How powerful are integer-valued martingales?

In the theory of algorithmic randomness, one of the central notions is that of computable randomness. An infinite binary sequence X is computably random if no recursive martingale (strategy) can win an infinite amount of money by betting on the values of the bits of X. In the classical model, the martingales considered are real-valued, that is, the bets made by the martingale can be arbitrary real numbers. In this paper, we investigate a more restricted model, where only integer-valued martingales are considered, and we study the class of random sequences induced by this model.Comment: Long version of the CiE 2010 paper

Lower bounds on the redundancy in computations from random oracles via betting strategies with restricted wagers

The Kučera–Gács theorem is a landmark result in algorithmic randomness asserting that every real is computable from a Martin-Löf random real. If the computation of the first n bits of a sequence requires n+h(n) bits of the random oracle, then h is the redundancy of the computation. Kučera implicitly achieved redundancy nlog⁡n while Gács used a more elaborate coding procedure which achieves redundancy View the MathML source. A similar bound is implicit in the later proof by Merkle and Mihailović. In this paper we obtain optimal strict lower bounds on the redundancy in computations from Martin-Löf random oracles. We show that any nondecreasing computable function g such that ∑n2−g(n)=∞ is not a general upper bound on the redundancy in computations from Martin-Löf random oracles. In fact, there exists a real X such that the redundancy g of any computation of X from a Martin-Löf random oracle satisfies ∑n2−g(n)<∞. Moreover, the class of such reals is comeager and includes a View the MathML source real as well as all weakly 2-generic reals. On the other hand, it has been recently shown that any real is computable from a Martin-Löf random oracle with redundancy g, provided that g is a computable nondecreasing function such that ∑n2−g(n)<∞. Hence our lower bound is optimal, and excludes many slow growing functions such as log⁡n from bounding the redundancy in computations from random oracles for a large class of reals. Our results are obtained as an application of a theory of effective betting strategies with restricted wagers which we develop

Monotonous betting strategies in warped casinos

Suppose that the outcomes of a roulette table are not entirely random, in the sense that there exists a successful betting strategy. Is there a successful `separable' strategy, in the sense that it does not use the winnings from betting on red in order to bet on black, and vice-versa? We study this question from an algorithmic point of view and observe that every strategy M can be replaced by a separable strategy which is computable from M and successful on any outcome-sequence where M is successful. We then consider the case of mixtures and show: (a) there exists an effective mixture of separable strategies which succeeds on every casino sequence with effective Hausdorff dimension less than 1/2; (b) there exists a casino sequence of effective Hausdorff dimension 1/2 on which no effective mixture of separable strategies succeeds. Finally we extend (b) to a more general class of strategies

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

Computability Theory

Computability and computable enumerability are two of the fundamental notions of mathematics. Interest in effectiveness is already apparent in the famous Hilbert problems, in particular the second and tenth, and in early 20th century work of Dehn, initiating the study of word problems in group theory. The last decade has seen both completely new subareas develop as well as remarkable growth in two-way interactions between classical computability theory and areas of applications. There is also a great deal of work on algorithmic randomness, reverse mathematics, computable analysis, and in computable structure theory/computable model theory. The goal of this workshop is to bring together researchers representing different aspects of computability theory to discuss recent advances, and to stimulate future work

Topics in Algorithmic Randomness and Computability Theory

This thesis establishes results in several different areas of computability theory. The first chapter is concerned with algorithmic randomness. A well-known approach to the definition of a random infinite binary sequence is via effective betting strategies. A betting strategy is called integer-valued if it can bet only in integer amounts. We consider integer-valued random sets, which are infinite binary sequences such that no effective integer-valued betting strategy wins arbitrarily much money betting on the bits of the sequence. This is a notion that is much weaker than those normally considered in algorithmic randomness. It is sufficiently weak to allow interesting interactions with topics from classical computability theory, such as genericity and the computably enumerable degrees. We investigate the computational power of the integer-valued random sets in terms of standard notions from computability theory. In the second chapter we extend the technique of forcing with bushy trees. We use this to construct an increasing ѡ-sequence 〈an〉 of Turing degrees which forms an initial segment of the Turing degrees, and such that each an₊₁ is diagonally noncomputable relative to an. This shows that the DNR₀ principle of reverse mathematics does not imply the existence of Turing incomparable degrees. In the final chapter, we introduce a new notion of genericity which we call ѡ-change genericity. This lies in between the well-studied notions of 1- and 2-genericity. We give several results about the computational power required to compute these generics, as well as other results which compare and contrast their behaviour with that of 1-generics

Optimal redundancy in computations from random oracles

It is a classic result in algorithmic information theory that every infinite binary sequence is computable from an infinite binary sequence which is random in the sense of Martin-Löf. Proved independently by Kuˇcera [Kuˇc85] and Gács [Gác86], this result answered a question by Charles Bennett and has seen numerous applications in the last 30 years. The optimal redundancy in such a coding process has, however, remained unknown. If the computation of the first n bits of a sequence requires n+g(n) bits of the random oracle, then g is the redundancy of the computation. Kuˇcera implicitly achieved redundancy n log n while Gács used a more elaborate block-coding procedure which achieved redundancy √n log n. Merkle and Mihailovi´c [MM04] provided a different presentation of Gács’ approach, without improving his redundancy bound. In this paper we devise a new coding method that achieves optimal logarithmic redundancy. For any computable non-decreasing function g such that Pi 2−g(i) is bounded we show that there is a coding process that codes any given infinite binary sequence into a Martin-Löf random infinite binary sequence with redundancy g. This redundancy bound is exponentially smaller than the previous bound of √n log n and is known to be the best possible by recent work [BLPT16], where it was shown that if Pi 2−g(i) diverges then there exists an infinite binary sequence X which cannot be computed by any Martin-Löf random infinite binary sequence with redundancy g. It follows that redundancy ǫ · log n in computation from a random oracle is possible for every infinite binary sequence, if and only if ǫ > 1