How Incomputable is the Separable Hahn-Banach Theorem?

Gherardi, Guido; Marcone, Alberto

doi:10.1016/j.entcs.2008.12.009

Cited by 37 publications

(87 citation statements)

References 16 publications

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“…it is complete among all limit computable operations with respect to Weihrauch reducibility and similarly LLPO is complete among all weakly computable operations [20,12]. Limit computable operations are exactly the effectively Σ 0 2 -measurable operations and these are exactly those operations that can be computed on a Turing machine that is allowed to revise its output.…”

Section: Realizability Of Theorems and Weihrauch Reducibilitymentioning

confidence: 99%

“…In our context, the most relevant approaches are constructive analysis as studied by Bishop, Bridges and Ishihara [1,15] and many others and reverse mathematics as proposed by Friedman and Simpson [48]. In computable analysis theorems have been classified according to the Borel complexity by the authors of this papers and others [5,19,10,20]:…”

Section: Realizability Of Theorems and Weihrauch Reducibilitymentioning

confidence: 99%

“…In fact the technical tool to express the relation of realizers to each other is a reducibility that Weihrauch introduced in the 1990s in two unpublished papers [50,51] and which since then has been studied by several others (see for instance [23,2,5,36,20,12,39]). Basically, the idea is to say that a singlevalued function F is Weihrauch reducible to G, in symbols F ≤ W G, if there are computable functions H and K such that F = H id, GK .…”

Section: Realizability Of Theorems and Weihrauch Reducibilitymentioning

confidence: 99%

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Effective Choice and Boundedness Principles in Computable Analysis

2011

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Abstract. In this paper we study a new approach to classify mathematical theorems according to their computational content. Basically, we are asking the question which theorems can be continuously or computably transferred into each other? For this purpose theorems are considered via their realizers which are operations with certain input and output data. The technical tool to express continuous or computable relations between such operations is Weihrauch reducibility and the partially ordered degree structure induced by it. We have identified certain choice principles such as co-finite choice, discrete choice, interval choice, compact choice and closed choice, which are cornerstones among Weihrauch degrees and it turns out that certain core theorems in analysis can be classified naturally in this structure. In particular, we study theorems such as the Intermediate Value Theorem, the Baire Category Theorem, the Banach Inverse Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem. We also explore how existing classifications of the Hahn-Banach Theorem and Weak Kőnig's Lemma fit into this picture. Well-known omniscience principles from constructive mathematics such as LPO and LLPO can also naturally be considered as Weihrauch degrees and they play an important role in our classification. Based on this we compare the results of our classification with existing classifications in constructive and reverse mathematics and we claim that in a certain sense our classification is finer and sheds some new light on the computational content of the respective theorems. Our classification scheme does not require any particular logical framework or axiomatic setting, but it can be carried out in the framework of classical mathematics using tools of topology, computability theory and computable analysis. We develop a number of separation techniques based on a new parallelization principle, on certain invariance properties of Weihrauch reducibility, on the Low Basis Theorem of Jockusch and Soare and based on the Baire Category Theorem. Finally, we present a number of metatheorems that allow to derive upper bounds for the classification of the Weihrauch degree of many theorems and we discuss the Brouwer Fixed Point Theorem as an example. §1. Introduction. The purpose of this paper is to propose a new approach to classify mathematical theorems according to their computational content and according to their logical complexity.2000 Mathematics Subject Classification. 03F60,03D30,03B30,03E15.

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Section: Realizability Of Theorems and Weihrauch Reducibilitymentioning

confidence: 99%

Section: Realizability Of Theorems and Weihrauch Reducibilitymentioning

confidence: 99%

Section: Realizability Of Theorems and Weihrauch Reducibilitymentioning

confidence: 99%

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Effective Choice and Boundedness Principles in Computable Analysis

2011

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“…But sometimes a solution of one problem can help to solve another one. Such "helping" between in general non-computable problems can be formalized by a reducibility relation as follows: For functions f, g between represented spaces define f ≤ W g if there are computable (partial) functions K, H on natural sets such that for every realizer [Sim99] has been discussed in [GM09,BG11b].…”

Section: Computability Via Representationsmentioning

confidence: 99%

Computability of the Radon-Nikodym Derivative

Hoyrup

Rojas

Weihrauch

2011

Models of Computation in Context

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We study the computational content of the Radon-Nokodym theorem from measure theory in the framework of the representation approach to computable analysis. We define computable measurable spaces and canonical representations of the measures and the integrable functions on such spaces. For functions f, g on represented sets, f is W-reducible to g if f can be computed by applying the function g at most once. Let RN be the Radon-Nikodym operator on the space under consideration and let EC be the non-computable operator mapping every enumeration of a set of natural numbers to its characteristic function. We prove that for every computable measurable space, RN is W-reducible to EC, and we construct a computable measurable space for which EC is W-reducible to RN.

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“…This reducibility (in its modern form) was introduced by Gherardi and Marcone [GM09] and Brattka and Gherardi [BG11b,BG11a] based on earlier work by Weihrauch on a reducibility between sets of functions analogous to Wadge reducibility [Wei92a,Wei92b].…”

Section: Introductionmentioning

confidence: 99%

Game Characterizations and Lower Cones in the Weihrauch Degrees

Nobrega

Pauly

2017

Unveiling Dynamics and Complexity

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Abstract. We introduce a parametrized version of the Wadge game for functions and show that each lower cone in the Weihrauch degrees is characterized by such a game. These parametrized Wadge games subsume the original Wadge game, the eraser and backtrack games as well as Semmes's tree games. In particular, we propose that the lower cones in the Weihrauch degrees are the answer to Andretta's question on which classes of functions admit game characterizations. We then discuss some applications of such parametrized Wadge games. Using machinery from Weihrauch reducibility theory, we introduce games characterizing every (transfinite) level of the Baire hierarchy via an iteration of a pruning derivative on countably branching trees.

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How Incomputable is the Separable Hahn-Banach Theorem?

Cited by 37 publications

References 16 publications

Effective Choice and Boundedness Principles in Computable Analysis

Effective Choice and Boundedness Principles in Computable Analysis

Computability of the Radon-Nikodym Derivative

Game Characterizations and Lower Cones in the Weihrauch Degrees

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