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Informationtheoretic Limitations of Formal Systems
 JOURNAL OF THE ACM
, 1974
"... An attempt is made to apply informationtheoretic computational complexity to metamathematics. The paper studies the number of bits of instructions that must be a given to a computer for it to perform finite and infinite tasks, and also the amount of time that it takes the computer to perform these ..."
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Cited by 45 (7 self)
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An attempt is made to apply informationtheoretic computational complexity to metamathematics. The paper studies the number of bits of instructions that must be a given to a computer for it to perform finite and infinite tasks, and also the amount of time that it takes the computer to perform these tasks. This is applied to measuring the difficulty of proving a given set of theorems, in terms of the number of bits of axioms that are assumed, and the size of the proofs needed to deduce the theorems from the axioms.
Sequential predictions based on algorithmic complexity
, 2004
"... This paper studies sequence prediction based on the monotone Kolmogorov complexity Km=−log m, i.e. based on universal deterministic/onepart MDL. m is extremely close to Solomonoff’s universal prior M, the latter being an excellent predictor in deterministic as well as probabilistic environments, wh ..."
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Cited by 6 (6 self)
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This paper studies sequence prediction based on the monotone Kolmogorov complexity Km=−log m, i.e. based on universal deterministic/onepart 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 onsequence; the offsequence convergence can be slow. In probabilistic environments, neither the posterior nor the losses converge,
A New Approach to Formal Language Theory by Kolmogorov Complexity
 Preprint, SIAM J. Comput
, 1995
"... We present a new approach to formal language theory using Kolmogorov complexity. The main results presented here are an alternative for pumping lemma(s), a new characterization for regular languages, and a new method to separate deterministic contextfree languages and nondeterministic contextfree ..."
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Cited by 3 (0 self)
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We present a new approach to formal language theory using Kolmogorov complexity. The main results presented here are an alternative for pumping lemma(s), a new characterization for regular languages, and a new method to separate deterministic contextfree languages and nondeterministic contextfree languages. The use of the new `incompressibility arguments' is illustrated by many examples. The approach is also successful at the high end of the Chomsky hierarchy since one can quantify nonrecursiveness in terms of Kolmogorov complexity. (This is a preliminary uncorrected version. The final version is the one published in SIAM J. Comput., 24:2(1995), 398410.) 1 Introduction It is feasible to reconstruct parts of formal language theory using algorithmic information theory (Kolmogorov complexity). We provide theorems on how to use Kolmogorov complexity as a concrete and powerful tool. We do not just want A preliminary version of part of this work was presented at the 16th International...
Algorithmic Randomness as Foundation of Inductive Reasoning and Artificial Intelligence
, 2010
"... This article is a personal account of the past, present, and future of algorithmic randomness, emphasizing its role in inductive inference and artificial intelligence. It is written for a general audience interested in science and philosophy. Intuitively, randomness is a lack of order or predictabil ..."
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This article is a personal account of the past, present, and future of algorithmic randomness, emphasizing its role in inductive inference and artificial intelligence. It is written for a general audience interested in science and philosophy. Intuitively, randomness is a lack of order or predictability. If randomness is the opposite of determinism, then algorithmic randomness is the opposite of computability. Besides many other things, these concepts have been used to quantify Ockham’s razor, solve the induction problem, and define intelligence. Contents 1 Why were you initially drawn to the study of computation