Relative Randomness for Martin-Löf Random Sets

Abstract

Let Γ be a set of functions on the natural numbers. We introduce a new randomness notion called semi Γ-randomness, which is associated with a Γ-indexed test. Fix a computable sequence {G _n } _n ∈ ω of all c.e. open sets. For any f ∈ Γ, {G _f(n)} _n ∈ ω is called a Γ-indexed test if μ(G _f(n)) ≤ 2^− n for all n. We prove that weak n-randomness is strictly stronger than semi $\Delta^0_n$-randomness, for n > 2. Moreover, we investigate the relationships between various definitions of randomness.