ICALP 2008: Automata, Languages and Programming pp 112-123

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

• Bernard Boigelot
• Julien Brusten
• Véronique Bruyère
Conference paper

DOI: 10.1007/978-3-540-70583-3_10

Part of the Lecture Notes in Computer Science book series (LNCS, volume 5126)
Cite this paper as:
Boigelot B., Brusten J., Bruyère V. (2008) On the Sets of Real Numbers Recognized by Finite Automata in Multiple Bases. In: Aceto L., Damgård I., Goldberg L.A., Halldórsson M.M., Ingólfsdóttir A., Walukiewicz I. (eds) Automata, Languages and Programming. ICALP 2008. Lecture Notes in Computer Science, vol 5126. Springer, Berlin, Heidelberg

## Abstract

This paper 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 Open image in new window. This result extends Cobham’s theorem, which characterizes the sets of integer numbers that are recognizable by finite automata in multiple bases.

In this paper, 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 Open image in new window. 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.