Inverse Star, Borders, and Palstars

TL;DR

This paper investigates the properties of an inverse operation to Kleene closure on closed languages, revealing limitations for context-free languages and establishing new relationships between unbordered words and prime palstars.

Contribution

It introduces the inverse Kleene closure operation for closed languages, explores its effects on regular and context-free languages, and links unbordered words with prime palstars.

Findings

Inverse Kleene closure preserves regularity but not for context-free languages.

Neither unbordered words nor prime palstars form context-free languages.

A new relationship between unbordered words and prime palstars is established.

Abstract

A language L is closed if L = L*. We consider an operation on closed languages, L-*, that is an inverse to Kleene closure. It is known that if L is closed and regular, then L-* is also regular. We show that the analogous result fails to hold for the context-free languages. Along the way we find a new relationship between the unbordered words and the prime palstars of Knuth, Morris, and Pratt. We use this relationship to enumerate the prime palstars, and we prove that neither the language of all unbordered words nor the language of all prime palstars is context-free.