Related papers: On the relation between representations and comput…
The Church-Turing thesis asserts that if a partial strings-to-strings function is effectively computable then it is computable by a Turing machine. In the 1930s, when Church and Turing worked on their versions of the thesis, there was a…
This survey paper examines the effective model theory obtained with the BSS model of real number computation. It treats the following topics: computable ordinals, satisfaction of computable infinitary formulas, forcing as a construction…
We develop synthetic notions of oracle computability and Turing reducibility in the Calculus of Inductive Constructions (CIC), the constructive type theory underlying the Coq proof assistant. As usual in synthetic approaches, we employ a…
This paper reviews connections between physics and computation, and explores their implications. The main topics are computational "hardness" of physical systems, computational status of fundamental theories, quantum computation, and the…
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…
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…
Significant advances in the development of computing devices based on quantum effects and the demonstration of their use to solve various problems have rekindled interest in the nature of the "quantum computational advantage." Although…
We provide a detailed study of two properties of spaces and pairs of spaces, the surjection property and the epsilon-surjection property, that were recently introduced to characterize the notion of computable type arising from computability…
The topic of this paper is the subtle interplay between countability and representations. In particular, we establish that the definition of countability of a certain set $X$ crucially hinges on the associated equivalence relation $=_{X}$.…
We explore in the framework of Quantum Computation the notion of computability, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm that exploits the quantum adiabatic processes is considered…
The familiar theories of physics have the feature that the application of the theory to make predictions in specific circumstances can be done by means of an algorithm. We propose a more precise formulation of this feature --- one based on…
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…
A universal Turing machine is a powerful concept - a single device can compute any function that is computable. A universal spin model, similarly, is a class of physical systems whose low energy behavior simulates that of any spin system.…
We study a new notion of reduction between structures called enumerable functors related to the recently investigated notion of computable functors. Our main result shows that enumerable functors and effective interpretability with the…
This article is a fundamental study in computable analysis. In the framework of Type-2 effectivity, TTE, we investigate computability aspects on finite and infinite products of effective topological spaces. For obtaining uniform results we…
We exhibit a sound and complete implicit-complexity formalism for functions feasibly computable by structural recursions over inductively defined data structures. Feasibly computable here means that the structural-recursive definition runs…
We investigate when a computable automorphism of a computable field can be effectively extended to a computable automorphism of its (computable) algebraic closure. We then apply our results and techniques to study effective embeddings of…
This work exposes which mechanisms and procesess in the Nature of evolution compute a function not computable by Turing machine. The computer with intelligence that is not higher than one bacteria population could have, but with efficency…
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…
Human societies continuously transform scattered information into collective judgments and coordinated action, whether through markets discovering prices, governments allocating resources, communities enforcing norms, or science converging…