THE NORMAL OPEN SET ALMOST EQUAL TO A REGULAR OMEGA LANGUAGE
Izumi Takeuti · 2002
Aregular omega language is the omega language which is recognised by aBuchi automaton. Anormal open set is an open set which is equal to the internal kernel of its closure. This paper shows that for each regular language, there exists anormal open set which is equal to the regular language except for the area of anull set as the error, and that the measure of the boundary of the normal open set is zero. A normal open set is an open set which is equal to the internal kernel of its closure. This property is natural for the shapes of real material objects. The main result of this paper is the following: for each regular language, there exists anormal open set which is equal to the regular language except for the area of anull set, and that the measure of the boundary is 0. This paper shows two ways of the proofs of the main theorem. One proves the theorem directly, and the other proves alittle extended theorem, which is the theorem for finite-state omega languages. All the regular omega languages are finite-state, although not all finite-state omega languages are regular. In both proofs, we will use decomposition lemmata. One is Lemma 5.2 for regular omega languages, and it appears in the literature (S98). The other is Lemma 6.8 for finite-state languages, and it is proved for the first time in this paper.