96 research outputs found
Kolmogorov complexity and computably enumerable sets
We study the computably enumerable sets in terms of the: (a) Kolmogorov
complexity of their initial segments; (b) Kolmogorov complexity of finite
programs when they are used as oracles. We present an extended discussion of
the existing research on this topic, along with recent developments and open
problems. Besides this survey, our main original result is the following
characterization of the computably enumerable sets with trivial initial segment
prefix-free complexity. A computably enumerable set is -trivial if and
only if the family of sets with complexity bounded by the complexity of is
uniformly computable from the halting problem
Computing halting probabilities from other halting probabilities
The halting probability of a Turing machine is the probability that the machine will halt if it starts with a random stream written on its one-way input tape. When the machine is universal, this probability is referred to as Chaitin's omega number, and is the most well known example of a real which is random in the sense of Martin-L\"{o}f. Although omega numbers depend on the underlying universal Turing machine, they are robust in the sense that they all have the same Turing degree, namely the degree of the halting problem. In this paper we give precise bounds on the redundancy growth rate that is generally required for the computation of an omega number from another omega number. We show that for each ϔ>1, any pair of omega numbers compute each other with redundancy ϔlogn. On the other hand, this is not true for ϔ=1. In fact, we show that for each omega number there exists another omega number which is not computable from the first one with redundancy logn. This latter result improves an older result of Frank Stephan
A Cappable Almost Everywhere Dominating Computably Enumerable Degree
AbstractWe show that there exists an almost everywhere (a.e.) dominating computably enumerable (c.e.) degree which is half of a minimal pair
Lower bounds on the redundancy in computations from random oracles via betting strategies with restricted wagers
The KuÄeraâGĂĄcs 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. KuÄera implicitly achieved redundancy nlogâĄn while GĂĄcs 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 MihailoviÄ. 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 ân2âg(n)=â 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 ân2âg(n)<â. 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 ân2âg(n)<â. Hence our lower bound is optimal, and excludes many slow growing functions such as logâĄn 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
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