Finite Automata, Digraph Connectivity, and Regular Expression Size

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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.