Effectful Mealy machines, which we introduce, are a generalization of Mealy machines with global effects determined by an effectful triple. We provide semantics of effectful Mealy machines in terms of both bisimilarity and traces: bisimilarity is characterized syntactically, via uniform feedback; traces are constructed coinductively in terms of streams. We prove that this framework characterizes standard causal processes and existing flavours of Mealy machine, bisimilarity, and trace equivalence. In the commutative case, we introduce a monoidal generalization of Raney's causal functions: monoidal causal processes.

翻译：本文引入的效应式Mealy机是Mealy机的一种推广形式，其全局效应由效应三元组决定。我们从双模拟等价与迹语义两个角度给出了效应式Mealy机的形式语义：双模拟等价通过统一反馈机制进行语法刻画；迹语义则基于流以共归纳方式构建。我们证明该框架能够刻画标准因果过程及现有各类Mealy机、双模拟等价与迹等价。在交换效应情形下，我们进一步提出了Raney因果函数的幺半群推广形式——幺半群因果过程。