Effective Bi-immunity and Randomness

Abstract

Abstract. We study the relationship between randomness and effective biimmunity. Greenberg and Miller have shown that for any oracle X, there are arbitrarily slow-growing DNR functions relative to X that compute no MartinLöf random set. We show that the same holds when Martin-Löf randomness is replaced with effective bi-immunity. It follows that there are sequences of effective Hausdorff dimension 1 that compute no effectively bi-immune set.We also establish an important difference between the two properties. The class Low(MLR, EBI) of oracles relative to which every Martin-Löf random is effectively bi-immune contains the jump-traceable sets, and is therefore of cardinality continuum.

Canonical immunity and genericity

“…The first author, together with Mushfeq Khan and Bjørn Kjos-Hanssen [1], have shown that every canonically immune set is immune and, moreover, that every Schnorr random is canonically immune with modulus of immunity i → i. The present paper provides some counterpoints to these observations and shows that the notion of canonical immunity is distinct from the standard notions of immunity and genericity in computability theory.…”

“…The existence of such a set follows from Theorem 7.4.2 in Downey-Hirschfeldt [4] and the fact that all Martin-Löf random reals are also Schnorr random. Such a set will perforce be canonically immune by [1,Theorem 5.2]. Finally, note that a set of positive density cannot be hyperimmune as its increasing enumeration will be bounded eventually by the computable function f (n) = 3n.…”

Index Sets of Universal Codes

“…Identifying P fin (ω) with 2 <ω , we could alternatively characterize a canonical numbering as a total computable function H : ω → 2 <ω such that each finite set is in the range of H. Definition 3.6. [1] A infinite set R ⊆ ω is canonically immune if, and only if, there is a total computable function h such that, for each canonical numbering H, and all but finitely many e ∈ ω, H(e) ⊆ R =⇒ |H(e)| ≤ h(e). Definition 3.7.…”

“…In the first place, an inspection of the proof Theorem 5.5 in [1] reveals that there is a Medvedev reduction of snr to ci. It follows from the definitions of ci and snr that the requisite topological and definability properties are satisfied in order to apply Theorem 3.2.…”