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

  • Bernard Boigelot
  • Julien Brusten
  • Véronique Bruyère
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5126)


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [BB07]
    Boigelot, B., Brusten, J.: A generalization of Cobham’s theorem to automata over real numbers. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 813–824. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  2. [BBR97]
    Boigelot, B., Bronne, L., Rassart, S.: An improved reachability analysis method for strongly linear hybrid systems. In: Grumberg, O. (ed.) CAV 1997. LNCS, vol. 1254, pp. 167–177. Springer, Heidelberg (1997)Google Scholar
  3. [BHMV94]
    Bruyère, V., Hansel, G., Michaux, C., Villemaire, R.: Logic and p-recognizable sets of integers. Bulletin of the Belgian Mathematical Society 1(2), 191–238 (1994)MathSciNetMATHGoogle Scholar
  4. [BJW05]
    Boigelot, B., Jodogne, S., Wolper, P.: An effective decision procedure for linear arithmetic over the integers and reals. ACM Transactions on Computational Logic 6(3), 614–633 (2005)CrossRefMathSciNetGoogle Scholar
  5. [BRW98]
    Boigelot, B., Rassart, S., Wolper, P.: On the expressiveness of real and integer arithmetic automata. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 152–163. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  6. [Büc62]
    Büchi, J.R.: On a decision method in restricted second order arithmetic. In: Proc. International Congress on Logic, Methodoloy and Philosophy of Science, pp. 1–12. Stanford University Press, Stanford (1962)Google Scholar
  7. [Cob69]
    Cobham, A.: On the base-dependence of sets of numbers recognizable by finite automata. Mathematical Systems Theory 3, 186–192 (1969)CrossRefMathSciNetMATHGoogle Scholar
  8. [EK06]
    Eisinger, J., Klaedtke, F.: Don’t care words with an application to the automata-based approach for real addition. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 67–80. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  9. [HW85]
    Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers, 5th edn. Oxford University Press, Oxford (1985)Google Scholar
  10. [McN66]
    McNaughton, R.: Testing and generating infinite sequences by a finite automaton. Information and Control 9(5), 521–530 (1966)CrossRefMathSciNetMATHGoogle Scholar
  11. [Per90]
    Perrin, D.: Finite automata. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science. Formal Models and Semantics, vol. B. Elsevier and MIT Press (1990)Google Scholar
  12. [PP04]
    Perrin, D., Pin, J.E.: Infinite words. Pure and Applied Mathematics, vol. 141. Elsevier, Amsterdam (2004)Google Scholar
  13. [Saf88]
    Safra, S.: On the complexity of ω-automata. In: Proc. 29th Symposium on Foundations of Computer Science, pp. 319–327. IEEE Computer Society, Los Alamitos (1988)Google Scholar
  14. [Var07]
    Vardi, M.: The Büchi complementation saga. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 12–22. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  15. [WB95]
    Wolper, P., Boigelot, B.: An automata-theoretic approach to Presburger arithmetic constraints. In: Mycroft, A. (ed.) SAS 1995. LNCS, vol. 983, pp. 21–32. Springer, Heidelberg (1995)Google Scholar
  16. [Wei99]
    Weispfenning, V.: Mixed real-integer linear quantifier elimination. In: Proc. ACM SIGSAM ISSAC, Vancouver, pp. 129–136. ACM Press, New York (1999)Google Scholar
  17. [Wil93]
    Wilke, T.: Locally threshold testable languages of infinite words. In: Enjalbert, P., Wagner, K.W., Finkel, A. (eds.) STACS 1993. LNCS, vol. 665, pp. 607–616. Springer, Heidelberg (1993)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Bernard Boigelot
    • 1
  • Julien Brusten
    • 1
  • Véronique Bruyère
    • 2
  1. 1.Institut MontefioreB28 Université de LiègeLiègeBelgium
  2. 2.Université de Mons-HainautMonsBelgium

Personalised recommendations