Computable analysis provides ways of representing points in a topological space, and therefore defining notion computable the space. In this article, we investigate when two topologies on same space induce different sets points. We first study purely version problem, which is to understand are not -homeomorphic. obtain characterization leading an effective version, prove that satisfying conditi...

We define and compare several probabilistic notions of computability for mappings from represented spaces (that are equipped with a measure or outer measure) into computable metric spaces. We thereby generalize definitions by [Ko 1991] and Parker (see [Parker 2003, Parker 2005, Parker 2006]), and furthermore introduce the new notion of computability in the mean. Some results employ a notion of ...

In this paper a distinction is made between Turing’s approach to computability, on the one hand, and his approach to mathematical reasoning and intelligence, on the other hand. Unlike Church’s approach to computability, which is top-down being based on the axiomatic method, Turing’s approach to computability is bottom-up, being based on an analysis of the actions of a human computer. It is argu...

We study the computability degree of real numbers arising as L 2 -Betti or -torsion groups, parametrised over Turing word problem.

The recently initiated approach called computability logic is a formal theory of interactive computation. It understands computational problems as games played by a machine against the environment, and uses logical formalism to describe valid principles of computability, with formulas representing computational problems and logical operators standing for operations on computational problems. Th...

Scientists apply digital computers to perform computations on natural numbers, nite strings, real numbers and more general objects like sets, functions and measures. While computability theory on many countable sets is well established and for computability on the real numbers several (unfortunately mutually non-equivalent) deenitions are being studied, in particular for measures no computabili...

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 for Hilbert’s tenth problem, which is equivalent to the Turing halting problem and is known to be mathematically noncomputable, is proposed where quantum continuous variables and quantum adiabatic evolution are employ...

A condition that guaranties the exponential decay of solutions initial-boundary value problem for damped wave equation is proved. method effective computability coefficient also presented.