Propagation of partial randomness

Abstract

Let f be a computable function from finite sequences of 0's and 1's to real numbers. We prove that strong f -randomness implies strong frandomness relative to a PA-degree. We also prove: if X is strongly f -random and Turing reducible to Y where Y is Martin-Löf random relative to Z, then X is strongly f -random relative to Z. In addition, we prove analogous propagation results for other notions of partial randomness, including non-K-triviality and autocomplexity. We prove that f -randomness relative to a PA-degree implies strong f -randomness, but f -randomness does not imply f -randomness relative to a PA-degree.

Solovay reducibility and continuity

“…When it is apparent from the context that q is a dyadic rational satisfying (3-6), we let k(q) denote the above-mentioned k. For example, 0.1010 has property (3-6) with k(0.1010) = 2. On the other hand, 0.1110 does not have property (3)(4)(5)(6). (1) α ≤ S β implies α ≤ qS β .…”

“…See Downey and Hirschfeld [3,Chapter 13] or Lutz [8] for a summary of partial randomness, also known as effective dimension. Some generalizations of partial randomness concepts have been discussed in Higuchi, Hudelson, Simpson and Yokoyama [5]. Here, we review the following terminology from [13].…”

Cone avoidance and randomness preservation

“…By [13,Theorem 8.16] it follows that X is not vehemently f -random, say X ∈ i U i where U i is uniformly Σ 0 1 and vwt f (U i ) ≤ 2 −i for all i. By [13,Lemma 8.15] let S i ⊆ {0, 1} * be uniformly r.e. such that U i ⊆ S i and pwt f (S i ) ≤ 2 −i+1 for all i.…”

“…For each τ ∈ {0, 1} * let V τ = {Y | τ ⊂ Φ Y }. Note that the indexed family V τ , τ ∈ {0, 1} * is a Levin system in the sense of [13,Definition 4.1]. By [13, Lemma 4.2] let c be a rational number such that ∀n (λ(V X↾n ) < 2 c−f (X↾n) ).…”

“…By [13, Lemma 4.2] let c be a rational number such that ∀n (λ(V X↾n ) < 2 c−f (X↾n) ). By [13,Lemma 4.3] let V τ ⊆ V τ , τ ∈ {0, 1} * be a Levin system such that ∀τ (λ( V τ ) ≤ 2 c−f (τ ) ) and ∀n ( V X↾n = V X↾n ). Note that λ( V ) = 1 and τ → λ( V τ ) is a left-r.e.…”