Randomness and lowness notions via open covers

One of the main lines of research in algorithmic randomness is that of lowness notions. Given a randomness notion R, we ask for which sequences A does relativization to A leave R unchanged (i.e., R^A = R)? Such sequences are call low for R. This question extends to a pair of randomness notions R and S, where S is weaker: for which A is S^A still weaker than R? In the last few years, many results have characterized the sequences that are low for randomness by their low computational strength. A few results have also given measure-theoretic characterizations of low sequences. For example, Kjos-Hanssen proved that A is low for Martin-L\"of randomness if and only if every A-c.e. open set of measure less than 1 can be covered by a c.e. open set of measure less than 1. In this paper, we give a series of results showing that a wide variety of lowness notions can be expressed in a similar way, i.e., via the ability to cover open sets of a certain type by open sets of some other type. This provides a unified framework that clarifies the study of lowness for randomness notions, and allows us to give simple proofs of a number of known results. We also use this framework to prove new results, including showing that the classes Low(MLR;SR) and Low(W2R;SR) coincide, answering a question of Nies. Other applications include characterizations of highness notions, a broadly applicable explanation for why low for randomness is the same as low for tests, and a simple proof that Low(W2R;S)=Low(MLR;S), where S is the class of Martin-L\"of, computable, or Schnorr random sequences. The final section gives characterizations of lowness notions using summable functions and convergent measure machines instead of open covers. We finish with a simple proof of a result of Nies, that Low(MLR) = Low(MLR; CR).Comment: This is a revised version of the APAL paper. In particular, a full proof of Proposition 24 is adde

Algorithmic Randomness

We consider algorithmic randomness in the Cantor space C of the infinite binary sequences. By an algorithmic randomness concept one specifies a set of elements of C, each of which is assigned the property of being random. Miscellaneous notions from computability theory are used in the definitions of randomness concepts that are essentially rooted in the following three intuitive randomness requirements: the initial segments of a random sequence should be effectively incompressible, no random sequence should be an element of an effective measure null set containing sequences with an “exceptional property”, and finally, considering betting games, in which the bits of a sequence are guessed successively, there should be no effective betting strategy that helps a player win an unbounded amount of capital on a random sequence. For various formalizations of these requirements one uses versions of Kolmogorov complexity, of tests, and of martingales, respectively. In case any of these notions is used in the definition of a randomness concept, one may ask in general for fundamental equivalent definitions in terms of the respective other two notions. This was a long-standing open question w.r.t. computable randomness, a central concept that had been introduced by Schnorr via martingales. In this thesis, we introduce bounded tests that we use to give a characterization of computable randomness in terms of tests. Our result was obtained independently of the prior test characterization of computable randomness due to Downey, Griffiths, and LaForte, who defined graded tests for their result. Based on bounded tests, we define bounded machines which give rise to a version of Kolmogorov complexity that we use to prove another characterization of computable randomness. This result, as in analog situations, allows for the introduction of interesting lowness and triviality properties that are, roughly speaking, “anti-randomness” properties. We define and study the notions lowness for bounded machines and bounded triviality. Using a theorem due to Nies, it can be shown that only the computable sequences are low for bounded machines. Further we show some interesting properties of bounded machines, and we demonstrate that every boundedly trivial sequence is K-trivial. Furthermore we define lowness for computable machines, a lowness notion in the setting of Schnorr randomness. We prove that a sequence is low for computable machines if and only if it is computably traceable. Gacs and independently Kucera proved a central theorem which states that every sequence is effectively decodable from a suitable Martin-Löf random sequence. We present a somewhat easier proof of this theorem, where we construct a sequence with the required property by diagonalizing against appropriate martingales. By a variant of that construction we prove that there exists a computably random sequence that is weak truth-table autoreducible. Further, we show that a sequence is computably enumerable self-reducible if and only if its associated real is computably enumerable. Finally we investigate interrelations between the Lebesgue measure and effective measures on C. We prove the following extension of a result due to Book, Lutz, and Wagner: A union of Pi-0-1 classes that is closed under finite variations has Lebesgue measure zero if and only if it contains no Kurtz random real. However we demonstrate that even a Sigma-0-2 class with Lebesgue measure zero need not be a Kurtz null class. Turning to Almost classes, we show among other things that every Almost class with respect to a bounded reducibility has computable packing dimension zero

Van Lambalgen's Theorem for uniformly relative Schnorr and computable randomness

We correct Miyabe's proof of van Lambalgen's Theorem for truth-table Schnorr randomness (which we will call uniformly relative Schnorr randomness). An immediate corollary is one direction of van Lambalgen's theorem for Schnorr randomness. It has been claimed in the literature that this corollary (and the analogous result for computable randomness) is a "straightforward modification of the proof of van Lambalgen's Theorem." This is not so, and we point out why. We also point out an error in Miyabe's proof of van Lambalgen's Theorem for truth-table reducible randomness (which we will call uniformly relative computable randomness). While we do not fix the error, we do prove a weaker version of van Lambalgen's Theorem where each half is computably random uniformly relative to the other