Generic computability, Turing degrees, and asymptotic density

Jockusch, Carl G.; Schupp, Paul E.

doi:10.1112/jlms/jdr051

Cited by 47 publications

(125 citation statements)

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“…These include the generic-case and coarse reducibilities defined by Jockusch and Schupp [29], as well as several ones introduced in [15]. One of these is infinite information reducibility, denoted by ii .…”

Section: Principles 23mentioning

confidence: 99%

On notions of computability-theoretic reduction between Π21 principles

Hirschfeldt

Jockusch

2016

J. Math. Log.

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Abstract. Several notions of computability theoretic reducibility between Π 1 2 principles have been studied. This paper contributes to the program of analyzing the behavior of versions of Ramsey's Theorem and related principles under these notions. Among other results, we show that for each n 3, there is an instance of RT n 2 all of whose solutions have PA degree over ∅ (n−2) , and use this to show that König's Lemma lies strictly between RT 2 2 and RT 3 2 under one of these notions. We also answer two questions raised by Dorais, Dzhafarov, Hirst, Mileti, and Shafer [ta] on comparing versions of Ramsey's Theorem and of the Thin Set Theorem with the same exponent but different numbers of colors. Still on the topic of the effect of the number of colors on the computable aspects of Ramsey theoretic properties, we show that for each m 2, there is an (m + 1)-coloring c of N such that every m-coloring of N has an infinite homogeneous set that does not compute any infinite homogeneous set for c, and connect this result with the notion of infinite information reducibility introduced by Dzhafarov and Igusa [ta]. Next, we introduce and study a new notion that provides a uniform version of the idea of implication with respect to ω-models of RCA 0 , and related notions that allow us to count how many applications of a principle P are needed to reduce another principle to P . Finally, we fill in a gap in the proof of Theorem 12.2 in Cholak, Jockusch, and Slaman [2001].

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Section: Principles 23mentioning

confidence: 99%

On notions of computability-theoretic reduction between Π21 principles

Hirschfeldt

Jockusch

2016

J. Math. Log.

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“…We thus have a full justification of the implications and nonimplications in Figure 1, but we can actually say more. Jockusch and Schupp [13,Theorem 2.22] built a c.e. set of density 1 with no computable subset of density 1.…”

Section: Comparing Notions Of Asymptotic Computabilitymentioning

confidence: 99%

“…Then C is c.e., C is coarsely computable by Theorem 2.19 of [13], and α(C) = 1 as in the proof of Proposition 2.3. Also, since B is not computable, C is not generically computable by Observation 2.11 of [13]. Hence, C is also not effectively densely computable, and so witnesses that (i) holds.…”

Section: Comparing Notions Of Asymptotic Computabilitymentioning

confidence: 99%

“…In addition to defining generic-case complexity, Kapovich, Myasnikov, Schupp, and Shpilrain [14] introduced the notion of generic computability. Jockusch and Schupp [13] then began to study it from a computability-theoretic viewpoint. They also defined coarse computability, although unbeknownst to them, that notion had already been considered by Terwijn [18,Section 3.3].…”

Section: Introductionmentioning

confidence: 99%

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Dense computability, upper cones, and minimal pairs

Astor

Hirschfeldt

Jockusch

2019

COM

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This paper concerns algorithms that give correct answers with (asymptotic) density 1. A dense description of a function g : ω → ω is a partial function f on ω such that {n : f (n) = g(n)} has density 1. We define g to be densely computable if it has a partial computable dense description f . Several previous authors have studied the stronger notions of generic computability and coarse computability, which correspond respectively to requiring in addition that g and f agree on the domain of f , and to requiring that f be total. Strengthening these two notions, call a function g effectively densely computable if it has a partial computable dense description f such that the domain of f is a computable set and f and g agree on the domain of f . We compare these notions as well as asymptotic approximations to them that require for each ε > 0 the existence of an appropriate description that is correct on a set of lower density of at least 1−ε. We determine which implications hold among these various notions of approximate computability and show that any Boolean combination of these notions is satisfied by a c.e. set unless it is ruled out by these implications. We define reducibilities corresponding to dense and effectively dense reducibility and show that their uniform and nonuniform versions are different. We show that there are natural embeddings of the Turing degrees into the corresponding degree structures, and that these embeddings are not surjective and indeed that sufficiently random sets have quasiminimal degree. We show that nontrivial upper cones in the generic, dense, and effective dense degrees are of measure 0 and use this fact to show that there are minimal pairs in the dense degrees.

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“…Question 4 (Jockusch and Schupp [9]; Igusa [7]; see also [5]). Are there minimal pairs in the (nonuniform or uniform) generic degrees?…”

Section: Definitionmentioning

confidence: 99%