On the Representation of Finite Automata

We give an unique string representation, up to isomorphism, for initially connected deterministic finite automata (ICDFA’s) with n states over an alphabet of k symbols. We show how to generate all these strings for each n and k, and how its enumeration provides an alternative way to obtain the exact number of ICDFA’s. 1 Motivation In symbolic manipulation environments for finite automata, it is important to have an adequate representation of automata and, dependent upon their use, several representations may be available. For example, for testing if two finite automata are isomorphic objects or for (random) generation of automata, the representation must be compact and somehow canonical. In the FAdo project [?, ?] a canonical form is used to test if two minimal DFA’s are isomorphic (i.e are the same up to renaming of states). In this paper we prove the correctness of that representation and show how it can be used for the exact enumeration and generation of initially connected deterministic finite automata (ICDFA). The problem of enumeration of finite automata was considered by several authors since early 1960s, in particular see Robinson [?], Harary and Palmer [?] and Liskovets [?] amongst many others. A survey may be found in Domaratzki et al. [?]. More recently, several authors examined related problems. Domaratzki et al. [?] studied the enumeration of distinct languages accepted by finite automata with n states; Nicaud [?], Champarnaud and Paranthoën [?, ?] and Bassino and Nicaud [?] analysed several aspects of the average behaviour of regular languages; Liskovets [?] and Domaratzki [?] gave (exact and asymptotic) enumerations of acyclic DFA’s and of finite languages. The paper is organised as follows. In the next section, we review some basic notions and introduce some notation. Section ?? describes a string representation for deterministic finite automata that is unique up to isomorphism for initially connected deterministic finite automata. Section ?? presents an efficient method to generate those strings. Section ?? shows how their enumeration provides an upper bound and the exact value for the number of ICDFA’s. Section ?? concludes with some final remarks. We address the reader attention to the longer version of this paper for some implementation issues and experimental results1. Work partially funded by Fundação para a Ciência e Tecnologia (FCT) and Program POSI. Corresponding author http://www.dcc.fc.up.pt/Pubs/TR05/dcc-2005-04.ps.gz