Asymptotic density, computable traceability, and 1-randomness

Abstract. For r ∈ [0, 1] we say that a set A ⊆ ω is coarsely computable at density r if there is a computable set C such that {n : C(n) = A(n)} has lower density at least r. Let γ(A) = sup{r : A is coarsely computable at density r}. We study the interactions of these concepts with Turing reducibility. For example, we show that if r ∈ (0, 1] there are sets A 0 , A 1 such that γ(A 0 ) = γ(A 1 ) = r where A 0 is coarsely computable at density r while A 1 is not coarsely computable at density r. We show that a real r ∈ [0, 1] is equal to γ(A) for some c.e. set A if and only if r is left-Σ 0 3 . A surprising result is that if G is a ∆ 0 2 1-generic set, and A T G with γ(A) = 1, then A is coarsely computable at density 1.

show abstract

“…Here it is not possible to replace 1 2 by any larger number, by Theorem 2.1. In [1], the following definition is made for Turing degrees a:…”

Section: Turing Degrees Coarse Computability and γmentioning

confidence: 99%

Asymptotic density and the coarse computability bound

Hirschfeldt

Jockusch

McNicholl

et al. 2016

COM

View full text Add to dashboard Cite

show abstract

“…Then Andrew et al [1] assign a value γ to each set of natural numbers. They use this to assign a value Γ to each Turing degree.…”

Section: Introductionmentioning

confidence: 99%

A Unifying Approach to the Gamma Question

Monin

Nies

2015

2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science

View full text Add to dashboard Cite

The Gamma question was formulated by Andrews et al. in "Asymptotic density, computable traceability and 1-randomness" (2013, available at http://www.math.wisc.edu/ ∼ lempp/papers/traceable.pdf). It is related to the recent notion of coarse computability which stems from complexity theory. The Gamma value of an oracle set measures to what extent each set computable with the oracle is approximable in the sense of density by a computable set. The closer to 1 this value is, the closer the oracle is to being computable. The Gamma question asks whether this value can be strictly in between 0 and 1/2. We say that an oracle is weakly Schnorr engulfing if it computes a Schnorr test that succeeds on all computable reals. We show that each non weakly Schnorr engulfing oracle has a Gamma value of at least 1/2. Together with a recent result of Kjos-Hanssen, Stephan, and Terwijn, this establishes new examples of such oracles. We also give a unifying approach to oracles with Gamma value 0. We say that an oracle is infinitely often equal with bound h if it computes a function that agrees infinitely often with each computable function bounded by h. We show that every oracle which is infinitely equal with bound 2 d n for d > 1, has a Gamma value of 0. This provides new examples of such oracles as well. We present a combinatorial characterization of being weakly Schnorr engulfing via traces, which is inspired by the study of cardinal characteristics in set theory.

show abstract

“…We again may assume that µ(F Φ ) > 2 3 , define S n as before, and conclude that µ(S n ) > 1 2 for density-1 many n. We then define the partial computable function d as before. If d(n)↑ or d(n)↓ = f (n), then the class of all X such that Φ X (n)↑ or Φ X (n)↓ = f (n) has measure greater than 1 2 , which implies that µ(S n ) < 1 2 , so there can be only density-0 many such n. Thus d is a computable dense description of f .…”

Section: Measure Upper Cones and Minimal Pairsmentioning

confidence: 99%

“…(There may be more than one i for which such a class exists, but we choose the first one we find.) If X ∈ F Φ , then Φ X is total and Φ X (n) ∈ {f (n), } for all n, so for each n, there are either more than measure- 1 3 many X such that Φ X (n) = f (n) or more than measure- 1 3 many X such that Φ X (n) = . Thus d is a total computable function and d(n) ∈ {f (n), } for all n. For each of the density-1 many n such that µ(S n ) 2 3 , there are at most measure- 1 3 many X such that Φ X (n) = , so d(n) = .…”

Section: Measure Upper Cones and Minimal Pairsmentioning

confidence: 99%

Dense computability, upper cones, and minimal pairs

Astor

Hirschfeldt

Jockusch

2019

COM

View full text Add to dashboard Cite

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.

show abstract

Asymptotic density, computable traceability, and 1-randomness

Cited by 11 publications

References 14 publications

Asymptotic density and the coarse computability bound

Asymptotic density and the coarse computability bound

A Unifying Approach to the Gamma Question

Dense computability, upper cones, and minimal pairs

Contact Info

Product

Resources

About