Finite Automata, Digraph Connectivity, and Regular Expression Size (Extended Abstract) Hermann Gruber 1 and Markus Holzer 2 1 Institut für Informatik, Ludwig-Maximilians-Universität München, Oettingenstraße 67, D-80538 München, Germany gruberh@tcs.ifi.lmu.de 2 Institut für Informatik, Technische Universität München, Boltzmannstraße 3, D-85748 Garching bei München, Germany holzer@in.tum.de Abstract. Recently lower bounds on the minimum required size for the conversion of deterministic finite automata into regular expressions and on the required size of regular expressions resulting from applying some basic language operations on them, were given by Gelade and Neven [8]. We strengthen and extend these results, obtaining lower bounds that are in part optimal, and, notably, the presented examples are over a binary alphabet, which is best possible. To this end, we develop a different, more versatile lower bound technique that is based on the star height of regular languages. It is known that for a restricted class of regular languages, the star height can be determined from the digraph underlying the transition structure of the minimal finite automaton accepting that language. In this way, star height is tied to cycle rank, a structural complexity measure for digraphs proposed by Eggan and Büchi, which measures the degree of connectivity of directed graphs. 1 Introduction One of the most basic theorems in formal language theory is that every regular expression can be effectively converted into an equivalent finite automaton, and vice versa [16]. While algorithms accomplishing these tasks have been known for a long time, there has been a renewed interest in these classical problems during the last few years. For instance, new algorithms for converting regular expressions into finite automata outperforming classical algorithms have been found only recently, as well as a matching lower bound of Ω(n · log 2 n) on the number of transitions required by any equivalent nondeterministic finite automaton (NFA). The lower bound is, however, only attained for growing alphabet size, and a better algorithm is known for constant alphabet size, see [26] for the current state of the art. In contrast, much less is known about the converse direction, namely of con- verting finite automata into regular expressions. Apart from the fundamental L. Aceto et al. (Eds.): ICALP 2008, Part II, LNCS 5126, pp. 39–50, 2008. c  Springer-Verlag Berlin Heidelberg 2008