ON INTERPRETATIONS OF PRESBURGER ARITHMETIC IN BÜCHI ARITHMETICS

Abstract

Büchi arithmetics BAn, \(n \geqslant 2\), are extensions of Presburger arithmetic with an unary functional symbol \({{V}_{n}}(x)\) denoting the largest power of n that divides x. Definability of a set in BAn is equivalent to its recognizability by a finite automaton receiving numbers in their n-ary expansion. We consider the interpretations of Presburger Arithmetic in the standard model of BAn and show that each such interpretation has an internal model isomorphic to the standard one. This answers a question by A. Visser on the interpretations of certain weak arithmetical theories in themselves.