Correspondence and Independence of Numerical Evaluations of Algorithmic Information Measures

We show that real-value approximations of Kolmogorov-Chaitin complexity K(s) using the algorithmic coding theorem, as calculated from the output frequency of a large set of small deterministic Turing machines with up to 5 states (and 2 symbols), is consistent with the number of instructions used by the Turing machines producing s, which in turn is consistent with strict integer-value program-size complexity (based on our knowledge of the smallest machine in terms of the number of instructions used). We also show that neither K(s) nor the number of instructions used manifests any correlation with Bennett's Logical Depth LD(s), other than what's predicted by the theory (shallow and non-random strings have low complexity under both measures). The agreement between the theory and the numerical calculations shows that despite the undecidability of these theoretical measures, the rate of convergence of approximations is stable enough to devise some applications. We announce a Beta version of an Online Algorithmic Complexity Calculator (OACC) implementing these methods

A Computable Measure of Algorithmic Probability by Finite Approximations with an Application to Integer Sequences

Given the widespread use of lossless compression algorithms to approximate algorithmic (Kolmogorov-Chaitin) complexity, and that lossless compression algorithms fall short at characterizing patterns other than statistical ones not different to entropy estimations, here we explore an alternative and complementary approach. We study formal properties of a Levin-inspired measure $m$ calculated from the output distribution of small Turing machines. We introduce and justify finite approximations $m_k$ that have been used in some applications as an alternative to lossless compression algorithms for approximating algorithmic (Kolmogorov-Chaitin) complexity. We provide proofs of the relevant properties of both $m$ and $m_k$ and compare them to Levin's Universal Distribution. We provide error estimations of $m_k$ with respect to $m$. Finally, we present an application to integer sequences from the Online Encyclopedia of Integer Sequences which suggests that our AP-based measures may characterize non-statistical patterns, and we report interesting correlations with textual, function and program description lengths of the said sequences.Comment: As accepted by the journal Complexity (Wiley/Hindawi

Approximations of Algorithmic and Structural Complexity Validate Cognitive-behavioural Experimental Results

We apply methods for estimating the algorithmic complexity of sequences to behavioural sequences of three landmark studies of animal behavior each of increasing sophistication, including foraging communication by ants, flight patterns of fruit flies, and tactical deception and competition strategies in rodents. In each case, we demonstrate that approximations of Logical Depth and Kolmogorv-Chaitin complexity capture and validate previously reported results, in contrast to other measures such as Shannon Entropy, compression or ad hoc. Our method is practically useful when dealing with short sequences, such as those often encountered in cognitive-behavioural research. Our analysis supports and reveals non-random behavior (LD and K complexity) in flies even in the absence of external stimuli, and confirms the "stochastic" behaviour of transgenic rats when faced that they cannot defeat by counter prediction. The method constitutes a formal approach for testing hypotheses about the mechanisms underlying animal behaviour.Comment: 28 pages, 7 figures and 2 table

Structure emerges faster during cultural transmission in children than in adults

How does children’s limited processing capacity affect cultural transmission of complex information? We show that over the course of iterated reproduction of two-dimensional random dot patterns transmission accuracy increased to a similar extent in 5- to 8-year-old children and adults whereas algorithmic complexity decreased faster in children. Thus, children require more structure to render complex inputs learnable. In line with the Less-Is-More hypothesis, we interpret this as evidence that children’s processing limitations affecting working memory capacity and executive control constrain the ability to represent and generate complexity, which, in turn, facilitates emergence of structure. This underscores the importance of investigating the role of children in the transmission of complex cultural traits

Coding-theorem Like Behaviour and Emergence of the Universal Distribution from Resource-bounded Algorithmic Probability

Previously referred to as `miraculous' in the scientific literature because of its powerful properties and its wide application as optimal solution to the problem of induction/inference, (approximations to) Algorithmic Probability (AP) and the associated Universal Distribution are (or should be) of the greatest importance in science. Here we investigate the emergence, the rates of emergence and convergence, and the Coding-theorem like behaviour of AP in Turing-subuniversal models of computation. We investigate empirical distributions of computing models in the Chomsky hierarchy. We introduce measures of algorithmic probability and algorithmic complexity based upon resource-bounded computation, in contrast to previously thoroughly investigated distributions produced from the output distribution of Turing machines. This approach allows for numerical approximations to algorithmic (Kolmogorov-Chaitin) complexity-based estimations at each of the levels of a computational hierarchy. We demonstrate that all these estimations are correlated in rank and that they converge both in rank and values as a function of computational power, despite fundamental differences between computational models. In the context of natural processes that operate below the Turing universal level because of finite resources and physical degradation, the investigation of natural biases stemming from algorithmic rules may shed light on the distribution of outcomes. We show that up to 60\% of the simplicity/complexity bias in distributions produced even by the weakest of the computational models can be accounted for by Algorithmic Probability in its approximation to the Universal Distribution.Comment: 27 pages main text, 39 pages including supplement. Online complexity calculator: http://complexitycalculator.com

On the Complexity and Behaviour of Cryptocurrencies Compared to Other Markets

We show that the behaviour of Bitcoin has interesting similarities to stock and precious metal markets, such as gold and silver. We report that whilst Litecoin, the second largest cryptocurrency, closely follows Bitcoin's behaviour, it does not show all the reported properties of Bitcoin. Agreements between apparently disparate complexity measures have been found, and it is shown that statistical, information-theoretic, algorithmic and fractal measures have different but interesting capabilities of clustering families of markets by type. The report is particularly interesting because of the range and novel use of some measures of complexity to characterize price behaviour, because of the IRS designation of Bitcoin as an investment property and not a currency, and the announcement of the Canadian government's own electronic currency MintChip.Comment: 16 pages, 11 figures, 4 table

Natural scene statistics mediate the perception of image complexity

Humans are sensitive to complexity and regularity in patterns. The subjective perception of pattern complexity is correlated to algorithmic (Kolmogorov-Chaitin) complexity as defined in computer science, but also to the frequency of naturally occurring patterns. However, the possible mediational role of natural frequencies in the perception of algorithmic complexity remains unclear. Here we reanalyze Hsu et al. (2010) through a mediational analysis, and complement their results in a new experiment. We conclude that human perception of complexity seems partly shaped by natural scenes statistics, thereby establishing a link between the perception of complexity and the effect of natural scene statistics

Optimal Uncertainty Quantification

We propose a rigorous framework for Uncertainty Quantification (UQ) in which the UQ objectives and the assumptions/information set are brought to the forefront. This framework, which we call Optimal Uncertainty Quantification (OUQ), is based on the observation that, given a set of assumptions and information about the problem, there exist optimal bounds on uncertainties: these are obtained as extreme values of well-defined optimization problems corresponding to extremizing probabilities of failure, or of deviations, subject to the constraints imposed by the scenarios compatible with the assumptions and information. In particular, this framework does not implicitly impose inappropriate assumptions, nor does it repudiate relevant information. Although OUQ optimization problems are extremely large, we show that under general conditions, they have finite-dimensional reductions. As an application, we develop Optimal Concentration Inequalities (OCI) of Hoeffding and McDiarmid type. Surprisingly, contrary to the classical sensitivity analysis paradigm, these results show that uncertainties in input parameters do not necessarily propagate to output uncertainties. In addition, a general algorithmic framework is developed for OUQ and is tested on the Caltech surrogate model for hypervelocity impact, suggesting the feasibility of the framework for important complex systems