Inside the Muchnik degrees I: Discontinuity, learnability and constructivism

Abstract

Abstract. 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 (Π 0 1 ) 2 -piecewise computability (where (Π 0 1 ) 2 denotes the difference of two Π 0 1 sets), error-bounded-learnability is equivalent to finite ∆ 0 2 -piecewise computability, and learnability is equivalent to countable Π 0 1 -piecewise computability (equivalently, countable Σ 0 2 -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.1. Summary 1.1. Introduction. Imagine the floor function, a real function that takes the integer part of an input. Although it seems easy to draw a rough graph of the floor function, it is not computable with respect to the standard real number representation [77], because computability automatically induces topological continuity. One way to study the floor function in computability theory is to "computabilize" it by changing the representation/topology of the real space (see, for instance, [79]). However, it is also important to enhance our knowledge of the noncomputability/discontinuity level of such seemingly computable functions without changing representation/topology. Our main objective is to study low levels of the arithmetical/Baire hierarchy of functions on Baire space from the viewpoint of approximate computability/continuity and piecewise computability/continuity.We postulate that a nearly computable function shall be, at the very least, nonuniformly computable, where a function f is said to be nonuniformly computable if for every input x, there exists an algorithm Ψ x that computes f (x) using x as an oracle, where we do not require the map x → Ψ x to be computable. The notion of nonuniform computability naturally arises in Computable Analysis [12,83]. However, of course, most nonuniformly computable discontinuous functions are far from being computable. Then, what type of discontinuous functions are recognized as being nearly computable? A nearly computable/continuous function has to be approximated using computable/continuous functions. For instance, a Baire function appears to be dynamically approximated by a sequence of continuous functions and a piecewise continuous (σ-continuous) function appears to be statically approximated by countably many continuous functions.There have been many challenges [15,82,81,80,83,78,79] in developing computability theory of (nonuniformly computable) discontinuous funct...

Weihrauch Degrees of Finding Equilibria in Sequential Games

“…This is the claim We will relate deciding the winner and finding a winning strategy for games induced by sets from some level of the difference hierarchy to the lessor limited principle of omniscience and the law of excluded middle for Σ 0 n -formulae of the corresponding level. These principles were studied in [1,10,24] (among others).…”

Fluctuations, effective learnability and metastability in analysis

“…The concept of mind change for Cauchy statements is also implicit in section 5.1 of [44] (Proof of Lemma 31.c). Effective learnability concepts for functionals F : D → N N (with D ⊆ N N ) have recently been investigated in [21].…”

Levels of Uniformity

“…For n = 0, the Medvedev lattice, this was shown by Medvedev [13], and for the Muchnik lattice this was shown by Muchnik [14]. For n = 1 this result is due to Higuchi and Kihara [4,Proposition 16], but the general proof below is our own.…”

“…The Medvedev and Muchnik lattices are also known to be Brouwer algebras. For M 1 this was shown in [4], and it is the only Brouwer algebra among the intermediate degree structures studied by Higuchi and Kihara. In section 3 we study maps between M n and M m for n = m. We show that the natural surjection from M n to M m for m > n preserves joins and meets, but not necessarily implications. On the other hand, we show that there are embeddings preserving joins and implications in the other direction.…”

“…Proof. We only prove that joins and implications are preserved, using the fact from the proof of [4,Corollary 42] that the map induced by α is well-defined, preserves the order and satisfies α(A)…”