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A generalization of the Parikh vector for finite and infinite words

  • Rani Siromoney
  • V. Rajkumar Dare
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 206)

Abstract

A metric is defined on the set of all finite and infinite words based on the difference between the occurrences of different letters of the alphabet. This induces a topology which coincides with the metric topology defined by Nivat [5]. Using this metric a vector is defined which gives rise to the position vector p showing the position in the word of each letter of the alphabet. The p-vector can be regarded as a generalization of the Parikh vector [6]. While the Parikh vector of a word enumerates the number of occurrences of each letter of the alphabet, the p-vector introduced in this paper indicates the positions of each letter of the alphabet in the word. Also, the Parikh vector is defined for finite words and the p-vector gives a generalization to infite words. The p-vector has nice mathematical properties. Characterizations of regular sets, context-free languages and a few families from Lindenmayer systems are given.

Keywords

Position Vector Regular Language Formal Language Theory Infinite Word Binary Alphabet 
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.

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References

  1. [1]
    C. Choffrut and K. Culik II, On extendibility of unavoidable sets, Lecture Notes in Computer Science 166 (Springer, Berlin, 1984) 326–338.Google Scholar
  2. [2]
    V.R. Dare and R. Siromoney, Subword topology, TR MATH 12/1985, Department of Mathematics, Madras Christian College, Madras.Google Scholar
  3. [3]
    M. Harrison, Introduction to Formal Languages Theory, (Addison-Wesley, Reading, Mass, 1978).Google Scholar
  4. [4]
    A. Nakamura and H. Ono, Pictures of functions and their acceptability by automata, Theoret. Comput. Sci. 23 (1983) 37–48.CrossRefGoogle Scholar
  5. [5]
    M. Nivat, Infinite words, infinite trees and infinite computations, Mathematical Centre Tracts 109 (1979) 1–52.Google Scholar
  6. [6]
    R.J. Parikh, On context-free languages, Journal of the Association for Computing Machinery 13 (1966) 570–581.Google Scholar
  7. [7]
    G. Rozenberg and A. Salomaa, The Mathematical Theory of L-systems (Academic Press, New York, 1980).Google Scholar
  8. [8]
    R. Siromoney, V.R. Dare and K.G. Subramanian, Infinite arrays and infinite computations, Theoret. Comput. Sci. 24 (1983) 195–205.CrossRefGoogle Scholar
  9. [9]
    R. Siromoney, K.G. Subramanian and V.R. Dare, On infinite arrays obtained by deterministic controlled table L-array systems, Theoret. Comput. Sci. 33 (1984) 3–11.CrossRefGoogle Scholar
  10. [10]
    R. Siromoney and V.R. Dare, On infinite words obtained by selective substitution grammars, Theoret. Comput. Sci. 39 (1985) (in press).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • Rani Siromoney
    • 1
  • V. Rajkumar Dare
    • 1
  1. 1.Department of MathematicsMadras Christian CollegeTambaram, Madras

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