Optimisation as an Algebraic Axiom in Cartesian Closed Categories: Applications to Dataflow Networks

Abstract

The aim of this paper is to present a framework that integrates optimisation into algebraic structures in Cartesian Closed Categories (CCCs). Traditional mathematical methods treated optimisation as an external process, which limited its foundational role in mathematics. Inspired by the finite nature hypothesis and Hilbert's sixth problem, which calls for the axiomatisation of physical principles, this study formalises optimisation as an intrinsic algebraic axiom in a way that aligns with Hilbert's vision of uniting mathematical and physical laws. The framework builds on Lawvere's categorical treatment of metric spaces and Birkhoff's HSP theorem, which the study uses to define an optimisation algebra. The study then provides proof that the class of such algebras forms a variety in universal algebra and demonstrates categorical soundness within CCCs. The proposed approach guarantees that optimisation is inherent within algebraic systems, establishing natural substructures of optimal elements and facilitating compositional reasoning in computational models. Applying this framework in dataflow networks demonstrates convergence to optimal steady states, enhancing resource utilisation and system efficiency. Future research includes using the framework in enriched categories, distributed systems and incorporating the operator in tools that can be used to solve real-world problems.