ABSTRACT Recent results on the internal, measure-theoretic structure of the exponential time complexity classes E and EXP are surveyed. The measure structure of these classes is seen to interact in informative ways with bi-immunity, complexity cores, polynomial-time reductions, completeness, circuit-size complexity, Kolmogorov complexity, natural proofs, pseudorandom generators, the density of hard languages, randomized complexity, and lowness. Possible implications for the structure of NP are also discussed. 1

The relationship between the Bayesian approach and the minimum description length approach is established. We sharpen and clarify the general modeling principles MDL and MML, abstracted as the ideal MDL principle and defined from Bayes's rule by means of Kolmogorov complexity. The basic condition under which the ideal principle should be applied is encapsulated as the Fundamental Inequality, which in broad terms states that the principle is valid when the data are random, relative to every contemplated hypothesis and also these hypotheses are random relative to the (universal) prior. Basically, the ideal principle states that the prior probability associated with the hypothesis should be given by the algorithmic universal probability, and the sum of the log universal probability of the model plus the log of the probability of the data given the model should be minimized. If we restrict the model class to the finite sets then application of the ideal principle turns into Kolmogorov's mi...

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 two-part codes consisting of the code for the statistic (the model summarizing the regularity, the meaningful information, in the data) and the model-to-data 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 non-stochastic 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 non-increase" law; (ii) it is shown that a function is a...

The likelihood principle states that the visual system prefers the most likely interpretation of a stimulus, whereas the simplicity principle states that it prefers the most simple interpretation. This study investi-gates how close these seemingly very different principles are by combining findings from classical, algorithmic, and structural information theory. It is argued that, in visual perception, the two principles are perhaps very different with respect to the viewpoint-independent aspects of perception but probably very close with respect to the viewpoint-dependent aspects which, moreover, seem decisive in everyday perception. This implies that either principle may have guided the evolution of visual systems and that the simplicity paradigm may provide perception models with the necessary quantitative specifications of the often plausible but also intuitive ideas provided by the likelihood paradigm. In visual perception research, an ongoing debate concerns the question of whether the likelihood principle (Von Helmholtz, 1909/1962) or the simplicity principle (Hochberg & McAlister, 1953) provides the best explanation of the human interpretation of visual stimuli. The phenomenon to be explained is, more specifi-cally, that human subjects usually show a clear preference for only

It is shown that the class of Kolmogorov-Loveland stochastic sequences is not closed under selecting subsequences by monotonic computable selection rules. This result gives a strong negative answer to the notorious open problem whether the Kolmogorov-Loveland stochastic sequences are closed under selecting subsequences by KolmogorovLoveland selection rules, i.e., by not necessarily monotonic partially computable selection rules. As a corollary, we obtain an easy proof for the previously known result that the Kolmogorov-Loveland stochastic sequences form a proper subclass of the Mises-Wald-Church stochastic sequences.

Consider a sequence of i.i.d. random variables, where each variable is refreshed (i.e., replaced by an independent variable with the same law) independently, according to a Poisson clock. At any fixed time t, the resulting sequence has the same law as at time 0, but there can be exceptional random times at which certain almost sure properties of the time 0 sequence are violated. We prove that there are no such exceptional times for the law of large numbers and the law of the iterated logarithm, so these laws are dynamically stable. However, there are times at which run lengths are exceptionally long, i.e., run tests are dynamically sensitive. We obtain a multifractal analysis of exceptional times for run lengths and for prediction. In particular, starting from an i.i.d. sequence of unbiased random bits, the random set of times t where ff log 2 (n) bits among the first n bits can be predicted from their predecessors, has Hausdorff dimension 1 \Gamma ff a.s. Finally, we consider simple random walk in the lattice Z d , and prove that transience is dynamically stable for d 5, and dynamically sensitive for d = 3; 4. Moreover, for d = 3; 4, the nonempty random set of exceptional times t where the walk is recurrent, has Hausdorff dimension (4 \Gamma d)=2 a.s. 1

After a brief review of ontic and epistemic descriptions, and of subjective, logical and statistical interpretations of probability, we summarize the traditional axiomatization of calculus of probability in terms of Boolean algebras and its set-theoretical realization in terms of Kolmogorov probability spaces. Since the axioms of mathematical probability theory say nothing about the conceptual meaning of “randomness ” one considers probability as property of the generating conditions of a process so that one can relate randomness with predictability (or retrodictability). In the measure-theoretical codification of stochastic processes genuine chance processes can be defined rigorously as so-called regular processes which do not allow a long-term prediction. We stress that stochastic processes are equivalence classes of individual point functions so that they do not refer to individual processes but only to an ensemble of statistically equivalent individual processes. Less popular but conceptually more important than statistical descriptions are individual descriptions which refer to individual chaotic processes. First, we review the individual description based on the generalized harmonic analysis by Norbert Wiener. It allows the definition of individual purely chaotic processes which can be interpreted as trajectories of regular statistical stochastic processes.

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Abstract The concept of randomness has been unjustly neglected in recent philosophical literature, and when philosophers have thought about it, they have usually acquiesced in views about the concept that are fundamentally flawed. After indicating the ways in which these accounts are flawed, I propose that randomness is to be understood as a special case of the epistemic concept of the unpredictability of a process. This proposal arguably captures the intuitive desiderata for the concept of randomness; at least it should suggest that the commonly accepted accounts cannot be the whole story and more philosophical attention needs to be paid. [R]andomness... is going to be a concept which is relative to our body of knowledge, which will somehow reflect what we know and what we don’t know.

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