Computer science and the fine structure of Borel sets

Abstract

(I) Wadge defined a natural refinement of the Borel hierarchy, now called the Wadge hierarchy WH. The fundamental properties of WH follow from the results of Kuratowski, Martin, Wadge and Louveau. We give a transparent restatement and proof of Wadge's main theorem. Our method is new for it yields a wide and unexpected extension: from Borel sets of reals to a class of natural but non Borel sets of infinite sequences. Wadge's theorem is quite ineffective and our generalization clearly worsens in this respect. Yet paradoxically our method is appropriate to effectivize this whole theory in the context discussed below. (II) Wagner defined on Büchi automata (accepting words of length ω) a hierarchy and proved for it an effective analog of Wadge's results. We extend Wagner's results to more general kinds of automata: counters, push-down automata and Büchi automata reading transfinite words. The notions and methods developed in (I) are quite useful for this extension, and we start to use them in order to look for extensions of the fundamental effective determinacy results of Büchi–Landweber, Rabin; and of Courcelle–Walukiewicz.