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24
The Structure of the Models of Decidable Monadic Theories of Graphs
, 1991
"... In this article the structure of the models of decidable (weak) monadic theories of planar graphs is investigated. It is shown that if the (weak) monadic theory of a class K of planar graphs is decidable, then the treewidth in the sense of Robertson and Seymour (1984) of the elements of K is univer ..."
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Cited by 47 (2 self)
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In this article the structure of the models of decidable (weak) monadic theories of planar graphs is investigated. It is shown that if the (weak) monadic theory of a class K of planar graphs is decidable, then the treewidth in the sense of Robertson and Seymour (1984) of the elements of K is universally bounded and there is a class T of trees such that the (weak) monadic theory of K is interpretable in the (weak) monadic theory of T.
Linear Time Computable Problems and FirstOrder Descriptions
, 1996
"... this article is a proof that each FO problem can be solved in linear time if only relational structures of bounded degree are considered. The basic idea of the proof is a localization technique based on a method that was originally developed by Hanf (Hanf 1965) to show that the elementary theories o ..."
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Cited by 31 (2 self)
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this article is a proof that each FO problem can be solved in linear time if only relational structures of bounded degree are considered. The basic idea of the proof is a localization technique based on a method that was originally developed by Hanf (Hanf 1965) to show that the elementary theories of two structures are equal under certain conditions, i.e., that two structures agree on all firstorder sentences. Fagin, Stockmeyer and Vardi (Fagin et al. 1993) developed a variant of this technique, which is applicable in descriptive complexity theory to classes of finite relational structures of uniformly bounded degree. Variants of this result can also be found in Gaifman (1982) (see also Thomas (1991)). The essential content of this result, which is also called the HanfSphere Lemma, is that two relational structures of bounded degree satisfy the same firstorder sentences of a certain quantifierrank if both contain, up to a certain number m, the same number of isomorphism types of substructures of a bounded radius r. In addition, a technique of model interpretability from Rabin (1965) (see also Arnborg et al. (1991), Seese (1992), Compton and Henson (1987) and Baudisch et al. (1982)) is adapted to descriptive complexity classes, and proved to be useful for reducing the case of an arbitrary class of relational structures to a class of structures consisting only of the domain and one binary irreflexive and symmetric relation, i.e., the class of simple graphs. It is shown that the class of simple graphs is lintimeuniversal with respect to firstorder logic, which shows that many problems on descriptive complexity classes, described in languages extending firstorder logic for arbitrary structures, can be reduced to problems on simple graphs. This paper is organized as f...
A New Method for Undecidability Proofs of First Order Theories
 Journal of Symbolic Computation
, 1992
"... this paper is to define a framework for such reduction proofs. The method proposed is illustrated by proving the undecidability of the theory of a term algebra modulo the axioms of associativity and commutativity and of the theory of a partial lexicographic path ordering. 1. Introduction ..."
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Cited by 30 (7 self)
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this paper is to define a framework for such reduction proofs. The method proposed is illustrated by proving the undecidability of the theory of a term algebra modulo the axioms of associativity and commutativity and of the theory of a partial lexicographic path ordering. 1. Introduction
Graphs and Decidable Transductions based on Edge Constraints
 In Proc. 19th Colloquium on Trees and Algebra in Programming, number 787 in LNCS
, 1994
"... Department of Computer Science, University of Aarhus, Ny Munkegade, DK8000 Aarhus, Denmark fklarlund,misg@daimi.aau.dk Abstract We give examples to show that not even cedNCE, the most general known notion of contextfree graph grammar, is suited for the specification of some common data stru ..."
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Cited by 27 (5 self)
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Department of Computer Science, University of Aarhus, Ny Munkegade, DK8000 Aarhus, Denmark fklarlund,misg@daimi.aau.dk Abstract We give examples to show that not even cedNCE, the most general known notion of contextfree graph grammar, is suited for the specification of some common data structures.
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
Width parameters beyond treewidth and their applications
 Computer Journal
, 2007
"... Besides the very successful concept of treewidth (see [Bodlaender, H. and Koster, A. (2007) Combinatorial optimisation on graphs of bounded treewidth. These are special issues on Parameterized Complexity]), many concepts and parameters measuring the similarity or dissimilarity of structures compare ..."
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Cited by 19 (0 self)
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Besides the very successful concept of treewidth (see [Bodlaender, H. and Koster, A. (2007) Combinatorial optimisation on graphs of bounded treewidth. These are special issues on Parameterized Complexity]), many concepts and parameters measuring the similarity or dissimilarity of structures compared to trees have been born and studied over the past years. These concepts and parameters have proved to be useful tools in many applications, especially in the design of efficient algorithms. Our presented novel look at the contemporary developments of these ‘width ’ parameters in combinatorial structures delivers—besides traditional treewidth and derived dynamic programming schemes—also a number of other useful parameters like branchwidth, rankwidth (cliquewidth) or hypertreewidth. In this contribution, we demonstrate how ‘width ’ parameters of graphs and generalized structures (such as matroids or hypergraphs), can be used to improve the design of parameterized algorithms and the structural analysis in other applications on an abstract level.
Complexity of Nonrecursive Logic Programs with Complex Values
 In Proceedings of the 17th ACM SIGACTSIGMODSIGART Symposium on Principles of Database Systems (PODS’98
, 1998
"... We investigate complexity of the SUCCESS problem for logic query languages with complex values: check whether a query defines a nonempty set. The SUCCESS problem for recursive query languages with complex values is undecidable, so we study the complexity of nonrecursive queries. By complex values we ..."
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Cited by 17 (2 self)
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We investigate complexity of the SUCCESS problem for logic query languages with complex values: check whether a query defines a nonempty set. The SUCCESS problem for recursive query languages with complex values is undecidable, so we study the complexity of nonrecursive queries. By complex values we understand values such as trees, finite sets, and multisets. Due to the wellknown correspondence between relational query languages and datalog, our results can be considered as results about relational query languages with complex values. The paper gives a complete complexity classification of the SUCCESS problem for nonrecursive logic programs over trees depending on the underlying signature, presence of negation, and range restrictedness. We also prove several results about finite sets and multisets. 1 Introduction A number of complexity results have been established for logic query languages. They are surveyed in [49, 18]. The major themes in these results are the complexity and expr...
Oracles and Quantifiers
, 1994
"... . We describe a general way of building logics with Lindstrom quantifiers, which capture regular complexity classes on ordered structures with polysize reductions. We then extend this method so as to accommodate complexity classes based on oracle Turing machines. Our main result shows an equivalence ..."
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Cited by 11 (4 self)
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. We describe a general way of building logics with Lindstrom quantifiers, which capture regular complexity classes on ordered structures with polysize reductions. We then extend this method so as to accommodate complexity classes based on oracle Turing machines. Our main result shows an equivalence between enhancing a logic with a Lindstr om quantifier and enhancing a complexity class with an oracle such that, if K is a set of structures, QK the associated Lindstrom quantifier and L a logic that captures a complexity class D, then the enhanced logic L[K] captures D K  the complexity class of machines in D using oracles for K. Our results are sensitive to the oracle computation model and hold in a natural modification of the unbounded model introduced by Buss [Bus88]. They do not hold in the, so called, space bounded oracle models or those that violate the `relativization thesis' of Buss. Our results generalize and extend previous results of Stewart [Ste93a, Ste93b] and Makowsky and...
The "Hardest" Natural Decidable Theory
 12th Annual IEEE Symp. on Logic in Computer Science (LICS'97)', IEEE
, 1997
"... We prove that any decision procedure for a modest fragment of L. Henkin's theory of pure propositional types [7, 12, 15, 11] requires time exceeding a tower of 2's of height exponential in the length of input. Until now the highest known lower bounds for natural decidable theories were at most linea ..."
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Cited by 10 (4 self)
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We prove that any decision procedure for a modest fragment of L. Henkin's theory of pure propositional types [7, 12, 15, 11] requires time exceeding a tower of 2's of height exponential in the length of input. Until now the highest known lower bounds for natural decidable theories were at most linearly high towers of 2's and since midseventies it was an open problem whether natural decidable theories requiring more than that exist [12, 2]. We give the affirmative answer. As an application of this today's strongest lower bound we improve known and settle new lower bounds for several problems in the simply typed lambda calculus. 1. Introduction In his survey paper [12] A. Meyer mentioned (p. 479), as a curious empirical observation, that all known natural decidable nonelementary problems require at most (upper bound) F (1; n) = exp 1 (n) = 2 2 \Delta \Delta \Delta 2 oe n Turing machine steps on inputs of length n to decide 1 . Until now the highest known lower bounds for natu...
Incremental Model Checking for Decomposable Structures
 In Proceedings of MFCS ’95, Lecture Notes in Computer Science
, 1995
"... . Assume we are given a transition system which is composed from several well identified components. We propose a method which allows us to reduce the model checking of formulas in the complex system to model checking of derived formulas in Monadic Second Order Logic in the components. Our method ap ..."
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Cited by 5 (2 self)
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. Assume we are given a transition system which is composed from several well identified components. We propose a method which allows us to reduce the model checking of formulas in the complex system to model checking of derived formulas in Monadic Second Order Logic in the components. Our method applies to all practically used hardware specification languages, although at a certain cost. The basic idea goes back to a method proposed first by Feferman and Vaught in 1959, which we adapt and generalize to the specific context of model checking of transition systems. Our method allows for a precise definition of incremental model checking. We give also estimates on when our incremental method starts to be better than traditional methods. 1 Introduction In hardware verification (but not only) we find the following situation: We are given a mathematical model of a finite state device in form of a finite relational structure A (transition system) and a formalized property OE. Usually OE is ...