The most efficient known construction of equation automaton is that due to Ziadi and Champarnaud. For a regular expression E, it requires O(|E| 2 ) time and space and is based on going from position automaton to equation automaton using c-continuations. This complexity is due to the sorting step that takes O(|E| 2 ) time used to identify the identical sub-expressions of E. In this paper, we present a more efficient construction of the equation automaton which avoids the sorting step and replaces it by a minimization of an acyclic finite deterministic automaton. We show that this minimization allows the identification of identical sub-expressions as well as the sorting step used in Champarnaud and Ziadi's approach. Using the minimization we get O(|E| + |E| · |E E |) time and space complexity where |E E | is the number of states of the equation automaton.

There exist two well-known quotients of the position automaton of a regular expression. The first one, called the equation automaton, was first introduced by Mirkin from the notion of prebase and has been redefined by Antimirov from the notion of partial derivative. The second one, due to Ilie and Yu and called the follow automaton, can be obtained by eliminating ε-transitions in an ε-NFA that is always smaller than the classical ε-NFAs (Thompson, Sippu and Soisalon–Soininen). Ilie and Yu discussed the difficulty of succeeding in a theoretical comparison between the size of the follow automaton and the size of the equation automaton and concluded that it is very likely necessary to realize experimental studies. In this paper we solve the theoretical question, by first defining a set of regular expressions, called normalized expressions, such that every regular expression can be normalized in linear time, and proving then that the equation automaton of a normalized expression is always smaller than its follow automaton.

In this article we generalize concepts of the position automaton and ZPC-structure to the regular [Formula: see text]-expressions. We show that the extended ZPC-structure can be built in linear time w.r.t. the size of the [Formula: see text]-expression and that the associated position automaton can be deduced from it in quadratic time.

The aim of this paper is to describe a quadratic algorithm for computing the equation K-automaton of a regular K-expression as defined by Lombardy and Sakarovitch. Our construction is based on an extension to regular K-expressions of the notion of c-continuation that we introduced to compute the equation automaton of a regular expression as a quotient of its position automaton.