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

Computability and analysis: the legacy of Alan Turing

We discuss the legacy of Alan Turing and his impact on computability and analysis.Comment: 49 page

Completion of Choice

We systematically study the completion of choice problems in the Weihrauch lattice. Choice problems play a pivotal role in Weihrauch complexity. For one, they can be used as landmarks that characterize important equivalences classes in the Weihrauch lattice. On the other hand, choice problems also characterize several natural classes of computable problems, such as finite mind change computable problems, non-deterministically computable problems, Las Vegas computable problems and effectively Borel measurable functions. The closure operator of completion generates the concept of total Weihrauch reducibility, which is a variant of Weihrauch reducibility with total realizers. Logically speaking, the completion of a problem is a version of the problem that is independent of its premise. Hence, studying the completion of choice problems allows us to study simultaneously choice problems in the total Weihrauch lattice, as well as the question which choice problems can be made independent of their premises in the usual Weihrauch lattice. The outcome shows that many important choice problems that are related to compact spaces are complete, whereas choice problems for unbounded spaces or closed sets of positive measure are typically not complete.Comment: 30 page

Intuitionistic implication makes model checking hard

We investigate the complexity of the model checking problem for intuitionistic and modal propositional logics over transitive Kripke models. More specific, we consider intuitionistic logic IPC, basic propositional logic BPL, formal propositional logic FPL, and Jankov's logic KC. We show that the model checking problem is P-complete for the implicational fragments of all these intuitionistic logics. For BPL and FPL we reach P-hardness even on the implicational fragment with only one variable. The same hardness results are obtained for the strictly implicational fragments of their modal companions. Moreover, we investigate whether formulas with less variables and additional connectives make model checking easier. Whereas for variable free formulas outside of the implicational fragment, FPL model checking is shown to be in LOGCFL, the problem remains P-complete for BPL.Comment: 29 pages, 10 figure

The Bolzano-Weierstrass Theorem is the Jump of Weak K\"onig's Lemma

We classify the computational content of the Bolzano-Weierstrass Theorem and variants thereof in the Weihrauch lattice. For this purpose we first introduce the concept of a derivative or jump in this lattice and we show that it has some properties similar to the Turing jump. Using this concept we prove that the derivative of closed choice of a computable metric space is the cluster point problem of that space. By specialization to sequences with a relatively compact range we obtain a characterization of the Bolzano-Weierstrass Theorem as the derivative of compact choice. In particular, this shows that the Bolzano-Weierstrass Theorem on real numbers is the jump of Weak K\"onig's Lemma. Likewise, the Bolzano-Weierstrass Theorem on the binary space is the jump of the lesser limited principle of omniscience LLPO and the Bolzano-Weierstrass Theorem on natural numbers can be characterized as the jump of the idempotent closure of LLPO. We also introduce the compositional product of two Weihrauch degrees f and g as the supremum of the composition of any two functions below f and g, respectively. We can express the main result such that the Bolzano-Weierstrass Theorem is the compositional product of Weak K\"onig's Lemma and the Monotone Convergence Theorem. We also study the class of weakly limit computable functions, which are functions that can be obtained by composition of weakly computable functions with limit computable functions. We prove that the Bolzano-Weierstrass Theorem on real numbers is complete for this class. Likewise, the unique cluster point problem on real numbers is complete for the class of functions that are limit computable with finitely many mind changes. We also prove that the Bolzano-Weierstrass Theorem on real numbers and, more generally, the unbounded cluster point problem on real numbers is uniformly low limit computable. Finally, we also discuss separation techniques.Comment: This version includes an addendum by Andrea Cettolo, Matthias Schr\"oder, and the authors of the original paper. The addendum closes a gap in the proof of Theorem 11.2, which characterizes the computational content of the Bolzano-Weierstra\ss{} Theorem for arbitrary computable metric space