Growth and irreducibility in path-incompressible trees

TL;DR

This paper investigates the properties of path-incompressible trees, focusing on their transformations, density, and the relationship between their paths and randomness, revealing new insights into their structure and computational capabilities.

Contribution

It introduces new results on the transformations, density, and branching properties of path-incompressible trees, advancing understanding of their computational and randomness-preserving features.

Findings

Some path-incompressible trees do not compute perfect path-random trees.

Sparse perfect path-incompressible trees can be effectively densified.

The branching density of path-random trees is characterized.

Abstract

We study effective randomness-preserving transformations of path-incompressible trees. Some path-incompressible trees with infinitely many paths do not compute perfect path-random trees with computable oracle-use. Sparse perfect path-incompressible trees can be effectively densified, almost surely. We characterize the branching density of path-random trees.