# Bialgebraic Semantics for String Diagrams

## TL;DR

This paper applies bialgebraic semantics to string diagrams, providing a compositional and well-behaved operational semantics that unify different interpretations and synchronization mechanisms across disciplines.

## Contribution

It introduces a bialgebraic framework for the semantics of string diagrams, enabling compositional analysis and linking different process interpretations.

## Key findings

- Bialgebraic semantics for string diagrams are well-behaved and compositional.
- The approach unifies different synchronization mechanisms in process theories.
- Provides semantics for models like signal flow graphs and Petri nets.

## Abstract

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) satisfies desirable properties: in particular, that it 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 bialgebraic semantics in terms of a distributive law over that monad. As a proof of concept, we provide bialgebraic compositional semantics for a versatile string diagrammatic language which has been used to model both signal flow graphs (control theory) and Petri nets (concurrency theory). Moreover, our approach reveals a correspondence between two different interpretations of the Frobenius equations on string diagrams and two synchronisation mechanisms for processes, \`a la Hoare and \`a la Milner.