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30
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
Optimal Ordered Problem Solver
, 2002
"... We present a novel, general, optimally fast, incremental way of searching for a universal algorithm that solves each task in a sequence of tasks. The Optimal Ordered Problem Solver (OOPS) continually organizes and exploits previously found solutions to earlier tasks, eciently searching not only the ..."
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Cited by 62 (20 self)
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We present a novel, general, optimally fast, incremental way of searching for a universal algorithm that solves each task in a sequence of tasks. The Optimal Ordered Problem Solver (OOPS) continually organizes and exploits previously found solutions to earlier tasks, eciently searching not only the space of domainspecific algorithms, but also the space of search algorithms. Essentially we extend the principles of optimal nonincremental universal search to build an incremental universal learner that is able to improve itself through experience.
Trivial Reals
"... Solovay showed that there are noncomputable reals ff such that H(ff _ n) 6 H(1n) + O(1), where H is prefixfree Kolmogorov complexity. Such Htrivial reals are interesting due to the connection between algorithmic complexity and effective randomness. We give a new, easier construction of an Htrivi ..."
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Cited by 57 (31 self)
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Solovay showed that there are noncomputable reals ff such that H(ff _ n) 6 H(1n) + O(1), where H is prefixfree Kolmogorov complexity. Such Htrivial reals are interesting due to the connection between algorithmic complexity and effective randomness. We give a new, easier construction of an Htrivial real. We also analyze various computabilitytheoretic properties of the Htrivial reals, showing for example that no Htrivial real can compute the halting problem. Therefore, our construction of an Htrivial computably enumerable set is an easy, injuryfree construction of an incomplete computably enumerable set. Finally, we relate the Htrivials to other classes of "highly nonrandom " reals that have been previously studied.
Algorithmic Statistics
 IEEE Transactions on Information Theory
, 2001
"... While Kolmogorov complexity is the accepted absolute measure of information content of an individual finite object, a similarly absolute notion is needed for the relation between an individual data sample and an individual model summarizing the information in the data, for example, a finite set (or ..."
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Cited by 52 (14 self)
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While Kolmogorov complexity is the accepted absolute measure of information content of an individual finite object, a similarly absolute notion is needed for the relation between an individual data sample and an individual model summarizing the information in the data, for example, a finite set (or probability distribution) where the data sample typically came from. The statistical theory based on such relations between individual objects can be called algorithmic statistics, in contrast to classical statistical theory that deals with relations between probabilistic ensembles. We develop the algorithmic theory of statistic, sufficient statistic, and minimal sufficient statistic. This theory is based on twopart codes consisting of the code for the statistic (the model summarizing the regularity, the meaningful information, in the data) and the modeltodata code. In contrast to the situation in probabilistic statistical theory, the algorithmic relation of (minimal) sufficiency is an absolute relation between the individual model and the individual data sample. We distinguish implicit and explicit descriptions of the models. We give characterizations of algorithmic (Kolmogorov) minimal sufficient statistic for all data samples for both description modes in the explicit mode under some constraints. We also strengthen and elaborate earlier results on the "Kolmogorov structure function" and "absolutely nonstochastic objects" those rare objects for which the simplest models that summarize their relevant information (minimal sucient statistics) are at least as complex as the objects themselves. We demonstrate a close relation between the probabilistic notions and the algorithmic ones: (i) in both cases there is an "information nonincrease" law; (ii) it is shown that a function is a...
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).
Logical Depth and Physical Complexity
 THE UNIVERSAL TURING MACHINE: A HALFCENTURY SURVEY
, 1988
"... Some mathematical and natural objects (a random sequence, a sequence of zeros, a perfect crystal, a gas) are intuitively trivial, while others (e.g. the human body, the digits of #) contain internal evidence of a nontrivial causal history. We formalize this ..."
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Cited by 48 (0 self)
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Some mathematical and natural objects (a random sequence, a sequence of zeros, a perfect crystal, a gas) are intuitively trivial, while others (e.g. the human body, the digits of #) contain internal evidence of a nontrivial causal history. We formalize this
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 Application Of Algorithmic Probability to Problems in Artificial Intelligence
 in Uncertainty in Artificial Intelligence, Kanal, L.N. and Lemmer, J.F. (Eds), Elsevier Science Publishers B.V
, 1986
"... INTRODUCTION We will cover two topics First, Algorithmic Probability  the motivation for defining it, how it overcomes di#culties in other formulations of probability, some of its characteristic properties and successful applications. Second, we will apply it to problems in A.I.  where it p ..."
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Cited by 30 (5 self)
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INTRODUCTION We will cover two topics First, Algorithmic Probability  the motivation for defining it, how it overcomes di#culties in other formulations of probability, some of its characteristic properties and successful applications. Second, we will apply it to problems in A.I.  where it promises to give near optimum search procedures for two very broad classes of problems. A strong motivation for revising classical concepts of probability has come from the analysis of human problem solving. When working on a di#cult problem, a person is in a maze in which he must make choices of possible courses of action. If the problem is a familiar one, the choices will all be easy. If it is not familiar, there can be much uncertainty in each choice, but choices must somehow be made. One basis for choice might be the probability of each choice leading to a quick solution  this probability being based on experience in this problem and in problems like it. A good reason for using proba
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 ...
On the relationship between complexity and entropy for Markov chains and regular languages
 Complex Systems
, 1991
"... Abstract. Using the pastfuture mutual information as a measure of complexity, the relation between the complexity and the Shannon entropy is determined analytically for sequences generated by Markov chains and regular languages. It is emphasized that, given an entropy value, there are many possible ..."
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Cited by 22 (2 self)
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Abstract. Using the pastfuture mutual information as a measure of complexity, the relation between the complexity and the Shannon entropy is determined analytically for sequences generated by Markov chains and regular languages. It is emphasized that, given an entropy value, there are many possible complexity values, and vice versa; that is, the relationship between complexity and entropy is not onetoone, but rather manytoone or onetomany. It is also emphasized that there are structures in the complexityversusentropy plots, and these structures depend on the details of a Markov chain or a regular language grammar. 1.