Partial Reversibility of One-Dimensional Cellular Automata

摘要

Reversibility is the property of very special cellular automata rules by which any path traversed in the configuration space can be traversed back by its inverse rule. Expanding this context, the notion of partial reversibility has been previously proposed in the literature, as an attempt to refer to rules as being more or less reversible than others, since some of the paths of non-reversible rules could be traversed back. The approach was couched in terms of a characterisation of the rule's pre-image pattern, that is, the number of pre-images of a rule for all configurations up to a given size, and their relative lexicographical ordering used to classify the rules in terms of their relative partial reversibility. Here, we reassess the original definition and define a measure that represents the reversibility degree of the rules, also based on their pre-image patterns, but now relying on the probability of correctly reverting each possible cyclic, finite length configuration, up to a maximum size. As a consequence, it becomes possible to look at partial reversibility in absolute terms, and not relatively to other rules, as well to infer the reversibility degrees for arbitrary lattice sizes, even in its limit to infinity. All the discussions are restricted to the elementary space, but are also applicable to any one-dimensional rule space.