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

Restricted Coding and Betting

One of the fundamental themes in the study of computability theory are oracle computations, i.e. the coding of one infinite binary sequence into another. A coding process where the prefixes of the coded sequence are coded such that the length difference of the coded and the coding prefix is bounded by a constant is known as cl-reducibility. This reducibility has received considerable attention over the last two decades due to its interesting degree structure and because it exhibits strong connections with algorithmic randomness. In the first part of this dissertation, we study a slightly relaxed version of cl-reducibility where the length difference is required to be bounded by some specific nondecreasing computable function~$h$. We show that in this relaxed model some of the classical results about cl-reducibility still hold in case the function $h$ grows slowly, at certain particular rates. Examples are the Yu-Ding theorem, which states that there is a pair of left-c.e. sequences that cannot be coded simultaneously by any left-c.e. sequence, as well as the Barmpalias-Lewis theorem that states that there is a left-c.e. sequence which cannot be coded by any random left-c.e. sequence. In case the bounding function~$h$ grows too fast, both results don't hold anymore. Betting strategies, which can be formulated equivalently in terms of martingales, are one of the main tools in the area of algorithmic randomness. A betting strategy is usually determined by two factors, the guessed outcome at every stage and the wager on it. In the second part of this dissertation we study betting strategies where one of these factors is restricted. First we study single-sided strategies, where the guessed outcome either is always 0 or is always 1. For computable strategies we show that single-sided strategies and usual strategies have the same power for winning, whereas the latter does not hold for strongly left-c.e. strategies, which are mixtures of computable strategies, even if we extend the class of single-sided strategies to the more general class of decidably-sided strategies. Finally, we study the case where the wagers are forced to have a certain granularity, i.e. must be multiples of some not necessarily constant betting unit. For usual strategies, wins can always be assumed to have the two following properties (a) ‘win with arbitrarily small initial capital’ and (b) ‘win by saving’. In a setting of variable granularity, where the betting unit shrinks over stages, we study how the shrinking rates interact with these two properties. We show that if the granularity shrinks fast, at certain particular rates,for such granular strategies both properties are preserved. For slower rates of shrinking, we show that neither property is preserved completely, however, a weaker version of property (a) still holds. In order to investigate property (b) in this case, we consider more restricted strategies where in addition the wager is bounded from above

Randomness and Computability

This thesis establishes significant new results in the area of algorithmic randomness. These results elucidate the deep relationship between randomness and computability. A number of results focus on randomness for finite strings. Levin introduced two functions which measure the randomness of finite strings. One function is derived from a universal monotone machine and the other function is derived from an optimal computably enumerable semimeasure. Gacs proved that infinitely often, the gap between these two functions exceeds the inverse Ackermann function (applied to string length). This thesis improves this result to show that infinitely often the difference between these two functions exceeds the double logarithm. Another separation result is proved for two different kinds of process machine. Information about the randomness of finite strings can be used as a computational resource. This information is contained in the overgraph. Muchnik and Positselsky asked whether there exists an optimal monotone machine whose overgraph is not truth-table complete. This question is answered in the negative. Related results are also established. This thesis makes advances in the theory of randomness for infinite binary sequences. A variant of process machines is used to characterise computable randomness, Schnorr randomness and weak randomness. This result is extended to give characterisations of these types of randomness using truthtable reducibility. The computable Lipschitz reducibility measures both the relative randomness and the relative computational power of real numbers. It is proved that the computable Lipschitz degrees of computably enumerable sets are not dense. Infinite binary sequences can be regarded as elements of Cantor space. Most research in randomness for Cantor space has been conducted using the uniform measure. However, the study of non-computable measures has led to interesting results. This thesis shows that the two approaches that have been used to define randomness on Cantor space for non-computable measures: that of Reimann and Slaman, along with the uniform test approach first introduced by Levin and also used by Gacs, Hoyrup and Rojas, are equivalent. Levin established the existence of probability measures for which all infinite sequences are random. These measures are termed neutral measures. It is shown that every PA degree computes a neutral measure. Work of Miller is used to show that the set of atoms of a neutral measure is a countable Scott set and in fact any countable Scott set is the set of atoms of some neutral measure. Neutral measures are used to prove new results in computability theory. For example, it is shown that the low computable enumerable sets are precisely the computably enumerable sets bounded by PA degrees strictly below the halting problem. This thesis applies ideas developed in the study of randomness to computability theory by examining indifferent sets for comeager classes in Cantor space. A number of results are proved. For example, it is shown that there exist 1-generic sets that can compute their own indifferent sets

Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)

The Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), Saarbr¨ucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), W¨urzburg (1993), Caen (1994), M¨unchen (1995), Grenoble (1996), L¨ubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..