The property of smallness for Π0 1 classes is introduced and is investigated with respect to Medvedev and Muchnik degree. It is shown that the property of containing a small Π0 1 class depends only on the Muchnik degree of a Π0 1 class. A comparison is made with the idea of thinness for Π0 1 classesmsthm.

The property of smallness for Π01 classes is introduced and is investigated with respect to Medvedev and Muchnik degree. It is shown that the property of containing a small Π01 class depends only on the Muchnik degree of a Π01 class. A comparison is made with the idea of thinness for Π01 classesmsthm.

We consider two axioms of second-order arithmetic. These axioms assert, in two different ways, that infinite but narrow binary trees always have infinite paths. We show that both axioms are strictly weaker than Weak König's Lemma, and incomparable in strength to the dual statement (WWKL) that wide binary trees have paths.

An infinite binary sequence is complex if the Kolmogorov complexity of its initial segments is bounded below by a computable function. We prove that a Π₁⁰ class P contains a complex element if and only if it contains a wtt-cover for the Cantor set. That is, if and only if for every Y⊆ω there is an X in P such that X≥wtt Y. We show that this is also equivalent to the Π₁⁰ class's being large in some sense. We give (...) an example of how this result can be used in the study of scattered linear orders. (shrink)

We review briefly the attempts to define random sequences. These attempts suggest two theorems: one concerning the number of subsequence selection procedures that transform a random sequence into a random sequence; the other concerning the relationship between definitions of randomness based on subsequence selection and those based on statistical tests.

We investigate the notion of hyperimmunity with respect to how it can be applied to Π{\sp 0}{\sb 1} classes and their Muchnik degrees. We show that hyperimmunity is a strong enough concept to prove the existence of Π{\sp 0}{\sb 1} classes with intermediate Muchnik degree—in contrast to Post's attempts to construct intermediate c.e. degrees.

A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if every member of Q Turing computes a member of P. We say that P is strongly reducible to Q if every member of Q Turing computes a member of P via a fixed Turing functional. The weak degrees and strong degrees are the equivalence classes of mass problems under weak and strong reducibility, respectively. We (...) focus on the countable distributive lattices P w and P s of weak and strong degrees of mass problems given by nonempty Π 1 0 subsets of 2 ω . Using an abstract Gödel/Rosser incompleteness property, we characterize the Π 1 0 subsets of 2 ω whose associated mass problems are of top degree in P w and P s , respectively. Let R be the set of Turing oracles which are random in the sense of Martin-Löf, and let r be the weak degree of R. We show that r is a natural intermediate degree within P w . Namely, we characterize r as the unique largest weak degree of a Π 1 0 subset of 2 ω of positive measure. Within P w we show that r is meet irreducible, does not join to 1, and is incomparable with all weak degrees of nonempty thin perfect Π 1 0 subsets of 2 ω . In addition, we present other natural examples of intermediate degrees in P w . We relate these examples to reverse mathematics, computational complexity, and Gentzen-style proof theory. (shrink)

Let f be a computable function from finite sequences of 0ʼs and 1ʼs to real numbers. We prove that strong f-randomness implies strong f-randomness relative to a PA-degree. We also prove: if X is strongly f-random and Turing reducible to Y where Y is Martin-Löf random relative to Z, then X is strongly f-random relative to Z. In addition, we prove analogous propagation results for other notions of partial randomness, including non-K-triviality and autocomplexity. We prove that f-randomness relative to a (...) PA-degree implies strong f-randomness, hence f-randomness does not imply f-randomness relative to a PA-degree. (shrink)

Let X be a compact metric space. A closed set K $\subseteq$ X is located if the distance function d(x, K) exists as a continuous real-valued function on X; weakly located if the predicate d(x, K) $>$ r is Σ 0 1 allowing parameters. The purpose of this paper is to explore the concepts of located and weakly located subsets of a compact separable metric space in the context of subsystems of second order arithmetic such as RCA 0 , WKL (...) 0 and ACA 0 . We also give some applications of these concepts by discussing some versions of the Tietze extension theorem. In particular we prove an RCA 0 version of this result for weakly located closed sets. (shrink)

We study the connection between Schnorr triviality and genericity. We show that while no 2-generic is Turing equivalent to a Schnorr trivial and no 1-generic is tt-equivalent to a Schnorr trivial, there is a 1-generic that is Turing equivalent to a Schnorr trivial. However, every such 1-generic must be high. As a corollary, we prove that not all K-trivials are Schnorr trivial. We also use these techniques to extend a previous result and show that the bases of cones of Schnorr (...) trivial Turing degrees are precisely those whose jumps are at least 0". (shrink)

We give several characterizations of Schnorr trivial sets, including a new lowness notion for Schnorr triviality based on truth-table reducibility. These characterizations allow us to see not only that some natural classes of sets, including maximal sets, are composed entirely of Schnorr trivials, but also that the Schnorr trivial sets form an ideal in the truth-table degrees but not the weak truth-table degrees. This answers a question of Downey, Griffiths and LaForte.

We examine the randomness and triviality of reals using notions arising from martingales and prefix-free machines.