Physical Computing: A Category Theoretic Perspective on Physical Computation and System Compositionality
Nima Dehghani, Gianluca Caterina
TL;DR
The paper tackles the lack of a precise formalism for physical computation by proposing a category-theoretic framework that couples physical systems and abstract representations via functors between categories ${\bf PhysProc}$ and ${\bf AbsProc}$. It formalizes computation as a pair of functors $(\mathcal{R_T},\widetilde{\mathcal{R}_T})$, and develops a rich compositional toolkit based on natural transformations and adjoint pairs to capture refinement, multiple realizability, and nested composition across scales. Key contributions include a rigorous, scalable definition of physical computation, a principled treatment of how different physical realizations can implement the same abstract computation, and a framework to analyze computation at multiple levels of abstraction, including biological and unconventional computing. The approach provides objective criteria to distinguish genuine computation from mere physical change, with broad implications for understanding computation in complex systems, neuroscience, and the limits of universal computation under principles like Church–Turing–Deutsch.
Abstract
This paper introduces a category theory-based framework to redefine physical computing in light of advancements in quantum computing and non-standard computing systems. By integrating classical definitions within this broader perspective, the paper rigorously recontextualizes what constitutes physical computing devices and processes. It demonstrates how the compositional nature and relational structures of physical computing systems can be coherently formalized using category theory. This approach not only encapsulates recent formalisms in physical computing but also offers a structured method to explore the dynamic interactions within these systems.