Classifying Regular Languages via Cascade Products of Automata

  • Marcus Gelderie
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6638)

Abstract

Building on the celebrated Krohn-Rhodes Theorem we characterize classes of regular languages in terms of the cascade decompositions of minimal DFA of languages in those classes. More precisely we provide characterizations for the classes of piecewise testable languages and commutative languages. To this end we use biased resets, which are resets in the classical sense, that can change their state at most once. Next, we introduce the concept of the scope of a cascade product of reset automata in order to capture a notion of locality inside a cascade product and show that there exist constant bounds on the scope for certain classes of languages. Finally we investigate the impact of biased resets in a product of resets on the dot-depth of languages recognized by this product. This investigation allows us to refine an upper bound on the dot-depth of a language, given by Cohen and Brzozowski.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Brzozowski, J.A., Fich, F.E.: Languages of J-trivial monoids. Journal of Computer and System Sciences 20(1), 32–49 (1980)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Cohen, R.S., Brzozowski, J.A.: Dot-depth of star-free events. Journal of Computer and System Sciences 5(1), 1–16 (1971)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Eilenberg, S.: Automata, Languages, and Machines. Pure and Applied Mathematics, vol. A. Elsevier, Burlington (1974)MATHGoogle Scholar
  4. 4.
    Eilenberg, S.: Automata, Languages, and Machines. Pure and Applied Mathematics, vol. B. Elsevier, Burlington (1974)MATHGoogle Scholar
  5. 5.
    Gelderie, M.: Classifying regular languages via cascade products of automata. Diploma thesis, RWTH Aachen University (2011)Google Scholar
  6. 6.
    Ginzburg, A.: Algebraic Theory of Automata. Academic Press, New York (1968)MATHGoogle Scholar
  7. 7.
    Krohn, K., Rhodes, J.: Algebraic theory of machines. I. Prime decomposition theorem for finite semigroups and machines. Trans. Amer. Math. Soc. 116, 450–464 (1965)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Meyer, A.R.: A note on star-free events. J. ACM 16(2), 220–225 (1969)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Pin, J.E.: Varieties Of Formal Languages. Plenum Publishing Co., New York (1986)CrossRefMATHGoogle Scholar
  10. 10.
    Pin, J.E.: Syntactic semigroups. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages. Word, Language, Grammar, vol. 1, pp. 679–746. Springer-Verlag New York, Inc., New York (1997)CrossRefGoogle Scholar
  11. 11.
    Schützenberger, M.P.: On finite monoids having only trivial subgroups. Information and Control 8(2), 190–194 (1965)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Simon, I.: Piecewise testable events. In: Proceedings of the 2nd GI Conference on Automata Theory and Formal Languages, pp. 214–222. Springer, London (1975)Google Scholar
  13. 13.
    Straubing, H.: On finite J-trivial monoids. Semigroup Forum 19, 107–110 (1980)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Straubing, H.: Finite Automata, Formal Logic, and Circuit Complexity. Birkhauser Verlag, Basel (1994)CrossRefMATHGoogle Scholar
  15. 15.
    Straubing, H., Thérien, D.: Partially ordered finite monoids and a theorem of I. Simon. J. Algebra 119(2), 393–399 (1988)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Thomas, W.: Classifying regular events in symbolic logic. Journal of Computer and System Sciences 25(3), 360–376 (1982)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Marcus Gelderie
    • 1
  1. 1.RWTH AachenAachenGermany

Personalised recommendations