On Interpretations in B\"uchi Arithmetics
Abstract
B\"uchi arithmetics BAn, n >= 2, are extensions of Presburger arithmetic with an unary functional symbol Vn(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 show that B\"uchi arithmetics BAn and BAm are bi-interpretable for any n,m. Furthermore, we establish that any interpretation of some structure in BAn is isomorphic to some one-dimensional interpretation; however, this isomorphism must not be BAn-definable.
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