Compositional Cyber-Physical Systems Modeling

Abstract

Engineering safe and secure cyber-physical systems requires system engineers to develop and maintain a number of model views, both dynamic and static, which can be seen as algebras. We posit that verifying the composition of requirement, behavioral, and architectural models using category theory gives rise to a strictly compositional interpretation of cyber-physical systems theory, which can assist in the modeling and analysis of safety-critical cyber-physical systems. APPLIED COMPOSITIONAL THINKINGLee [2], among others, recognized early in the development of the field of cyber-physical systems that there is a need for developing competing methods to hybrid systems and process algebras. While this is true, an important observation is that both these formalisms form algebras. In fact, the design of cyber-physical systems involves the study of different algebras (Figure 1).There is significant research in developing these individual algebras and implementing composition within a particular algebra. However, there is still an open problem about how to relate those paradigms that in practice represent individual models and to examine the behavior of the system as a whole must be composed too. Compositional cyber-physical systems theory [3], [4] uses category theory to transform data from one algebra to another and to ultimately relate them formally, such that we can compose across domains. This provides one solution to the open problem of composition between formal methods and their corresponding model views in cyberphysical system design [5], [6].Recently all these three areas of control [10], contracts [4], and co-design [11, chapter 4] have been described, generalized, and unified with fun-

Category-Theoretic Formulation of the Model-Based Systems Architecting Cognitive-Computational Cycle

“…Graphs of models are representable in the broader hypergraph category, which enables of model analysis, composition, comparison, and utilization [15]. Representing a system as a category with component types and inter-component relations as morphisms, with Ologs (knowledge representation structures that capture non-mathematical, freely-defined relations among objects as morphisms) to specify design constructs, current-state situations, constraints, and requirements [51,52] Iterative co-design of electro-mechanical functions in a cyber-physical system architecture [53][54][55][56] hierarchical requirements engineering, gradually evolving a system architecture; formal and verifiable system design [57][58][59] Structural and functional composition of system models [60] Using operads-categorical structures that map multi-object compositions to a single object-for hierarchical decomposition, design synthesis, separation of syntax from semantics, and semantic reasoning about complex systems [61,62] Model-Driven Software Engineering…”

Categorical Semantics of Cyber-Physical Systems Theory

“…Compositional theory can assist with model conformance [58] and model federation at large [31]. Complementary works using category theory have shown small but useful examples of categorical modeling of systems and how they can facilitate model conformance in [8,13,14,50].…”

“…Recall that the image of a function is the subset of its codomain where all elements of the domain get mapped to, namely for an arbitrary : → , ( ) = { ∈ | ∃ ∈ such that ( ) = }. In the end, using the above constructions of the twostep process exhibited in (15), the explicit description of the resulting contract is precisely equation (8).…”

EPHS: A Port-Hamiltonian Modelling Language

“…In applied category theory research, the formalization of such diagrams based on monoidal categories and their underlying operads has been a major topic, see e.g. Baez and Erberle (2015); Baez and Pollard (2017); Baez and Fong (2018); Baez et al (2021); Bakirtzis et al (2021) and references therein. We focus on the operadic perspective because it is much closer to how we visually express and implement the language.…”