Turi and Plotkin's bialgebraic semantics is an abstract approach to specifying the operational semantics of a system, by means of a distributive law between its syntax (encoded as a monad) and its dynamics (an endofunctor). This setup is instrumental in showing that a semantic specification (a coalgebra) is compositional.

In this work, we use the bialgebraic approach to derive well-behaved structural operational semantics of string diagrams, a graphical syntax that is increasingly used in the study of interacting systems across different disciplines. Our analysis relies on representing the two-dimensional operations underlying string diagrams in various categories as a monad, and their semantics as a distributive law for that monad.

As a proof of concept, we provide bialgebraic semantics for a versatile string diagrammatic language which has been used to model both signal flow graphs (control theory) and Petri nets (concurrency theory).

Turi 和 Plotkin 的双代数语义是一种抽象方法，用于指定系统的操作语义，通过其语法（编码为 monad）和动态（一个内函子）之间的分配律。此设置有助于表明语义规范（余代数）是组合的。

在这项工作中，我们使用双代数方法来推导出字符串图的良好结构操作语义，这是一种图形语法，越来越多地用于跨学科交互系统的研究。我们的分析依赖于将各种类别中字符串图的二维操作表示为一个 monad，并将它们的语义表示为该 monad 的分配规律。

作为概念证明，我们为通用字符串图语言提供双代数语义，该语言已用于对信号流图（控制理论）和 Petri 网（并发理论）进行建模。