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

Abstract

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 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. Keywords Algorithmic complexity · Randomness · Subjective probability · Coding theorem method Randomness and complexity are two concepts which are intimately related and are both central to numerous recent developments in various fields, including finance (Taufemback et al., 2011, Brandouy et al., 2012, linguistics (Gruber, 2010;Naranan, 2011), neuropsychology (Machado et al., 2010;Fernández et al., 2011Fernández et al., , 2012, psychiatry (Yang and Tsai, 2012;Takahashi 2013), genetics (Yagil, 2009;Ryabko et al. 2013), sociology (Elzinga, 2010 and the behavioral sciences (Watanabe et al., 2003;Scafetta et al., 2009). In psychology, randomness and complexity have recently attracted interest, following the realization that they could shed light on a diversity of previously undeciphered behaviors and mental processes. It has been found, for instance, that the subjective difficulty of a concept is directly related to its "boolean complexity", defined as the shortest logical description of a concept (Feldman, 2000(Feldman, , 2003(Feldman, , 2006. In the same vein, visual detection of shapes has been shown to be related to contour complexity (Wilder et al., 2011).More generally, perceptual organization itself has been described as based on simplicity or, equivalently, likelihood (Chater, 1996;Chater and Vitányi, 2003), in a model reconciling the complexity approach (perception is organized to minimize complexity) and a probability approach (perception is organized to maximize likelihood), very much in line with our view in this paper. Even the perception of similarity may be viewed through the lens of (conditional) complexity (Hahn et al., 2003).Randomness and complexity also play an important role in modern approaches to selecting the "best" among a set of candidate models (i.e., model selection; e.g., Myung et Behav Res (2016) 48:314-329 315 al., 2006Kellen et al., 2013), as discussed in more detail below in the section called "Relationship to complexity based model selection".Complexity can also shed light on short term memory storage and recall, more specifically, on the process underlying chunking. It is well known that the short term memory span lies b...

The simplicity principle in perception and cognition

“…Note that the actual value of the K(S) is uncomputable. b For long strings, it can be approximated by effective string-compression algorithms such as Lempel-Ziv (see Ref 10) implemented in the common utility gzip, 14 meaning that the Kolmogorov complexity of a long string S is approximately the length, in characters, of the gzipped version of S. For shorter strings, such approximations are in principle less reliable, though recent work by Gauvrit et al 15 has for the first time provided practical techniques for estimating the Kolmogorov complexity of short strings, opening an intriguing research avenue for evaluating the role of complexity in psychological models. conveys information given by − log p i , which quantifies the degree of "surprise" or unexpectedness entailed by the message.…”

The Information-Theoretic and Algorithmic Approach to Human, Animal, and Artificial Cognition

“…Using these methods [Gauvrit et al 2014b], we can compute that, with a prior of 0.5, the probability that the string HHHHHTTTTT is random amounts to 0.58, whereas the probability that HTTHTHHHTT is random amounts to 0.83, thus confirming the common intuition that the latter is "more random" than the former.…”

“…Humans too display biases in the same algorithmic direction, from their motion trajectories [Peng et al 2014] to their perception of reality [Chater 1999]. Indeed, we have shown that cognition, including visual perception and the generation of subjective randomness, shows a bias that can be accounted for with the seminal concept of algorithmic probability [Gauvrit et al 2014a, Gauvrit et al 2014b, Gauvrit et al 2014c, Kempe et al 2015, Mathy et al 2014]. Using a computer to look at human behavior in a novel fashion, specifically by using a reverse Turing test where what is assessed is the human mind and an "average" Turing machine or computer program implementing any possible compression algorithm, we will show that the human mind behaves more like a machine.…”

“…ACSS provides researchers with a user-friendly tool [Gauvrit et al 2014b] for assessing the algorithmic complexity of very short strings. Despite the huge sample (billions) of Turing machines used to build m(s) and K(s) using the Coding theorem method, two strings of length 12 were never produced, yet ACSS can safely be used for strings up to length 12 by assigning the missing two the greatest complexity in the set plus 1.…”

On Measuring the Complexity of Networks: Kolmogorov Complexity versus Entropy

“…In our experiments we have used the acss R package [42] which implements the Coding Theorem Method [34,43] and the Block Decomposition Method [35].…”