Combinatorial complexity of regular languages

Abstract

We study combinatorial complexity (or counting function) of regular languages, describing these functions in three ways. First, we classify all possible asymptotically tight upper bounds of these functions up to a multiplicative constant, relating each particular bound to certain parameters of recognizing automata. Second, we show that combinatorial complexity equals, up to an exponentially small term, to a function constructed from a finite number of polynomials and exponentials. Third, we describe oscillations of combinatorial complexity for factorial, prefix-closed, and arbitrary regular languages. Finally, we construct a fast algorithm for calculating the growth rate of complexity for regular languages, and apply this algorithm to approximate growth rates of complexity of power-free languages, improving all known upper bounds for growth rates of such languages. © 2008 Springer-Verlag Berlin Heidelberg.