Physical Computing: A Category Theoretic Perspective on Physical Computation and System Compositionality

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. Keywords: Computation, Computability, Physical Computation, Category Theory 1. Introduction Roots of computability trace back to Leibniz, who invented a mechanical calculator for automating the evaluation of mathematical expressions [23, 20, 9]. At the Paris international conference for mathematics, David Hilbert extended Leibniz's fascination by proposing the Entscheidungsproblem (the decision problem), questioning the existence of an "effective procedure" to determine the truth or falsity of mathematical statements [29]. Alan Turing and Alonso Church independently demonstrated the impossibility of resolving the Entscheidungsproblem. This discovery, known as the "Church-Turing thesis", posited that no effective procedure (or "algorithm" in contemporary terms) can Present address: McGovern Institute for Brain Research, MIT, Cambridge, 02139, MA, USA Prior to the emergence of algorithm(s), procedural calculation through a finite number of exact, finite instructions was labeled "effective procedure (or effective calculation).