Inside the Muchnik degrees I: Discontinuity, learnability and constructivism

Every computable function has to be continuous. To develop computability theory of discontinuous functions, we study low levels of the arithmetical hierarchy of nonuniformly computable functions on Baire space. First, we classify nonuniformly computable functions on Baire space from the viewpoint of learning theory and piecewise computability. For instance, we show that mind-change-bounded learnability is equivalent to finite (Π10)2-piecewise computability (where (Π10)2 denotes the difference of two Π10 sets), error-bounded learnability is equivalent to finite Δ20-piecewise computability, and learnability is equivalent to countable Π10-piecewise computability (equivalently, countable Σ20-piecewise computability). Second, we introduce disjunction-like operations such as the coproduct based on BHK-like interpretations, and then, we see that these operations induce Galois connections between the Medvedev degree structure and associated Medvedev/ Muchnik-like degree structures. Finally, we interpret these results in the context of the Weihrauch degrees and Wadge-like games.