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Languages of Dot-Depth 3/2

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Abstract

We show that level 3/2 of the dot-depth hierarchy is decidable. More precisely, we identify a pattern  \(\mathbb{B}\) such that the following holds: If F is a deterministic finite automaton that accepts L, then L belongs to level 3/2 of the dot-depth hierarchy if and only if F does not have \(\mathbb{B}\) as a subgraph in its transition graph. The latter condition can be tested in nondeterministic logarithmic space.

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References

  1. Arfi, M.: Polynomial operations on rational languages. In: Proceedings 4th Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science, vol. 247, pp. 198–206. Springer, New York (1987)

    Google Scholar 

  2. Arfi, M.: Opérations polynomiales et hiérarchies de concaténation. Theor. Comput. Sci. 91, 71–84 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  3. Brzozowski, J.A., Knast, R.: The dot-depth hierarchy of star-free languages is infinite. J. Comput. Syst. Sci. 16, 37–55 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cohen, R.S., Brzozowski, J.A.: Dot-depth of star-free events. J. Comput. Syst. Sci. 5, 1–16 (1971)

    MATH  MathSciNet  Google Scholar 

  5. Eilenberg, S.: Automata, Languages and Machines, vol. B. Academic, New York (1976)

    MATH  Google Scholar 

  6. Glaßer, C.: A normal form for classes of concatenation hierarchies. Technical Report 216, Inst. für Informatik, Univ. Würzburg (1998)

  7. Glaßer, C., Schmitz, H.: Decidable hierarchies of starfree languages. In: Proceedings 20th Conference on the Foundations of Software Technology and Theoretical Computer Science. Lecture Notes in Computer Science, vol. 1974, pp. 503–515. Springer, New York (2000)

    Google Scholar 

  8. Glaßer, C., Schmitz, H.: Languages of dot-depth 3/2. In: Proceedings 17th Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science, vol. 1770, pp. 555–566. Springer, New York (2000)

    Google Scholar 

  9. Glaßer, C., Schmitz, H.: Level 5/2 of the Straubing-Thérien-hierarchy for two-letter alphabets. In: Proceedings 5th International Conference on Developments in Language Theory. Lecture Notes in Computer Science, vol. 2295, pp. 251–261. Springer, New York (2001)

    Chapter  Google Scholar 

  10. Immerman, N.: Nondeterministic space is closed under complementation. SIAM J. Comput. 17, 935–938 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  11. McNaughton, R.: Algebraic decision procedures for local testablility. Math. Syst. Theor. 8, 60–76 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  12. McNaughton, R., Papert, S.: Counterfree Automata. MIT Press, Cambridge (1971)

    Google Scholar 

  13. Pin, J.-E.: Syntactic semigroups. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. I, pp. 679–746. Springer, New York (1996)

    Google Scholar 

  14. Pin, J.-E.: Bridges for concatenation hierarchies. In: Proceedings 25th International Colloquium on Automata, Languages and Programming. Lecture Notes in Computer Science, vol. 1443, pp. 431–442. Springer, New York (1998)

    Chapter  Google Scholar 

  15. Pin, J.-E., Weil, P.: Polynomial closure and unambiguous product. Theory Comput. Syst. 30, 383–422 (1997)

    MATH  MathSciNet  Google Scholar 

  16. Pin, J.E., Weil, P.: The wreath product principle for ordered semigroups. Commun. Algebra 30, 5677–5713 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  17. Schützenberger, M.P.: On finite monoids having only trivial subgroups. Inf. Control 8, 190–194 (1965)

    Article  MATH  Google Scholar 

  18. Simon, I.: Hierarchies of events with dot-depth one. Ph.D. thesis, University of Waterloo (1972)

  19. Straubing, H.: A generalization of the Schützenberger product of finite monoids. Theor. Comput. Sci. 13, 137–150 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  20. Straubing, H.: Finite semigroup varieties of the form V*D. J. Pure Appl. Algebra 36, 53–94 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  21. Straubing, H.: Semigroups and languages of dot-depth two. Theor. Comput. Sci. 58, 361–378 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  22. Szelepcsényi, R.: The method of forcing for nondeterministic automata. Bull. EATCS 33, 96–100 (1987)

    MATH  Google Scholar 

  23. Thérien, D.: Classification of finite monoids: the language approach. Theor. Comput. Sci. 14, 195–208 (1981)

    Article  MATH  Google Scholar 

  24. Thomas, W.: Classifying regular events in symbolic logic. J. Comput. Syst. Sci. 25, 360–376 (1982)

    Article  MATH  Google Scholar 

  25. Thomas, W.: An application of the Ehrenfeucht–Fraïssé game in formal language theory. Soc. Math. France Mém. 16(2), 11–21 (1984)

    MATH  Google Scholar 

  26. Wagner, K.W.: Leaf language classes. In: Proceedings International Conference on Machines, Computations, and Universality. Lecture Notes in Computer Science, vol. 3354. Springer, New York (2004)

    Google Scholar 

  27. Weil, P.: Algebraic recognizability of languages. In: Proceedings 29th Mathematical Foundations of Computer Science. Lecture Notes in Computer Science, vol. 3153, pp. 149–175. Springer, New York (2004)

    Google Scholar 

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Authors and Affiliations

  1. Theoretische Informatik, Universität Würzburg, 97074, Würzburg, Germany

    Christian Glaßer

  2. Theoretische Informatik, Fachhochschule Trier, 54208, Trier, Germany

    Heinz Schmitz

Authors
  1. Christian Glaßer
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  2. Heinz Schmitz
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Correspondence to Christian Glaßer.

Additional information

A preliminary version of this paper was presented at the conference STACS 2000.

C. Glaßer supported by Studienstiftung des Deutschen Volkes.

H. Schmitz supported by Deutsche Forschungsgemeinschaft (DFG), grant Wa 847/4-1.

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Glaßer, C., Schmitz, H. Languages of Dot-Depth 3/2. Theory Comput Syst 42, 256–286 (2008). https://doi.org/10.1007/s00224-007-9002-0

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