Deleting Powers in Words

TL;DR

This paper investigates the language of words reducible to the empty word through iterative deletion of powers, revealing regularity in binary squares, non-regularity in larger alphabets, and characterization via linear index and indexed grammars.

Contribution

It demonstrates the regularity of the language for binary squares, non-regularity for larger alphabets, and provides grammar-based characterizations for the general case.

Findings

Language is regular for deleting squares in binary words.

Language is not regular for deleting squares over larger alphabets.

The language can be generated by a linear index grammar and an indexed grammar.

Abstract

We consider the language consisting of all words such that it is possible to obtain the empty word by iteratively deleting powers. It turns out that in the case of deleting squares in binary words this language is regular, and in the case of deleting squares in words over a larger alphabet the language is not regular. However, for deleting squares over any alphabet we find that this language can be generated by a linear index grammar which is a mildly context sensitive grammar formalism. In the general case we show that this language is generated by an indexed grammar.