Graded Monad Coalgebras for Continuous-Time Transition Systems
This work provides a foundational framework for modeling continuous-time systems in coalgebra, benefiting researchers in semantics of computation and stochastic processes.
The paper introduces graded coalgebras of graded monads to model continuous-time transition systems, developing their theory including graded distributive laws and conditions for terminal coalgebras. It defines branching-time and trace semantics, linking them to Feller-Dynkin processes, and develops coalgebraic modal logics with invariance and expressivity criteria.
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.