Constructing the Maximum Prefix-Closed Subset for a Set of –ω-Words Defined by a –ω-Regular Expression

In this paper, we present a method of constructing the maximum prefix-closed subset of a set of – ω -words R defined by a – ω -regular expression. This method is based on constructing a labeled graph called a graph of elementary extensions whose vertices are some – ω -regular subsets of the set R. With every graph vertex, we associate a linear equation over sets of – ω -words. Thus, the graph of elementary extensions determines a system of linear equations. As a result of solving this system of equations, for each vertex of the graph, we obtain the maximum prefix-closed with respect to R subset of the set of – ω -words corresponding to this vertex. The union of such subsets that correspond to all initial vertices of the graph, that is, the vertices constructing the graph starts with, is a maximum prefix-closed subset of the given set R . A method of constructing such a graph is proposed, which we called a method of incomplete intersections. We also present a method to solve the system of equations determined by the graph of elementary extensions.