Wadge Reducibility and Infinite Computations

Introduction Investigation of the infinite behavior of computing devices is of great interest for computer science because many hardware and software concurrent systems (like processors or operating systems) may not terminate. The study of infinite computations is important for several branches of theoretical computer science, including verification and synthesis of reactive computing systems [th90], stream computability [wa85] and computability in analysis [wei00]. In contrast with the theory of terminating computations, the study of infinite computations crucially depends on topological considerations. In particular, it needs notions and techniques for analyzing the complexity of sets and functions on (topological) spaces. The search for natural notions of complexity usually results in structures known as hierarchies and reducibilities. Introduction In his remarkable PhD thesis [wad72,wad84], W.W. Wadge introduced an important notion of reducibility on spaces now known as the Wadge reducibility. For subsets A, B of a space X , we say that A is Wadge reducible to B (in symbols A ≤ W B) if A = f −1 (B) for a continuous function f on X. Along with a deep study of Wadge reducibility on the Baire space, the thesis [wad84] contains a discussion of the relationship of Wadge reducibility to infinite computations based on the well known important fact that computable functions on natural spaces are continuous. Soon it became clear that Wadge reducibility is a central notion of descriptive set theory providing a useful tool for analyzing the infinite computations. Introduction In many cases, the infinite behavior of a device is captured by the notion of ω-language recognized by the device. Independently of W.W. Wadge, K.W. Wagner [wag76, wag79] characterized the Wadge reducibility and some of its effective versions on the class of regular ω-languages, i.e. on the subsets of Cantor space recognized by finite automata. This class of regular ω-languages was already well recognized as fundamental for the field of specification, verification and synthesis of finite-state systems. The paper [wag79] initiated a systematic study of infinite behavior of different computing devices based on Wadge reducibility and its effective versions. In this way Wadge reducibility became intimately related to the theory of infinite computations. Introduction Here we discuss some results (including very recent ones) about Wadge reducibility and some of its variations in some spaces and its applications to the study of infinite computations. We mention some important techniques used in this field, in particular the so called Gale-Stewart games …