On the Average Complexity of Strong Star Normal Form

Abstract

For regular expressions in (strong) star normal form a large set of efficient algorithms is known, from conversions into finite automata to characterisations of unambiguity. In this paper we study the average complexity of this class of expressions using analytic combinatorics. As it is not always feasible to obtain explicit expressions for the generating functions involved, here we show how to get the required information for the asymptotic estimates with an indirect use of the existence of Puiseux expansions at singularities. We study, asymptotically and on average, the alphabetic size, the size of the ε-follow automaton and the ratio of these expressions to standard regular expressions.

On Average Behaviour of Regular Expressions in Strong Star Normal Form

“…A preliminary version of this paper was presented in Ref. [6]. In the next section, we review some basics on regular expressions and NFAs.…”

A Local Limit Property for Pattern Statistics in Bicomponent Stochastic Models

“…It turns out that evaluating the frequency of patterns from a regular expression in a random text generated by a Markovian model can be reduced to determining the frequency of a single symbol in a word over a binary alphabet generated by a rational stochastic model [11,3]. Moreover, it is well-known that the average number of occurrences of symbols in words of regular and context-free languages plays a relevant role in the analysis of the descriptional complexity of languages and computational models [5,6]. Clearly the limit distributions of these quantities (also in the local form) yield a more complete information and in particular they allow to evaluate their dispersion around the average values.…”