Weihrauch-completeness for layerwise computability

Layerwise computability is an effective counterpart to continuous functions that are almosteverywhere defined. This notion was introduced by Hoyrup and Rojas [17]. A function defined on Martin-Löf random inputs is called layerwise computable, if it becomes computable if each input is equipped with some bound on the layer where it passes a fixed universal Martin-Löf test. Interesting examples of functions that are layerwise computable but not computable are obtained e.g. from Birkhoff’s theorem or the study of algorithmically random Brownian motion (more below). Weihrauch reducibility [5, 4] is a framework to compare the extent of non-computability of multivalued functions. It has been proposed with a meta-mathematical investigation of the constructive content of existence theorem in mathematics in mind. However, it has also been fruitfully employed to study (effective) function classes such as (effective) Borel measurability [3] or piecewise continuity (computability) and (effective) ∆2-measurability [33]. Our interest in this paper is in problems that are Weihrauch-complete for layerwise computability, i.e. problems that are layerwise computable, and every layerwise computable problem