Inverse Star, Borders, and Palstars

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 contextfree 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. 1 Inverse star Let L be a language such that L = L. Then, following [3], we say that L is closed. Brzozowski [2] studied the the “smallest” language M such that L = M. Definition 1. For closed languages L, define