2. Sets, measure, and martingales 4 2.1. Sets and measure 4 2.2. Martingales 5

We study computably enumerable reals (i.e. their left cut is computably enumerable) in terms of their spectra of representations and presentations.

. 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 0 . This contrasts with a result of Kucera and Terwijn [5] on sets that are low for the class of Martin-Lof random reals. The Cantor space 2 ! is the set of infinite binary sequences; these are called reals and are identified with subsets of !. If oe 2 2 !! , that is, oe is a finite binary sequence, we denote by [oe] the set of reals that extend oe. These form a basis of clopen sets for the usual discrete topology on 2 ! . Write joej for the length of oe 2 2 !! . The Lebesgue measure on 2 ! is defined by stipulating that [oe] = 2 \Gammajoej . With every set U ` 2 !! we associate the open set S oe2U [oe]. When it is convenient, we confuse U with the open set associated to it, in particular we write U for the measure of the open set correspondi...

A binary sequence A = A(0)A(1) ... is called i.o. Turing-autoreducible if A is reducible to itself via an oracle Turing machine that never queries its oracle at the current input, outputs either A(x) or a don't-know symbol on any given input x, and outputs A(x) for infinitely many x. If in addition the oracle Turing machine terminates on all inputs and oracles, A is called i.o. truth-table-autoreducible.

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

It is shown that the class of Kolmogorov-Loveland stochastic sequences is not closed under selecting subsequences by monotonic computable selection rules. This result gives a strong negative answer to the notorious open problem whether the Kolmogorov-Loveland stochastic sequences are closed under selecting subsequences by KolmogorovLoveland selection rules, i.e., by not necessarily monotonic partially computable selection rules. As a corollary, we obtain an easy proof for the previously known result that the Kolmogorov-Loveland stochastic sequences form a proper subclass of the Mises-Wald-Church stochastic sequences.

Abstract. We study ∆2 reals x in terms of how they can be approximated symmetrically by a computable sequence of rationals. We deal with a natural notion of ‘approximation representation ’ and study how these are related computationally for a fixed x. This is a continuation of earlier work; it aims at a classification of ∆2 reals based on approximation and it turns out to be quite different than the existing ones (based on information content etc.) 1.

Abstract. We show that there is a low T-upper bound for the class of Ktrivial sets, namely those which are weak from the point of view of algorithmic randomness. This result is a special case of a more general characterization of ideals in ∆0 2 T-degrees for which there is a low T-upper bound. 1.

Much of the recent research on algorithmic randomness has focused on randomness for Lebesgue measure. While, from a computability theoretic point of view, the picture remains unchanged if one passes to arbitrary computable measures, interesting phenomena occur if one studies the the set of reals which are random for an arbitrary (continuous) probability measure or a generalized Hausdorff measure on Cantor space. This paper tries to give a survey of some of the research that has been done on randomness for non-Lebesgue measures.