Average Symmetry and Complexity of Binary Sequences

Abstract

The concept of complexity as average symmetry is here formalised by introducing a general expression dependent on the relevant symmetry and a related discrete set of transformations. This complexity has hybrid features of both statistical complexities and of those related to algorithmic complexity. Like the former, random objects are not the most complex while they still are more complex than the more symmetric ones (as in the latter). By applying this definition to the particular case of rotations of binary sequences, we are able to find a precise expression for it. In particular, we then analyse the behaviour of this measure in different well-known automatic sequences, where we find interesting new properties. A generalisation of the measure to statistical ensembles is also presented and applied to the case of i.i.d. random sequences and to the equilibrium configurations of the one-dimensional Ising model. In both cases, we find that the complexity is continuous and differentiable as a function of the relevant parameters and agrees with the intuitive requirements we were looking for.