Advertisement

Operations on Weakly Recognizing Morphisms

  • Lukas Fleischer
  • Manfred Kufleitner
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9777)

Abstract

Weakly recognizing morphisms from free semigroups onto finite semigroups are a classical way for defining the class of \(\omega \)-regular languages, i.e., a set of infinite words is weakly recognizable by such a morphism if and only if it is accepted by some Büchi automaton. We consider the descriptional complexity of various constructions for weakly recognizing morphisms. This includes the conversion from and to Büchi automata, the conversion into strongly recognizing morphisms, and complementation. For some problems, we are able to give more precise bounds in the case of binary alphabets or simple semigroups.

Notes

Acknowledgments

We thank the anonymous referees for several useful suggestions which helped to improve the presentation of this paper.

References

  1. 1.
    Arnold, A.: A syntactic congruence for rational \(\omega \)-languages. Theoret. Comput. Sci. 39, 333–335 (1985)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Büchi, J.R.: Weak second-order arithmetic and finite automata. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 6, 66–92 (1960)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chrobak, M.: Finite automata and unary languages. Theoret. Comput. Sci. 47(2), 149–158 (1986)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Devadze, H.M.: Generating sets of the semigroup of all binary relations in a finite set. Doklady Akademii Nauk BSSR 12, 765–768 (1968)MathSciNetMATHGoogle Scholar
  5. 5.
    Fleischer, L., Kufleitner, M.: Efficient algorithms for morphisms over omega-regular languages. In: Proceedings of the FSTTCS 2015. LIPIcs, vol. 45, pp. 112–124. Dagstuhl Publishing (2015)Google Scholar
  6. 6.
    Holzer, M., König, B.: On deterministic finite automata and syntactic monoid size. Theoret. Comput. Sci. 327(3), 319–347 (2004)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Kim, K.H., Roush, F.W.: Two-generator semigroups of binary relations. J. Math. Psychol. 17(3), 236–246 (1978)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Konieczny, J.: A proof of Devadze’s theorem on generators of the semigroup of boolean matrices. Semigroup Forum 83(2), 281–288 (2011)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Pécuchet, J.: Variétés de semis groupes et mots infinis. In: Proceedings of the STACS 1986, pp. 180–191 (1986)Google Scholar
  10. 10.
    Perrin, D., Pin, J.-É.: Infinite Words. Pure and Applied Mathematics, vol. 141. Elsevier, Amsterdam (2004)MATHGoogle Scholar
  11. 11.
    Pin, J.-É.: Varieties of Formal Languages. North Oxford Academic, London (1986)CrossRefMATHGoogle Scholar
  12. 12.
    Sakoda, W.J., Sipser, M.: Nondeterminism and the size of two way finite automata. In: Proceedings of the STOC 1978, pp. 275–286. ACM Press (1978)Google Scholar
  13. 13.
    Thomas, W.: Automata on infinite objects. In: Handbook of Theoretical Computer Science, chap. 4, pp. 133–191. Elsevier (1990)Google Scholar
  14. 14.
    Yan, Q.: Lower bounds for complementation of omega-automata via the full automata technique. Logical Methods Comput. Sci. 4(1), 1–20 (2008)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2016

Authors and Affiliations

  1. 1.FMIUniversity of StuttgartStuttgartGermany

Personalised recommendations