## Journal of Symbolic Logic

### On Decidable Extensions of Presburger Arithmetic: From A. Bertrand Numeration Systems to Pisot Numbers

Francoise Point

#### Abstract

We study extensions of Presburger arithmetic with a unary predicate R and we show that under certain conditions on R, R is sparse (a notion introduced by A. L. Semenov) and the theory of $\langle\mathbb{N}, +, R\rangle$ is decidable. We axiomatize this theory and we show that in a reasonable language, it admits quantifier elimination. We obtain similar results for the structure $\langle\mathbb{Q},+, R\rangle$.

#### Article information

Source
J. Symbolic Logic, Volume 65, Issue 3 (2000), 1347-1374.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183746185

Mathematical Reviews number (MathSciNet)
MR1791380

Zentralblatt MATH identifier
0971.03013

JSTOR