Transformation Between Regular Expressions and omega-Automata

Abstract

We propose a new definition of regular expressions for describing languages of omega-words, called infinity-regular expressions. These expressions are obtained by adding to the standard regular expression on finite words an operator infinity that acts similar to the Kleene-star but can be iterated finitely or infinitely often (as opposed to the omega-operator from standard omega-regular expressions, which has to be iterated infinitely often). We show that standard constructions between automata and regular expressions for finite words can smoothly be adapted to infinite words in this setting: We extend the Glushkov construction yielding a simple translation of infinity-regular expressions into parity automata, and we show how to translate parity automata into infinity-regular expressions by the classical state elimination technique, where in both cases the nesting of the * and the infinity operators corresponds to the priority range used in the parity automaton. We also briefly discuss the concept of deterministic expressions that directly transfers from standard regular expressions to infinity-regular expressions.