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29
Minimum Message Length and Kolmogorov Complexity
 Computer Journal
, 1999
"... this paper is to describe some of the relationships among the different streams and to try to clarify some of the important differences in their assumptions and development. Other studies mentioning the relationships appear in [1, Section IV, pp. 10381039], [2, sections 5.2, 5.5] and [3, p. 465] ..."
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Cited by 104 (25 self)
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this paper is to describe some of the relationships among the different streams and to try to clarify some of the important differences in their assumptions and development. Other studies mentioning the relationships appear in [1, Section IV, pp. 10381039], [2, sections 5.2, 5.5] and [3, p. 465]
The Dimensions of Individual Strings and Sequences
 INFORMATION AND COMPUTATION
, 2003
"... 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 ..."
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Cited by 93 (10 self)
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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
The Speed Prior: A New Simplicity Measure Yielding NearOptimal Computable Predictions
 Proceedings of the 15th Annual Conference on Computational Learning Theory (COLT 2002), Lecture Notes in Artificial Intelligence
, 2002
"... Solomonoff's optimal but noncomputable method for inductive inference assumes that observation sequences x are drawn from an recursive prior distribution p(x). Instead of using the unknown p() he predicts using the celebrated universal enumerable prior M() which for all exceeds any recursive p() ..."
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Cited by 51 (20 self)
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Solomonoff's optimal but noncomputable method for inductive inference assumes that observation sequences x are drawn from an recursive prior distribution p(x). Instead of using the unknown p() he predicts using the celebrated universal enumerable prior M() which for all exceeds any recursive p(), save for a constant factor independent of x. The simplicity measure M() naturally implements "Occam's razor " and is closely related to the Kolmogorov complexity of . However, M assigns high probability to certain data that are extremely hard to compute. This does not match our intuitive notion of simplicity. Here we suggest a more plausible measure derived from the fastest way of computing data. In absence of contrarian evidence, we assume that the physical world is generated by a computational process, and that any possibly infinite sequence of observations is therefore computable in the limit (this assumption is more radical and stronger than Solomonoff's).
Hierarchies Of Generalized Kolmogorov Complexities And Nonenumerable Universal Measures Computable In The Limit
 INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE
, 2000
"... The traditional theory of Kolmogorov complexity and algorithmic probability focuses on monotone Turing machines with oneway writeonly output tape. This naturally leads to the universal enumerable SolomonoLevin measure. Here we introduce more general, nonenumerable but cumulatively enumerable m ..."
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Cited by 38 (20 self)
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The traditional theory of Kolmogorov complexity and algorithmic probability focuses on monotone Turing machines with oneway writeonly output tape. This naturally leads to the universal enumerable SolomonoLevin 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.
The Fastest And Shortest Algorithm For All WellDefined Problems
, 2002
"... An algorithm M is described that solves any welldefined problem p as quickly as the fastest algorithm computing a solution to p, save for a factor of 5 and loworder additive terms. M optimally distributes resources between the execution of provably correct psolving programs and an enumeration of ..."
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Cited by 35 (7 self)
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An algorithm M is described that solves any welldefined problem p as quickly as the fastest algorithm computing a solution to p, save for a factor of 5 and loworder additive terms. M optimally distributes resources between the execution of provably correct psolving programs and an enumeration of all proofs, including relevant proofs of program correctness and of time bounds on program runtimes. M avoids Blum's speedup 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.
Exploring the Predictable
, 2002
"... Details of complex event sequences are often not predictable, but their reduced abstract representations are. I study an embedded active learner that can limit its predictions to almost arbitrary computable aspects of spatiotemporal events. It constructs probabilistic algorithms that (1) control in ..."
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Cited by 23 (10 self)
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Details of complex event sequences are often not predictable, but their reduced abstract representations are. I study an embedded active learner that can limit its predictions to almost arbitrary computable aspects of spatiotemporal events. It constructs probabilistic algorithms that (1) control interaction with the world, (2) map event sequences to abstract internal representations (IRs), (3) predict IRs from IRs computed earlier. Its goal is to create novel algorithms generating IRs useful for correct IR predictions, without wasting time on those learned before. This requires an adaptive novelty measure which is implemented by a coevolutionary scheme involving two competing modules collectively designing (initially random) algorithms representing experiments. Using special instructions, the modules can bet on the outcome of IR predictions computed by algorithms they have agreed upon. If their opinions dier then the system checks who's right, punishes the loser (the surprised one), and rewards the winner. An evolutionary or reinforcement learning algorithm forces each module to maximize reward. This motivates both modules to lure each other into agreeing upon experiments involving predictions that surprise it. Since each module essentially can veto experiments it does not consider profitable, the system is motivated to focus on those computable aspects of the environment where both modules still have confident but different opinions. Once both share the same opinion on a particular issue (via the loser's learning process, e.g., the winner is simply copied onto the loser), the winner loses a source of reward  an incentive to shift the focus of interest onto novel experiments. My simulations include an example where surprisegeneration of this kind helps to speed up ...
Algorithmic Complexity and Stochastic Properties of Finite Binary Sequences
, 1999
"... This paper is a survey of concepts and results related to simple Kolmogorov complexity, prefix complexity and resourcebounded complexity. We also consider a new type of complexity statistical complexity closely related to mathematical statistics. Unlike other discoverers of algorithmic complexit ..."
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Cited by 17 (0 self)
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This paper is a survey of concepts and results related to simple Kolmogorov complexity, prefix complexity and resourcebounded complexity. We also consider a new type of complexity statistical complexity closely related to mathematical statistics. Unlike other discoverers of algorithmic complexity, A. N. Kolmogorov's leading motive was developing on its basis a mathematical theory more adequately substantiating applications of probability theory, mathematical statistics and information theory. Kolmogorov wanted to deduce properties of a random object from its complexity characteristics without use of the notion of probability. In the first part of this paper we present several results in this direction. Though the subsequent development of algorithmic complexity and randomness was different, algorithmic complexity has successful applications in a traditional probabilistic framework. In the second part of the paper we consider applications to the estimation of parameters and the definition of Bernoulli sequences. All considerations have finite combinatorial character. 1.
The New AI: General & Sound & Relevant for Physics
, 2003
"... Most traditional artificial intelligence (AI) systems of the past 50 years are either very limited, or based on heuristics, or both. The new millennium, however, has brought substantial progress in the field of theoretically optimal and practically feasible algorithms for prediction, search, inducti ..."
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Cited by 15 (9 self)
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Most traditional artificial intelligence (AI) systems of the past 50 years are either very limited, or based on heuristics, or both. The new millennium, however, has brought substantial progress in the field of theoretically optimal and practically feasible algorithms for prediction, search, inductive inference based on Occam's razor, problem solving, decision making, and reinforcement learning in environments of a very general type. Since inductive inference is at the heart of all inductive sciences, some of the results are relevant not only for AI and computer science but also for physics, provoking nontraditional predictions based on Zuse's thesis of the computergenerated universe.
OpenEnded Artificial Evolution
, 2001
"... Of all the issues discussed at Alife VII: Looking Forward, Looking Backward, the issue of whether it was possible to create an arti cial life system that exhibits openended evolution of novelty is by far the biggest. ..."
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Cited by 8 (3 self)
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Of all the issues discussed at Alife VII: Looking Forward, Looking Backward, the issue of whether it was possible to create an arti cial life system that exhibits openended evolution of novelty is by far the biggest.
The Kolmogorov sampler
, 2002
"... iid 2 Given noisy observations Xi = θi + Zi, i =1,...,n, with noise Zi ∼ N(0,σ), we wish to recover the signal θ with small meansquared error. We consider the Minimum Kolmogorov Complexity Estimator (MKCE), defined roughly as the nvector ˆ θ(X) solving the problem min Y K(Y) subject to �X − Y �2 l ..."
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Cited by 7 (1 self)
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iid 2 Given noisy observations Xi = θi + Zi, i =1,...,n, with noise Zi ∼ N(0,σ), we wish to recover the signal θ with small meansquared error. We consider the Minimum Kolmogorov Complexity Estimator (MKCE), defined roughly as the nvector ˆ θ(X) solving the problem min Y K(Y) subject to �X − Y �2 l 2 n ≤ σ2 · n, where K(Y) denotes the length of the shortest computer program that can compute the finiteprecision nvector Y.Inwords, this is the simplest object that fits the data to within the lackoffit between θ and X that would be expected on statistical grounds. Suppose that the θi are successive samples from a stationary ergodic process obeying