Algorithmic complexity for psychology: A user-friendly implementation of the coding theorem method

Kolmogorov-Chaitin complexity has long been believed to be impossible to approximate when it comes to short sequences (e.g. of length 5-50). However, with the newly developed \emph{coding theorem method} the complexity of strings of length 2-11 can now be numerically estimated. We present the theoretical basis of algorithmic complexity for short strings (ACSS) and describe an R-package providing functions based on ACSS that will cover psychologists' needs and improve upon previous methods in three ways: (1) ACSS is now available not only for binary strings, but for strings based on up to 9 different symbols, (2) ACSS no longer requires time-consuming computing, and (3) a new approach based on ACSS gives access to an estimation of the complexity of strings of any length. Finally, three illustrative examples show how these tools can be applied to psychology.Comment: to appear in "Behavioral Research Methods", 14 pages in journal format, R package at http://cran.r-project.org/web/packages/acss/index.htm

Algorithmic Complexity for Short Binary Strings Applied to Psychology: A Primer

Since human randomness production has been studied and widely used to assess executive functions (especially inhibition), many measures have been suggested to assess the degree to which a sequence is random-like. However, each of them focuses on one feature of randomness, leading authors to have to use multiple measures. Here we describe and advocate for the use of the accepted universal measure for randomness based on algorithmic complexity, by means of a novel previously presented technique using the the definition of algorithmic probability. A re-analysis of the classical Radio Zenith data in the light of the proposed measure and methodology is provided as a study case of an application.Comment: To appear in Behavior Research Method

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

Kolmogorov Complexity in perspective. Part I: Information Theory and Randomnes

We survey diverse approaches to the notion of information: from Shannon entropy to Kolmogorov complexity. Two of the main applications of Kolmogorov complexity are presented: randomness and classification. The survey is divided in two parts in the same volume. Part I is dedicated to information theory and the mathematical formalization of randomness based on Kolmogorov complexity. This last application goes back to the 60's and 70's with the work of Martin-L\"of, Schnorr, Chaitin, Levin, and has gained new impetus in the last years.Comment: 40 page

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