### Corrigendum to our paper: How Expressions Can Code for Automata

Sylvain Lombardy, Jacques Sakarovitch
2010 RAIRO - Theoretical Informatics and Applications
We correct a mistake made in a previous paper in the construction of an automaton from a rational expression. We used there the definition of derivation of expression given by Antimirov, and this definition has to be further adapted for our purpose. In [7], we were considering the following problem: Is it possible to build an algorithm Ω such that for any rational expression E computed from an automaton A -i.e. E = Φ(A) where Φ is the state elimination method for instance -the following holds:
more » ... e following holds: A = Ω(E) ? We did not solve the problem completely, but we have identified two constructions that are good candidates to be the core components of such an algorithm Ω. The first one is the construction of an automaton ∆(E) from an expression E, the second one is the computation of the minimal co-quotient Υ(B) of an automaton B. There is no problem with the minimal co-quotient but the definition we gave for ∆(E) was faulty. This can be observed for instance on the following example. Let A 1 be the automaton of Figure 1 (a), and let E 1 = (a + b + 1) [a(a + b)] * be the expression computed from A 1 , which we write E 1 = (a + b + 1)F 1 with F 1 = [a(a + b)] * ; Figure 1 (b) shows ∆(E 1 ), whose co-quotient is not isomorphic to A 1 . As we shall see, it is not difficult to recover correct definitions and a true statement (Theorem 1.6) for the key result (Theorem 3.5) in the original paper.