Hidden potentialities

Abstract

The search for “complexity signatures” in natural laws is a main concern for researchers working in many different fields, going from physics to biology. Very simple laws are able to produce unforeseen behaviors, and from their “simplicity” sometimes it is not possible to predict anything about the potential complexities they are able to produce when applied to random initial conditions. Here elementary cellular automata (ECA) are used to illustrate this idea. In fact, using a recently developed approach to establish a correspondence between ECA rules and logical functions that constitute their structure (logical spectra) we analyzed ECA groups represented by automata 120 and 164. The automaton governed by rule 120 generates complex patterns under certain random initial conditions, whereas the one corresponding to rule 164 is known as a simple law when it is evaluated at random initial conditions. However, slight changes in the initial conditions of both ECA produce dramatic “simplicity–complexity transitions”, as it is shown here. These examples show that complexity, even in very simple formal systems, results from a subtle interplay between the structure of the laws and the initial conditions. Moreover, they draw attention to the importance of investigating if analogous comportments hold in large-scale natural phenomena as the yet enigmatic genotype–phenotype mapping.