From Algebra to Logic: There and Back Again The Story of a Hierarchy

(Invited Paper)
  • Pascal Weil
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8633)


This is a survey about a collection of results about a (double) hierarchy of classes of regular languages, which occurs in a natural fashion in a number of contexts. One of these occurrences is given by an alternated sequence of deterministic and co-deterministic closure operations, starting with the piecewise testable languages. Since these closure operations preserve varieties of languages, this defines a hierarchy of varieties, and through Eilenberg’s variety theorem, a hierarchy of pseudo-varieties (classes of finite monoids that are defined by pesudo-identities). The point of this excursion through algebra is that it provides reasonably simple decision algorithms for the membership problem in the corresponding varieties of languages. Another interesting point is that the hierarchy of pseudo-varieties bears a formal resemblance with another hierarchy, the hierarchy of varieties of idempotent monoids, which was much studied in the 1970s and 1980s and is by now well understood. This resemblance provides keys to a combinatorial characterization of the different levels of our hierarchies, which turn out to be closely related with the so-called rankers, a specification mechanism which was introduced to investigate the two-variable fragment of the first-order theory of the linear order. And indeed the union of the varieties of languages which we consider coincides with the languages that can be defined in that fragment. Moreover, the quantifier alternation hierarchy within that logical fragment is exactly captured by our hierarchy of languages, thus establishing the decidability of the alternation hierarchy.

There are other combinatorial and algebraic approaches of the same logical hierarchy, and one recently introduced by Krebs and Straubing also establishes decidability. Yet the algebraic operations involved are seemingly very different, an intriguing problem…


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Almeida, J.: Finite Semigroups and Universal Algebra. World Scientific, Singapore (1994)zbMATHGoogle Scholar
  2. 2.
    Gerhard, J.: The lattice of equational classes of idempotent semigroups. Journal of Algebra 15, 195–224 (1970)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Gerhard, J., Petrich, M.: Varieties of bands revisited. Proceedings of the London Mathematical Society 58(3), 323–350 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Krebs, A., Straubing, H.: An effective characterization of the alternation hierarchy in two-variable logic. In: FSTTCS 2012. LIPIcs, vol. 18, pp. 86–98. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl (2012), doi:10.4230/LIPIcs.FSTTCS.2012.86Google Scholar
  5. 5.
    Krohn, K., Rhodes, J., Tilson, B.: Homomorphisms and semilocal theory. In: Arbib, M. (ed.) The Algebraic Theory of Machines, Languages and Semigroups. Academic Press (1965)Google Scholar
  6. 6.
    Kufleitner, M., Weil, P.: On the lattice of sub-pseudovarieties of DA. Semigroup Forum 81, 243–254 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Kufleitner, M., Weil, P.: The FO 2 alternation hierarchy is decidable. In: CSL 2012. LIPIcs, vol. 16, pp. 426–439. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl (2012), doi:10.4230/LIPIcs.CSL.2012.426Google Scholar
  8. 8.
    Kufleitner, M., Weil, P.: On logical hierarchies within FO 2-definable languages. Logical Methods in Computer Science 8(3:11), 1–30 (2012)MathSciNetGoogle Scholar
  9. 9.
    Pin, J.E.: Propriétés syntactiques du produit non ambigu. In: de Bakker, J.W., van Leeuwen, J. (eds.) ICALP 1980. LNCS, vol. 85, pp. 483–499. Springer, Heidelberg (1980)CrossRefGoogle Scholar
  10. 10.
    Pin, J.É.: Varieties of Formal Languages. North Oxford Academic, London (1986)CrossRefzbMATHGoogle Scholar
  11. 11.
    Rhodes, J., Steinberg, B.: The \(\mathfrak{q}\)-theory of finite semigroups. Springer Monographs in Mathematics. Springer, New York (2009)CrossRefGoogle Scholar
  12. 12.
    Schützenberger, M.P.: Sur le produit de concaténation non ambigu. Semigroup Forum 13, 47–75 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Schwentick, T., Thérien, D., Vollmer, H.: Partially-ordered two-way automata: A new characterization of DA. In: Kuich, W., Rozenberg, G., Salomaa, A. (eds.) DLT 2001. LNCS, vol. 2295, pp. 239–250. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  14. 14.
    Simon, I.: Piecewise testable events. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 214–222. Springer, Heidelberg (1975)Google Scholar
  15. 15.
    Straubing, H.: Finite automata, formal logic, and circuit complexity. Birkhäuser, Boston (1994)CrossRefzbMATHGoogle Scholar
  16. 16.
    Straubing, H.: Algebraic Characterization of the Alternation Hierarchy in FO 2[ < ] on Finite Words. In: CSL 2011. LIPIcs, vol. 12, pp. 525–537. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl (2011), doi:10.4230/LIPIcs.CSL.2011.525Google Scholar
  17. 17.
    Weil, P.: Some results on the dot-depth hierarchy. Semigroup Forum 46, 352–370 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Weis, P., Immerman, N.: Structure theorem and strict alternation hierarchy for FO2 on words. Logical Methods in Computer Science 5, 1–23 (2009)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Pascal Weil
    • 1
    • 2
  1. 1.CNRS, LaBRI, UMR 5800TalenceFrance
  2. 2.LaBRI, UMR 5800Univ. BordeauxTalenceFrance

Personalised recommendations