A Galois connection between Turing jumps and limits

Limit computable functions can be characterized by Turing jumps on the input side or limits on the output side. As a monad of this pair of adjoint operations we obtain a problem that characterizes the low functions and dually to this another problem that characterizes the functions that are computable relative to the halting problem. Correspondingly, these two classes are the largest classes of functions that can be pre or post composed to limit computable functions without leaving the class of limit computable functions. We transfer these observations to the lattice of represented spaces where it leads to a formal Galois connection. We also formulate a version of this result for computable metric spaces. Limit computability and computability relative to the halting problem are notions that coincide for points and sequences, but even restricted to continuous functions the former class is strictly larger than the latter. On computable metric spaces we can characterize the functions that are computable relative to the halting problem as those functions that are limit computable with a modulus of continuity that is computable relative to the halting problem. As a consequence of this result we obtain, for instance, that Lipschitz continuous functions that are limit computable are automatically computable relative to the halting problem. We also discuss 1-generic points as the canonical points of continuity of limit computable functions, and we prove that restricted to these points limit computable functions are computable relative to the halting problem. Finally, we demonstrate how these results can be applied in computable analysis

An extension of Chaitin's halting probability \Omega to a measurement operator in an infinite dimensional quantum system

This paper proposes an extension of Chaitin's halting probability \Omega to a measurement operator in an infinite dimensional quantum system. Chaitin's \Omega is defined as the probability that the universal self-delimiting Turing machine U halts, and plays a central role in the development of algorithmic information theory. In the theory, there are two equivalent ways to define the program-size complexity H(s) of a given finite binary string s. In the standard way, H(s) is defined as the length of the shortest input string for U to output s. In the other way, the so-called universal probability m is introduced first, and then H(s) is defined as -log_2 m(s) without reference to the concept of program-size. Mathematically, the statistics of outcomes in a quantum measurement are described by a positive operator-valued measure (POVM) in the most general setting. Based on the theory of computability structures on a Banach space developed by Pour-El and Richards, we extend the universal probability to an analogue of POVM in an infinite dimensional quantum system, called a universal semi-POVM. We also give another characterization of Chaitin's \Omega numbers by universal probabilities. Then, based on this characterization, we propose to define an extension of \Omega as a sum of the POVM elements of a universal semi-POVM. The validity of this definition is discussed. In what follows, we introduce an operator version \hat{H}(s) of H(s) in a Hilbert space of infinite dimension using a universal semi-POVM, and study its properties.Comment: 24 pages, LaTeX2e, no figures, accepted for publication in Mathematical Logic Quarterly: The title was slightly changed and a section on an operator-valued algorithmic information theory was adde

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

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

Functional first order definability of LRTp

The language LRTp is a non-deterministic language for exact real number computation. It has been shown that all computable rst order relations in the sense of Brattka are denable in the language. If we restrict the language to single-valued total relations (e.g. functions), all polynomials are denable in the language. This paper is an expanded version of [12] in which we show that the non-deterministic version of the limit operator, which allows to dene all computable rst order relations, when restricted to single-valued total inputs, produces single-valued total outputs. This implies that not only the polynomials are denable in the language but also allcomputable rst order functions

On the topological aspects of the theory of represented spaces

Represented spaces form the general setting for the study of computability derived from Turing machines. As such, they are the basic entities for endeavors such as computable analysis or computable measure theory. The theory of represented spaces is well-known to exhibit a strong topological flavour. We present an abstract and very succinct introduction to the field; drawing heavily on prior work by Escard\'o, Schr\"oder, and others. Central aspects of the theory are function spaces and various spaces of subsets derived from other represented spaces, and -- closely linked to these -- properties of represented spaces such as compactness, overtness and separation principles. Both the derived spaces and the properties are introduced by demanding the computability of certain mappings, and it is demonstrated that typically various interesting mappings induce the same property.Comment: Earlier versions were titled "Compactness and separation for represented spaces" and "A new introduction to the theory of represented spaces