|Reference : On the Sets of Real Numbers Recognized by Finite Automata in Multiple Bases|
|Scientific congresses and symposiums : Paper published in a journal|
|Engineering, computing & technology : Computer science|
|On the Sets of Real Numbers Recognized by Finite Automata in Multiple Bases|
|Boigelot, Bernard [Université de Liège - ULg > Dép. d'électric., électron. et informat. (Inst.Montefiore) > Informatique >]|
|Brusten, Julien [Université de Liège - ULg > Dép. d'électric., électron. et informat. (Inst.Montefiore) > Informatique >]|
|Bruyère, Véronique [Université de Mons-Hainaut - UMH > Institut d'Informatique > > >]|
|Lecture Notes in Computer Science|
|35th International Colloquium on Automata, Languages and Programming|
|July 7-11, 2008|
|[en] Automata ; Real numbers ; Mixed real-integer arithmetic ; Cobham's theorem|
|[en] 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 (R, Z, +, <). 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 (R, Z, +, <). 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.
|Fonds de la Recherche Scientifique (Communauté française de Belgique) - F.R.S.-FNRS ; Interuniversity Attraction Poles program MoVES of the Belgian Federal Science Office ; Grant 2.4530.02 of the Belgian Fund for Scientific Research (F.R.S.-FNRS)|
|The original publication is available at www.springerlink.com|
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