The traditional theory of Kolmogorov complexity and algorithmic probability focuses on monotone Turing machines with one-way write-only output tape. This naturally leads to the universal enumerable Solomono-Levin measure. Here we introduce more general, nonenumerable but cumulatively enumerable measures (CEMs) derived from Turing machines with lexicographically nondecreasing output and random input, and even more general approximable measures and distributions computable in the limit. We obtain a natural hierarchy of generalizations of algorithmic probability and Kolmogorov complexity, suggesting that the "true" information content of some (possibly in nite) bitstring x is the size of the shortest nonhalting program that converges to x and nothing but x on a Turing machine that can edit its previous outputs. Among other things we show that there are objects computable in the limit yet more random than Chaitin's "number of wisdom" Omega, that any approximable measure of x is small for any x lacking a short description, that there is no universal approximable distribution, that there is a universal CEM, and that any nonenumerable CEM of x is small for any x lacking a short enumerating program. We briey mention consequences for universes sampled from such priors.

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.

Abstract. Decision theory formally solves the problem of rational agents in uncertain worlds if the true environmental probability distribution is known. Solomonoff’s theory of universal induction formally solves the problem of sequence prediction for unknown distributions. We unify both theories and give strong arguments that the resulting universal AIξ model behaves optimally in any computable environment. The major drawback of the AIξ model is that it is uncomputable. To overcome this problem, we construct a modified algorithm AIξ tl, which is still superior to any other time t and length l bounded agent. The computation time of AIξ tl is of the order t·2 l. 1

This paper studies sequence prediction based on the monotone Kolmogorov complexity Km=-log m, i.e. based on universal MDL. m is extremely close to Solomonoff's prior M, the latter being an excellent predictor in deterministic as well as probabilistic environments, where performance is measured in terms of convergence of posteriors or losses. Despite this closeness to M, it is difficult to assess the prediction quality of m, since little is known about the closeness of their posteriors, which are the important quantities for prediction. We show that for deterministic computable environments, the "posterior" and losses of m converge, but rapid convergence could only be shown on-sequence; the off-sequence behavior is unclear. In probabilistic environments, neither the posterior nor the losses converge, in general.

Decision theory formally solves the problem of rational agents in uncertain worlds if the true environmental prior probability distribution is known. Solomonoff's theory of universal induction formally solves the problem of sequence prediction for unknown prior distribution. We combine both ideas and get a parameterless theory of universal Artificial Intelligence. We give strong arguments that the resulting AIXI model is the most intelligent unbiased agent possible. We outline for a number of problem classes, including sequence prediction, strategic games, function minimization, reinforcement and supervised learning, how the AIXI model can formally solve them. The major drawback of the AIXI model is that it is uncomputable. To overcome this problem, we construct a modified algorithm AIXItl, which is still effectively more intelligent than any other time t and space l bounded agent. The computation time of AIXItl is of the order t·2^l. Other discussed topics are formal definitions of intelligence order relations, the horizon problem and relations of the AIXI theory to other AI approaches.