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.

翻译：函子余代数能够刻画一大类必须在离散步骤中演化的转移系统。我们引入分次单子的分次余代数，并提出将其用于建模连续时间转移系统。我们发展了分次余代数的理论——包括分次单子之间的分次分配律——并给出了终端余代数存在的条件。我们定义了分支时间语义和迹语义，并将其与近期关于费勒-登金过程的研究联系起来。最后，我们为两种过程语义开发了余代数模态逻辑，并给出了不变性和表达性的状态准则。