
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Cited by 34 (1 self)
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We consider the restriction of the guarded fragment to the twovariable case where, in addition, binary relations may be specified as transitive. We show that (i) this very restricted form of the guarded fragment without equality is undecidable and that (ii) when allowing nonunary relations to occur only in guards, the logic becomes decidable. The latter subclass of the guarded fragment is the one that occurs naturally when translating multimodal logics of the type K4, S4 or S5 into rstorder logic. We also show that the loosely guarded fragment without equality and with a single transitive relation is undecidable.
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.
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
Canonical Graph Shapes
 Programming Languages and Systems — European Symposium on Programming (ESOP
, 2004
"... Graphs are an intuitive model for states of a (software) system that include pointer structures  for instance, objectoriented programs. ..."
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Cited by 21 (9 self)
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Graphs are an intuitive model for states of a (software) system that include pointer structures  for instance, objectoriented programs.
Two Variable FirstOrder Logic over Ordered Domains
 Journal of Symbolic Logic
, 1998
"... The satisfiability problem for the twovariable fragment of firstorder logic is investigated over finite and infinite linearly ordered, respectively wellordered domains, as well as over finite and infinite domains in which one or several designated binary predicates are interpreted as arbitrary wel ..."
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Cited by 14 (0 self)
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The satisfiability problem for the twovariable fragment of firstorder logic is investigated over finite and infinite linearly ordered, respectively wellordered domains, as well as over finite and infinite domains in which one or several designated binary predicates are interpreted as arbitrary wellfounded relations. It is shown that FO 2 over ordered, respectively wellordered, domains or in the presence of one wellfounded relation, is decidable for satisfiability as well as for finite satisfiability. Actually the complexity of these decision problems is essentially the same as for plain unconstrained FO 2 , namely nondeterministic exponential time. In contrast FO 2 becomes undecidable for satisfiability and for finite satisfiability, if a sufficiently large number of predicates are required to be interpreted as orderings, wellorderings, or as arbitrary wellfounded relations. This undecidability result also entails the undecidability of the natural common extension of FO 2 an...
A Survey of Decidable FirstOrder Fragments and Description Logics
 Journal of Relational Methods in Computer Science
, 2004
"... The guarded fragment and its extensions and subfragments are often considered as a framework for investigating the properties of description logics. There are also other, some less wellknown, decidable fragments of firstorder logic which all have in common that they generalise the standard tran ..."
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Cited by 9 (2 self)
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The guarded fragment and its extensions and subfragments are often considered as a framework for investigating the properties of description logics. There are also other, some less wellknown, decidable fragments of firstorder logic which all have in common that they generalise the standard translation of to firstorder logic. We provide a short survey of some of these fragments and motivate why they are interesting with respect to description logics, mentioning also connections to other nonclassical logics.