On the Sets of Real Numbers Recognized by Finite Automata in Multiple Bases

TL;DR

This paper explores the recognition of real number sets by finite automata across multiple bases, extending Cobham's theorem, and characterizes the sets recognizable by different automata classes in various base conditions.

Contribution

It generalizes Cobham's theorem to multiplicatively independent bases for real numbers and characterizes sets recognizable by Muller automata in multiple bases.

Findings

Sets recognizable in multiple bases are definable in the first order additive theory of reals and integers.

Muller automata do not satisfy Cobham's theorem in the context of real numbers.

Recognizable sets by Muller automata in multiple bases are also recognizable by weak deterministic automata.

Abstract

This article studies the expressive power of finite automata recognizing sets of real numbers encoded in positional notation. We consider Muller automata as well as the restricted class of weak deterministic automata, used as symbolic set representations in actual applications. In previous work, it has been established that the sets of numbers that are recognizable by weak deterministic automata in two bases that do not share the same set of prime factors are exactly those that are definable in the first order additive theory of real and integer numbers. This result extends Cobham's theorem, which characterizes the sets of integer numbers that are recognizable by finite automata in multiple bases. In this article, we first generalize this result to multiplicatively independent bases, which brings it closer to the original statement of Cobham's theorem. Then, we study the sets of reals recognizable by Muller automata in two bases. We show with a counterexample that, in this setting, Cobham's theorem does not generalize to multiplicatively independent bases. Finally, we prove that the sets of reals that are recognizable by Muller automata in two bases that do not share the same set of prime factors are exactly those definable in the first order additive theory of real and integer numbers. These sets are thus also recognizable by weak deterministic automata. This result leads to a precise characterization of the sets of real numbers that are recognizable in multiple bases, and provides a theoretical justification to the use of weak automata as symbolic representations of sets.