Abstract
This work studies the properties of finite automata recognizing vectors with real components, encoded positionally in a given integer numeration base. Such automata are used, in particular, as symbolic data structures for representing sets definable in the first-order theory 〈ℝ, ℤ, + , ≤ 〉, i.e., the mixed additive arithmetic of integer and real variables. They also lead to a simple decision procedure for this arithmetic.
In previous work, it has been established that the sets definable in 〈ℝ, ℤ, + , ≤ 〉 can be handled by a restricted form of infinite-word automata, weak deterministic ones, regardless of the chosen numeration base. In this paper, we address the reciprocal property, proving that the sets of vectors that are simultaneously recognizable in all bases, by either weak deterministic or Muller automata, are those definable in 〈ℝ, ℤ, + , ≤ 〉. This result can be seen as a generalization to the mixed integer and real domain of Semenov’s theorem, which characterizes the sets of integer vectors recognizable by finite automata in multiple bases. It also extends to multidimensional vectors a similar property recently established for sets of numbers.
As an additional contribution, the techniques used for obtaining our main result lead to valuable insight into the internal structure of automata recognizing sets of vectors definable in 〈ℝ, ℤ, + , ≤ 〉. This structure might be exploited in order to improve the efficiency of representation systems and decision procedures for this arithmetic.
This work is supported by the Interuniversity Attraction Poles program MoVES of the Belgian Federal Science Policy Office, by the grant 2.4530.02 of the Belgian Fund for Scientific Research (F.R.S.-FNRS), and by the French project ANR-06-SETI-001 AVERISS.
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References
Presburger, M.: Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. In: Comptes Rendus du Premier Congrès des Mathématiciens des Pays Slaves, Warsaw, pp. 92–101 (1929)
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)
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.: Symbolic methods for exploring infinite state spaces. PhD thesis, Université de Liège (1998)
Cobham, A.: On the base-dependence of sets of numbers recognizable by finite automata. Mathematical Systems Theory 3, 186–192 (1969)
Semenov, A.: Presburgerness of predicates regular in two number systems. Siberian Mathematical Journal 18, 289–299 (1977)
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)
The Liège Automata-based Symbolic Handler (LASH), http://www.montefiore.ulg.ac.be/~boigelot/research/lash/
Becker, B., Dax, C., Eisinger, J., Klaedtke, F.: LIRA: Handling constraints of linear arithmetics over the integers and the reals. In: Damm, W., Hermanns, H. (eds.) CAV 2007. LNCS, vol. 4590, pp. 307–310. Springer, Heidelberg (2007)
Boigelot, B., Brusten, J., Bruyère, V.: On the sets of real numbers recognized by finite automata in multiple bases. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 112–123. Springer, Heidelberg (2008)
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)
van Leeuwen, J. (ed.): Handbook of Theoretical Computer Science. Formal Models and Semantics, vol. B. Elsevier and MIT Press (1990)
Vardi, M.: The Büchi complementation saga. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 12–22. Springer, Heidelberg (2007)
Löding, C.: Efficient minimization of deterministic weak ω-automata. Information Processing Letters 79(3), 105–109 (2001)
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)
Perrin, D., Pin, J.: Infinite words. Pure and Applied Mathematics, vol. 141. Elsevier, Amsterdam (2004)
Boigelot, B., Brusten, J.: A generalization of Cobham’s theorem to automata over real numbers. Theoretical Computer Science (2009) (in press)
Ferrante, J., Rackoff, C.: The Computational Complexity of Logical Theories. Lecture Notes in Mathematics, vol. 718. Springer, Heidelberg (1979)
Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers, 5th edn. Oxford University Press, Oxford (1985)
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)
Bryant, R.: Symbolic Boolean manipulation with ordered binary decision diagrams. ACM Computing Surveys 24(3), 293–318 (1992)
Weispfenning, V.: Mixed real-integer linear quantifier elimination. In: Proc. ACM SIGSAM ISSAC, Vancouver, pp. 129–136. ACM Press, New York (1999)
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)
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, pp. 147–156. IEEE Computer Society, Los Alamitos (2005)
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Boigelot, B., Brusten, J., Leroux, J. (2009). A Generalization of Semenov’s Theorem to Automata over Real Numbers. In: Schmidt, R.A. (eds) Automated Deduction – CADE-22. CADE 2009. Lecture Notes in Computer Science(), vol 5663. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02959-2_34
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