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

📝 Abstract

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.

💡 Analysis

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.

📄 Content

Complexity for short strings 1 Running head: COMPLEXITY FOR SHORT STRINGS Algorithmic Complexity for Short Binary Strings Applied to Psychology: A Primer Nicolas Gauvrit CHART (PARIS-reasoning), University of Paris VIII and EPHE, Paris, France Hector Zenil Unit of Computational Medicine, Center for Molecular Medicine, Karolinska Institute, Stockholm, Sweden Jean-Paul Delahaye Laboratoire d’Informatique Fondamentale de Lille, France Fernando Soler-Toscano Grupo de L´ogica, Lenguaje e Informaci´on, Universidad de Sevilla, Spain. Corresponding author: Nicolas Gauvrit, ngauvrit@me.com arXiv:1106.3059v3 [cs.CC] 9 Dec 2013 Complexity for short strings 2 Abstract As human randomness production has come to be more closely studied and used to assess executive functions (especially inhibition), many normative measures for assessing the degree to which a sequence is random-like have been suggested. However, each of these measures focus on one feature of randomness, leading researchers to have to use multiple measures. Although algorithmic complexity has been suggested as a means for overcoming this inconvenience, it has never been used because standard Kolmogorov complexity is inapplicable to short strings (e.g. of length l ≤50), due to both computational and theoretical limitations. Here we describe a novel technique (the “Coding theorem method”) based on the calculation of a universal distribution, which yields an objective and universal measure of algorithmic complexity for short strings that approximates Kolmogorov-Chaitin complexity. Complexity for short strings 3 Algorithmic Complexity for Short Binary Strings Applied to Psychology: A Primer The production of randomness by humans requires high-level cognitive abilities such as sustained attention and inhibition, and is impaired by poor working memory. Unlike other frontal neuropsychological tests, random generation tasks possess specific features of interest: their demand on executive functions, especially inhibition processes, is high; and more importantly, training does not reduce this demand through automatization (Towse & Cheshire, 2007). On the contrary, generating a random-like sequence requires continuous avoidance of any routine, thus preempting any automatized success. Random generation tasks have been widely used in the last few decades to assess working memory, especially (sustained) inhibitive abilities (Miyake et al., 2000), in normal subjects as well as in patients suffering from a wide variety of pathologies. In normal subjects, random generation varies with personal characteristics or states, such as belief in the paranormal (Brugger, Landis, & Regard, 1990) or cultural background (Vandewiele, D’Hondt, Didillon, Iwawaki, & Mwamwendat, 1986; Strenge, Lesmana, & Suryani, 2009). It affords insight into the cognitive effects of aging (Heuer, Janczyk, & Kunde, 2010), hemispheric neglect (Loetscher & Brugger, 2008), schizophrenia (Chan, Hui, Chiu, Chan, & Lam, 2011), aphasia (Proios, Asaridou, & Brugger, 2008), and Down syndrome (Rinehart, Bradshaw, Moss, Brereton, & Tonge, 2006). As a rule, random generation tasks involve generating a random-like sequence of digits (Loetscher, Bockisch, & Brugger, 2009), nouns (Heuer et al., 2010), words (Taylor, Salmon, Monsch, & Brugger, 2005) or heads-or-tails (Hahn & Warren, 2009). Some authors have also offered a choice of more neutral items, such as dots, e.g., in the classical Mittenecker test (Mittenecker, 1958). Formally, however, these cases all amount to producing sequences of bits, that is 0 or 1 digits, since any object can be coded in this way. In most studies, the sequence length lies between 5 and 50 items. Measuring the Complexity for short strings 4 formal “randomness” of a given short sequence (say of length 5 to 50) is thus a crucial challenge. Apart from any objective and formal definition of randomness, researchers regularly use a variety of indices, none of which is sufficient by itself because of the profound limitations they all exhibit. Recently for instance, Schulter, Mittenecker and Papousek (2010) provided software calculating the most widely used of such measures applied to the case of the Mittenecker Pointing Test, together with a comprehensive overview of the usual coefficients of randomness in behavioral and cognitive research. These tools provide a new way to describe how a given sequence differs from a truly random one. However, it is not fully satisfactory: multiple unsatisfactory measures do not result in a satisfactory description. The usual measures of randomness The most common coefficients used to assess the quality of a pseudo-random production may be classified into three large varieties according to their main focus. Departure from uniformity The simplest coefficients–even though they may rely upon sophisticated theories–are based on the mere distribution of outcomes, and are therefore independent of the order of the outcomes. In brief, they amount to the calculation of a distance between

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