A Formal Definition of Scale-dependent Complexity and the Multi-scale Law of Requisite Variety

Ashby's law of requisite variety allows a comparison of systems with their environments, providing a necessary (but not sufficient) condition for system efficacy: a system must possess at least as much complexity as any set of environmental behaviors that require distinct responses from the system. However, the complexity of a system depends on the level of detail, or scale, at which it is described. Thus, the complexity of a system can be better characterized by a complexity profile (complexity as a function of scale) than by a single number. It would therefore be useful to have a multi-scale generalization of Ashby's law that requires that a system possess at least as much complexity as the relevant set of environmental behaviors *at each scale*. We construct a formalism for a class of complexity profiles that is the first, to our knowledge, to exhibit this multi-scale law of requisite variety. This formalism not only provides a characterization of multi-scale complexity but also generalizes the single constraint on system behaviors provided by Ashby's law to an entire class of multi-scale constraints. We show that these complexity profiles satisfy a sum rule, which reflects the important tradeoff between smaller- and larger-scale degrees of freedom, and we extend our results to subdivided systems and systems with a continuum of components.