Related papers: Categorifying computable reducibilities
In this paper we study Weihrauch reducibility for multi-valued functions on represented spaces. We call the corresponding equivalence classes Weihrauch degrees and we show that the corresponding partial order induces a lower semi-lattice…
The basic notions of quantum mechanics are formulated in terms of separable infinite dimensional Hilbert space $\mathcal{H}$. In terms of the Hilbert lattice $\mathcal{L}$ of closed linear subspaces of $\mathcal{H}$ the notions of state and…
This paper compares different representations (in the sense of computable analysis) of a number of function spaces that are of interest in analysis. In particular subspace representations inherited from a larger function space are compared…
This article is a fundamental study in computable measure theory. We use the framework of TTE, the representation approach, where computability on an abstract set X is defined by representing its elements with concrete "names", possibly…
We introduce two notions of effective reducibility for set-theoretical statements, based on computability with Ordinal Turing Machines (OTMs), one of which resembles Turing reducibility while the other is modelled after Weihrauch…
We initiate the study of computable presentations of real and complex C*-algebras under the program of effective metric structure theory. With the group situation as a model, we develop corresponding notions of recursive presentations and…
We identify a notion of reducibility between predicates, called instance reducibility, which commonly appears in reverse constructive mathematics. The notion can be generally used to compare and classify various principles studied in…
A major part of computability theory focuses on the analysis of a few structures of central importance. As a tool, the method of coding with first-order formulas has been applied with great success. For instance, in the c.e. Turing degrees,…
The history of computability theory and and the history of analysis are surprisingly intertwined since the beginning of the twentieth century. For one, \'Emil Borel discussed his ideas on computable real number functions in his introduction…
Computable reducibility is a well-established notion that allows to compare the complexity of various equivalence relations over the natural numbers. We generalize computable reducibility by introducing degree spectra of reducibility and…
Although there is a somewhat standard formalization of computability on countable sets given by Turing machines, the same cannot be said about uncountable sets. Among the approaches to define computability in these sets, order-theoretic…
In this paper, we introduce a foundation for computable model theory of rational Pavelka logic (an extension of {\L}ukasiewicz logic) and continuous logic, and prove effective versions of some theorems in model theory. We show how to reduce…
In this article, we investigate the arithmetical hierarchy from the perspective of realizability theory. An experimental observation in classical computability theory is that the notion of degrees of unsolvability for natural arithmetical…
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…
Our main result is the equivalence of two notions of reducibility between structures. One is a syntactical notion which is an effective version of interpretability as in model theory, and the other one is a computational notion which is a…
We introduce a notion of realizability with ordinal Turing machines based on recognizability rather than computability, i.e., the ability to uniquely identify an object. We show that the arising concept of $r$-realizabilty has the property…
Choosing an encoding over binary strings for input/output to/by a Turing Machine is usually straightforward and/or inessential for discrete data (like graphs), but delicate -- heavily affecting computability and even more computational…
There is a way of assigning a realizability notion to each degree of incomputability. In our setting, we make use of Weihrauch degrees (degrees of incomputability/discontinuity of partial multi-valued functions) to obtain Lifschitz-like…
Weihrauch reducibility is a notion of reducibility between computational problems that is useful to calibrate the uniform computational strength of a multivalued function. It complements the analysis of mathematical theorems done in reverse…
Every computable function has to be continuous. To develop computability theory of discontinuous functions, we study low levels of the arithmetical hierarchy of nonuniformly computable functions on Baire space. First, we classify…