Growth and irreducibility in path-incompressible trees

TL;DR

This work investigates how randomness properties in path-incompressible trees behave under effective transformations that modify topological density, specifically branching. Using a framework of hitting-costs, envelope families, and tree-functionals, it proves a partial negative answer to whether density can be increased without loss of path-incompressibility, constructing a path-incompressible proper tree that cannot compute any path-incompressible perfect tree within computable oracle-use. It also establishes precise density criteria (via a computable $oldsymbol{ ell}$) under which $oldsymbol{ ext{ell-perfect}}$ path-random and path-incompressible trees are equivalent, and shows a method to densify sparse perfect path-incompressible trees to denser ones with high probability. These results illuminate the limits of randomness extraction in tree-structured data and provide a robust methodology for analyzing transformations between closed sets of random points in Cantor space. The findings have implications for compactness in fragments of second-order arithmetic and related areas of algorithmic randomness and structure of trees.

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.

Theorems & Definitions (46)

• Definition 1.1
• Theorem 1.2
• Proposition 1.3
• Theorem 1.4
• Theorem 1.5: Informal
• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• Proposition : Proposition \ref{['jlTkH88T3Z']}