A Unifying Approach to the Gamma Question

Monin, Benoît; Nies, André

doi:10.1109/lics.2015.60

Cited by 13 publications

(17 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For the operator IOE, separations for some rather special cases of functions g,h were obtained in [17]. We answer the full question for IOE in the affirmative.…”

Section: A Pair Of Dual Mass Problems For Functionsmentioning

confidence: 74%

“…They then applied the framework of Rupprecht [22], in the notation of Brooke-Taylor et al [4]. This led to cardinal characteristics d(p), the least size of a set G of bit sequences so that for each bit sequence x there is a bit sequence y in G such that (x ⇔ y) > p. Dualising this both in computability and in set theory, they introduced for each 0 ≤ p < 1/2 the highness property B(p), the class of oracles A that compute a bit sequence Y such that for each computable sequence X, we have (X ⇔ Y ) > p, and the analogous cardinal characteristic b(p), the least size of a set F of bit sequences so that for each bit sequence y, there is a bit sequence x in F such that (x ⇔ y) ≤ p. [17], we will show that all the highness properties D(p) coincide for 0 < p < 1/2, and similarly for the highness properties B(p). Since Γ(A) < p ⇒ A ∈ D(p), we reobtain Monin's result that Γ(A) < 1/2 implies Γ(A) = 0.…”

Section: Dualitymentioning

confidence: 99%

“…A proper hierarchy of problems IOE(h) in the weak degrees. Recall that by IOE(h) we denote the mass problem of functions f such that ∃ ∞ n [f(n) = r(n)] for each computable function r < h. In this section we study how the Muchnik degree of IOE(h) depends on the function h. In [17] the authors obtained the following two results: Theorem 5.1. ( [17], Theorem IV.1) Let c ≥ 2 be any integer, which we view as a constant function.…”

Section: 9mentioning

confidence: 99%

“…Monin [19] answered their question in the negative, and also characterized the degrees with Γ-value < 1/2. He built on some initial work of the present authors [17] involving functions that agree with each computable function infinitely often.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Muchnik Degrees and Cardinal Characteristics

Monin¹,

Nies²

2020

J. symb. log.

Self Cite

View full text Add to dashboard Cite

A mass problem is a set of functions → . For mass problems C,D, one says that C is Muchnik reducible to D if each function in C is computed by a function in D. In this paper we study some highness properties of Turing oracles, which we view as mass problems. We compare them with respect to Muchnik reducibility and its uniform strengthening, Medvedev reducibility.For p ∈ [0,1] let D(p) be the mass problem of infinite bit sequences y (i.e., {0,1}-valued functions) such that for each computable bit sequence x, the bit sequence x ↔ y has asymptotic lower density at most p (where x ↔ y has a 1 in position n iff x(n) = y(n)). We show that all members of this family of mass problems parameterized by a real p with 0 < p < 1/2 have the same complexity in the sense of Muchnik reducibility. We prove this by showing Muchnik equivalence of the problems D(p) with the mass problem IOE(2 2 n ); here for an order function h, the mass problem IOE(h) consists of the functions f that agree infinitely often with each computable function bounded by h. This result also yields a new version of the proof to of the affirmative answer to the "Gamma question" due to the first author: Γ(A) < 1/2 implies Γ(A) = 0 for each Turing oracle A.As a dual of the problem D(p), define B(p), for 0 ≤ p < 1/2, to be the set of bit sequences y such that (x ↔ y) > p for each computable set x. We prove that the Medvedev (and hence Muchnik) complexity of the mass problems B(p) is the same for all p ∈ (0,1/2), by showing that they are Medvedev equivalent to the mass problem of functions bounded by 2 2 n that are almost everywhere different from each computable function.Next, together with Joseph Miller, we obtain a proper hierarchy of the mass problems of type IOE: we show that for any order function g there exists a faster growing order function h such that IOE(h) is strictly above IOE(g) in the sense of Muchnik reducibility.We study cardinal characteristics in the sense of set theory that are analogous to the highness properties above. For instance, d(p) is the least size of a set G of bit sequences such that for each bit sequence x there is a bit sequence y in G so that (x ↔ y) > p. We prove within ZFC all the coincidences of cardinal characteristics that are the analogs of the results above. §1. Introduction. It is of fundamental interest in computability theory to determine the inherent computational complexity of an object, such as an infinite bit sequence, or more generally a function f on the natural numbers. To determine this complexity, one can place the object within classes of objects that all have a similar complexity. Among such classes, we will focus on highness properties. They specify a sense in which the object in question is computationally powerful.The Γ-value of an infinite bit sequence A, introduced by Andrews et al.[1], is a real in between 0 and 1 that in a sense measures how well all oracle sets in its Turing degree can be approximated by computable sequences. For each p ∈ (0,1], "Γ(A) < p" is a highness property of A. The values 0,1/2...

show abstract

“…For the operator IOE, separations for some rather special cases of functions g,h were obtained in [17]. We answer the full question for IOE in the affirmative.…”

Section: A Pair Of Dual Mass Problems For Functionsmentioning

confidence: 74%

Section: Dualitymentioning

confidence: 99%

Section: 9mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Muchnik Degrees and Cardinal Characteristics

Monin¹,

Nies²

2020

J. symb. log.

Self Cite

View full text Add to dashboard Cite

show abstract

“…This highness notion of an oracle set A was introduced by Monin and Nies in [37], where it was called "h-infinitely often equal". The notion also corresponds to a cardinal characteristic, namely dp‰hq which is a bounded version of the well-known characteristic dp‰˚q.…”

Section: Analogs Of Cardinal Characteristicsmentioning

confidence: 99%