Graded Monad Coalgebras for Continuous-Time Transition Systems

TL;DR

This paper introduces graded coalgebras of graded monads to model continuous-time transition systems, extending coalgebraic theory and semantics to continuous dynamics.

Contribution

It develops the theory of graded coalgebras, including distributive laws and terminal coalgebras, and links them to continuous-time process semantics and modal logics.

Findings

Established conditions for the existence of terminal coalgebras.

Linked coalgebraic semantics to Feller-Dynkin processes.

Developed modal logics for process semantics and invariance.

Abstract

Functor coalgebras capture a wide range of transition systems that must however evolve in discrete steps. We introduce graded coalgebras of graded monads and propose them to model continuous-time transition systems. We develop the theory of graded coalgebras-including graded distributive laws between graded monads-and we give conditions for the existence of terminal coalgebras. We define both branching-time and trace semantics, linking them to recent work on Feller-Dynkin processes. Finally, we develop coalgebraic modal logics for both process semantics and state criteria for invariance and expressivity.