The Von Neumann Threshold of Self-Reproducing Systems

This paper is devoted to the study of systems of entities that are capable of generating other entities of the same kind and, possibly, self-reproducing. The main technical issue addressed is to quantify the requirements that such entities must meet to be able to produce a progeny that is not degenerate, i.e., that has the same reproductive capability as the progenitor. A novel theory that allows an explicit quantification of these requirements is presented. The notion of generation rank of an entity is introduced, and it is proved that the generation process, in most cases, is degenerative in that it strictly and irreversibly decreases the generation rank from parent to descendant. It is also proved that there exists a threshold of rank such that this degeneracy can be avoided if and only if the entity has a generation rank that meets that threshold — this is the von Neumann rank threshold. Based on this threshold, an information threshold is derived, which quantifies the minimum amount of information that must be provided to specify an entity such that its descendants are not degenerate. Furthermore, a complexity threshold is obtained, which quantifies the minimum length of the description of that entity in a given language. Examples that illustrate the theory are provided.