Categorical Semantics of Cyber-Physical Systems Theory

Abstract

Cyber-physical systems require the construction and management of various models to assure their correct, safe, and secure operation. These various models are necessary because of the coupled physical and computational dynamics present in cyberphysical systems. However, to date the different model views of cyber-physical systems are largely related informally, which raises issues with the degree of formal consistency between those various models of requirements, system behavior, and system architecture. We present a category-theoretic framework to make different types of composition explicit in the modeling and analysis of cyber-physical systems, which could assist in verifying the system as a whole. This compositional framework for cyber-physical systems gives rise to unified system models, where system behavior is hierarchically decomposed and related to a system architecture using the systems-as-algebras paradigm. As part of this paradigm, we show that an algebra of (safety) contracts generalizes over the state of the art, providing more uniform mathematical tools for constraining the behavior over a richer set of composite cyber-physical system models, which has the potential of minimizing or eliminating hazardous behavior.

Structured versus Decorated Cospans

“…Thinking about systems and processes categorically dates back to early works by Lawvere [35], Bunge-Fiore [13], Joyal-Nielsen-Winskel [31], Katis-Sabadini-Walters [32] and others. Spivak and others have used wiring diagrams and sheaves to capture compositional features of dynamical systems [10,42,46]. Another approach uses signal flow diagrams and other string diagrams [5,11,21] to understand systems behaviorally, following ideas of Willems [48].…”

Structured versus Decorated Cospans

“…Thinking about systems and processes categorically dates back to early works by Lawvere [35], Bunge-Fiore [11], Joyal-Nielsen-Winskel [30], Katis-Sabadini-Walters [31] and others. Spivak and others have used wiring diagrams and sheaves to capture compositional features of dynamical systems, [8,41,45]. Another approach uses signal flow diagrams and other string diagrams [3,9,19] to understand systems behaviorally, following ideas of Willems [47].…”

Yoneda Hacking: The Algebra of Attacker Actions

“…Additionally, this security framework is part of a alternate paradigm of systems modeling that has its foundations in categorical modeling. In this framework it is possible to provide formal traces requirements, behaviors, and architectures [18] but also describe a vast amount of dynamical systems with applications to robotics, event-based systems, and hybrid systems, to name a few [45], [46], [47], [48].…”

“…This paper attempts to develop the foundations of security within a compositional CPS theory modeling framework [18], a flavor of what Lee calls dynamical computational systems theory [19]. Formal composition rules could overcome some of the new challenges CPS introduce to security [20], [21].…”