Generalized Benford's Law as a Lie Detector.

The first significant (leftmost nonzero) digit of seemingly random numbers often appears to conform to a logarithmic distribution, with more 1s than 2s, more 2s than 3s, and so forth, a phenomenon known as Benford's law. When humans try to produce random numbers, they often fail to conform to this distribution. This feature grounds the so-called Benford analysis, aiming at detecting fabricated data. A generalized Benford's law (GBL), extending the classical Benford's law, has been defined recently. In two studies, we provide some empirical support for the generalized Benford analysis, broadening the classical Benford analysis. We also conclude that familiarity with the numerical domain involved as well as cognitive effort only have a mild effect on the method's accuracy and can hardly explain the positive results provided here.

Human behavioral complexity peaks at age 25.

Random Item Generation tasks (RIG) are commonly used to assess high cognitive abilities such as inhibition or sustained attention. They also draw upon our approximate sense of complexity. A detrimental effect of aging on pseudo-random productions has been demonstrated for some tasks, but little is as yet known about the developmental curve of cognitive complexity over the lifespan. We investigate the complexity trajectory across the lifespan of human responses to five common RIG tasks, using a large sample (n = 3429). Our main finding is that the developmental curve of the estimated algorithmic complexity of responses is similar to what may be expected of a measure of higher cognitive abilities, with a performance peak around 25 and a decline starting around 60, suggesting that RIG tasks yield good estimates of such cognitive abilities. Our study illustrates that very short strings of, i.e., 10 items, are sufficient to have their complexity reliably estimated and to allow the documentation of an age-dependent decline in the approximate sense of complexity.

Calculating Kolmogorov complexity from the output frequency distributions of small Turing machines.

Drawing on various notions from theoretical computer science, we present a novel numerical approach, motivated by the notion of algorithmic probability, to the problem of approximating the Kolmogorov-Chaitin complexity of short strings. The method is an alternative to the traditional lossless compression algorithms, which it may complement, the two being serviceable for different string lengths. We provide a thorough analysis for all Σ(n=1)(11) 2(n) binary strings of length n<12 and for most strings of length 12≤n≤16 by running all ~2.5 x 10(13) Turing machines with 5 states and 2 symbols (8 x 22(9) with reduction techniques) using the most standard formalism of Turing machines, used in for example the Busy Beaver problem. We address the question of stability and error estimation, the sensitivity of the continued application of the method for wider coverage and better accuracy, and provide statistical evidence suggesting robustness. As with compression algorithms, this work promises to deliver a range of applications, and to provide insight into the question of complexity calculation of finite (and short) strings. Additional material can be found at the Algorithmic Nature Group website at http://www.algorithmicnature.org. An Online Algorithmic Complexity Calculator implementing this technique and making the data available to the research community is accessible at http://www.complexitycalculator.com.

Algorithmic complexity for short binary strings applied to psychology: a primer.

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 randomlike have been suggested. However, each of these measures focuses 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.