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.
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References
Avigad, J., Yin, Y.: Quantifier elimination for the reals with a predicate for the powers of two. Theoretical Computer Science 370, 48–59 (2007)
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)
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)
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)
Boigelot, B.: Symbolic methods for exploring infinite state spaces. PhD thesis, Université de Liège (1998)
Brusten, J.: Etude des propriétés des RVA. Graduate thesis, Université de Liège (May 2006)
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)
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)
Cobham, A.: On the base-dependence of sets of numbers recognizable by finite automata. Mathematical Systems Theory 3, 186–192 (1969)
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)
Latour, L.: Presburger arithmetic: from automata to formulas. PhD thesis, Université de Liège (2005)
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)
Semenov, A.L.: Presburgerness of predicates regular in two number systems. Siberian Mathematical Journal 18, 289–299 (1977)
van den Dries, L.: The field of reals with a predicate for the powers of two. Manuscripta Mathematica 54, 187–195 (1985)
Villemaire, R.: The theory of 〈ℕ, + , V k , V l 〉 is undecidable. Theoretical Computer Science 106(2), 337–349 (1992)
Wolper, P., Boigelot, B.: An automata-theoretic approach to Presburger arithmetic constraints. In: Mycroft, A. (ed.) SAS 1995. LNCS, vol. 983, Springer, Heidelberg (1995)
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)
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)
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Boigelot, B., Brusten, J. (2007). A Generalization of Cobham’s Theorem to Automata over Real Numbers. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds) Automata, Languages and Programming. ICALP 2007. Lecture Notes in Computer Science, vol 4596. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73420-8_70
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DOI: https://doi.org/10.1007/978-3-540-73420-8_70
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