Word Problems for 2-Homogeneous Monoids and Symmetric Logspace

  • Markus Lohrey
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2136)


We prove that the word problem for every monoid presented by a fixed 2-homogeneous semi-Thue system can be solved in log-space, which generalizes a result of Lipton and Zalcstein for free groups. The uniform word problem for the class of all 2-homogeneous semi-Thue systems is shown to be complete for symmetric log-space.


Binary Relation Word Problem Logarithmic Space Thue System Oracle Gate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S.I. Adjan. Defining relations and algorithmic problems for groups and semigroups, volume 85 of Proceedings of the Steklov Institute of Mathematics. American Mathematical Society, 1967.Google Scholar
  2. 2.
    C. Alvarez and R. Greenlaw. A compendium of problems complete for symmetric logarithmic space. Electronic Colloquium on Computational Complexity, Report No. TR96-039, 1996.Google Scholar
  3. 3.
    R. Armoni, A. Ta-Shma, A. Widgerson, and S. Zhou. An O(log(n)4/3) space algorithm for (s, t) connectivity in undirected graphs. Journal of the Association for Computing Machinery, 47(2):294–311, 2000.MATHGoogle Scholar
  4. 4.
    D. A. M. Barrington and J. Corbet. On the relative complexity of some languages in NC1. Information Processing Letters, 32:251–256, 1989.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    D. A. M. Barrington, N. Immerman, and H. Straubing. On uniformity within NC1. Journal of Computer and System Sciences, 41:274–306, 1990.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    R. Book and F. Otto. String-Rewriting Systems. Springer, 1993.Google Scholar
  7. 7.
    R. V. Book. Confluent and other types of Thue systems. Journal of the Association for Computing Machinery, 29(1):171–182, 1982.MATHMathSciNetGoogle Scholar
  8. 8.
    R. V. Book. Homogeneous Thue systems and the Church-Rosser property. Discrete Mathematics, 48:137–145, 1984.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    R. V. Book, M. Jantzen, B. Monien, C. P. O’Dunlaing, and C. Wrathall. On the complexity of word problems in certain Thue systems. In J. Gruska and M. Chytil, editors, Proceedings of the 10rd Mathematical Foundations of Computer Science (MFCS’81), Strbské Pleso (Czechoslovakia), number 118 in Lecture Notes in Computer Science, pages 216–223. Springer, 1981.Google Scholar
  10. 10.
    S. R. Buss. The Boolean formula value problem is in ALOGTIME. In Proceedings of the 19th Annual Symposium on Theory of Computing (STOC 87), pages 123–131. ACM Press, 1987.Google Scholar
  11. 11.
    S. A. Cook. A taxonomy of problems with fast parallel algorithms. Information and Control, 64:2–22, 1985.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    N. Immerman. Languages that capture complexity classes. SIAM Journal on Computing, 16(4):760–778, 1987.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    M. Jantzen. Confluent string rewriting. In EATCS Monographs on theoretical computer science, volume 14. Springer, 1988.Google Scholar
  14. 14.
    H. R. Lewis and C. H. Papadimitriou. Symmetric space-bounded computation. Theoretical Computer Science, 19(2):161–187, 1982.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    R. J. Lipton and Y. Zalcstein. Word problems solvable in logspace. Journal of the Association for Computing Machinery, 24(3):522–526, 1977.MATHMathSciNetGoogle Scholar
  16. 16.
    M. Lohrey. Word problems and confluence problems for restricted semi-Thue systems. In L. Bachmair, editor, Proceedings of the 11th International Conference on Rewrite Techniques and Applications (RTA 2000), Norwich (UK), number 1833 in Lecture Notes in Computer Science, pages 172–186. Springer, 2000.Google Scholar
  17. 17.
    R. C. Lyndon and P. E. Schupp. Combinatorial Group Theory. Springer, 1977.Google Scholar
  18. 18.
    A. Markov. On the impossibility of certain algorithms in the theory of associative systems. Doklady Akademii Nauk SSSR, 55,58:587–590, 353–356, 1947.Google Scholar
  19. 19.
    N. Nisan and A. Ta-Shma. Symmetric logspace is closed under complement. Chicago Journal of Theoretical Computer Science, 1995.Google Scholar
  20. 20.
    F. Otto and L. Zhang. Decision problems for finite special string-rewriting systems that are confluent on some congruence class. Acta Informatica, 28:477–510, 1991.MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    C. H. Papadimitriou. Computational Complexity. Addison Wesley, 1994.Google Scholar
  22. 22.
    E. Post. Recursive unsolvability of a problem of Thue. Journal of Symbolic Logic, 12(1):1–11, 1947.CrossRefMathSciNetGoogle Scholar
  23. 23.
    D. Robinson. Parallel Algorithms for Group Word Problems. PhD thesis, University of California, San Diego, 1993.Google Scholar
  24. 24.
    W. L. Ruzzo. On uniform circuit complexity. Journal of Computer and System Sciences, 22:365–383, 1981.MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    I. Stewart. Complete problems for symmetric logspace involving free groups. Information Processing Letters, 40:263–267, 1991.MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    I. Stewart. Refining known results on the generalized word problem for free groups. International Journal of Algebra and Computation, 2:221–236, 1992.MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    H. Vollmer. Introduction to Circuit Complexity. Springer, 1999.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Markus Lohrey
    • 1
  1. 1.Institut für InformatikUniversität StuttgartStuttgartGermany

Personalised recommendations