The ambiguity of primitive words

  • H. Petersen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 775)


A word is primitive if it is not a proper power of a shorter word. We prove that the set Q of primitive words over an alphabet is not an unambiguous context-free language. This strengthens the previous result that Q cannot be deterministic context-free. Further we show that the same holds for the set L of Lyndon words. We describe 2DPDA accepting Q and L which imply efficient decidability on the RAM model of computation and we analyse the number of comparisons required for deciding Q. Finally we give a new proof showing a related language not to be context-free, which only relies on properties of semi-linear and regular sets.


Normal Order Formal Language Theory Proper Power Lyndon Word Primitive Word 
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  1. [1]
    T. M. Apostol: Introduction to Analytic Number Theory, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo (1976, 3rd printing 1986).Google Scholar
  2. [2]
    K.S. Booth: Lexicographically least circular substrings, Inform. Process. Lett. 10 (4) (1980) pp. 240–242.Google Scholar
  3. [3]
    N. Chomsky, M. P. Schützenberger: The algebraic theory of context-free languages, Computer Programming and Formal Systems (P. Brafford, D. Hirschberg eds.), North-Holland, Amsterdam (1963), pp. 118–161.Google Scholar
  4. [4]
    L. Comtet: Calcul pratique des coefficients de Taylor d'une fonction algébrique, L'Enseignement Mathématique 10 (1964) pp. 267–270.Google Scholar
  5. [5]
    S. A. Cook: Linear time simulation of deterministic two-way pushdown automata, Inf. Proc. 71, North-Holland (1972) pp. 75–80.Google Scholar
  6. [6]
    V. Diekert, personal communication (1993).Google Scholar
  7. [7]
    P. Dömösi, S. Horváth, M. Ito: Formal languages and primitive words, Publ. Math., Debrecen 42 3–4 (1993) pp. 315–321.Google Scholar
  8. [8]
    P. Dömösi, S. Horváth, M. Ito, L. Kászonyi, M. Katsura: Formai languages consisting of primitive words, Proc. FCT93, LNCS 710, Springer-Verlag, Berlin, Heidelberg, New York (1993) pp. 194–203.Google Scholar
  9. [9]
    Ph. Flajolet: Analytic models and ambiguity of context-free languages, TCS 49 (1987) pp. 283–309.Google Scholar
  10. [10]
    S. Ginsburg: The Mathematical Theory of Context-Free Languages, McGraw-Hill, New York (1966).Google Scholar
  11. [11]
    S. Ginsburg, E. H. Spanier: Bounded Algol-like languages, Trans. AMS 113 (1964) pp. 333–368.Google Scholar
  12. [12]
    J.-P. Haas: Theorie und Anwendungen Semilinearer Vektormengen, Universität Hamburg, Fachbereich Informatik, Bericht Nr. 130 (1987).Google Scholar
  13. [13]
    G. H. Hardy, E. M. Wright: An Introduction to the Theory of Numbers, Oxford University Press, London (1938, reprinted 1968).Google Scholar
  14. [14]
    M. A. Harrison: Introduction to Formal Language Theory, Addison-Wesley, Reading Mass. (1978).Google Scholar
  15. [15]
    J. E. Hopcroft, J. D. Ullman: Introduction to Automata Theory, Languages, and Computation, Addison-Wesley, Reading Mass. (1979).Google Scholar
  16. [16]
    C. S. Iliopoulos, W. F. Smyth: Optimal algorithms for computing the canonical form of a circular string, TCS 92 (1992) pp. 87–105.Google Scholar
  17. [17]
    M. Ito, M. Katsura, H. J. Shyr, S. S. Yu: Automata accepting primitive words, Semigroup Forum 37 (1988) pp. 45–50.Google Scholar
  18. [18]
    D. Knuth, J. Morris, V. Pratt: Fast pattern matching in strings, SIAM J. Comp. 6 (1977) pp. 323–350.Google Scholar
  19. [19]
    M. Lothaire: Combinatorics on Words, Addison-Wesley, Reading Mass. (1983).Google Scholar
  20. [20]
    R. C. Lyndon, M. P. Schützenberger: The equation a M=b N c P in a free group, Michigan Math. J. 9 (1962) pp. 289–298.Google Scholar
  21. [21]
    R. J. Parikh: On context-free languages, JACM 13 (1966) pp. 570–581.Google Scholar
  22. [22]
    H. Petersen: Remarks on the power of 2-DPDA, Universität Hamburg, Fachbereich Informatik, Bericht Nr. 162.Google Scholar
  23. [23]
    H. J. Shyr, G. Thierrin: Disjunctive languages and codes, Proc. FCT77, LNCS 56, Springer-Verlag, Berlin, Heidelberg, New York (1977) pp. 171–176.Google Scholar
  24. [24]
    R. P. Stanley: Generating functions, MAA Studies in Math. (G.-C. Rota ed.), Vol. 17: Studies in Combinatorics, MAA (1978) pp. 100–141.Google Scholar
  25. [25]
    R. Stearns, J. Hartmanis, P. M. Lewis II: Hierarchies of memory limited computations, in: Proc. 6th Annual Symp. on Switching Circuit Theory and Logical Design (1965) pp. 179–190.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • H. Petersen
    • 1
  1. 1.Fachbereich InformatikUniversität HamburgDeutschland

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