An Analogy between Cardinal Characteristics and Highness Properties of Oracles

Abstract

WSPC Proceedings -9in x 6inBrendleBrookeNgNies˙Cichon˙recursion˙theory page 2 2 We present an analogy between cardinal characteristics from set theory and highness properties from computability theory, which specify a sense in which a Turing oracle is computationally strong. While this analogy was first studied explicitly by Rupprecht (Effective correspondents to cardinal characteristics in Cichoń's diagram, PhD thesis, University of Michigan, 2010), many prior results can be viewed from this perspective. After a comprehensive survey of the analogy for characteristics from Cichoń's diagram, we extend it to Kurtz randomness and the analogue of the Specker-Eda number.

From Eventually Different Functions to Pandemic Numberings

“…If R k is really infinite then t k is total and so f (e) ≥ t k (e) for some (infinitely many) e. So we define D e,k to consist of the first h( e, k ) elements of R k , if any, as found during a search time of f ( e, k ) or even just f (e). By Corollary 7 and Theorem 11, the dualism between immunity and pandemics is the same, Muchnik-degree-wise, as that between, in the notation of Brendle et al [7], -eventually different functions, b( = * ), and functions that are infinitely often equal to each recursive function, d( = * ). Remark 1.…”

Effective localization number: Building k-surviving degrees

“…Before starting the construction we can assume one more hypothesis: for all τ ∈ T s there exist τ 0 , ..., τ k in T s extending τ and n ∈ ω such that ϕ τ i s (ℓ) converges for all i < k + 1 and all ℓ < n. Also, we need that ϕ τ i s ↾n = ϕ τ j s ↾n for all i = j < k + 1. 2 If there is a τ extending r s such that the above hypothesis is false, that means that ϕ A s can have at most k different values as long as τ is an initial segment of A. Therefore, defining T s+1 the subtree of T s extending τ , we can find a computable tree U s with at most k branches such that ϕ A s , with A ∈ [T s+1 ], is always one of those branches.…”

Covering the recursive sets

“…Brendle, Brook-Taylor, Ng and Nies [1] called the notion of (weakly) Schnorr covering in their paper (weakly) Schnorr engulfing. In this paper, we will use the original terminology of Rupprecht [19,20].…”

“…Brendle, Brooke-Taylor, Ng and Nies [1,Question 4.1], posed three questions, (7), (8) and (9). In this section, we will provide the answers to the questions (7) and (9).…”

“…It is easy to see that there is an effective list of all admissible truth-table reductions. Condition (1) includes that all Φ E i for even i follow finite sets; this is needed in order to avoid that the construction of the set B gets stuck; it is easy to obtain this condition by considering a join of a given truth-table reduction with a default one computing all finite sets. The inclusion of the partial reductions is there in order to account for the fact that there is no recursive enumeration of all total recursive functions and thus also no recursive enumeration of all total truth-table reductions.…”