Algorithmically random functions and effective capacities

TL;DR

This paper explores the properties of algorithmically random functions, especially online random functions, and their connection to computable capacities and random closed sets, revealing their limitations and characterizations.

Contribution

It introduces the concept of online random functions, analyzes their range properties, and connects them to known classes of random closed sets, advancing the understanding of algorithmic randomness in functions.

Findings

Online random functions are neither onto nor one-to-one.

Necessary conditions on the range of online random functions involve initial segment complexity.

A family of online partial random functions has ranges exactly matching certain random closed sets.

Abstract

We continue the investigation of algorithmically random functions and closed sets, and in particular the connection with the notion of capacity. We study notions of random continuous functions given in terms of a family of computable measures called symmetric Bernoulli measures. We isolate one particular class of random functions that we refer to as online random functions $F$, where the value of $y(n)$ for $y = F(x)$ may be computed from the values of $x(0),\dots,x(n)$. We show that random online functions are neither onto nor one-to-one. We give a necessary condition on the members of the ranges of online random functions in terms of initial segment complexity and the associated computable capacity. Lastly, we introduce the notion of online \emph{partial} Martin-L\"of random function on $2^\omega$ and give a family of online partial random functions the ranges of which are precisely the random closed sets introduced by Barmpalias, Brodhead, Cenzer, Dashti, and Weber.