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.

💡 Research Summary

The paper addresses a longstanding methodological problem in the assessment of human‑generated random sequences, which are widely used to probe executive functions such as inhibition and working memory. Traditional approaches rely on a suite of statistical indices—symbol‑frequency based uniformity measures (e.g., Symbol Redundancy), normality assessments that examine dyads, triads and higher‑order n‑grams (e.g., Context Redundancy, Coefficient of Constraint), and gap‑based metrics that look at distances between repeated symbols. While each of these indices captures a specific bias (over‑use of a symbol, alternation bias, cycling bias, etc.), they all share two critical shortcomings: (1) they are blind to simple algorithmic regularities that nevertheless produce sequences with the same statistical profile (e.g., Champernowne or Copeland‑Erdős sequences), and (2) they lack a unifying theoretical foundation, forcing researchers to combine several measures to obtain a partial picture of randomness.

To overcome these limitations, the authors propose using algorithmic (Kolmogorov‑Chaitin) complexity as a universal, single‑value measure of randomness. Because Kolmogorov complexity is uncomputable in general, they adopt the “Coding theorem method” (CTM), which exploits the relationship between algorithmic probability m(s) (the probability that a random program outputs string s) and complexity K(s) via the coding theorem K(s) ≈ –log₂ m(s). By exhaustively enumerating all small Turing machines (up to five states, two symbols) and recording the frequencies with which each possible binary string of length 1–12 is produced, they empirically approximate m(s) for short strings. This yields a lookup table of estimated complexities that is applicable to any binary sequence within the examined length range.

The authors then re‑analyse the classic Radio Zenith data, a set of human‑generated binary strings previously evaluated with Symbol Redundancy and alternation measures. Using the CTM‑derived complexity values, they demonstrate that (i) complexity declines with age and is markedly lower in participants with frontal‑lobe damage, (ii) complexity correlates only weakly with uniformity measures and moderately (inversely) with alternation indices, and (iii) the complexity metric captures deviations from true randomness that are invisible to the traditional indices, such as hidden algorithmic regularities.

Three major contributions emerge from this work. First, it provides a practical, theoretically grounded tool for quantifying randomness in short strings, eliminating the need for a battery of ad‑hoc statistics. Second, by linking complexity to algorithmic probability, the method offers a natural bridge to Bayesian models of subjective probability, allowing researchers to compute the likelihood that a given human‑produced sequence arose from a random process. Third, the complexity table can be employed in developmental studies (tracking the maturation of random‑generation abilities) and clinical assessments (detecting executive dysfunction) because it yields a single, comparable score across individuals and across sequence lengths.

The paper concludes by acknowledging that while algorithmic complexity supplies a global assessment of randomness, it does not replace domain‑specific analyses that target particular cognitive biases. Nonetheless, it represents a significant step toward a unified, mathematically rigorous framework for evaluating human randomness production, and it opens avenues for extending the approach to longer strings, multi‑symbol alphabets, and intervention studies that use complexity feedback to train more random behaviour.