The Effective Sequence of Uniformities and its Limit as a Methodology in Computable Analysis

Yasugi, Mariko; Mori, Takakazu; Tsujii, Yoshiki

doi:10.4288/jafpos1956.15.99

Annals of the Japan Association for Philosophy of Science

2007

DOI: 10.4288/jafpos1956.15.99

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The Effective Sequence of Uniformities and its Limit as a Methodology in Computable Analysis

Mariko Yasugi¹,

Takakazu Mori²,

Yoshiki Tsujii³

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2014

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“…There have been many challenges [15,82,81,80,83,78,79] in developing computability theory of (nonuniformly computable) discontinuous functions using the notion of learnability (dynamicalapproximation) and piecewise computability (statical-approximation). Indeed, one can show the equivalence of effective learnability and Π 0 1 -piecewise computability: the class of functions that are computable with finitely many mind changes is exactly the class of functions that are decomposable into countably many computable functions with Π 0 1 domains.…”

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confidence: 99%

Inside the Muchnik degrees I: Discontinuity, learnability and constructivism

Higuchi

Kihara

2014

Annals of Pure and Applied Logic

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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...

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Inside the Muchnik degrees I: Discontinuity, learnability and constructivism

Higuchi

Kihara

2014

Annals of Pure and Applied Logic

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The Effective Sequence of Uniformities and its Limit as a Methodology in Computable Analysis

Cited by 1 publication

References 12 publications

Inside the Muchnik degrees I: Discontinuity, learnability and constructivism

Inside the Muchnik degrees I: Discontinuity, learnability and constructivism

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