arXiv subjects

Willem Fouché

Publications and source records attributed to Willem Fouché.

3 recordsLinked to original sources

arXiv2019

Randomized Computation of Continuous Data: Is Brownian Motion Computable?

We consider randomized computation of continuous data in the sense of Computable Analysis. Our first contribution formally confirms that it is no loss of generality to take as sample space the Cantor space of infinite FAIR coin flips. This extends [Schröder&Simpson'05] and [Hoyrup&Rojas'09] considering sequences of suitably and adaptively BIASED coins. Our second contribution is concerned with 1D Brownian Motion (aka Wiener Process), a probability distribution on the space of continuous functions f:[0,1]->R with f(0)=0 whose computability has been conjectured [Davie&Fouché'13; arXiv:1409.4667,S6]. We establish that this (higher-type) random variable is computable iff some/every computable family of moduli of continuity (as ordinary random variables) has a computable probability distribution with respect to the Wiener Measure.

math.NA↗

arXiv2018

Weihrauch-completeness for layerwise computability

We introduce the notion of being Weihrauch-complete for layerwise computability and provide several natural examples related to complex oscillations, the law of the iterated logarithm and Birkhoff's theorem. We also consider hitting time operators, which share the Weihrauch degree of the former examples but fail to be layerwise computable.

cs.LO↗

arXiv2007

Deutsch's Universal Quantum Turing Machine (Revisited)

Deutsch, Feynman, and Manin viewed quantum computing as a kind of universal physical simulation procedure. Much of the writing about quantum Turing machines has shown how these machines can simulate an arbitrary unitary transformation on a finite number of qubits. This interesting problem has been addressed most famously in a paper by Deutsch, and later by Bernstein and Vazirani. Quantum Turing machines form a class closely related to deterministic and probabilistic Turing machines and one might hope to find a universal machine in this class. A universal machine is the basis of a notion of programmability. The extent to which universality has in fact been established by the pioneers in the field is examined and a key notion in theoretical computer science (universality) is scrutinised. In a forthcoming paper, the authors will also consider universality in the quantum gate model.

quant-ph↗