Browsing by Author "Greenberg, Noam"
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- ItemComputability Theory (hybrid meeting)(Zürich : EMS Publ. House, 2021) Greenberg, Noam; Kalimullin, Iskander; Soskova, MariyaOver the last decade computability theory has seen many new and fascinating developments that have linked the subject much closer to other mathematical disciplines inside and outside of logic. This includes, for instance, work on enumeration degrees that has revealed deep and surprising relations to general topology, the work on algorithmic randomness that is closely tied to symbolic dynamics and geometric measure theory. Inside logic there are connections to model theory, set theory, effective descriptive set theory, computable analysis and reverse mathematics. In some of these cases the bridges to seemingly distant mathematical fields have yielded completely new proofs or even solutions of open problems in the respective fields. Thus, over the last decade, computability theory has formed vibrant and beneficial interactions with other mathematical fields. The goal of this workshop was to bring together researchers representing different aspects of computability theory to discuss recent advances, and to stimulate future work.
- ItemK-triviality, Oberwolfach randomness, and differentiability(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2012) Bienvenu, Laurent; Greenberg, Noam; Kucera, Antonín; Nies, André; Turetsky, DanWe show that a Martin-Löf random set for which the effective version of the Lebesgue density theorem fails computes every K-trivial set. Combined with a recent result by Day and Miller, this gives a positive solution to the ML-covering problem (Question 4.6 in Randomness and computability: Open questions. Bull. Symbolic Logic, 12(3):390-410, 2006). On the other hand, we settle stronger variants of the covering problem in the negative. We show that any witness for the solution of the covering problem, namely an incomplete random set which computes all K-trivial sets, must be very close to being Turing complete. For example, such a random set must be LR-hard. Similarly, not every K-trivial set is computed by the two halves of a random set. The work passes through a notion of randomness which characterises computing K-trivial sets by random sets. This gives a "smart" K-trivial set, all randoms from whom this set is computed have to compute all K-trivial sets.