Effective Choice and Boundedness Principles in Computable Analysis

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 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 and others. We also explore how existing classifications of the Hahn-Banach Theorem and Weak K"onig's Lemma fit into this picture. 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. 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

Effective Choice and Boundedness Principles in Computable Analysis

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 on closed sets 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. Well-known omniscience principles from constructive mathematics such as $LPO$ and $LLPO$ can naturally be considered as Weihrauch degrees and they play an important role in our classification. 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. 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

Calibrating the complexity of combinatorics: reverse mathematics and Weihrauch degrees of some principles related to Ramsey’s theorem

In this thesis, we study the proof-theoretical and computational strength of some combinatorial principles related to Ramsey's theorem: this will be accomplished chiefly by analyzing these principles from the points of view of reverse mathematics and Weihrauch complexity. We start by studying a combinatorial principle concerning graphs, introduced by Bill Rival and Ivan Sands as a form of ``inside-outside'' Ramsey's theorem: we will determine its reverse mathematical strength and present the result characterizing its Weihrauch degree. Moreover, we will study a natural restriction of this principle, proving that it is equivalent to Ramsey's theorem. We will then move to a related result, this time concerning countable partial orders, again introduced by Rival and Sands: we will give a thorough reverse mathematical investigation of the strength of this theorem and of its original proof. Moreover, we will be able to generalize it, and this generalization will itself be presented in the reverse mathematical perspective. After this, we will focus on two forms of Ramsey's theorem that can be considered asymmetric. First, we will focus on a restriction of Ramsey's theorem to instances whose solutions have a predetermined color, studying it in reverse mathematics and from the point of view of the complexity of the solutions in a computability theoretic sense. Next, we move to a classical result about partition ordinals, which will undergo the same type of analysis. Finally, we will present some results concerning a recently introduced operator on the Weihrauch degrees, namely the first-order part operator: after presenting an alternative characterization of it, we will embark on the study the result of its applications to jumps of Weak Kőnig's Lemma