Unifying Theories of Time with Generalised Reactive Processes

TL;DR

This paper extends Hoare and He's reactive process theory by introducing an abstract trace algebra, enabling the modeling of hybrid systems with both discrete and continuous traces, and verifies key laws in Isabelle/HOL.

Contribution

It generalizes reactive process theory using an abstract trace algebra to include continuous-time traces and hybrid systems.

Findings

Successfully modeled hybrid systems with discrete and continuous traces.

Reconstructed reactive process laws in a generic algebraic setting.

Mechanically verified laws in Isabelle/HOL proof assistant.

Abstract

Hoare and He's theory of reactive processes provides a unifying foundation for the formal semantics of concurrent and reactive languages. Though highly applicable, their theory is limited to models that can express event histories as discrete sequences. In this paper, we show how their theory can be generalised by using an abstract trace algebra. We show how the algebra, notably, allows us to also consider continuous-time traces and thereby facilitate models of hybrid systems. We then use this algebra to reconstruct the theory of reactive processes in our generic setting, and prove characteristic laws for sequential and parallel processes, all of which have been mechanically verified in the Isabelle/HOL proof assistant.