α‎-c.a. functions

This chapter discusses the notion of α‎-c.a. functions. The main issue is to properly define what is meant by a computable function o from N to α‎, which is required for the definition of α‎-computable approximations. Naturally, to deal with an ordinal α‎ computably, one needs a notation for this ordinal, or more generally, a computable well-ordering of order-type α‎. To form the basis of a solid hierarchy, the notion of α‎-c.a. should not depend on which well-ordering one takes, rather it should only depend on its order-type. Thus, one cannot consider all computable copies of α‎. Rather, one restricts one's self to a class of particularly well-behaved well-orderings, in a way that ensures that they are all computably isomorphic. Having defined α‎-c.a. functions, the chapter relates these functions to iterations of the bounded jump (the jump inside the weak truth-table degrees).

Anti-Complex Sets and Reducibilities with Tiny Use

In contrast with the notion of complexity, a set A is called anti-complex if the Kolmogorov complexity of the initial segments of A chosen by a recursive function is always bounded by the identity function. We show that, as for complexity, the natural arena for examining anti-complexity is the weak-truth table degrees. In this context, we show the equivalence of anti-complexity and other lowness notions such as r.e. traceability or being weak truth-table reducible to a Schnorr trivial set. A set A is anti-complex if and only if it is reducible to another set B with tiny use, whereby we mean that the use function for reducing A to B can be made to grow arbitrarily slowly, as gauged by unbounded nondecreasing recursive functions. This notion of reducibility is then studied in its own right, and we also investigate its range and the range of its uniform counterpart.

Schnorr trivial sets and truth-table reducibility

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.