Generating all Circular Shifts by Context-Free Grammars in Chomsky Normal Form

Let {a1, a2, . . . , an} be an alphabet of n symbols and let Cn be the language of circular shifts of the word a1a2 · · · an; so Cn = {a1a2 · · · an−1an, a2a3 · · · ana1, . . . , ana1 · · · an−2an−1}. We discuss a few families of context-free grammars Gn (n ≥ 1) in Chomsky normal form such that Gn generates Cn. The grammars in these families are investigated with respect to their descriptional complexity, i.e., we determine the number of nonterminal symbols ν(n) and the number of rules π(n) of Gn as functions of n. These ν and π happen to be functions bounded by low-degree polynomials, particularly when we focus our attention to unambiguous grammars. Finally, we introduce a family of minimal unambiguous grammars for which ν and π are linear.