Theory of Computing Systems

, Volume 49, Issue 2, pp 283–305 | Cite as

Compressed Word Problems in HNN-extensions and Amalgamated Products



It is shown that the compressed word problem for an HNN-extension 〈H,tt −1 at=ϕ(a) (aA)〉 with A finite is polynomial time Turing-reducible to the compressed word problem for the base group H. An analogous result for amalgamated free products is shown as well.


Algorithms for compressed strings Straight-line programs Word problems for groups HNN-extensions 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bertoni, A., Choffrut, C., Radicioni, R.: Literal shuffle of compressed words. In: Proceeding of the 5th IFIP International Conference on Theoretical Computer Science (IFIP TCS 2008), Milan (Italy), pp. 87–100. Springer, Berlin (2008) CrossRefGoogle Scholar
  2. 2.
    Boone, W.W.: The word problem. Ann. Math. (2) 70, 207–265 (1959) MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dehn, M.: Über die Toplogie des dreidimensionalen Raumes. Math. Ann. 69, 137–168 (1910). In German MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Dicks, W., Dunwoody, M.J.: Groups Acting on Graphs. Cambridge University Press, Cambridge (1989) MATHGoogle Scholar
  5. 5.
    Gasieniec, L., Karpinski, M., Plandowski, W., Rytter, W.: Efficient algorithms for Lempel-Ziv encoding (extended abstract). In: Karlsson, R.G., Lingas, A. (eds.) Proceedings of the 5th Scandinavian Workshop on Algorithm Theory (SWAT 1996), Reykjavík (Iceland). Lecture Notes in Computer Science, vol. 1097, pp. 392–403. Springer, Berlin (1996) Google Scholar
  6. 6.
    Hagenah, C.: Gleichungen mit regulären Randbedingungen über freien Gruppen. PhD thesis, University of Stuttgart, Institut für Informatik (2000) Google Scholar
  7. 7.
    Higman, G., Neumann, B.H., Neumann, H.: Embedding theorems for groups. J. Lond. Math. Soc. (2) 24, 247–254 (1949) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Howie, J.M.: Embedding theorems for semigroups. Q. J. Math. Oxford (2) 14, 254–258 (1963) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Kapovich, I., Myasnikov, A., Schupp, P., Shpilrain, V.: Generic-case complexity, decision problems in group theory, and random walks. J. Algebra 264(2), 665–694 (2003) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Lifshits, Y.: Processing compressed texts: A tractability border. In: Ma, B., Zhang, K. (eds.) Proceedings of the 18th Annual Symposium on Combinatorial Pattern Matching (CPM 2007), London (Canada). Lecture Notes in Computer Science, vol. 4580. Springer, Berlin (2007) Google Scholar
  11. 11.
    Lohrey, M.: Word problems and membership problems on compressed words. SIAM J. Comput. 35(5), 1210–1240 (2006) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Lohrey, M., Schleimer, S.: Efficient computation in groups via compression. In: Proceedings of Computer Science in Russia (CSR 2007), Ekaterinburg (Russia). Lecture Notes in Computer Science, vol. 4649, pp. 249–258. Springer, Berlin (2007) Google Scholar
  13. 13.
    Lohrey, M., Sénizergues, G.: Theories of HNN-extensions and amalgamated products. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) Proceedings of the 33st International Colloquium on Automata, Languages and Programming (ICALP 2006), Venice (Italy). Lecture Notes in Computer Science, vol. 4052, pp. 681–692. Springer, Berlin (2006) Google Scholar
  14. 14.
    Lohrey, M., Sénizergues, G.: Rational subsets in HNN-extensions and amalgamated products. Int. J. Algebra Comput. 18(1), 111–163 (2008) MATHCrossRefGoogle Scholar
  15. 15.
    Lyndon, R.C., Schupp, P.E.: Combinatorial Group Theory. Springer, Berlin (1977) MATHGoogle Scholar
  16. 16.
    Macdonald, J.: Compressed words and automorphisms in fully residually free groups. Int. J. Algebra Comput. 20(3), 343–355 (2010) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Miyazaki, M., Shinohara, A., Takeda, M.: An improved pattern matching algorithm for strings in terms of straight-line programs. In: Apostolico, A., Hein, J. (eds.) Proceedings of the 8th Annual Symposium on Combinatorial Pattern Matching (CPM 97), Aarhus (Denmark). Lecture Notes in Computer Science, pp. 1–11. Springer, Berlin (1997) Google Scholar
  18. 18.
    Myasnikov, A., Shpilrain, V., Ushakov, A.: Group-based Cryptography. Birkhäuser, Boston (2008) MATHGoogle Scholar
  19. 19.
    Novikov, P.S.: On the algorithmic unsolvability of the word problem in group theory. Trans. Am. Math. Soc. II. Ser. 9, 1–122 (1958) MATHGoogle Scholar
  20. 20.
    Plandowski, W.: Testing equivalence of morphisms on context-free languages. In: van Leeuwen, J. (ed.) Second Annual European Symposium on Algorithms (ESA’94), Utrecht (The Netherlands). Lecture Notes in Computer Science, vol. 855, pp. 460–470. Springer, Berlin (1994) Google Scholar
  21. 21.
    Plandowski, W., Rytter, W.: Application of Lempel-Ziv encodings to the solution of word equations. In: Proceedings of the 25th International Colloquium on Automata, Languages and Programming (ICALP 1998). Lecture Notes in Computer Science, vol. 1443, pp. 731–742. Springer, Berlin (1998) CrossRefGoogle Scholar
  22. 22.
    Plandowski, W., Rytter, W.: Complexity of language recognition problems for compressed words. In: Karhumäki, J., Maurer, H.A., Paun, G., Rozenberg, G. (eds.) Jewels are Forever, Contributions on Theoretical Computer Science in Honor of Arto Salomaa, pp. 262–272. Springer, Berlin (1999) Google Scholar
  23. 23.
    Schleimer, S.: Polynomial-time word problems. Comment. Math. Helv. 83(4), 741–765 (2008) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Stallings, J.R.: Group Theory and Three-Dimensional Manifolds. Yale Mathematical Monographs, vol. 4. Yale University Press, New Haven (1971) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Institut für InformatikUniversität LeipzigLeipzigGermany

Personalised recommendations