Advertisement

A Logical Descriptor for Regular Languages via Stone Duality

  • Stefano Aguzzoli
  • Denisa Diaconescu
  • Tommaso Flaminio
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8687)

Abstract

In this paper we introduce a class of descriptors for regular languages arising from an application of the Stone duality between finite Boolean algebras and finite sets. These descriptors, called classical fortresses, are object specified in classical propositional logic and capable to accept exactly regular languages. To prove this, we show that the languages accepted by classical fortresses and deterministic finite automata coincide. Classical fortresses, besides being propositional descriptors for regular languages, also turn out to be an efficient tool for providing alternative and intuitive proofs for the closure properties of regular languages.

Keywords

regular languages finite automata propositional logic Stone duality 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Blok, W., Pigozzi, D.: Algebraizable logics. Memoirs of The American Mathematical Society, vol. 77. American Mathematical Society (1989)Google Scholar
  2. 2.
    Brzozowski, J.A.: Canonical regular expressions and minimal state graphs for definite events. In: Mathematical Theory of Automata. MRI Symposia Series, vol. 12, pp. 529–561. Polytechnic Press, Polytechnic Institute of Brooklyn (1962)Google Scholar
  3. 3.
    Büchi, J.R.: Weak second-order arithmetic and finite automata. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 6, 66–92 (1960)CrossRefzbMATHGoogle Scholar
  4. 4.
    Burris, S., Sankappanavar, H.P.: A course in Universal Algebra. Springer (1981)Google Scholar
  5. 5.
    Cintula, P., Hájek, P., Noguera, C.: Handbook of Mathematical Fuzzy Logic, vol. 2. College Publications (2011)Google Scholar
  6. 6.
    Elgot, C.C.: Decision problems of finite automata design and related arithmetics. Trans. Amer. Math. Soc. 98, 21–51 (1961)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hansen, H.H., Panangaden, P., Rutten, J.J.M.M., Bonchi, F., Bonsangue, M.M., Silva, A.: Algebra-coalgebra duality in Brzozwski’s minimization algorithm. ACM Transactions on Computational Logic (to appear)Google Scholar
  8. 8.
    Gehrke, M.: Duality and recognition. In: Murlak, F., Sankowski, P. (eds.) MFCS 2011. LNCS, vol. 6907, pp. 3–18. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  9. 9.
    Gehrke, M., Grigorieff, S., Pin, J.-É.: Duality and equational theory of regular languages. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 246–257. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  10. 10.
    Hopcroft, J.E., Motwani, R., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation, 2nd edn. Addison-Wesley (2000)Google Scholar
  11. 11.
    Johnstone, P.T.: Stone Spaces. Cambridge University Press (1982)Google Scholar
  12. 12.
    Kleene, S.C.: Representation of events in nerve nets and finite automata. In: Shannon, C.E., McCarthy, J. (eds.) Automata Studies, pp. 3–42. Princeton University Press (1956)Google Scholar
  13. 13.
    Weyuker, E.J., Davis, M.E., Sigal, R.: Computability, complexity, and languages: Fundamentals of theoretical computer science. Academic Press, Boston (1994)Google Scholar
  14. 14.
    Trakhtenbrot, B.A.: Finite automata and the logic of oneplace predicates. Siberian Math. J. 3, 103–131 (1962)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Stefano Aguzzoli
    • 1
  • Denisa Diaconescu
    • 2
  • Tommaso Flaminio
    • 3
  1. 1.Department of Computer ScienceUniversity of MilanItaly
  2. 2.Department of Computer Science, Faculty of Mathematics and Computer ScienceUniversity of BucharestRomania
  3. 3.DiSTA - Department of Theoretical and Applied ScienceUniversity of InsubriaItaly

Personalised recommendations