Generating All Circular Shifts by Context-Free Grammars in Greibach Normal Form

For each alphabet Σn = {a1, a2, . . . , an}, linearly ordered by a1 < a2 < · · · < an, let Cn be the language of circular or cyclic shifts over Σn, i.e., Cn = {a1a2 · · · an−1an, a2a3 · · · ana1, . . . , ana1 · · · an−2an−1}. We study a few families of context-free grammars Gn (n ≥ 1) in Greibach normal form such that Gn generates Cn. The members of these grammar families are investigated with respect to the following descriptional complexity measures: the number of nonterminals ν(n), the number of rules π(n) and the number of leftmost derivations δ(n) of Gn. As in the case of Chomsky normal form, these ν, π and δ are functions bounded by low-degree polynomials. However, the question whether there exists a family of grammars that is minimal with respect to all these measures remains open.