Quotient Complexities of Atoms of Regular Languages

  • Janusz Brzozowski
  • Hellis Tamm
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7410)

Abstract

An atom of a regular language L with n (left) quotients is a non-empty intersection of uncomplemented or complemented quotients of L, where each of the n quotients appears in a term of the intersection. The quotient complexity of L, which is the same as the state complexity of L, is the number of quotients of L. We prove that, for any language L with quotient complexity n, the quotient complexity of any atom of L with r complemented quotients has an upper bound of 2n − 1 if r = 0 or r = n, and \(1+\sum_{k=1}^{r} \sum_{h=k+1}^{k+n-r} C_{h}^{n} \cdot C_{k}^{h}\) otherwise, where \(C_j^i\) is the binomial coefficient. For each \(n\geqslant 1\), we exhibit a language whose atoms meet these bounds.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Brzozowski, J.: Canonical regular expressions and minimal state graphs for definite events. In: Proceedings of the Symposium on Mathematical Theory of Automata. MRI Symposia Series, vol. 12, pp. 529–561. Polytechnic Press, Polytechnic Institute of Brooklyn, N.Y. (1963)Google Scholar
  2. 2.
    Brzozowski, J.: Quotient complexity of regular languages. J. Autom. Lang. Comb. 15(1/2), 71–89 (2010)Google Scholar
  3. 3.
    Brzozowski, J., Tamm, H.: Theory of Átomata. In: Mauri, G., Leporati, A. (eds.) DLT 2011. LNCS, vol. 6795, pp. 105–116. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  4. 4.
    Dénes, J.: On transformations, transformation semigroups and graphs. In: Erdös, P., Katona, G. (eds.) Theory of Graphs. Proceedings of the Colloquium on Graph Theory held at Tihany, 1966, pp. 65–75. Akadémiai Kiado (1968)Google Scholar
  5. 5.
    Piccard, S.: Sur les fonctions définies dans les ensembles finis quelconques. Fund. Math. 24, 298–301 (1935)Google Scholar
  6. 6.
    Salomaa, A., Wood, D., Yu, S.: On the state complexity of reversals of regular languages. Theoret. Comput. Sci. 320, 315–329 (2004)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Yu, S.: State complexity of regular languages. J. Autom. Lang. Comb. 6, 221–234 (2001)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Janusz Brzozowski
    • 1
  • Hellis Tamm
    • 2
  1. 1.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Institute of CyberneticsTallinn University of TechnologyTallinnEstonia

Personalised recommendations