A Hitchhiker's Guide to descriptional complexity through analytic combinatorics

Abstract

Nowadays, increasing attention is being given to the study of the descriptional complexity in the average case. Although the underlying theory for such a study seems intimidating, one can obtain interesting results in this area without too much effort. In this gentle introduction we take the reader on a journey through the basic analytical tools of that theory, giving some illustrative examples using regular expressions. Additionally, new asymptotic average-case results for several ε-NFA constructions are presented, in a unified framework. It turns out that, asymptotically, and in the average case, the complexity gap between the several constructions is significantly larger than in the worst case. Furthermore, one of the ε-NFA constructions approaches the corresponding ε-free NFA construction, asymptotically and on average.