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

Alexander F. Siegenfeld; Yaneer Bar-Yam

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arXiv · 2206.04896

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

Alexander F. Siegenfeld·Yaneer Bar-Yam

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Abstract

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, to account for the dependence of a system's complexity on the level of detail -- or scale -- of its description, a multi-scale generalization of Ashby's law is needed. We define a class of complexity profiles (complexity as a function of scale) that is the first, to our knowledge, to exhibit a multi-scale law of requisite variety. This formalism provides a characterization of multi-scale complexity and generalizes the law of requisite variety's single constraint on system behaviors to a class of multi-scale constraints. We show that these complexity profiles satisfy a sum rule, which reflects a tradeoff between smaller- and larger-scale degrees of freedom, and we extend our results to subdivided systems and systems with a continuum of components.

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Alexander F. Siegenfeld, Yaneer Bar-Yam. 2022-06-10. A Formal Definition of Scale-dependent Complexity and the Multi-scale Law of Requisite Variety. https://arxiv.org/abs/2206.04896

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