English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We study the computational content of various theorems with reversemathematical strength around Arithmetical Transfinite Recursion($\mathsf{ATR}_0$) from the point of view of computability-theoreticreducibilities, in particular Weihrauch reducibility. Our first main resultstates that it is equally hard to construct an embedding between two givenwell-orderings, as it is to construct a Turing jump hierarchy on a givenwell-ordering. This answers a question of Marcone. We obtain a similar resultfor Fra\"iss\'e's conjecture restricted to well-orderings. We then turn ourattention to K\"onig's duality theorem, which generalizes K\"onig's theoremabout matchings and covers to infinite bipartite graphs. Our second main resultshows that the problem of constructing a K\"onig cover of a given bipartitegraph is roughly as hard as the following "two-sided" version of theaforementioned jump hierarchy problem: given a linear ordering $L$, constructeither a jump hierarchy on $L$ (which may be a pseudohierarchy), or an infinite$L$-descending sequence. We also obtain several results relating the aboveproblems with choice on Baire space (choosing a path on a given ill-foundedtree) and unique choice on Baire space (given a tree with a unique path,produce said path).

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