Polynomial Life: the Structure of Adaptive Systems

Toby St Clere Smithe
(Topos Institute)

We extend our earlier work on the compositional structure of cybernetic systems in order to account for the embodiment of such systems. All their interactions proceed through their bodies' boundaries: sensations impinge on their surfaces, and actions correspond to changes in their configurations. We formalize this morphological perspective using polynomial functors. The 'internal universes' of systems are shown to constitute an indexed category of statistical games over polynomials; their dynamics form an indexed category of behaviours. We characterize 'active inference doctrines' as indexed functors between such categories, resolving a number of open problems in our earlier work, and pointing to a formalization of the 'free energy principle' as adjoint to such doctrines. We illustrate our framework through fundamental examples from biology, including homeostasis, morphogenesis, and autopoiesis, and suggest a formal connection between spatial navigation and the process of proof.

In Kohei Kishida: Proceedings of the Fourth International Conference on Applied Category Theory (ACT 2021), Cambridge, United Kingdom, 12-16th July 2021, Electronic Proceedings in Theoretical Computer Science 372, pp. 133–148.
Summary of work in progress; comments welcome.
Published: 3rd November 2022.

ArXived at: https://dx.doi.org/10.4204/EPTCS.372.10 bibtex PDF
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