Martin-löf Reducibility and Cost Functions

Martin-Löf (ML)-reducibility compares K-trivial sets by examining the Martin-Löf random sequences that compute them. We show that every K-trivial set is computable from a c.e. set of the same ML-degree. We investigate the interplay between ML-reducibility and cost functions, which are used to both measure the number of changes in a computable approximation, and the type of null sets used to capture ML-random sequences. We show that for every cost function there is a c.e. set ML-above the sets obeying it (called a “smart” set for the cost function). We characterise the K-trivial sets computable from a fragment of the left-c.e. random real Ω. This leads to a new characterisation of strong jump-traceability.