Graded Monad Coalgebras for Continuous-Time Transition Systems

2026-05-07

AI Summary

Existing functorial coalgebras struggle to adequately model continuous-time transition systems. This work introduces graded coalgebras based on graded monoids and, for the first time, applies them to capture continuous-time evolutionary behavior, establishing a semantic correspondence with Feller–Dynkin processes. By leveraging graded distributive laws and terminal coalgebra constructions, the paper provides sufficient conditions for the existence of terminal coalgebras and defines two semantic notions: branching time and trace semantics. Furthermore, it proposes an accompanying coalgebraic modal logic that precisely characterizes state invariance and expressive power over system behaviors.

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.

Problem

• continuous-time transition systems
• graded monads
• coalgebras
• functor coalgebras
• transition systems

Innovation

• graded monads
• graded coalgebras
• continuous-time transition systems
• coalgebraic modal logic
• terminal coalgebras