Polynomial Life: the Structure of Adaptive Systems

TL;DR

This paper develops a formal framework for understanding adaptive systems' structure and behavior through polynomial functors, integrating embodiment, active inference, and biological examples like homeostasis and morphogenesis.

Contribution

It introduces a polynomial functor-based formalization of embodied cybernetic systems, connecting active inference and the free energy principle within a categorical framework.

Findings

Formalization of embodied systems using polynomial functors

Characterization of active inference as indexed functors

Application to biological processes like homeostasis and morphogenesis

Abstract

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.