On Average Behaviour of Regular Expressions in 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 [Formula: see text]-follow automaton and of the position automaton, as well as the ratio and the size of these expressions to standard regular expressions.

On the Uniform Distribution of Regular Expressions

“…Second, the results of Koechlin et al do not imply that asymptotic estimates obtained by considering the whole set of regular expressions are different from those obtained by using a more refined set with less equivalent expressions. For instance, some results obtained for expressions in strong star normal form coincide with the ones for standard regular expressions [2]. In order to further sustain the above claim, in this paper we consider the set R of regular expressions avoiding an absorbing pattern which extends the pattern in the example above and was the one considered by Koechlin et al It is shown that, although the set R is significantly smaller than the set RE, the asymptotic estimates for the size of the Glushkov automaton on these sets is the same.…”