A constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences. This constructive dimension is used to assign every individual (infinite, binary) sequence S a dimension, which is a real number dim(S) in the interval [0, 1]. Sequences that

An algorithm M is described that solves any well-defined problem p as quickly as the fastest algorithm computing a solution to p, save for a factor of 5 and low-order additive terms. M optimally distributes resources between the execution of provably correct p-solving programs and an enumeration of all proofs, including relevant proofs of program correctness and of time bounds on program runtimes. M avoids Blum's speed-up theorem by ignoring programs without correctness proof. M has broader applicability and can be faster than Levin's universal search, the fastest method for inverting functions save for a large multiplicative constant. An extension of Kolmogorov complexity and two novel natural measures of function complexity are used to show that the most efficient program computing some function f is also among the shortest programs provably computing f.

Several new relations between universal Solomonoff sequence prediction and informed prediction and general probabilistic prediction schemes will be proved. Among others, they show that the number of errors in Solomonoff prediction is finite for computable prior probability, if finite in the informed case, where the prior is known. Deterministic variants will also be studied. The most interesting result is that the deterministic variant of Solomonoff prediction is optimal compared to any other probabilistic or deterministic prediction scheme apart from additive square root corrections only. This makes it well suited even for difficult prediction problems, where it does not suffice when the number of errors is minimal to within some factor greater than one. Solomonoff's original bound and the ones presented here complement each other in a useful way.

Supergales, generalizations of supermartingales, have been used by Lutz (2002) to define the constructive dimensions of individual binary sequences. Here it is shown that gales, the corresponding generalizations of martingales, can be equivalently used to define constructive dimension. 1

In the context of the dynamical systems of classical mechanics, we introduce two new notions called \algorithmic ne-grain and coarse-grain entropy". The ne-grain algorithmic entropy is, on the one hand, a simple variant of the randomness tests of Martin-Lof (and others) and is, on the other hand, a connecting link between description (Kolmogorov) complexity, Gibbs entropy and Boltzmann entropy.

We investigate the initial segment complexity of random reals. Let K(...

Introduction Let B = f0; 1g, B n = f0; 1g n , B = [ n0 B n , B 1 = f0; 1g 1 . Let be a probability measure on B 1 . x 1 1 2 B 1 is dened to be typical for if for any u 2 B , lim n!1 u (x n 1 ) n juj + 1 = (u); where u (x n 1 ) is the number of occurrences of u

this paper the general notion of an algorithmic problem (see [7] for such discussion), as our paper is devoted to very specic problems. The plain Kolmogorov complexity, K(x), is the Kolmogorov complexity of the problem \print x". Likewise the conditional Kolmogorov complexity, dened as K(xjy) = minfl(p) j p(y) = xg; is the complexity of the problem \given y print x"

this paper we only consider the context of concept learning : Let X be a set called the instance space. A concept is a subset of X . Usually concepts are identied with their indicating function (by abuse of notations c(x) = 1 x 2 c) A concept class is a set C 2

The algorithmic theory of randomness is well developed when the underlying space is the set of finite or infinite sequences and the underlying probability distribution is the uniform distribution or a computable distribution. These restrictions seem artificial. Some progress has been made to extend...