International Journal of Foundations of Computer Science Vol. 30, Nos. 6 & 7 (2019) 899–920 c  World Scientific Publishing Company DOI: 10.1142/S0129054119400227 On Average Behaviour of Regular Expressions in Strong Star Normal Form Sabine Broda * , Ant´ onio Machiavelo † , Nelma Moreira ‡ and Rog´ erio Reis § CMUP & DM-DCC, Faculdade de Ciˆ encias da Universidade do Porto Rua do Campo Alegre, 4169-007 Porto, Portugal * sbb@dcc.fc.up.pt † ajmachia@fc.up.pt ‡ nam@dcc.fc.up.pt § rvr@dcc.fc.up.pt Received 26 November 2017 Accepted 3 December 2018 Communicated by C. Cˆampeanu and G. Pighizzini 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 of the position automaton, as well as the ratio and the size of these expressions to standard regular expressions. Keywords : Regular expressions; star normal form; conversions into finite automata; analytic combinatorics; asymptotic average case complexity; Puiseux series. 1. Introduction A regular expression α is in strong star normal form (ssnf ) if for any subexpression of the form β ⋆ or β + ε the language represented by β does not include the empty word, ε. The broader notion of star normal form was introduced by Br¨ uggemann- Klein [7] as a step to improve the construction of the position automaton from a regular expression from cubic to quadratic time. Transforming a regular expression into this normal form can be achieved in linear time, and moreover the position automaton resulting from that normal form coincides with the one of the original expression. In the same paper, the star normal form was also used to characterize certain types of unambiguous expressions. The position automaton construction [11] ‡ Corresponding author. 899 Int. J. Found. Comput. Sci. 2019.30:899-920. Downloaded from www.worldscientific.com by 34.228.24.229 on 07/03/20. Re-use and distribution is strictly not permitted, except for Open Access articles.