Search

We explore the Weihrauch degree of the problems ``find a bad sequence in a non-well quasi order'' ($\mathsf{BS}$) and ``find a descending sequence in an ill-founded linear order'' ($\mathsf{DS}$). We prove that $\mathsf{DS}$ is strictly Weihrauch reducible to $\mathsf{BS}$, correcting our mistaken claim in [arXiv:2010.03840]. This is done by separating their respective first-order parts. On the other hand, we show that $\mathsf{BS}$ and $\mathsf{DS}$ have the same finitary and deterministic parts, confirming that $\mathsf{BS}$ and $\mathsf{DS}$ have very similar uniform computational strength. We prove that K\"onig's lemma $\mathsf{KL}$ and the problem $\mathsf{wList}_{2^{\mathbb{N}},\leq\omega}$ of enumerating a given non-empty countable closed subset of $2^{\mathbb{N}}$ are not Weihrauch reducible to $\mathsf{DS}$ or $\mathsf{BS}$, resolving two main open questions raised in [arXiv:2010.03840]. We also answer the question, raised in [arXiv:1804.10968], on the existence of a ``parallel quotient'' operator, and study the behavior of $\mathsf{BS}$ and $\mathsf{DS}$ under the quotient with some known problems., Comment: This is an extended version of the homonymous paper published in: Twenty Years of Theoretical and Practical Synergies. CiE 2024. Lecture Notes in Computer Science, vol 14773, pp. 339-350

Halin [1965] proved that if a graph has $n$ many pairwise disjoint rays for each $n$ then it has infinitely many pairwise disjoint rays. We analyze the complexity of this and other similar results in terms of computable and proof theoretic complexity. The statement of Halin's theorem and the construction proving it seem very much like standard versions of compactness arguments such as K\"{o}nig's Lemma. Those results, while not computable, are relatively simple. They only use arithmetic procedures or, equivalently, finitely many iterations of the Turing jump. We show that several Halin type theorems are much more complicated. They are among the theorems of hyperarithmetic analysis. Such theorems imply the ability to iterate the Turing jump along any computable well ordering. Several important logical principles in this class have been extensively studied beginning with work of Kreisel, H. Friedman, Steel and others in the 1960s and 1970s. Until now, only one purely mathematical example was known. Our work provides many more and so answers Question 30 of Montalb\'{a}n's Open Questions in Reverse Mathematics [2011]. Some of these theorems including ones in Halin [1965] are also shown to have unusual proof theoretic strength as well., Comment: 51 pages, 4 figures, Version E

Let At denote the set of infinite sequences of effective dimension t. We determine both how close and how far an infinite sequence of dimension s can be from one of dimension t, measured using the Besicovitch pseudometric. We also identify classes of sequences for which these infima and suprema are realized as minima and maxima. When t < s, we find d(X,At) is minimized when X is a Bernoulli p-random, where H(p)=s, and maximized when X belongs to a class of infinite sequences that we call s-codewords. When s < t, the situation is reversed., Comment: 28 pages, 3 figures

In this work we investigate the Weihrauch degree of the problem $\mathsf{DS}$ of finding an infinite descending sequence through a given ill-founded linear order, which is shared by the problem $\mathsf{BS}$ of finding a bad sequence through a given non-well quasi-order. We show that $\mathsf{DS}$, despite being hard to solve (it has computable inputs with no hyperarithmetic solution), is rather weak in terms of uniform computational strength. To make the latter precise, we introduce the notion of the deterministic part of a Weihrauch degree. We then generalize $\mathsf{DS}$ and $\mathsf{BS}$ by considering $\boldsymbol{\Gamma}$-presented orders, where $\boldsymbol{\Gamma}$ is a Borel pointclass or $\boldsymbol{\Delta}^1_1$, $\boldsymbol{\Sigma}^1_1$, $\boldsymbol{\Pi}^1_1$. We study the obtained $\mathsf{DS}$-hierarchy and $\mathsf{BS}$-hierarchy of problems in comparison with the (effective) Baire hierarchy and show that they do not collapse at any finite level., Comment: Added errata. The problems $\mathsf{DS}$ and $\mathsf{BS}$ are not Weihrauch-equivalent, and the separation has been proved in arXiv:2401.11807. Please check the errata for the full list of changes

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).

We study the positions in the Weihrauch lattice of parallel products of various combinatorial principles related to Ramsey's theorem. Among other results, we obtain an answer to a question of Brattka, by showing that Ramsey's theorem for pairs ($\mathsf{RT}^2_2$) is strictly Weihrauch below the parallel product of the stable Ramsey's theorem for pairs and the cohesive principle ($\mathsf{SRT}^2_2 \times \mathsf{COH}$)., Comment: 30 pages