Reference : On the Sets of Real Numbers Recognized by Finite Automata in Multiple Bases
Scientific journals : Article
Engineering, computing & technology : Computer science
http://hdl.handle.net/2268/74868
On the Sets of Real Numbers Recognized by Finite Automata in Multiple Bases
English
Boigelot, Bernard mailto [Université de Liège - ULg > Dép. d'électric., électron. et informat. (Inst.Montefiore) > Informatique >]
Brusten, Julien mailto [Université de Liège - ULg > Dép. d'électric., électron. et informat. (Inst.Montefiore) > Informatique >]
Bruyère, Véronique mailto [Université de Mons-Hainaut - UMH > Institut d'Informatique > > >]
2010
Logical Methods in Computer Science
6
1
1-17
Yes
International
[en] Automata ; Reals numbers ; Semenov's theorem ; Mixed real-integer arithmetic
[en] 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.
Interuniversity Attraction Poles program MoVES of the Belgian Federal Science Policy Office ; Grant 2.4530.02 of the Belgian Fund for Scientific Research (F.R.S.-FNRS)
Researchers ; Professionals ; Students
http://hdl.handle.net/2268/74868
10.2168/LMCS-6(1:6)2010

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