Abstract. Several classes of diagonally non-recursive (DNR) functions are characterized in terms of Kolmogorov complexity. In particular, a set of natural numbers A can wtt-compute a DNR function iff there is a nontrivial recursive lower bound on the Kolmogorov complexity of the initial segments of A. Furthermore, A can Turing compute a DNR function iff there is a nontrivial A-recursive lower bound on the Kolmogorov complexity of the initial segements of A. A is PA-complete, that is, A can compute a {0, 1}-valued DNR function, iff A can compute a function F such that F (n) is a string of length n and maximal C-complexity among the strings of length n. A ≥T K iff A can compute a function F such that F (n) is a string of length n and maximal H-complexity among the strings of length n. Further characterizations for these classes are given. The existence of a DNR function in a Turing degree is equivalent to the failure of the Recursion Theorem for this degree; thus the provided results characterize those Turing degrees in terms of Kolmogorov complexity which do no longer permit the usage of the Recursion Theorem. 1.

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. The following theorems on the structure inside nonrecursive truthtable degrees are established: Degtev's result that the number of bounded truth-table degrees inside a truth-table degree is at least two is improved by showing that this number is infinite. There are even infinite chains and antichains of bounded truth-table degrees inside the truth-table degrees which implies an affirmative answer to a question of Jockusch whether every truthtable degree contains an infinite antichain of many-one degrees. Some but not all truth-table degrees have a least bounded truth-table degree. The technique to construct such a degree is used to solve an open problem of Beigel, Gasarch and Owings: there are Turing degrees (constructed as hyperimmunefree truth-table degrees) which consist only of 2-subjective sets and do therefore not contain any objective set. Furthermore a truth-table degree consisting of three positive degrees is constructed where one positive degree consists of enum...

Abstract. We show that the existence of a nontrivial proper subspace of a vector space of dimension greater than one (over an infinite field) is equivalent to WKL0 over RCA0, and that the existence of a finite-dimensional nontrivial proper subspace of such a vector space is equivalent to ACA0 over RCA0. 1.

Abstract In this paper, we give several characterizations of Schnorr trivial sets, including a new lowness notion for Schnorr triviality based on truth-table reducibility. These characterizations enable 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. 1 Introduction One of the major achievements in the study of Martin-L"of randomness is Nies's discovery [14] that the following conditions on a set A coincide. * A is low for Martin-L"of randomness, that is, every set that is Martin-L"of random

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Abstract. We prove that van Lambalgen’s Theorem fails for both Schnorr randomness and computable randomness. To characterize randomness, various definitions of randomness for individual elements of Cantor space have been introduced. The most popular (and maybe the most important) definitions of randomness are Martin-Löf randomness, Schnorr randomness and computable randomness. We use µ to denote Lebesgue measure on Cantor space 2ω. Definition 0.1 (Martin-Löf [7]). (i) Given a set X ⊆ ω, anX-Martin-Löf test is a computable collection {Vn: n ∈ N} of computably enumerable open sets such that µ(Vn) ≤ 2−n. (ii) Given a set X ⊆ ω, asetYis said to pass the X-Martin-Löf test if Y /∈ n∈N Vn. (iii) Given a set X,asetYis said to be X-ML-random if it passes all X-Martin-Löf tests.

Abstract. Demuth tests generalize Martin-Löf tests (Gm)m∈N in that one can exchange the m-th component for a computably bounded number of times. A set Z ⊆ N fails a Demuth test if Z is in infinitely many final versions of the Gm. If we only allow Demuth tests such that Gm ⊇ Gm+1 for each m, we have weak Demuth randomness. We show that a weakly Demuth random set can be high, yet not superhigh. Next, any c.e. set Turing below a Demuth random set is strongly jump-traceable. We also prove a basis theorem for non-empty Π 0 1 classes P. It extends the Jockusch-Soare basis theorem that some member of P is computably dominated. We use the result to show that some weakly 2-random set does not compute a 2-fixed point free function. 1.

Abstract. Demuth tests generalize Martin-Löf tests (Gm)m∈N in that one can exchange the m-th component a computably bounded number of times. A set Z ⊆ N fails a Demuth test if Z is in infinitely many final versions of the Gm. If we only allow Demuth tests such that Gm ⊇ Gm+1 for each m, we have weak Demuth randomness. We show that a weakly Demuth random set can be high and ∆ 0 2, yet not superhigh. Next, any c.e. set Turing below a Demuth random set is strongly jump-traceable. We also prove a basis theorem for non-empty Π 0 1 classes P. It extends the Jockusch-Soare basis theorem that some member of P is computably dominated. We use the result to show that some weakly 2-random set does not compute a 2-fixed point free function. 1.