# A generalization of the Parikh vector for finite and infinite words

## 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## Preview

Unable to display preview. Download preview PDF.

## References

- [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]V.R. Dare and R. Siromoney, Subword topology, TR MATH 12/1985, Department of Mathematics, Madras Christian College, Madras.Google Scholar
- [3]M. Harrison, Introduction to Formal Languages Theory, (Addison-Wesley, Reading, Mass, 1978).Google Scholar
- [4]A. Nakamura and H. Ono, Pictures of functions and their acceptability by automata, Theoret. Comput. Sci. 23 (1983) 37–48.CrossRefGoogle Scholar
- [5]M. Nivat, Infinite words, infinite trees and infinite computations, Mathematical Centre Tracts 109 (1979) 1–52.Google Scholar
- [6]R.J. Parikh, On context-free languages, Journal of the Association for Computing Machinery 13 (1966) 570–581.Google Scholar
- [7]G. Rozenberg and A. Salomaa, The Mathematical Theory of L-systems (Academic Press, New York, 1980).Google Scholar
- [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]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]R. Siromoney and V.R. Dare, On infinite words obtained by selective substitution grammars, Theoret. Comput. Sci. 39 (1985) (in press).Google Scholar