Andrew E. M. Lewis

A c.e. real that cannot be sw-computed by any Ω number

The strong weak truth table (sw) reducibility was suggested by Downey, Hirschfeldt, and LaForte as a measure of relative randomness, 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 geometry. We study the sw-degrees of c.e. reals and construct a c.e. real which has no random c.e. real (i.e., Ω number) sw-above it.

Randomness and the linear degrees of computability

We show that there exists a real α such that, for all reals β, if α is linear reducible to β (α≤lβ, previously denoted as α≤swβ) then β≤Tα. In fact, every random real satisfies this quasi-maximality property. As a corollary we may conclude that there exists no l-complete Δ2 real. Upon realizing that quasi-maximality does not characterize the random reals–there exist reals which are not random but which are of quasi-maximal l-degree – it is then natural to ask whether maximality could provide such a characterization. Such hopes, however, are in vain since no real is of maximal l-degree.

Chaitin’s halting probability and the compression of strings using oracles

If a computer is given access to an oracle—the characteristic function of a set whose membership relation may or may not be algorithmically calculable—this may dramatically affect its ability to compress information and to determine structure in strings, which might otherwise appear random. This leads to the basic question, ‘given an oracle A, how many oracles can compress information at most as well as A?’ This question can be formalized using Kolmogorov complexity. We say that B≤LKA if there exists a constant c such that KA(σ)<KB(σ)+c for all strings σ, where KX denotes the prefix-free Kolmogorov complexity relative to oracle X. The formal counterpart to the previous question now is, ‘what is the cardinality of the set of ≤LK-predecessors of A?’ We completely determine the number of oracles that compress at most as well as any given oracle A, by answering a question of Miller (Notre Dame J. Formal Logic, 2010, 50, 381–391), which also appears in Nies (Computability and randomness. Oxford, UK: Oxford University Press, 2009. Problem 8.1.13); the class of ≤LK-predecessors of a set A is countable if and only if Chaitins halting probability Ω is Martin-Löf random relative to A.

A Weakly 2-Random Set That Is Not Generalized Low

A guiding question in the study of weak 2-randomness is whether weak 2-randomness is closer to 1-randomness, or closer to 2-randomness. Recent research indicates that the first alternative holds. We add further evidence in this direction by showing that, in contrast to the case for 2-randomness, a weakly 2-random set can fail to be generalized low.

The First Order Theories of the Medvedev and Muchnik Lattices

We show that the first order theories of the Medevdev lattice and the Muchnik lattice are both computably isomorphic to the third order theory of the natural numbers.

A single minimal complement for the c.e. degrees

We show that there exists a minimal (Turing) degree such that for all non-zero c.e. degrees , . Since is minimal this means that complements all c.e. degrees other than and . Since every -c.e. degree bounds a non-zero c.e. degree, complements every -c.e. degree other than and .

Working with the LR degrees

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 propose a methodology for studying the LR degrees and present a number of recent results of ours, including sketches of their proofs.

THE HYPERSIMPLE-FREE C.E. WTT DEGREES ARE DENSE IN THE C.E. WTT DEGREES

We show that in the c.e. weak truth table degrees if b < c then there is an a which contains no hypersimple set and b < a < c. We also show that for every w < c in the c.e. wtt degrees such that w is hypersimple, there is a hypersimple a such that w < a < c. On the other hand, we know that there are intervals which contain no hypersimple set.

The jump classes of minimal covers

We work in

Randomness, lowness and degrees

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