By applying descriptive set theory to the Wagner's fine structure of regular w-languages we get quite different proofs of his results and obtain new results.
Theoretical Computer Science ELSEVIER
Theoretical Computer Science 191 (1998) 37-59
Fine hierarchy of regular co-languages Victor
’
Selivanov
Novosibirsk Pedagogical University. Novosibirsk 630126, Russia
Communicated February 1995; received in revised form August 1996: accepted October 1996 Communicated
by M. Nivat
Abstract By applying descriptive set theory to the Wagner’s fine structure of regular w-languages we get quite different proofs of his results and obtain new results. We give an automata-free description of the fine structure. We present also a simple property of a deterministic Muller automaton equivalent to the condition that the corresponding regular o-language belongs to any given level of the fine structure. Our results and proofs demonstrate deep interconnections between descriptive set theory and the theory of w-languages.
1. Introduction
and discussion
Regular o-languages were introduced by J.R. Biichi in the 1960s and studied by many people including B.A. Trakhtenbrot, R. McNaughton and M.O. Rabin. The subject quickly developed into a rich topic with several deep applications. Much information
and references on the subject may be found e.g. in [ 18,15,22]. We assume acquaintance with some basic concepts, notation and results in this field, all of them may be found in the cited papers. One branch of the discussed
topic
deals
with the classifications
of regular
o-
languages by means of topology, hierarchies and reducibilities. A series of papers culminated with the paper [22] giving in a sense the finest possible classification. We will revise the paper of K. Wagner by giving new, quite different proofs of his results. Our approach leads to strengthenings of some results from [21,22, l] and to several new results. Let us start with recalling some notation and terminology. Fix a finite alphabet X containing more than one symbol (for simplicity we may assume that X = {X 1x 1). Let X* and X0 denote, respectively, the sets of all words and of all w-words (i.e. sequences CI: o +X) over X. For n
