Π10-Encodability and Omniscient Reductions

Abstract

A set of integers A is computably encodable if every infinite set of integers has an infinite subset computing A. By a result of Solovay, the computably encodable sets are exactly the hyperarithmetic ones. In this paper, we extend this notion of computable encodability to subsets of the Baire space and we characterize the Π 0 1 encodable compact sets as those who admit a non-empty Σ 1 1 subset. Thanks to this equivalence, we prove that weak weak König's lemma is not strongly computably reducible to Ramsey's theorem. This answers a question of Hirschfeldt and Jockusch. Π 0 1 ENCODABILITY AND OMNISCIENT REDUCTIONS 3 Π 0 1 ENCODABILITY AND OMNISCIENT REDUCTIONS 9

Lawvere-Tierney topologies for computability theorists

“…For instance, f is computable reducible to g in the sense of [14,19] if and only if f is one-query bilayered Turing reducible to (g | Advice N ) (see Definition 4.5) if we properly extend the above notions to the context of N N -computability. The notion of omniscient computable/Weihrauch reducibility [26,15,13] can also be explained in the bilayer context.…”

Some results concerning the SRT 2 2 vs. COH problem

“…Intuitively, we can think of P as being a combinatorial consequence of Q. This notion was first isolated and studied by Monin and Patey [27] under the name of omniscient computable reducibility.…”

“…The second is due to Cholak, Jockusch, and Slaman [4,Section 4]. Both have found wide application in the literature (e.g., [6,10,11,12,14,27,32,34]). And while both methods use Mathias forcing with similar conditions, the combinatorial cores of the two approaches are somewhat different, and this fact is reflected in their effectivity.…”

“…Over the past several years, work on the effective content of variants of Ramsey's theorem, including towards a solution of the above problem, has driven much of the progress in computable combinatorics and the reverse mathematics of combinatorial principles. It has also been one of the main impetuses for the fruitful and growing intersection of these subjects with computable analysis (see, e.g., [1,3,7,11,15,18,21,22,27,28,31,34]). In this paper, we study several aspects of this problem.…”

Ramsey’s theorem and products in the Weihrauch degrees

“…The latter is a tool that has been widely deployed in computable analysis and complexity theory; see the recent survey article by Brattka, Gherardi, and Pauly [4]. Recently it has gained prominence also in the study of computable combinatorics, and it is currently seeing a surge of activity; see, e.g., [1,10,13,15,18,22,24,25,28,30,31,32]. See also Brattka [2] for an updated bibliography.…”