Measurability and Computability

Weihrauch-completeness for layerwise computability

Layerwise computability is an effective counterpart to continuous functions that are almosteverywhere defined. This notion was introduced by Hoyrup and Rojas [17]. A function defined on Martin-Löf random inputs is called layerwise computable, if it becomes computable if each input is equipped with some bound on the layer where it passes a fixed universal Martin-Löf test. Interesting examples of...

Computing the Noncomputable

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 which is equivalent to the Turing halting problem and known to be mathematically noncomputable. Generalised quantum algorithms are also considered for some other mathematical noncom...

An Application of Martin-Löf Randomness to Effective Probability Theory

In this paper we provide a framework for computable analysis of measure, probability and integration theories. We work on computable metric spaces with computable Borel probability measures. We introduce and study the framework of layerwise computability which lies on Martin-Löf randomness and the existence of a universal randomness test. We then prove characterizations of effective notions of ...

Notions of Probabilistic Computability on Represented Spaces

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 ...

Existence and Measurability of the Solution of the Stochastic Differential Equations Driven by Fractional Brownian Motion