Context-free sets of infinite words

Abstract

In this paper we give some new results about context-free sets of infinite words. The presentation will be a generalization of McNaughton's approach in [7], where he analyzed regular sets of infinite words. However, our extension to the regular case is not straightforward and thus distinguishes from the approach given in [4]. Some of the results given below originate from two papers by Nivat [9,10], others are unpublished supplementary results due to Nivat and Boasson. We recall from [9] that to each context-free grammar G one can associate an operator Ĝ, which has a unique fixed point over finite words and a greatest fixed point over finite and infinite words, each of them being the vector of languages generated by the non-terminals of G. We then show that any context-free set of infinite words can be obtained by a substitution of some context-free languages into a regular set of infinite words. In the sequel the notions of adherence and center of context-free languages are introduced and analyzed to establish a link between the infinite words and the language generated by a grammar.