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We say that A ≤LR B if every B-random number is A-random. Intuitively this means that if oracle A can identify some patterns on some real γ, oracle B can also find patterns on γ. In other words, B is at least as good as A for this purpose. We study the structure of the LR degrees globally and locally (i.e. restricted to the computably enumerable degrees)… (More)

We show that the identity bounded Turing degrees of computably enumerable sets are not dense.

We say that A ≤ LR B if every B-random set is A-random with respect to Martin-Löf randomness. We study this reducibility and its interactions with the Turing reducibility, Π 0 1 classes, hyperimmunity and other recursion theoretic notions. A natural variant of the Turing reducibility from the point of view of Martin-Löf randomness is the LR reducibility… (More)

The strong weak truth table (sw) reducibility was suggested by Downey, Hirschfeldt, and LaForte as a measure of relative random-ness, alternative to the Solovay reducibility. It also occurs naturally in proofs in classical computability theory as well as in the recent work of Soare, Nabutovsky and Weinberger on applications of computability to differential… (More)

Lipschitz continuity is used as a tool for analyzing the relationship between incomputability and randomness. Having presented a simpler proof of one of the major results in this area—the theorem of Yu and Ding that there exists no cl-complete c.e. real—we go on to consider the global theory. The existential theory of the cl degrees is decidable but this… (More)

We study the Medvedev degrees of mass problems with distinguished topo-logical properties, such as denseness, closedness, or discreteness. We investigate the sublattices generated by these degrees; the prime ideal generated by the dense degrees and its complement, a prime filter; the filter generated by the nonzero closed degrees and the filter generated by… (More)

We show that there exists a real α such that, for all reals β, if α is linear reducible to β (α ≤ β, previously denoted α ≤sw β) then β ≤T α. In fact, every random real satisfies this quasi-maximality property. As a corollary we may conclude that there exists no-complete ∆2 real. Upon realizing that quasi-maximality does not characterize the random… (More)

We prove that every diagonally noncomputable function computes a set A which is bi-immune, meaning that neither A nor its complement has an infinite computably enumerable subset.

We prove a number of results in effective randomness, using methods in which Π 0 1 classes play an essential role. Amongst many others, 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 there are computably enumerable (c.e.) sets with maximum initial segment Kolmogorov complexity amongst all c.e. sets (with respect to both the plain and the prefix-free version of Kolmogorov complexity). These c.e. sets belong to the weak truth table degree of the halting problem, but not every weak truth table complete set has maximum initial… (More)