Formalization of the responsive and formal design process using category theory

In this paper, we formalize the Responsive and Formal Design (RFD) process using category theory. The RFD process combines Model-Based Systems Engineering (MBSE) to manage system modeling complexity and formal methods to ensure that designs are verifiably correct against their requirements. It consists of a set of levels of abstraction. Each level of abstraction represents a set of requirements and its associated models, simulations, and the relationship between them. Abstraction and refinement functions relate different levels of representation. In this paper, we represent and analyze the RFD process using category theory. Category theory provides us a means (using a collection of objects and arrows) to represent each level of abstractions and communications between them. We represent each level of abstraction using a pullback categorical structure (define the objects and morphisms). The facts and theorems in one level of abstraction will be passed to another via a refinement or an abstraction functor (function). The two functors operate in an inverse (adjoint) relationship. This means refinement traceability is met in the design process fundamentally. Additionally, since adjunction is a weaker relation than any other relations (such as equality, isomorphism, and equivalence) between two categories, it is a relatively better option for relaxing the design space. Finally, we introduce an idea of defining a category of refinements (i.e. a category of functors) and interpreting the universal properties.