# Uniform definition of sets using relations and complement of Presburger Arithmetic

2 Automates et Applications [LIAFA]
IRIF (UMR_8243) - Institut de Recherche en Informatique Fondamentale
Abstract : In 1996, Michaux and Villemaire considered integer relations $R$ which are not definable in Presburger Arithmetic. That is, not definable in first-order logic over integers with the addition function and the order relation (FO[N,+,<]-definable relations). They proved that, for each such $R$, there exists a FO[N,+,<,$R$]-formula $\nu_{R}(x)$ which defines a set of integers which is not ultimately periodic, i.e. not FO[N,+,<]-definable. It is proven in this paper that the formula $\nu(x)$ can be chosen such that it does not depend on the interpretation of $R$. It is furthermore proven that $\nu(x)$ can be chosen such that it defines an expanding set. That is, an infinite set of integers such that the distance between two successive elements is not bounded.
Domain :

https://hal.archives-ouvertes.fr/hal-01676721
Contributor : Arthur Milchior <>
Submitted on : Saturday, January 6, 2018 - 6:05:02 AM
Last modification on : Friday, April 10, 2020 - 5:27:58 PM

### Identifiers

• HAL Id : hal-01676721, version 1
• ARXIV : 1611.03839

### Citation

Arthur Milchior. Uniform definition of sets using relations and complement of Presburger Arithmetic. 2018. ⟨hal-01676721⟩

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