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.

This work was partially supported by the Board of Research in Nuclear Science, Department of Atomic Energy.

This work was partially supported by the University Grants Commission under the scheme ‘Financial Assistance to College Teachers for Minor Projects’.