A Formal Definition of Scale-dependent Complexity and the Multi-scale Law of Requisite Variety
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866915333695078400 |
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| author | Siegenfeld, Alexander F. Bar-Yam, Yaneer |
| author_facet | Siegenfeld, Alexander F. Bar-Yam, Yaneer |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_04896 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A Formal Definition of Scale-dependent Complexity and the Multi-scale Law of Requisite Variety Siegenfeld, Alexander F. Bar-Yam, Yaneer Physics and Society 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. |
| title | A Formal Definition of Scale-dependent Complexity and the Multi-scale Law of Requisite Variety |
| topic | Physics and Society |
| url | https://arxiv.org/abs/2206.04896 |