Theory of Computing Systems

, Volume 61, Issue 2, pp 352–370 | Cite as

The Half-Levels of the FO2 Alternation Hierarchy

  • Lukas Fleischer
  • Manfred KufleitnerEmail author
  • Alexander Lauser


The alternation hierarchy in two-variable first-order logic FO2[<] over words was shown to be decidable by Kufleitner and Weil, and independently by Krebs and Straubing. We consider a similar hierarchy, reminiscent of the half levels of the dot-depth hierarchy or the Straubing-Thérien hierarchy. The fragment \({{\Sigma }^{2}_{m}}\) of FO2 is defined by disallowing universal quantifiers and having at most m−1 nested negations. The Boolean closure of \({{\Sigma }^{2}_{m}}\) yields the m th level of the FO2-alternation hierarchy. We give an effective characterization of \({{\Sigma }^{2}_{m}}\), i.e., for every integer m one can decide whether a given regular language is definable in \({{\Sigma }^{2}_{m}}\). Among other techniques, the proof relies on an extension of block products to ordered monoids.


Regular language Finite monoid Positive variety First-order logic 



This work was supported by the German Research Foundation (DFG) under grant DI 435/5-2.


  1. 1.
    Almeida, J., Weil, P.: Profinite categories and semidirect products. J. Pure Appl. Algebra 123(1–3), 1–50 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Büchi, J.R.: Weak second-order arithmetic and finite automata. Z. Math. Logik Grundlagen Math. 6, 66–92 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cohen, R.S., Brzozowski, J.A.: Dot-depth of star-free events. J. Comput. Syst. Sci. 5(1), 1–16 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Diekert, V., Gastin, P., Kufleitner, M.: A survey on small fragments of first-order logic over finite words. Int. J. Found. Comput. Sci. 19(3), 513–548 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ebbinghaus, H.-D., Flum, J.: Finite Model Theory. Perspectives in Mathematical Logic. Springer (1995)Google Scholar
  6. 6.
    Eilenberg, S.: Automata, Languages, and Machines, vol. B. Academic Press, New York and London (1976)Google Scholar
  7. 7.
    Elgot, C.C.: Decision problems of finite automata design and related arithmetics. Trans. Amer. Math. Soc. 98, 21–51 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fleischer, L., Kufleitner, M., Lauser, A.: Block products and nesting negations in FO 2. In: Proceedings of CSR 2014 of LNCS, vol. 8476, p 2014. SpringerGoogle Scholar
  9. 9.
    Glaßer, Ch., Schmitz, H.: Languages of dot-depth 3/2. Theory Comput. Syst. 42(2), 256–286 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kamp, J.A.W.: Tense Logic and the Theory of Linear Order. PhD thesis, University of California (1968)Google Scholar
  11. 11.
    Knast, R.: A semigroup characterization of dot-depth one languages. RAIRO, Inf. Théor. 17(4), 321–330 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Krebs, A., Straubing, H.: An effective characterization of the alternation hierarchy in two-variable logic. In: Proceedings of FSTTCS 2012 of LIPIcs, vol. 18, p 2012. Dagstuhl PublishingGoogle Scholar
  13. 13.
    Kufleitner, M., Lauser, A.: Languages of dot-depth one over infinite words. In: Proceedings of LICS 2011, pp 23–32. IEEE Computer Society (2011)Google Scholar
  14. 14.
    Kufleitner, M., Lauser, A.: Around dot-depth one. Int. J. Comput. Sci. 23 (6), 1323–1339 (2012)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Kufleitner, M., Lauser, A.: The join levels of the Trotter-Weil hierarchy are decidable. In: Proceedings of MFCS 2012 of LNCS, vol. 7464, pp 603–614. Springer (2012)Google Scholar
  16. 16.
    Kufleitner, M., Lauser, A.: The join of the varieties of R-trivial and L-trivial monoids via combinatorics on words. Discrete Math. Theor. Comput. Sci. 14(1), 141–146 (2012)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Kufleitner, M., Lauser, A.: Quantifier alternation in two-variable first-order logic with successor is decidable. In: Proceedings of STACS 2013 of LIPIcs, vol. 20, p 2013. Dagstuhl PublishingGoogle Scholar
  18. 18.
    Kufleitner, M., Weil, P.: The FO2 alternation hierarchy is decidable. In: Proceedings of CSL 2012 of LIPIcs, vol. 16, pp 426–439. Dagstuhl Publishing (2012)Google Scholar
  19. 19.
    McNaughton, R., Papert, S.: Counter-Free Automata. The MIT Press (1971)Google Scholar
  20. 20.
    Pin, J.-É.: A variety theorem without complementation. In: Russian Mathematics (Iz. VUZ), vol. 39, pp 80–90 (1995)Google Scholar
  21. 21.
    Pin, J.-É., Weil, P.: Polynomial closure and unambiguous product. Theory Comput. Syst. 30(4), 383–422 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Pin, J.-É., Weil, P.: Semidirect products of ordered semigroups. Commun. Algebra 30(1), 149–169 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Pin, J.-É., Weil, P.: The wreath product principle for ordered semigroups. Commun. Algebra 30(12), 5677–5713 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Place, Th., Zeitoun, M.: Going higher in the first-order quantifier alternation hierarchy on words. In: Proceedings of ICALP 2014 of LNCS, vol. 8573, pp 342–353. Springer (2014)Google Scholar
  25. 25.
    Rhodes, J., Tilson, B.: The kernel of monoid morphisms. J. Pure Appl. Algebra 62(3), 227–268 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Rhodes, J., Weil, P.: Decomposition techniques for finite semigroups, using categories II. J. Pure Appl. Algebra 62(3), 285–312 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Schützenberger, M.P.: On finite monoids having only trivial subgroups. Inf. Control 8, 190–194 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Schwentick, Th., Thérien, D., Vollmer, H.: Partially-ordered two-way automata: A new characterization of DA. In: Proceedings of DLT 2001 of LNCS, vol. 2295, pp 239–250. Springer (2002)Google Scholar
  29. 29.
    Simon, I.: Piecewise testable events. In: 2nd GI Conference of Automation, Theoretical Formation Languages of LNCS, vol. 33, pp 214–222. Springer (1975)Google Scholar
  30. 30.
    Straubing, H.: Families of recognizable sets corresponding to certain varieties of finite monoids. J. Pure Appl. Algebra 15(3), 305–318 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Straubing, H.: Aperiodic homomorphisms and the concatenation product of recognizable sets. J. Pure Appl. Algebra 15(3), 319–327 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Straubing, H.: A generalization of the Schützenberger product of finite monoids. Theor. Comput. Sci. 13, 137–150 (1981)CrossRefzbMATHGoogle Scholar
  33. 33.
    Straubing, H.: Finite semigroup varieties of the form VD. J. Pure Appl. Algebra 36(1), 53–94 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Straubing, H.: Finite Automata, Formal Logic, and Circuit Complexity. Birkhäuser (1994)Google Scholar
  35. 35.
    Straubing, H.: Algebraic characterization of the alternation hierarchy in FO2[<] on finite words. In: Proceedings of CSL 2011 of LIPIcs, vol. 12, pp 525–537. Dagstuhl Publishing (2011)Google Scholar
  36. 36.
    Tesson, P., Thérien, D.: Diamonds are forever: The variety DA. In: Proceedings of Semigroups, Algorithms, Automata and Languages 2001, pp 475–500. World Scientific (2002)Google Scholar
  37. 37.
    Thérien, D.: Classification of finite monoids: The language approach. Theor. Comput. Sci. 14(2), 195–208 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Thérien, D.: Two-sided wreath product of categories. J. Pure Appl. Algebra 74(3), 307–315 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Thérien, D., Wilke, Th.: Over words, two variables are as powerful as one quantifier alternation. In: Proceedings of STOC 1998, pp 234–240. ACM Press (1998)Google Scholar
  40. 40.
    Thomas, W.: Classifying regular events in symbolic logic. J. Comput. Syst. Sci. 25, 360–376 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Trakhtenbrot, B.A.: Finite automata and logic of monadic predicates. Dokl. Akad. Nauk SSSR 140, 326–329 (1961). in RussianGoogle Scholar
  42. 42.
    Weil, P.: Closure of varieties of languages under products with counter. J. Comput. Syst. Sci. 45(3), 316–339 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Weis, Ph., Immerman, N.: Structure theorem and strict alternation hierarchy for FO2 on words. Log. Methods Comput. Sci. 5(3), 1–23 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Lukas Fleischer
    • 1
  • Manfred Kufleitner
    • 1
    Email author
  • Alexander Lauser
    • 1
  1. 1.FMIUniversity of StuttgartStuttgartGermany

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