Layerwise computability and image randomness

TL;DR

This paper extends the theory of algorithmic randomness to layerwise computable mappings, establishing a natural equivalence between randomness of an object and its preimage under such mappings, with implications for image distribution.

Contribution

It generalizes the known folklore result from computable to layerwise computable mappings, providing a broader framework for randomness preservation.

Findings

Proves the equivalence of randomness with respect to image distribution and preimages for layerwise computable mappings.

Discusses quantitative aspects of the generalized result.

Extends classical results in algorithmic randomness theory.

Abstract

Algorithmic randomness theory starts with a notion of an individual random object. To be reasonable, this notion should have some natural properties; in particular, an object should be random with respect to image distribution if and only if it has a random preimage. This result (for computable distributions and mappings, and Martin-L\"of randomness) was known for a long time (folklore); in this paper we prove its natural generalization for layerwise computable mappings, and discuss the related quantitative results.