Collapsing Words: A Progress Report

  • Dmitry S. Ananichev
  • Ilja V. Petrov
  • Mikhail V. Volkov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3572)

Abstract

A word w over a finite alphabet Σ is n-collapsing if for an arbitrary DFA \({\mathcal A}=\langle Q,\Sigma,\delta\rangle\), the inequality |δ(Q,w)| ≤ |Q| − n holds provided that |δ(Q,u)| ≤ |Q| − n for some word u∈Σ +  (depending on \({\mathcal A}\)). We overview some recent results related to this notion. One of these results implies that the property of being n-collapsing is algorithmically recognizable for any given positive integer n.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Almeida, J., Volkov, M.V.: Profinite identities for finite semigroups whose subgroups belong to a given pseudovariety. J. Algebra Appl. 2, 137–163 (2003)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Almeida, J., Volkov, M.V.: Subword complexity of profinite words and subgroups of free profinite semigroups. Int. J. Algebra Comp. (accepted)Google Scholar
  3. 3.
    Ananichev, D.S., Cherubini, A., Volkov, M.V.: Image reducing words and subgroups of free groups. Theor. Comput. Sci. 307(1), 77–92 (2003)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Ananichev, D.S., Cherubini, A., Volkov, M.V.: An inverse automata algorithm for recognizing 2-collapsing words. In: Ito, M., Toyama, M. (eds.) DLT 2002. LNCS, vol. 2450, pp. 270–282. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  5. 5.
    Ananichev, D.S., Volkov, M.V.: Collapsing words vs. synchronizing words. In: Kuich, W., Rozenberg, G., Salomaa, A. (eds.) DLT 2001. LNCS, vol. 2295, pp. 166–174. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  6. 6.
    Ananichev, D.S., Petrov, I.V.: Quest for short synchronizing words and short collapsing words. In: Proc. 4th Int. Conf. on WORDS, Univ. of Turku, Turku, pp. 411–418 (2003)Google Scholar
  7. 7.
    Bergman, C., Slutzki, G.: Complexity of some problems concerning varieties and quasi-varieties of algebras. SIAM J. Comput. 30, 359–382 (2000)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Burris, S., Lawrence, J.: Results on the equivalence problem for finite groups. Dept. Pure Math., Univ. of Waterloo (preprint)Google Scholar
  9. 9.
    Černý, J.: Poznámka k homogénnym eksperimentom s konecnými automatami. Mat.-Fyz. Cas. Slovensk. Akad. Vied 14, 208–216 (1964) (in Slovak)MATHGoogle Scholar
  10. 10.
    Frankl, P.: An extremal problem for two families of sets. Eur. J. Comb. 3, 125–127 (1982)MATHMathSciNetGoogle Scholar
  11. 11.
    Lawrence, J.: The complexity of the equivalence problem for nonsolvable groups. Dept. Pure Math., Univ. of Waterloo (preprint)Google Scholar
  12. 12.
    Margolis, S.W., Pin, J.-E., Volkov, M.V.: Words guaranteeing minimum image. Int. J. Foundations Comp. Sci. 15, 259–276 (2004)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Mateesku, A., Salomaa, A.: Aspects of classical language theory. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, Word. Language, Grammar, vol. I, pp. 175–251. Springer, Heidelberg (1997)Google Scholar
  14. 14.
    Moore, E.: Gedanken-experiments with sequential machines. In: Shannon, C.E., McCarthy, J. (eds.) Automata Studies Ann. Math. Studies, vol. 34, pp. 129–153. Princeton Univ. Press, Princeton (1956)Google Scholar
  15. 15.
    Pin, J.-E.: Utilisation de l’algèbre linéaire en théorie des automates. In: Actes du 1er Colloque AFCET-SMF de Mathématiques Appliquées, AFCET, Tome II, pp. 85–92 (1978) (in French)Google Scholar
  16. 16.
    Pin, J.-E.: On two combinatorial problems arising from automata theory. Ann. Discrete Math. 17, 535–548 (1983)MATHGoogle Scholar
  17. 17.
    Pöschel, R., Sapir, M.V., Sauer, N., Stone, M.G., Volkov, M.V.: Identities in full transformation semigroups. Algebra Universalis 31, 580–588 (1994)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Pribavkina, E.V.: On some properties of the language of 2-collapsing words. In: De Felice, C., Restivo, A. (eds.) DLT 2005. LNCS, vol. 3572, pp. 374–384. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  19. 19.
    Reilly, N.R., Zhang, S.: Decomposition of the lattice of pseudovarieties of finite semigroups induced by bands. Algebra Universalis 44, 217–239 (2000)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Sauer, N., Stone, M.G.: Composing functions to reduce image size. Ars Combinatoria 31, 171–176 (1991)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Dmitry S. Ananichev
    • 1
  • Ilja V. Petrov
    • 1
  • Mikhail V. Volkov
    • 1
  1. 1.Department of Mathematics and MechanicsUral State UniversityEkaterinburgRussia

Personalised recommendations