Bulletin of Symbolic Logic

Effective choice and boundedness principles in computable analysis

Vasco Brattka and Guido Gherardi

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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.

Article information

Source
Bull. Symbolic Logic Volume 17, Issue 1 (2011), 73-117.

Dates
First available: 5 January 2011

Permanent link to this document
http://projecteuclid.org/euclid.bsl/1294186663

Digital Object Identifier
doi:10.2178/bsl/1294186663

Zentralblatt MATH identifier
05868865

Mathematical Reviews number (MathSciNet)
MR2760117

Subjects
Primary: 03F60: Constructive and recursive analysis [See also 03B30, 03D45, 03D78, 26E40, 46S30, 47S30] 03D30: Other degrees and reducibilities 03B30, 03E15

Keywords
Computable analysis constructive analysis reverse mathematics effective descriptive set theory

Citation

Brattka, Vasco; Gherardi, Guido. Effective choice and boundedness principles in computable analysis. Bulletin of Symbolic Logic 17 (2011), no. 1, 73--117. doi:10.2178/bsl/1294186663. http://projecteuclid.org/euclid.bsl/1294186663.


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