The set A is low for (Martin-Löf) randomness if each random set is already random relative to A. A is K-trivial if the prefix complexity K of each initial segment of A is minimal, namely ∀n K(A n) ≤… (More)

We compare various notions of algorithmic randomness. First we consider relativized randomness. A set is n-random if it is Martin-Löf random relative to ∅(n−1). We show that a set is 2-random if and… (More)

We study and compare two combinatorial lowness notions: strong jump-traceability and wellapproximability of the jump, by strengthening the notion of jump-traceability and ω-r.e. for sets of natural… (More)

Let R be a notion of algorithmic randomness for individual subsets of N. We say B is a base for R randomness if there is a Z >T B such that Z is R random relative to B. We show that the bases for… (More)

We prove that any Chaitin Ω number (i.e., the halting probability of a universal self-delimiting Turing machine) is wtt-complete, but not tt-complete. In this way we obtain a whole class of natural… (More)

In this paper we investigate computable models of א1-categorical theories and Ehrenfeucht theories. For instance, we give an example of an א1categorical but not א0-categorical theory T such that all… (More)

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… (More)