Alternation Hierarchies of First Order Logic with Regular Predicates

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9210)


We investigate the decidability of the definability problem for fragments of first order logic over finite words enriched with regular numerical predicates. In this paper, we focus on the quantifier alternation hierarchies of first order logic. We obtain that deciding this problem for each level of the alternation hierarchy of both first order logic and its two-variable fragment when equipped with all regular numerical predicates is not harder than deciding it for the corresponding level equipped with only the linear order.

Relying on some recent results, this proves the decidability for each level of the alternation hierarchy of the two-variable first order fragment while in the case of the first order logic the question remains open for levels greater than two.

The main ingredients of the proofs are syntactic transformations of first-order formulas as well as the infinitely testable property, a new algebraic notion on varieties that we define.


  1. 1.
    Almeida, J.: A syntactical proof of locality of DA. Internat. J. Algebra Comput. 6(2), 165–177 (1996)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Barrington, D.A.M., Compton, K., Straubing, H., Thérien, D.: Regular languages in \({\rm{NC}}^1\). J. Comput. System Sci. 44(3), 478–499 (1992)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Büchi, J.R.: Weak second-order arithmetic and finite automata. Z. Math. Logik Grundlagen Math. 6, 66–92 (1960)Google Scholar
  4. 4.
    Chaubard, L., Pin, J.-É., Straubing, H.: First order formulas with modular predicates. In: LICS, pp. 211–220. IEEE (2006)Google Scholar
  5. 5.
    Cohen, R.S., Brzozowski, J.A.: Dot-depth of star-free events. J. Comput. Syst. Sci. 5(1), 1–16 (1971)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Dartois, L., Paperman, C.: Two-variable first order logic with modular predicates over words. In: Portier, N., Wilke, T. (eds.) STACS, pp. 329–340. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Dagstuhl (2013)Google Scholar
  7. 7.
    Diekert, V., Gastin, P., Kufleitner, M.: A survey on small fragments of first-order logic over finite words. Internat. J. Found. Comput. Sci. 19(3), 513–548 (2008)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Ésik, Z., Ito, M.: Temporal logic with cyclic counting and the degree of aperiodicity of finite automata. Acta Cybernet. 16(1), 1–28 (2003)MathSciNetMATHGoogle Scholar
  9. 9.
    Knast, R.: A semigroup characterization of dot-depth one languages. RAIRO Inform. Théor. 17(4), 321–330 (1983)MathSciNetMATHGoogle Scholar
  10. 10.
    Krebs, A., Straubing, H.: An effective characterization of the alternation hierarchy in two-variable logic. In: FSTTCS, pp. 86–98 (2012)Google Scholar
  11. 11.
    Kufleitner, M., Lauser, A.: Quantifier alternation in two-variable first-order logic with successor is decidable. In: STACS, pp. 305–316 (2013)Google Scholar
  12. 12.
    Kufleitner, M., Weil, P.: The \({\rm {FO}}^2\) alternation hierarchy is decidable. In: Computer Science Logic 2012, Volume 16 of LIPIcs. Leibniz Int. Proc. Inform., pp. 426–439. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern (2012)Google Scholar
  13. 13.
    McNaughton, R., Papert, S.: Counter-free Automata. The M.I.T. Press, Cambridge (1971)MATHGoogle Scholar
  14. 14.
    Nerode, A.: Linear automaton transformation. Proc. AMS 9, 541–544 (1958)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Péladeau, P.: Logically defined subsets of \({ N}^k\). Theoret. Comput. Sci. 93(2), 169–183 (1992)CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Perrin, D., Pin, J.É.: First-order logic and star-free sets. J. Comput. System Sci. 32(3), 393–406 (1986)CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Pin, J.-É.: Syntactic semigroups. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 1, pp. 679–746. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  18. 18.
    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
  19. 19.
    Schützenberger, M.P.: On finite monoids having only trivial subgroups. Inf. Control 8, 190–194 (1965)CrossRefMATHGoogle Scholar
  20. 20.
    Simon, I.: Piecewise testable events. In: Brakhage, H. (ed.) Automata Theory and Formal Languages (Second GI Conference Kaiserslautern, 1975). LNCS, vol. 33, pp. 214–222. Springer, Heidelberg (1975)CrossRefGoogle Scholar
  21. 21.
    Straubing, H.: A generalization of the Schützenberger product of finite monoids. Theoret. Comput. Sci. 13(2), 137–150 (1981)CrossRefMathSciNetMATHGoogle Scholar
  22. 22.
    Straubing, H.: Finite semigroup varieties of the form \(V\ast D\). J. Pure Appl. Algebra 36(1), 53–94 (1985)CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    Straubing, H.: Finite Automata, Formal Logic, and Circuit Complexity. Birkhäuser Boston Inc., Boston (1994)CrossRefMATHGoogle Scholar
  24. 24.
    Thérien, D.: Classification of finite monoids: the language approach. Theoret. Comput. Sci. 14(2), 195–208 (1981)CrossRefMathSciNetMATHGoogle Scholar
  25. 25.
    Thérien, D., Wilke, T.: Over words, two variables are as powerful as one quantifier alternation. In: STOC 1998 (Dallas. TX), pp. 234–240. ACM, New York (1999)Google Scholar
  26. 26.
    Thomas, W.: Classifying regular events in symbolic logic. J. Comput. System Sci. 25(3), 360–376 (1982)CrossRefMathSciNetMATHGoogle Scholar
  27. 27.
    Tilson, B.: Categories as algebra: an essential ingredient in the theory of monoids. J. Pure Appl. Algebra 48(1–2), 83–198 (1987)CrossRefMathSciNetMATHGoogle Scholar
  28. 28.
    Weis, P., Immerman, N.: Structure theorem and strict alternation hierarchy for \({\rm {FO}}^2\) on words. Log. Methods Comput. Sci. 5(3:3:4), 23 (2009)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.University of WarsawWarsawPoland
  2. 2.École Centrale MarseilleMarseilleFrance
  3. 3.LIFUMR7279, Aix-Marseille Université and CNRSMarseilleFrance

Personalised recommendations