Inverse star, borders, and palstars

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 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. Inverse starLet 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

Counting the Palstars

“…Consider a palstar of length 2n > 0. Either it is a prime palstar, and by [7] there are u k (n) = u k (n)p k (0) of them, or it is the concatenation of two or more prime palstars. In the latter case, consider the length of this first factor; it can potentially be 2i for 1 ≤ i ≤ n. Removing this first factor, what is left is also a palstar.…”

“…Otherwise a word is unbordered. Rampersad et al [7] recently gave a bijection between the unbordered strings of length n and the prime palstars of length 2n. As a consequence they obtained a formula for the number of prime palstars.…”

Unique decipherability in formal languages

“…Clearly PALSTAR is a context-free language (CFL). We see that UD(PALSTAR) = PRIMEPALSTAR, which was proven in [11] to be non-context-free.…”

Factorization in Formal Languages

“…Clearly L is a CFL. Then uf(L) = PRIMEPALSTAR, which was proven in [3] to be non-contextfree. (Here PRIMEPALSTAR is the set of all elements of PALSTAR that cannot be written as the product of two or more elements of PALSTAR.)…”