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Notions of locality and their logical characterizations over finite models

Published online by Cambridge University Press:  12 March 2014

Lauri Hella ,
Leonid Libkin  and
Juha Nurmonen
Lauri Hella
Affiliation:
Department of Mathematics, P.O. Box 4 (Yliopistonkatu 5), 00014 University of Helsinki, Finland, E-mail: lauri.hella@helsinki.fi
Leonid Libkin
Affiliation:
Bell Laboratories, 600 Mountain Avenue, Murray Hill, NJ 07974, USA, E-mail: libkin@bell-labs.com
Juha Nurmonen
Affiliation:
Department of Mathematics and Computer Science, University of Leicester, University Road, Leicester Lei 7RH, UK, E-mail: j.nurmonen@mcs.le.ac.uk
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Abstract

Many known tools for proving expressibility bounds for first-ordér 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 first-order case. We use them to derive expressibility bounds for first-order logic with unary quantifiers and counting. We also characterize the notions of locality on structures of small degree.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 4 , December 1999 , pp. 1751 - 1773
Copyright
Copyright © Association for Symbolic Logic 1999

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