Inclusion is undecidable for pattern languages

Extended abstract
  • Tao Jiang
  • Arto Salomaa
  • Kai Salomaa
  • Sheng Yu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 700)


The inclusion problem for (nonerasing) pattern languages was raised by Angluin [1] in 1980. It has been open ever since. In this paper, we settle this open problem and show that inclusion is undecidable for (both erasing and nonerasing) pattern languages. In addition, we show that a special case of the inclusion problem, i.e., the inclusion problem for terminal-free erasing pattern languages, is decidable.


patterns pattern languages inclusion problems equivalence problems decidability 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    D. Angluin, “Finding patterns common to a set of strings”, Journal of Computer and System Sciences 21 (1980) 46–62.Google Scholar
  2. [2]
    D. Angluin, “Inductive inference of formal languages from positive data”, Information and Control 45 (1980) 117–135.Google Scholar
  3. [3]
    B.S. Baker and R.V. Book, “Reversal-bounded multipushdown machines”, Journal of Computer and System Sciences 8 (1974) 315–332.Google Scholar
  4. [4]
    D. R. Bean, A. Ehrenfeucht, and G. F. McNulty, “Avoidable patterns in strings of symbols”, Pacific Journal of Mathematics 85 (1979) 261–294.Google Scholar
  5. [5]
    T. Harju and J. Karhumäki, “The equivalence problem of multitape finite automata”, Theoretical Computer Science 78 (1991) 347–355.Google Scholar
  6. [6]
    M. A. Harrison, Introduction to Formal Language Theory, Addison-Wesley, Reading, 1978.Google Scholar
  7. [7]
    O.H. Ibarra, “Reversal-bounded multicounter machines and their decision problems”, Journal of the Association for Computing Machinery 25 (1978) 116–133.Google Scholar
  8. [8]
    O. Ibarra and T. Jiang “Learning regular languages from counterexamples”, Journal of Computer and System Sciences, 43 (1991) 299–316.Google Scholar
  9. [9]
    T. Jiang, E. Kinber, A. Salomaa, K. Salomaa, S. Yu, “Pattern languages with and without erasing”, to appear in the International Journal of Computer Mathematics.Google Scholar
  10. [10]
    M.L. Minsky, “Recursive unsolvability of Post's problem of ‘Tag’ and other topics in theory of Turing machines”, Annals of Mathematics 74 (1961) 437–455.Google Scholar
  11. [11]
    N. Tanida and T. Yokomori, “Polynomial-time identification of strictly regular languages in the limit”, IEICE Trans. Inf. and Syst. E75-D (1992) 125–132.Google Scholar
  12. [12]
    A. Thue, “über unendliche Zeichenreihen”, Norske Vid. Selsk. Skr., I Mat. Nat. Kl, Christiania 7 (1906) 1–22.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Tao Jiang
    • 1
  • Arto Salomaa
    • 2
  • Kai Salomaa
    • 3
  • Sheng Yu
    • 3
  1. 1.Dept. of Computer ScienceMcMaster UniversityHamiltonCanada
  2. 2.Academy of Finland and Mathematics Dept.University of TurkuTurkuFinland
  3. 3.Dept. of Computer ScienceUniversity of Western OntarioLondonCanada

Personalised recommendations