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21
Reachability is harder for directed than for undirected finite graphs
 Journal of Symbolic Logic
, 1990
"... Abstract. Although it is known that reachability in undirected finite graphs can be expressed by an existential monadic secondorder sentence, our main result is that this is not the case for directed finite graphs (even in the presence of certain “builtin ” relations, such as the successor relatio ..."
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Cited by 71 (8 self)
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Abstract. Although it is known that reachability in undirected finite graphs can be expressed by an existential monadic secondorder sentence, our main result is that this is not the case for directed finite graphs (even in the presence of certain “builtin ” relations, such as the successor relation). The proof makes use of EhrenfeuchtFrai’sse games, along with probabilistic arguments. However, we show that for directed finite graphs with degree at most k, reachability is expressible by an existential monadic secondorder sentence. $1. Introduction. If s and t denote distinguished points in a directed (resp. undirected) graph, then we say that a graph is (s, t)connected if there is a directed (undirected) path from s to t. We sometimes refer to the problem of deciding whether a given directed (undirected) graph with two given points sand t is (s, t)connected as the directed (undirected) reachability problem.
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...
Logics For ContextFree Languages
, 1995
"... We define matchings, and show that they capture the essence of contextfreeness. More precisely, we show that the class of contextfree languages coincides with the class of those sets of strings which can be defined by sentences of the form 9 b', where ' is first order, b is a binary predicate sym ..."
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Cited by 23 (5 self)
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We define matchings, and show that they capture the essence of contextfreeness. More precisely, we show that the class of contextfree languages coincides with the class of those sets of strings which can be defined by sentences of the form 9 b', where ' is first order, b is a binary predicate symbol, and the range of the second order quantifier is restricted to the class of matchings. Several variations and extensions are discussed.
On Winning Ehrenfeucht Games and Monadic NP
 Annals of Pure and Applied Logic
, 1996
"... Inexpressibility results in Finite Model Theory are often proved by showing that Duplicator, one of the two players of an Ehrenfeucht game, has a winning strategy on certain structures. In this article a new method is introduced that allows, under certain conditions, the extension of a winning strat ..."
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Cited by 21 (3 self)
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Inexpressibility results in Finite Model Theory are often proved by showing that Duplicator, one of the two players of an Ehrenfeucht game, has a winning strategy on certain structures. In this article a new method is introduced that allows, under certain conditions, the extension of a winning strategy of Duplicator on some small parts of two finite structures to a global winning strategy. As applications of this technique it is shown that (*) Graph Connectivity is not expressible in existential monadic secondorder logic (MonNP), even in the presence of a builtin linear order, (*) Graph Connectivity is not expressible in MonNP even in the presence of arbitrary builtin relations of degree n^o(1), and (*) the presence of a builtin linear order gives MonNP more expressive power than the presence of a builtin successor relation.
The Closure of Monadic NP
 Journal of Computer and System Sciences
, 1997
"... It is a wellknown result of Fagin that the complexity class NP coincides with the class of ..."
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Cited by 21 (0 self)
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It is a wellknown result of Fagin that the complexity class NP coincides with the class of
Graph Connectivity and Monadic NP
 In Proc. 35th IEEE Symp. on Foundations of Computer Science
, 1994
"... Ehrenfeucht games are a useful tool in proving that certain properties of finite structures are not expressible by formulas of a certain type. In this paper a new method is introduced that allows the extension of a local winning strategy for Duplicator, one of the two players in Ehrenfeucht games, t ..."
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Cited by 18 (8 self)
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Ehrenfeucht games are a useful tool in proving that certain properties of finite structures are not expressible by formulas of a certain type. In this paper a new method is introduced that allows the extension of a local winning strategy for Duplicator, one of the two players in Ehrenfeucht games, to a global winning strategy. As an application it is shown that Graph Connectivity cannot be expressed by existential secondorder formulas, where the secondorder quantification is restricted to unary relations (Monadic NP), even in the presence of a builtin linear order. This settles an open problem from [1] and [11]. As a second application it is stated, that, on the other hand, the presence of a linear order increases the power of Monadic NP more than the presence of a successor relation. 1 Introduction Fagin [8] showed that the complexity class NP coincides with the class of all sets of finite structures that can be characterized by existential secondorder formulas (\Sigma 1 1 ). Th...
Graph Connectivity, Monadic NP and Builtin Relations of Moderate Degree
 IN PROC. 22ND INTERNATIONAL COLLOQ. ON AUTOMATA, LANGUAGES, AND PROGRAMMING
, 1995
"... It has been conjectured [FSV93] that an existential secondoder formula, in which the secondorder quantification is restricted to unary relations (i.e. a Monadic NP formula), cannot express Graph Connectivity even in the presence of arbitrary builtin relations. In this paper it is shown that Graph ..."
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Cited by 16 (8 self)
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It has been conjectured [FSV93] that an existential secondoder formula, in which the secondorder quantification is restricted to unary relations (i.e. a Monadic NP formula), cannot express Graph Connectivity even in the presence of arbitrary builtin relations. In this paper it is shown that Graph Connectivity cannot be expressed by Monadic NP formulas in the presence of arbitrary builtin relations of degree n^o(1). The result is obtained by using a simplified version of a method introduced in [Sch94] that allows the extension of a local winning strategy for Duplicator, one of the two players in Ehrenfeucht games, to a global winning strategy.
On winning strategies in EhrenfeuchtFraïssé games
 Theoretical Computer Science
, 1997
"... We present a powerful and versatile new sufficient condition for the second player (the ..."
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Cited by 14 (3 self)
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We present a powerful and versatile new sufficient condition for the second player (the
How to Define a Linear Order on Finite Models
, 1997
"... We carry out a systematic investigation of the definability of linear order on classes of finite rigid structures. We obtain upper and lower bounds for the expressibility of linear order in various logics that have been studied extensively in finite model theory, such as least fixpoint logic LFP, pa ..."
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Cited by 14 (1 self)
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We carry out a systematic investigation of the definability of linear order on classes of finite rigid structures. We obtain upper and lower bounds for the expressibility of linear order in various logics that have been studied extensively in finite model theory, such as least fixpoint logic LFP, partial fixpoint logic PFP, infinitary logic L ! 1! with a finite number of variables, as well as the closures of these logics under implicit definitions. Moreover, we show that the upper and lower bounds established here can not be made substantially tighter, unless outstanding conjectures in complexity theory are resolved at the same time. Research of L. Hella was partially supported by a grant from the University of Helsinki. y Research of Ph. Kolaitis was partially supported by a 1993 John Simon Guggenheim Fellowship and by NSF Grants No. CCR9108631, CCR9307758, and INT9024681 z Research of K. Luosto was partially supported by a grant from the Emil Aaltonen Foundation. 1 Intro...
Subclasses of Binary NP
, 1996
"... Binary NP consists of all sets of finite structures which are expressible in existential second order logic with second order quantification restricted to relations of arity 2. We look at semantical restrictions of binary NP, where the second order quantifiers range only over certain classes of rela ..."
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Cited by 10 (5 self)
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Binary NP consists of all sets of finite structures which are expressible in existential second order logic with second order quantification restricted to relations of arity 2. We look at semantical restrictions of binary NP, where the second order quantifiers range only over certain classes of relations. We consider mainly three types of classes of relations: unary functions, order relations and graphs with degree bounds. We show that many of these restrictions have the same expressive power and establish a 4level strict hierarchy, represented by sets, permutations, unary functions and arbitrary binary relations, respectively.