Randomness and Determinism, is It Possible to Quantify These Notions?

Abstract

The article presents results obtained in attempts to quantify randomness characteristics for real numerical sequences or strings, using relative entropy. The characterization of the randomness of a series of real numbers is proposed to guide researchers in investigating phenomena towards deterministic or stochastic models. A numerical string's relative entropy is calculated using the histograms corresponding to the analysed strings, compared to the maximum entropy for the same histogram. It is shown that the entropy values have an asymptotic behaviour, but the relative entropy decreases with the increase in the number of histogram classes. Compared to other methods of characterizing the randomness of strings, which are not many, most of them being based on statistical tests, the method proposed in this article determines a better resolution for the classification of strings and, in addition, it can designate them as belonging to a class of randomness similar to that of some known strings, such as finite substrings of prime numbers, pseudorandom strings generated by common programs, trigonometric strings, etc. The attempt to quantify the randomness of real numerical strings, the results of which are presented in this article, is a first step in characterizing the randomness of experimental numerical strings, this being the final goal of the investigations.