Theory of Computing Systems

, Volume 62, Issue 3, pp 682–738 | Cite as

The Word Problem for Omega-Terms over the Trotter-Weil Hierarchy

  • Manfred Kufleitner
  • Jan Philipp WächterEmail author


For two given ω-terms α and β, the word problem for ω-terms over a variety V asks whether α = β in all monoids in V. We show that the word problem for ω-terms over each level of the Trotter-Weil Hierarchy is decidable. More precisely, for every fixed variety in the Trotter-Weil Hierarchy, our approach yields an algorithm in nondeterministic logarithmic space (NL). In addition, we provide deterministic polynomial time algorithms which are more efficient than straightforward translations of the NL-algorithms. As an application of our results, we show that separability by the so-called corners of the Trotter-Weil Hierarchy is witnessed by ω-terms (this property is also known as ω-reducibility). In particular, the separation problem for the corners of the Trotter-Weil Hierarchy is decidable.


Regular language Finite monoid Pseudoidentity Omega-term Separation problem Trotter-Weil Hierarchy FO2 alternation hierarchy 


  1. 1.
    Almeida, J.: Implicit operations on finite \(\mathcal {J}\)-trivial semigroups and a conjecture of I. Simon. J. Pure Applied Algebra 69(3), 205–218 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Almeida, J.: Finite Semigroups and Universal Algebra. World Scientific (1994)Google Scholar
  3. 3.
    Almeida, J.: Finite semigroups: an introduction to a unified theory of pseudovarieties. In: dos Gomes Moreira da Cunha, G.M., da Silva, P.V.A., Pin, J.É. (eds.) Semigroups, Algorithms, Automata and Languages, pp. 3–64. World Scientific (2002)Google Scholar
  4. 4.
    Almeida, J., Zeitoun, M.: An automata-theoretic approach to the word problem for ω-terms over R. Theor. Comput. Sci. 370(1), 131–169 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Diekert, V., Gastin, P., Kufleitner, M.: A survey on small fragments of first-order logic over finite words. Int. J. Found. Comput. Sci. 19, 513–548 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Eilenberg, S.: Automata, Languages, and Machines, vol. B. Academic press (1976)Google Scholar
  7. 7.
    Gerhard, J., Petrich, M.: Varieties of bands revisited. Proc. Lond. Math. Soc. 58(3), 323–350 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hall, T., Weil, P.: On radical congruence systems. Semigroup Forum 59(1), 56–73 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hopcroft, J., Karp, R.: A linear algorithm for testing equivalence of finite automata. Cornell University, Tech. rep. (1971)Google Scholar
  10. 10.
    Huschenbett, M., Kufleitner, M.: Ehrenfeucht-Fraïssé games on omega-terms. In: Mayr, E.W., Portier, N. (eds.) STACS 2014, Proceedings, LIPIcs, vol. 25, pp 374–385. Dagstuhl Publishing, Dagstuhl, Germany (2014)Google Scholar
  11. 11.
    Krohn, K., Rhodes, J.L., Tilson, B.: Homomorphisms and semilocal theory. In: Arbib, M.A. (ed.) Algebraic Theory of Machines, Languages, and Semigroups, chap. 8, pp 191–231. Academic Press, New York and London (1968)Google Scholar
  12. 12.
    Kufleitner, M., Lauser, A., Widmayer, P.: The join levels of the Trotter-Weil Hierarchy are decidable. In: Rovan, B., Sassone, V. (eds.) MFCS 2012, Proceedings, LNCS, vol. 7464, pp. 603–614. Springer (2012)Google Scholar
  13. 13.
    Kufleitner, M., Weil, P.: On the lattice of sub-pseudovarieties of DA. Semigroup Forum 81, 243–254 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kufleitner, M., Weil, P.: The F O 2 alternation hierarchy is decidable. In: Cégielski, P., Durand, A. (eds.) CSL 2012, Proceedings, LIPIcs, vol. 16, pp. 426–439. Dagstuhl Publishing, Dagstuhl, Germany (2012)Google Scholar
  15. 15.
    Kufleitner, M., Weil, P.: On logical hierarchies within F O 2-definable languages. Logical Methods Comput. Sci. 8(3), 1–30 (2012)CrossRefzbMATHGoogle Scholar
  16. 16.
    Lodaya, K., Pandya, P.K., Shah, S.S.: Marking the chops: an unambiguous temporal logic. In: IFIP TCS 2008, Proceedings, IFIP, pp. 461–476. Springer (2008)Google Scholar
  17. 17.
    McCammond, J.P.: Normal forms for free aperiodic semigroups. Int. J. Algebra Comput. 11(5), 581–625 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Moura, A.: The word problem for ω-terms over DA. Theor. Comput. Sci. 412(46), 6556–6569 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Perrin, D., Pin, J.É.: Infinite words, Pure and Applied Mathematics, vol. 141. Elsevier, Amsterdam (2004)Google Scholar
  20. 20.
    Pin, J.É.: Varieties of Formal Languages. North Oxford Academic Publishers Ltd, Oxford (1986)CrossRefzbMATHGoogle Scholar
  21. 21.
    Place, T.H., van Rooijen, L., Zeitoun, M.: Separating regular languages by piecewise testable and unambiguous languages MFCS 2013, Proceedings, pp. 729–740. Springer (2013)Google Scholar
  22. 22.
    Schützenberger, M.: On finite monoids having only trivial subgroups. Inf. Control. 8, 190–194 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Schwentick, T.H., Thérien, D., Vollmer, H.: Partially-ordered two-way automata: A new characterization of DA. In: Kuich, W., Rozenberg, G., Salomaa, A. (eds.) DLT 2001, Proceedings, LNCS, vol. 2295, pp. 239–250. Springer (2002)Google Scholar
  24. 24.
    Simon, I.: Piecewise testable events. In: Automata Theory and Formal Languages, 2nd GI Conference ., LNCS, vol. 33, pp. 214–222. Springer (1975)Google Scholar
  25. 25.
    Simon, I.: Factorization forests of finite height. Theor. Comput. Sci. 72(1), 65–94 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Tesson, P., Thérien, D.: Diamonds are forever: The variety DA. In: dos Gomes Moreira da Cunha, G.M., da Silva, P.V.A., Pin, J.É. (eds.) Semigroups, Algorithms, Automata and Languages, pp. 475–500. World Scientific (2002)Google Scholar
  27. 27.
    Thérien, D., Wilke, T.H.: Over words, two variables are as powerful as one quantifier alternation. In: Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, pp. 234–240. ACM (1998)Google Scholar
  28. 28.
    Trotter, P., Weil, P.: The lattice of pseudovarieties of idempotent semigroups and a non-regular analogue. Algebra Univers. 37(4), 491–526 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    van Rooijen, L., Zeitoun, M.: The separation problem for regular languages by piecewise testable languages. arXiv:1303.2143 (2013)
  30. 30.
    Weis, Ph., Immerman, N.: Structure Theorem and Strict Alternation Hierarchy for F O 2 on Words. Logical Methods in Computer Science 5(3), 1–23 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Institut für Formale Methoden der InformatikUniversity of StuttgartStuttgartGermany

Personalised recommendations