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On the Expressive Power of Datalog: Tools and a Case Study
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 1990
"... We study here the language Datalog(6=), which is the query language obtained from Datalog by allowing equalities and inequalities in the bodies of the rules. We view Datalog(6=) as a fragment of an innitary logic L ! and show that L ! can be characterized in terms of certain twoperson pebble ga ..."
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Cited by 52 (9 self)
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We study here the language Datalog(6=), which is the query language obtained from Datalog by allowing equalities and inequalities in the bodies of the rules. We view Datalog(6=) as a fragment of an innitary logic L ! and show that L ! can be characterized in terms of certain twoperson pebble games. This characterization provides us with tools for investigating the expressive power of Datalog(6=). As a case study, we classify the expressibility of fixed subgraph homeomorphism queries on directed graphs. Fortune et al. [FHW80] classied the computational complexity of these queries by establishing two dichotomies, which are proper only if P 6= NP. Without using any complexitytheoretic assumptions, we show here that the two dichotomies are indeed proper in terms of expressibility in Datalog(6=).
Local Properties of Query Languages
"... predeterminedportionoftheinput.Examplesincludeallrelationalcalculusqueries. everyrelationalcalculus(rstorder)queryislocal,thegeneralresultsprovedforlocalqueriescan manyeasyinexpressibilityproofsforlocalqueries.Wethenconsideracloselyrelatedproperty, namely,theboundeddegreeproperty.Itdescribestheoutp ..."
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Cited by 33 (21 self)
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predeterminedportionoftheinput.Examplesincludeallrelationalcalculusqueries. everyrelationalcalculus(rstorder)queryislocal,thegeneralresultsprovedforlocalqueriescan manyeasyinexpressibilityproofsforlocalqueries.Wethenconsideracloselyrelatedproperty, namely,theboundeddegreeproperty.Itdescribestheoutputsoflocalqueriesonstructuresthat locallylook\simple."Everyquerythatislocalisshowntohavetheboundeddegreeproperty.Since Westartbyprovingageneralresultdescribingoutputsoflocalqueries.Thisresultleadsto toapplythanEhrenfeuchtFrassegames.Wealsoshowthatsomegeneralizationsofthebounded degreepropertythatwereconjecturedtohold,failforrelationalcalculus. beviewedas\otheshelf"strategiesforprovinginexpressibilityresults,whichareofteneasier maintenanceofviews,andshowthatSQLandrelationalcalculusareincapableofmaintainingthe gregates,whichisessentiallyplainSQL,hastheboundeddegreeproperty,thusansweringaques tionthathasbeenopenforseveralyears.Consequently,rstorderquerieswithHartigorRescher quantiersalsohavetheboundeddegreeproperty.Finally,weapplyourresultstoincremental Wethenprovethatthelanguageobtainedfromrelationalcalculusbyaddinggroupingandag
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...
Notions of Locality and Their Logical Characterizations Over Finite Models
, 1997
"... Many known tools for proving expressibility bounds for firstorder logic are based on one of several locality properties. In this paper we characterize the relationship between those notions of locality. We note that Gaifman's locality theorem gives rise to two notions: one deals with sentences a ..."
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Cited by 25 (17 self)
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Many known tools for proving expressibility bounds for firstorder logic are based on one of several locality properties. In this paper we characterize the relationship between those notions of locality. We note that Gaifman's locality theorem gives rise to two notions: one deals with sentences and one with open formulae. We prove that the former implies Hanf's notion of locality, which in turn implies Gaifman's locality for open formulae. Each of these implies the bounded degree property, which is one of the easiest tools for proving expressibility bounds. These results apply beyond the firstorder case. We use them to derive expressibility bounds for firstorder logic with unary quantifiers and counting. We also characterize the notions of locality on structures of small degree. 1 Introduction It is well known that firstorder logic has limited expressive power. Typically, inexpressibility proofs are based on either a compactness argument, or EhrenfeuchtFraiss'e games. In ...
On Winning Strategies With Unary Quantifiers
 J. Logic and Computation
, 1996
"... A combinatorial argument for two finite structures to agree on all sentences with bounded quantifier rank in firstorder logic with any set of unary generalized quantifiers, is given. It is known that connectivity of finite structures is neither in monadic \Sigma 1 1 nor in L !! (Q u ), where Q ..."
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Cited by 25 (6 self)
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A combinatorial argument for two finite structures to agree on all sentences with bounded quantifier rank in firstorder logic with any set of unary generalized quantifiers, is given. It is known that connectivity of finite structures is neither in monadic \Sigma 1 1 nor in L !! (Q u ), where Q u is the set of all unary generalized quantifiers. Using this combinatorial argument and a combination of secondorder EhrenfeuchtFra iss'e games developed by Ajtai and Fagin, we prove that connectivity of finite structures is not in monadic \Sigma 1 1 with any set of unary quantifiers, even if sentences are allowed to contain builtin relations of moderate degree. The combinatorial argument is also used to show that no class (if it is not in some sense trivial) of finite graphs defined by forbidden minors, is in L !! (Q u ). Especially, the class of planar graphs is not in L !! (Q u ). 1. Introduction The expressive power of firstorder logic L !! is rather limited. This is beca...
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.
The complexity of graph connectivity
, 1992
"... In this paper we survey the major developments in understanding the complexity of the graph connectivity problem in several computational models, and highlight some challenging open problems. 1 ..."
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Cited by 23 (1 self)
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In this paper we survey the major developments in understanding the complexity of the graph connectivity problem in several computational models, and highlight some challenging open problems. 1
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
FiniteModel Theory  A Personal Perspective
 Theoretical Computer Science
, 1993
"... Finitemodel theory is a study of the logical properties of finite mathematical structures. This paper is a very personalized view of finitemodel theory, where the author focuses on his own personal history, and results and problems of interest to him, especially those springing from work in his Ph ..."
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Cited by 20 (0 self)
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Finitemodel theory is a study of the logical properties of finite mathematical structures. This paper is a very personalized view of finitemodel theory, where the author focuses on his own personal history, and results and problems of interest to him, especially those springing from work in his Ph.D. thesis. Among the topics discussed are: