Abstract
Martin—Löf (ML)-reducibility compares the complexity of K-trivial sets of natural numbers by examining the Martin—Löf random sequences that compute them. One says that a K-trivial set A is ML-reducible to a K-trivial set B if every ML-random computing B also computes A. 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 intended to capture ML-random sequences. We show that for every cost function there is a c.e. set that is ML-complete among the sets obeying it. We characterise the K-trivial sets computable from a fragment of the left-c.e. random real Ω given by a computable set of bit positions. This leads to a new characterisation of strong jump-traceability.
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This research was started during a retreat at the Research Centre Coromandel.
Greenberg and Nies were supported by the Marsden Fund of New Zealand.
Greenberg was also supported by a Rutherford Discovery Fellowship from the Royal Society of New Zealand.
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Greenberg, N., Miller, J.S., Nies, A. et al. Martin–Löf reducibility and cost functions. Isr. J. Math. (2023). https://doi.org/10.1007/s11856-023-2565-x
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DOI: https://doi.org/10.1007/s11856-023-2565-x