# Some computability-theoretic reductions between principles around $\mathsf{ATR}_0$

## TL;DR

This paper explores the computational complexity of principles around $\

## Contribution

It establishes equivalences in computational difficulty between various theorems related to $\

## Key findings

- Constructing an embedding between well-orderings is as hard as building a Turing jump hierarchy.
- The problem of finding a K"onig cover is as complex as a two-sided jump hierarchy problem.
- Connections are made between these problems and choice principles on Baire space.

## Abstract

We study the computational content of various theorems with reverse mathematical strength around Arithmetical Transfinite Recursion ($\mathsf{ATR}_0$) from the point of view of computability-theoretic reducibilities, in particular Weihrauch reducibility. Our first main result states that it is equally hard to construct an embedding between two given well-orderings, as it is to construct a Turing jump hierarchy on a given well-ordering. This answers a question of Marcone. We obtain a similar result for Fra\"iss\'e's conjecture restricted to well-orderings. We then turn our attention to K\"onig's duality theorem, which generalizes K\"onig's theorem about matchings and covers to infinite bipartite graphs. Our second main result shows that the problem of constructing a K\"onig cover of a given bipartite graph is roughly as hard as the following "two-sided" version of the aforementioned jump hierarchy problem: given a linear ordering $L$, construct either a jump hierarchy on $L$ (which may be a pseudohierarchy), or an infinite $L$-descending sequence. We also obtain several results relating the above problems with choice on Baire space (choosing a path on a given ill-founded tree) and unique choice on Baire space (given a tree with a unique path, produce said path).