On the Uniform Computational Content of Computability Theory

Abstract

We demonstrate that the Weihrauch lattice can be used to classify the uniform computational content of computability-theoretic properties as well as the computational content of theorems in one common setting. The properties that we study include diagonal non-computability, hyperimmunity, complete consistent extensions of Peano arithmetic, 1-genericity, Martin-Löf randomness, and cohesiveness. The theorems that we include in our case study are the low basis theorem of Jockusch and Soare, the Kleene-Post theorem, and Friedberg's jump inversion theorem. It turns out that all the aforementioned properties and many theorems in computability theory, including all theorems that claim the existence of some Turing degree, have very little uniform computational content: they are located outside of the upper cone of binary choice (also known as LLPO); we call problems with this property indiscriminative. Since practically all theorems from classical analysis whose computational content has been classified are discriminative, our observation could yield an explanation for why theorems and results in computability theory typically have very few direct consequences in other disciplines such as analysis. A notable exception in our case study is the low basis theorem which is discriminative. This is perhaps why it is considered to be one of the most applicable theorems in computability theory. In some cases a bridge between the indiscriminative world and the discriminative world of classical mathematics can be established via a suitable residual operation and we demonstrate this in the case of the cohesiveness problem and the problem of consistent complete extensions of Peano arithmetic. Both turn out to be the quotient of two discriminative problems.

On the Uniform Computational Content of Ramsey’s Theorem

“…We can at least say something. We have COH ≤ W lim by [8,Proposition 12.10] and by Corollary 4.10 and Lemma 4.9 we know that SRT 2,2 ≤ W RT ′ 1,2 × lim. Since lim is idempotent we obtain the following corollary.…”

“…DNC N ≤ sW RT ′ 1,2 = D 2,2 . More information on the uniform content and relations between the problems DNC, PA, WKL, WWKL, MLR, COH and other problems can be found in [8].…”

“…It follows from Corollary 5.23 below that in Corollary 3.15 C 2 cannot be replaced by ACC N , as defined in[8].…”

On the Uniform Computational Content of the Baire Category Theorem

“…We recall that in [14] a problem f was called ω-discriminative if ACC N ≤ W f and ω-indiscriminative otherwise. Here ACC N is the problem C N restricted to dom(ACC N ) = {A : |N \ A| ≤ 1}; hence the name "all-or-co-unique choice".…”

“…On the other hand, BCT 0 and BCT 2 are both ω-indiscriminative and hence also indiscriminative: since BCT 0 and BCT 2 are each densely realized, 3 by the Baire Category Theorem (1.2) itself, this follows from [14,Proposition 4.3]. Moreover, 3 A notion introduced in [14], which roughly speaking, says that the image of BCT 0 and BCT 2 is densely covered over all realizers. every jump of BCT 0 or BCT 2 is also ω-indiscriminative, since it is merely a property of the image.…”