Advertisement

An Algebraic Characterization of Strictly Piecewise Languages

  • Jie Fu
  • Jeffrey Heinz
  • Herbert G. Tanner
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6648)

Abstract

This paper provides an algebraic characterization of the Strictly Piecewise class of languages studied by Rogers et al. 2010. These language are a natural subclass of the Piecewise Testable languages (Simon 1975) and are relevant to natural language. The algebraic characterization highlights a similarity between the Strictly Piecewise and Strictly Local languages, and also leads to a procedure which can decide whether a regular language L is Strictly Piecewise in polynomial time in the size of the syntactic monoid for L.

Keywords

Regular Language Sink State Free Semigroup Transformation Semigroup Algebraic Characterization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Anderson, J.A.: Automata Theory with Modern Applications. Cambridge University Press, Cambridge (2006)CrossRefzbMATHGoogle Scholar
  2. 2.
    Clifford, A.: The Algebraic Theory of Semigroups. American Mathematical Society, Providence (1967)zbMATHGoogle Scholar
  3. 3.
    García, P., Ruiz, J.: Learning k-testable and k-piecewise testable languages from positive data. Grammars 7, 125–140 (2004)Google Scholar
  4. 4.
    Green, J.A.: On the structure of semigroups. The Annals of Mathematics 54(1), 163–172 (1951)CrossRefzbMATHGoogle Scholar
  5. 5.
    Haines, L.H.: On free moniods partially ordered by embedding. Journal of Combinatorial Theory 6, 94–98 (1969)CrossRefzbMATHGoogle Scholar
  6. 6.
    Heinz, J.: Learning long-distance phonotactics. Linguistic Inquiry 41(4), 623–661 (2010)CrossRefGoogle Scholar
  7. 7.
    Higman, G.: Ordering by divisibility in abstract algebras. Proceedings of the London Mathematical Society 3(2), 326–336 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Holzer, M., König, B.: Regular languages, sizes of syntactic monoids, graph colouring, state complexity results, and how these topics are related to each other. EATCS Bulletin 83, 139–155 (2004)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Hopcroft, J.E.: An n log n algorithm for minimizing states in a finite automaton. Tech. rep., Stanford, CA, USA (1971)Google Scholar
  10. 10.
    McNaughton, R., Papert, S.: Counter-Free Automata. MIT Press, Cambridge (1971)zbMATHGoogle Scholar
  11. 11.
    Pin, J.É.: Syntactic Semigroups. In: Rozenberg, G., Salomaa, A. (eds.), vol. 1, Springer, Heidelberg (1997)Google Scholar
  12. 12.
    Rogers, J., Heinz, J., Bailey, G., Edlefsen, M., Visscher, M., Wellcome, D., Wibel, S.: On languages piecewise testable in the strict sense. In: Ebert, C., Jäger, G., Michaelis, J. (eds.) MOL 10. LNCS, vol. 6149, pp. 255–265. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  13. 13.
    Simon, I.: Piecewise testable events. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 214–222. Springer, Heidelberg (1975)Google Scholar
  14. 14.
    Thierrin, G.: Convex languages. In: ICALP 1972. pp. 481–492 (1972)Google Scholar
  15. 15.
    Watanabe, T., Nakamura, A.: On the transformation semigroups of finite automata. Journal of Computer and System Sciences 26(1), 107–138 (1983)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Jie Fu
    • 1
  • Jeffrey Heinz
    • 1
  • Herbert G. Tanner
    • 1
  1. 1.University of DelawareUSA

Personalised recommendations