On Decidability of Intermediate Levels of Concatenation Hierarchies

  • Jorge Almeida
  • Jana Bartoňová
  • Ondřej Klíma
  • Michal KuncEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9168)


It is proved that if definability of regular languages in the \(\Sigma _n\) fragment of the first-order logic on finite words is decidable, then it is decidable also for the \(\Delta _{n+1}\) fragment. In particular, the decidability for \(\Delta _5\) is obtained. More generally, for every concatenation hierarchy of regular languages, it is proved that decidability of one of its half levels implies decidability of the intersection of the following half level with its complement.


Binary Relation Transitive Closure Regular Language Positive Variety Continuous Homomorphism 
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  1. 1.
    Almeida, J.: Some algorithmic problems for pseudovarieties. Publ. Math. Debrecen 54(Suppl), 531–552 (1999)MathSciNetGoogle Scholar
  2. 2.
    Almeida, J.: Profinite semigroups and applications. In: Kudryavtsev, V.B., Rosenberg, I.G. (eds.) Structural Theory of Automata, Semigroups and Universal Algebra, pp. 1–45. Springer (2005)Google Scholar
  3. 3.
    Almeida, J., Klíma, O.: New decidable upper bound of the second level in the Straubing-Thérien concatenation hierarchy of star-free languages. Discrete Math. Theor. Comput. Sci. 12, 41–58 (2010)Google Scholar
  4. 4.
    Branco, M.J.J., Pin, J.É.: Equations defining the polynomial closure of a lattice of regular languages. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part II. LNCS, vol. 5556, pp. 115–126. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  5. 5.
    Brzozowski, J.A., Cohen, R.S.: Dot-depth of star-free events. J. Comput. System Sci. 5, 1–15 (1971)MathSciNetCrossRefGoogle Scholar
  6. 6.
    McNaughton, R., Papert, S.: Counter-Free Automata. MIT Press (1971)Google Scholar
  7. 7.
    Pin, J.-É.: Syntactic semigroups. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, Chapter 10. Springer (1997)Google Scholar
  8. 8.
    Pin, J.É., Straubing, H., Thérien, D.: Locally trivial categories and unambiguous concatenation. J. Pure Appl. Algebra 52, 297–311 (1988)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Pin, J.É., Weil, P.: Profinite semigroups, Mal’cev products and identities. J. Algebra 182, 604–626 (1996)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Pin, J.É., Weil, P.: Polynomial closure and unambiguous product. Theory Comput. Systems 30, 383–422 (1997)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Place, T.: Separating regular languages with two quantifier alternations. In: Proc. LICS (2015), to appearGoogle Scholar
  12. 12.
    Place, T., Zeitoun, M.: Going higher in the first-order quantifier alternation hierarchy on words. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014, Part II. LNCS, vol. 8573, pp. 342–353. Springer, Heidelberg (2014) Google Scholar
  13. 13.
    Reiterman, J.: The Birkhoff theorem for finite algebras. Algebra Universalis 14, 1–10 (1982)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Rhodes, J., Steinberg, B.: The q-theory of Finite Semigroups. Springer (2009)Google Scholar
  15. 15.
    Schützenberger, M.-P.: On finite monoids having only trivial subgroups. Inform. and Control 8, 190–194 (1965)CrossRefGoogle Scholar
  16. 16.
    Straubing, H.: Finite semigroup varieties of the form \(V * D\). J. Pure Appl. Algebra 36, 53–94 (1985)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Thomas, W.: Classifying regular events in symbolic logic. J. Comput. System Sci. 25, 360–376 (1982)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Jorge Almeida
    • 1
  • Jana Bartoňová
    • 2
  • Ondřej Klíma
    • 2
  • Michal Kunc
    • 2
    Email author
  1. 1.CMUP, Dep. Matemática, Faculdade de CiênciasUniversidade do PortoPortoPortugal
  2. 2.Department of Mathematics and StatisticsMasaryk UniversityBrnoCzech Republic

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