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

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