Lowness Properties of Reals and Hyper-Immunity

Reals which Compute Little André

We investigate combinatorial lowness properties of sets of natural numbers (reals). The real A is super-low if A′ ≤tt ∅′, and A is jump-traceable if the values of {e}A(e) can be effectively approximated in a sense to be specified. We investigate those properties, in particular showing that super-lowness and jump-traceability coincide within the r.e. sets but none of the properties implies the o...

Reals which compute little

We investigate combinatorial lowness properties of sets of natural numbers (reals). The real A is super-low if A′ ≤tt ∅′, and A is jump-traceable if the values of {e}A(e) can be effectively approximated in a sense to be specified. We investigate those properties, in particular showing that super-lowness and jump-traceability coincide within the r.e. sets but none of the properties implies the o...

Lowness Properties of Reals and Randomness

CDMTCS Research Report Series Higher Randomness Notions and Their Lowness Properties

We study randomness notions given by higher recursion theory, establishing the relationships Π1-randomness ⊂ Π1-Martin-Löf randomness ⊂ ∆1randomness = ∆1-Martin-Löf randomness. We characterize the set of reals that are low for ∆1 randomness as precisely those that are ∆1 -traceable. We prove that there is a perfect set of such reals.

Higher Randomness Notions and Their Lowness Properties

We study randomness notions given by higher recursion theory, establishing the relationships Π1-randomness ⊂ Π1-Martin-Löf randomness ⊂ ∆1randomness = ∆1-Martin-Löf randomness. We characterize the set of reals that are low for ∆1 randomness as precisely those that are ∆ 1 1 -traceable. We prove that there is a perfect set of such reals.