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Journals and Conferences
Consider a Martin-Löf random ∆2 set Z. We give lower bounds for the number of changes of Zs n for computable approximations of Z. We show that each nonempty Π 1 class has a low member Z with a computable approximation that changes only o(2) times. We prove that each superlow ML-random set already satisfies a stronger randomness notion called balanced… (More)
We develop an analogy between cardinal characteristics from set theory and highness properties from computability theory, which specify a sense in which a Turing oracle is computationally strong. We focus on characteristics from Cichoń’s diagram.
In this paper we show that there is no minimal bound for jump traceability. In particular, there is no single order function such that strong jump traceability is equivalent to jump traceability for that order. The uniformity of the proof method allows us to adapt the technique to showing that the index set of the c.e. strongly jump traceables
We introduce a natural strengthening of prompt simplicity which we call strong promptness, and study its relationship with existing lowness classes. This notion provides a ≤wtt version of superlow cuppability. We show that every strongly prompt c.e. set is superlow cuppable. Unfortunately, strong promptness is not a Turing degree notion, and so cannot… (More)
We examine the sequences A that are low for dimension, i.e., those for which the effective (Hausdorff) dimension relative to A is the same as the unrelativized effective dimension. Lowness for dimension is a weakening of lowness for randomness, a central notion in effective randomness. By considering analogues of characterizations of lowness for randomness,… (More)
We prove a number of results in effective randomness, using methods in which Π1 classes play an essential role. The results proved include the fact that every PA Turing degree is the join of two random Turing degrees, and the existence of a minimal pair of LR degrees below the LR degree of the halting problem.
We show that if a point in a computable probability space X satisfies the ergodic recurrence property for a computable measure-preserving T : X → X with respect to effectively closed sets, then it also satisfies Birkhoff’s ergodic theorem for T with respect to effectively closed sets. As a corollary, every Martin-Löf random sequence in the Cantor space… (More)
We study computably enumerable equivalence relations (ceers), under the reducibility R ≤ S if there exists a computable function f such that x R y if and only if f(x) S f(y), for every x, y. We show that the degrees of ceers under the equivalence relation generated by ≤ form a bounded poset that is neither a lower semilattice, nor an upper semilattice, and… (More)
We show that every real low for Demuth randomness is of hyperimmune-free degree.