The Kucera-Gacs theorem is a landmark result in algorithmic randomness asserting that every real is computable from a Martin-Lof random real. If the computation of the first @math bits of a sequence requires @math bits of the random oracle, then @math is the redundancy of the computation. Kucera implicitly achieved redundancy @math while Gacs used a more elaborate coding procedure which achieves redundancy @math . A similar upper bound is implicit in the later proof by Merkle and Mihailovic. In this paper we obtain strict optimal lower bounds on the redundancy in computations from Martin-Lof random oracles. We show that any nondecreasing computable function @math such that @math is not a general upper bound on the redundancy in computations from Martin-Lof random oracles. In fact, there exists a real @math such that the redundancy @math of any computation of @math from a Martin-Lof random oracle satisfies @math . Moreover, the class of such reals is comeager and includes a @math real as well as all weakly 2-generic reals. This excludes many slow growing functions such as @math from bounding the redundancy in computations from random oracles for a large class of reals. On the other hand it was recently shown that if @math then @math is a general upper bound for the redundancy in computations of any real from some Martin-Lof random oracle. Our results are obtained as an application of a theory of effective betting strategies with restricted wagers which we develop.
Asymptotic conditions on the redundancy @math in computations from random oracles such as the ones in Theorem , have been used with respect to Chaitin's @math in Tadaki @cite_1 and Barmpalias, Fang and Lewis-Pye @cite_6 . However the latter work only refers to computations of computably enumerable sets and reals and does not have essential connections with the present work, except perhaps for some apparent analogy of the statements proved.
@cite_1 @cite_6
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Chaitin [G. J. Chaitin, J. Assoc. Comput. Mach. , vol. 22, pp. 329---340, 1975] introduced his Ω number as a concrete example of random real. The real Ω is defined as the probability that an optimal computer halts, where the optimal computer is a universal decoding algorithm used to define the notion of program-size complexity. Chaitin showed Ω to be random by discovering the property that the first n bits of the base-two expansion of Ω solve the halting problem of the optimal computer for all binary inputs of length at most n . In the present paper we investigate this property from various aspects. It is known that the base-two expansion of Ω and the halting problem are Turing equivalent. We consider elaborations of both the Turing reductions which constitute the Turing equivalence. These elaborations can be seen as a variant of the weak truth-table reduction, where a computable bound on the use function is explicitly specified. We thus consider the relative computational power between the base-two expansion of Ω and the halting problem by imposing the restriction to finite size on both the problems. We characterise the asymptotic upper bounds on the use of Chaitin's ź in oracle computations of halting probabilities (i.e. c.e. reals). We show that the following two conditions are equivalent for any computable function h such that h ( n ) - n is non-decreasing: (1) h ( n ) - n is an information content measure, i.e. the series ź n 2 n - h ( n ) converges, (2) for every c.e. real α there exists a Turing functional via which ź computes α with use bounded by h. We also give a similar characterisation with respect to computations of c.e. sets from ź, by showing that the following are equivalent for any computable non-decreasing function g: (1) g is an information-content measure, (2) for every c.e. set A, ź computes A with use bounded by g. Further results and some connections with Solovay functions (studied by a number of authors 38,3,26,11) are given. Optimal oracle-use of Chaitin's Omega number.Qualification and optimization of the completeness of Chaitin's Omega.Tight bounds on oracle use in the computably enumerable sets and reals.