On Interpretations in B\"uchi Arithmetics

TL;DR

This paper explores the interpretability and definability properties of B"uchi arithmetics, showing bi-interpretability between different bases and characterizing interpretations within these systems.

Contribution

It proves bi-interpretability of BA_n and BA_m for any n,m and characterizes interpretations in BA_n as essentially one-dimensional, though not necessarily BA_n-definable.

Findings

BA_n and BA_m are bi-interpretable for any n,m

Any interpretation in BA_n is isomorphic to a one-dimensional interpretation

Interpretations in BA_n are not always BA_n-definable

Abstract

B\"uchi arithmetics BA_n, n >= 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 BA_n is equivalent to its recognizability by a finite automaton receiving numbers in their n-ary expansion. We show that B\"uchi arithmetics BA_n and BA_m are bi-interpretable for any n,m. Furthermore, we establish that any interpretation of some structure in BA_n is isomorphic to some one-dimensional interpretation; however, this isomorphism must not be BA_n-definable.