(Non-)Equivalence of Universal Priors

Ray Solomonoff invented the notion of universal induction featuring an aptly termed "universal" prior probability function over all possible computable environments. The essential property of this prior was its ability to dominate all other such priors. Later, Levin introduced another construction --- a mixture of all possible priors or `universal mixture'. These priors are well known to be equivalent up to multiplicative constants. Here, we seek to clarify further the relationships between these three characterisations of a universal prior (Solomonoff's, universal mixtures, and universally dominant priors). We see that the the constructions of Solomonoff and Levin define an identical class of priors, while the class of universally dominant priors is strictly larger. We provide some characterisation of the discrepancy.Comment: 10 LaTeX pages, 1 figur

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 ϵ>1, any pair of omega numbers compute each other with redundancy ϵlogn. On the other hand, this is not true for ϵ=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 note on the differences of computably enumerable reals

We show that given any non-computable left-c.e. real α there exists a left-c.e. real β such that α≠β+γ for all left-c.e. reals and all right-c.e. reals γ. The proof is non-uniform, the dichotomy being whether the given real α is Martin-Loef random or not. It follows that given any universal machine U, there is another universal machine V such that the halting probability of U is not a translation of the halting probability of V by a left-c.e. real. We do not know if there is a uniform proof of this fact

Random reals à la Chaitin with or without prefix-freeness

AbstractWe give a general theorem that provides examples of n-random reals à la Chaitin, for every n≥1; these are halting probabilities of partial computable functions that are universal by adjunction for the class of all partial computable functions, The same result holds for the class functions of partial computable functions with prefix-free domain. Thus, the usual technical requirement of prefix-freeness on domains is an option which we show to be non-critical when dealing with universality by adjunction. We also prove that the condition of universality by adjunction (which, though particular, is a very natural case of optimality) is essential in our theorem

Optimal asymptotic bounds on the oracle use in computations from Chaitin’s Omega

Chaitin’s number is the halting probability of a universal prefix-free machine, and although it depends on the underlying enumeration of prefix-free machines, it is always Turing-complete. It can be observed, in fact, that for every computably enumerable (c.e.) real �, there exists a Turing functional via which computes �, and such that the number of bits of that are needed for the computation of the first n bits of � (i.e. the use on argument n) is bounded above by a computable function h(n) = n + o (n). We characterise the asymptotic upper bounds on the use of Chaitin’s 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)

Differences of halting probabilities

We study the differences of Martin-Löf random left-c.e. reals and show that for each pair of such reals α,β there exists a unique number r>0 such that qα−β is a Martin-Löf random left-c.e. real for each positive rational q>r and a Martin-Löf random right-c.e. real for each positive rational q<r. Based on this result we develop a theory of differences of halting probabilities, which answers a number of questions about Martin-Löf random left-c.e. reals, including one of the few remaining open problems from the list of open questions in algorithmic randomness [21]. The halting probability of a prefix-free machine M restricted to a set X is the probability that the machine halts and outputs an element of X . Becher, Figueira, Grigorieff, and Miller asked whether ΩU(X) is Martin-Löf random when U is universal and X is a View the MathML source set. We apply our theory of differences of halting probabilities to give a positive answer

Randomness and universal machines

10.1016/j.jco.2006.05.001Journal of Complexity226738-75