# Leaf management

## TL;DR

This paper investigates the complexity of leaf set problems in trees within reverse mathematics and Weihrauch frameworks, establishing equivalences among various principles and their restrictions to leaf-set trees.

## Contribution

It demonstrates that several leaf-related principles are equivalent to a key comprehension principle and are strongly Weihrauch equivalent, clarifying their computational and logical relationships.

## Key findings

- All principles are equivalent to ${
m{ ext{Pi}}^1_1}$-${
m{ ext{CA}}}_0$ over ${
m{ ext{RCA}}}_0$.
- The principles are strongly Weihrauch equivalent.
- Restrictions to leaf-set trees preserve the equivalence.

## Abstract

Finding the set of leaves for an unbounded tree is a nontrivial process in both the Weihrauch and reverse mathematics settings. Despite this, many combinatorial principles for trees are equivalent to their restrictions to trees with leaf sets. For example, let ${\widehat{\sf{WF}}}$ denote the problem of choosing which trees in a sequence are well-founded, and let ${{\sf{PK}}}$ denote the problem of finding the perfect kernel of a tree. Let ${\widehat{\sf{WF}}}_L$ and ${{\sf{PK}}}_L$ denote the restrictions of these principles to trees with leaf sets. Then ${\widehat{\sf{WF}}}$, ${\widehat{\sf{WF}}}_L$, ${{\sf{PK}}}$, and ${{\sf{PK}}}_L$ are all equivalent to ${\Pi^1_1 {\rm -} {\sf{CA}}_0}$ over ${{\sf{RCA}}_0}$, and all strongly Weihrauch equivalent.