Effective Choice and Boundedness Principles in Computable Analysis

Abstract

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.

Total Representations

“…For recent results on similar reducibilities on arbitrary quasi-Polish spaces see [MSS12] In [Wei92, Her93, Wei00] some notions of reducibility for functions on spaces were introduced which turned out useful for understanding the non-computability and non-continuity of interesting decision problems in computable analysis [Her96,BG11a] and constructive mathematics [BG11]. In particular, the following notions of reducibilities between functions f : X → Z, g : Y → Z on topological spaces were introduced:…”

“…For this purpose people usually prefer to use complete sets in suitable effective hierarchies like those discussed in the previous section. Another way to "improve" the algebraic structure of, say, Weihrauch degrees is to extend the Weihrauch reducibility to multi-valued functions [Wei92,Wei00,BG11,BG11a]. In this way one obtains algebraically more regular degree structures which are applicable to the complexity of many interesting problems related to Constructive Analysis.…”

Computability of the Radon-Nikodym Derivative

“…Furthermore, EC is complete for effectively Σ 0 2 -measurable functions (in the Borel hierarchy) with respect to ≤ W [Bra05, Theorem 7.6]. Many non-computable problems from Analysis are equivalent to EC [Wei92a,BG11b,BG11a].…”

“…In this setting, the Radon-Nikodym operator RN mapping λ and µ to the function h is not computable. We characterize its degree of non-computability in the ≤ W hierarchy of problems on represented sets [Wei92a,BG11b,BG11a]. Let EC be the non-computable operator mapping every enumeration of a set of natural numbers to its characteristic function.…”

Game Characterizations and Lower Cones in the Weihrauch Degrees

“…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].…”