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Relativized logspace and generalized quantifiers over finite ordered structures

Published online by Cambridge University Press:  12 March 2014

Georg Gottlob
Georg Gottlob*
Affiliation:
Technische Universität Wien, Institut für Informationssysteme, Paniglgasse 16, A-1040 Wien, Austria, E-mail: gottlob@dbai.tuwien.ac.at
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Abstract

We here examine the expressive power of first order logic with generalized quantifiers over finite ordered structures. In particular, we address the following problem: Given a family Q of generalized quantifiers expressing a complexity class C, what is the expressive power of first order logic FO(Q) extended by the quantifiers in Q? From previously studied examples, one would expect that FO(Q) captures LC, i.e., logarithmic space relativized to an oracle in C. We show that this is not always true. However, after studying the problem from a general point of view, we derive sufficient conditions on C such that FO(Q) captures LC. These conditions are fulfilled by a large number of relevant complexity classes, in particular, for example, by NP. As an application of this result, it follows that first order logic extended by Henkin quantifiers captures LNP. This answers a question raised by Blass and Gurevich [Ann. Pure Appl. Logic, vol. 32, 1986]. Furthermore we show that for many families Q of generalized quantifiers (including the family of Henkin quantifiers), each FO(Q)-formula can be replaced by an equivalent FO(Q)-formula with only two occurrences of generalized quantifiers. This generalizes and extends an earlier normal-form result by I. A. Stewart [Fundamenta Inform, vol. 18, 1993].

Keywords

generalized quantifiersHenkin quantifiersfinite model theoryfinite structuresexpressive powercomplexityoraclesrelativizationlogarithmic spacelogspacebounded queries
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 2 , June 1997 , pp. 545 - 574
Copyright
Copyright © Association for Symbolic Logic 1997

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