641 research outputs found
Statistical Complexity Analysis of Turing Machine tapes with Fixed Algorithmic Complexity Using the Best-Order Markov Model
Sources that generate symbolic sequences with algorithmic nature may differ in statistical complexity because they create structures that follow algorithmic schemes, rather than generating symbols from a probabilistic function assuming independence. In the case of Turing machines, this means that machines with the same algorithmic complexity can create tapes with different statistical complexity. In this paper, we use a compression-based approach to measure global and local statistical complexity of specific Turing machine tapes with the same number of states and alphabet. Both measures are estimated using the best-order Markov model. For the global measure, we use the Normalized Compression (NC), while, for the local measures, we define and use normal and dynamic complexity profiles to quantify and localize lower and higher regions of statistical complexity. We assessed the validity of our methodology on synthetic and real genomic data showing that it is tolerant to increasing rates of editions and block permutations. Regarding the analysis of the tapes, we localize patterns of higher statistical complexity in two regions, for a different number of machine states. We show that these patterns are generated by a decrease of the tape's amplitude, given the setting of small rule cycles. Additionally, we performed a comparison with a measure that uses both algorithmic and statistical approaches (BDM) for analysis of the tapes. Naturally, BDM is efficient given the algorithmic nature of the tapes. However, for a higher number of states, BDM is progressively approximated by our methodology. Finally, we provide a simple algorithm to increase the statistical complexity of a Turing machine tape while retaining the same algorithmic complexity. We supply a publicly available implementation of the algorithm in C++ language under the GPLv3 license. All results can be reproduced in full with scripts provided at the repository.Peer reviewe
Towards a Universal Theory of Artificial Intelligence based on Algorithmic Probability and Sequential Decision Theory
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 distribution. We unify both theories and give
strong arguments that the resulting universal AIXI model behaves optimal in any
computable environment. The major drawback of the AIXI model is that it is
uncomputable. To overcome this problem, we construct a modified algorithm
AIXI^tl, which is still superior to any other time t and space l bounded agent.
The computation time of AIXI^tl is of the order t x 2^l.Comment: 8 two-column pages, latex2e, 1 figure, submitted to ijca
Towards a universal theory of artificial intelligence based on algorithmic probability and sequential decisions
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, which is still superior to any other time t and length l bounded agent. The computation time of AIξtl is of the order t·2 l.This work was supported by SNF grant 2000-61847.00 to Jürgen Schmidhuber
Universal Algorithmic Intelligence: A mathematical top->down approach
Sequential 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 parameter-free theory of universal Artificial Intelligence. We give
strong arguments that the resulting AIXI model is the most intelligent unbiased
agent possible. We outline how the AIXI model can formally solve a number of
problem classes, including sequence prediction, strategic games, function
minimization, reinforcement and supervised learning. The major drawback of the
AIXI model is that it is uncomputable. To overcome this problem, we construct a
modified algorithm AIXItl that is still effectively more intelligent than any
other time t and length l bounded agent. The computation time of AIXItl is of
the order t x 2^l. The discussion includes formal definitions of intelligence
order relations, the horizon problem and relations of the AIXI theory to other
AI approaches.Comment: 70 page
Sequential Predictions based on Algorithmic Complexity
This paper studies sequence prediction based on the monotone Kolmogorov
complexity Km=-log m, i.e. based on universal deterministic/one-part MDL. m is
extremely close to Solomonoff's universal 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 convergence can
be slow. In probabilistic environments, neither the posterior nor the losses
converge, in general.Comment: 26 pages, LaTe
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