RAIRO - Theoretical Informatics and Applications (RAIRO: ITA)

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RAIRO-Theor. Inf. Appl. 42, 121-135 (2008)
DOI: 10.1051/ita:2007047

Deciding whether a relation defined in Presburger logic can be defined in weaker logics

Christian Choffrut

LIAFA, Université de Paris 7, 2 place Jussieu, 75251 Paris Cedex 05; cc@liafa.jussieu.fr

(Published online: 18 January 2008)

Abstract
We consider logics on $\mathbb{Z}$ and $\mathbb{N}$ which are weaker than Presburger arithmetic and we settle the following decision problem: given a k-ary relation on $\mathbb{Z}$ and $\mathbb{N}$ which are first order definable in Presburger arithmetic, are they definable in these weaker logics? These logics, intuitively, are obtained by considering modulo and threshold counting predicates for differences of two variables.

Mathematics Subject Classification. 03B10, 68Q70

Key words: Presburger arithmetic -- first order logic -- decidability

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