We prove that there are uncountably many sets that are low for the class of Schnorr random reals. We give a purely recursion theoretic characterization of these sets and show that they all have Turing degree incomparable to 0'. This contrasts with a result of Kucera and Terwijn [5] on sets that are low for the class of Martin-Lof random reals.

Simpson introduced the lattice of Π0 1 classes under Medvedev reducibility. Questions regarding completeness in are related to questions about measure and randomness. We present a solution to a question of Simpson about Medvedev degrees of Π0 1 classes of positive measure that was independently solved by Simpson and Slaman. We then proceed to discuss connections to constructive logic. In particular we show that the dual of does not allow an implication operator (i.e. that is not a Heyting algebra). We (...) also discuss properties of the class of PA-complete sets that are relevant in this context. (shrink)

It is shown that if A and C are sets of degrees uniformly recursive in 0' with $\mathbf{0} \nonin \mathscr{C}$ then there is a degree b with b' = 0', b ∪ c = 0' for every c ∈ C, and $\mathbf{a} \nleq \mathbf{b}$ for every a ∈ A ∼ {0}. The proof is given as an oracle construction recursive in 0'. It follows that any nonrecursive degree below 0' can be joined to 0' by a degree strictly below 0'. (...) Also, if $\mathbf{a ' and a" = 0" then there is a degree b such that a ∪ b = 0' and a ∩ b = 0. (shrink)

It is shown that the nonrecursive predecessors of a 1-generic degree $ are all 1-generic. As a corollary, it is shown that the 1-generic degrees are not densely ordered.

We survey the theory of Mucnik and Medvedev degrees of subsets of $^{\omega}{\omega}$with particular attention to the degrees of $\Pi_{1}^{0}$ subsets of $^{\omega}2$. Sections 1-6 present the major definitions and results in a uniform notation. Sections 7-6 present proofs, some more complete than others, of the major results of the subject together with much of the required background material.

Volume II of Classical Recursion Theory describes the universe from a local (bottom-up or synthetical) point of view, and covers the whole spectrum, from the recursive to the arithmetical sets. The first half of the book provides a detailed picture of the computable sets from the perspective of Theoretical Computer Science. Besides giving a detailed description of the theories of abstract Complexity Theory and of Inductive Inference, it contributes a uniform picture of the most basic complexity classes, ranging from small (...) time and space bounds to the elementary functions, with a particular attention to polynomial time and space computability. It also deals with primitive recursive functions and larger classes, which are of interest to the proof theorist. The second half of the book starts with the classical theory of recursively enumerable sets and degrees, which constitutes the core of Recursion or Computability Theory. Unlike other texts, usually confined to the Turing degrees, the book covers a variety of other strong reducibilities, studying both their individual structures and their mutual relationships. The last chapters extend the theory to limit sets and arithmetical sets. The volume ends with the first textbook treatment of the enumeration degrees, which admit a number of applications from algebra to the Lambda Calculus. The book is a valuable source of information for anyone interested in Complexity and Computability Theory. The student will appreciate the detailed but informal account of a wide variety of basic topics, while the specialist will find a wealth of material sketched in exercises and asides. A massive bibliography of more than a thousand titles completes the treatment on the historical side. (shrink)

We prove that there are uncountably many sets that are low for the class of Schnorr random reals. We give a purely recursion theoretic characterization of these sets and show that they all have Turing degree incomparable to 0'. This contrasts with a result of Kučera and Terwijn [5] on sets that are low for the class of Martin-Löf random reals.

We answer a question of Ambos-Spies and Kuˇcera in the affirmative. They asked whether, when a real is low for Schnorr randomness, it is already low for Schnorr tests.