Computable analysis

1993 ◽  
pp. 145-153
Author(s):  
Vladimir Uspensky ◽  
Alexei Semenov
Keyword(s):  
Computable Analysis

On the computability of a construction of Brownian motion

2013 ◽  
Vol 23 (6) ◽  
pp. 1257-1265 ◽  
Author(s):  
GEORGE DAVIE ◽  
WILLEM L. FOUCHÉ
Keyword(s):  
Brownian Motion ◽  
Lower Bound ◽  
Rate Of Convergence ◽  
Upper Bound ◽  
Binary Sequence ◽  
Computable Analysis

We examine a construction due to Fouché in which a Brownian motion is constructed from an algorithmically random infinite binary sequence. We show that although the construction is provably not computable in the sense of computable analysis, a lower bound for the rate of convergence is computable in any upper bound for the compressibilty of the sequence, making the construction layerwise computable.

Further Topics in Computable Analysis

2017 ◽  
pp. 50-74
Author(s):  
Marian B. Pour-El ◽  
J. Ian Richards
Keyword(s):  
Computable Analysis

Signal Recovery by Linear Estimation

2019 ◽  
pp. 260-385
Author(s):  
John Stillwell
Keyword(s):  
Human Activities ◽  
Mathematical Concept ◽  
Rational Numbers ◽  
Precise Definition ◽  
Computable Analysis ◽  
Signal Recovery ◽  
Linear Estimation ◽  
Definition Of ◽  
Computable Sequence ◽  
Informal Description

This chapter develops the basic results of computability theory, many of which are about noncomputable sequences and sets, with the goal of revealing the limits of computable analysis. Two of the key examples are a bounded computable sequence of rational numbers whose limit is not computable, and a computable tree with no computable infinite path. Computability is an unusual mathematical concept, because it is most easily used in an informal way. One often talks about it in terms of human activities, such as making lists, rather than by applying a precise definition. Nevertheless, there is a precise definition of computability, so this informal description of computations can be formalized.

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