A Generalization of Cobham’s Theorem to Automata over Real Numbers

  • Bernard Boigelot
  • Julien Brusten
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4596)

Abstract

This paper studies the expressive power of finite-state automata recognizing sets of real numbers encoded positionally. It is known that the sets that are definable in the first-order additive theory of real and integer variables 〈ℝ, ℤ, + , < 〉 can all be recognized by weak deterministic Büchi automata, regardless of the encoding base r > 1. In this paper, we prove the reciprocal property, i.e., that a subset of ℝ that is recognizable by weak deterministic automata in every base r > 1 is necessarily definable in 〈ℝ, ℤ, + , < 〉. This result generalizes to real numbers the well-known Cobham’s theorem on the finite-state recognizability of sets of integers. Our proof gives interesting insight into the internal structure of automata recognizing sets of real numbers, which may lead to efficient data structures for handling these sets.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Avigad, J., Yin, Y.: Quantifier elimination for the reals with a predicate for the powers of two. Theoretical Computer Science 370, 48–59 (2007)MATHCrossRefMathSciNetGoogle Scholar
  2. 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. 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)MATHMathSciNetGoogle Scholar
  4. 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. Boigelot, B.: Symbolic methods for exploring infinite state spaces. PhD thesis, Université de Liège (1998)Google Scholar
  6. Brusten, J.: Etude des propriétés des RVA. Graduate thesis, Université de Liège (May 2006)Google Scholar
  7. 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
  8. 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
  9. Cobham, A.: On the base-dependence of sets of numbers recognizable by finite automata. Mathematical Systems Theory 3, 186–192 (1969)MATHCrossRefMathSciNetGoogle Scholar
  10. Kupferman, O., Vardi, M.Y.: Complementation constructions for nondeterministic automata on infinite words. In: Halbwachs, N., Zuck, L.D. (eds.) TACAS 2005. LNCS, vol. 3440, pp. 206–221. Springer, Heidelberg (2005)Google Scholar
  11. Latour, L.: Presburger arithmetic: from automata to formulas. PhD thesis, Université de Liège (2005)Google Scholar
  12. Leroux, J.: A polynomial time Presburger criterion and synthesis for number decision diagrams. In: Proc. 20th LICS, Chicago, June 2005, pp. 147–156. IEEE Computer Society Press, Los Alamitos (2005)Google Scholar
  13. Semenov, A.L.: Presburgerness of predicates regular in two number systems. Siberian Mathematical Journal 18, 289–299 (1977)MATHCrossRefGoogle Scholar
  14. van den Dries, L.: The field of reals with a predicate for the powers of two. Manuscripta Mathematica 54, 187–195 (1985)CrossRefMathSciNetGoogle Scholar
  15. Villemaire, R.: The theory of 〈ℕ, + , V k, V l 〉 is undecidable. Theoretical Computer Science 106(2), 337–349 (1992)MATHCrossRefMathSciNetGoogle Scholar
  16. Wolper, P., Boigelot, B.: An automata-theoretic approach to Presburger arithmetic constraints. In: Mycroft, A. (ed.) SAS 1995. LNCS, vol. 983, Springer, Heidelberg (1995)Google Scholar
  17. Wolper, P., Boigelot, B.: Verifying systems with infinite but regular state spaces. In: Vardi, M.Y. (ed.) CAV 1998. LNCS, vol. 1427, pp. 88–97. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  18. 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 2007

Authors and Affiliations

  • Bernard Boigelot
    • 1
  • Julien Brusten
    • 1
  1. 1.Institut Montefiore, B28, Universit‘e de Li‘ege, B-4000 Li‘egeBelgium

Personalised recommendations