On Interpretations in B\"uchi Arithmetics
Alexnader Zapryagaev

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.
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Taxonomy
Topicssemigroups and automata theory · Computability, Logic, AI Algorithms · Advanced Algebra and Logic
