Acyclic automata and small expressions using multi-tilde-bar operators

A regular expression with n occurrences of symbol can be converted into an equivalent automaton with (n+1) states, the so-called Glushkov automaton of the expression. Conversely, it is possible to decide whether a given (n+1)-state automaton is a Glushkov one and, if so, to convert it back to an equivalent regular expression of width n. Our goal is to extend the class of automata for which such a linear retranslation is possible. We define new regular operators, called multi-tilde-bars, allowing us to simultaneously apply a multi-tilde operator and a multi-bar one to a list of expressions. The main results are that a multi-tilde-bar expression of width n can be converted into an (n+1)-state position-like automaton and that any acyclic n-state automaton can be turned into an extended expression of width O(n).