Bialgebraic Semantics for String Diagrams
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 wellbehaved 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 twodimensional 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, à la Hoare and à la Milner.
 Publication:

arXiv eprints
 Pub Date:
 June 2019
 DOI:
 10.48550/arXiv.1906.01519
 arXiv:
 arXiv:1906.01519
 Bibcode:
 2019arXiv190601519B
 Keywords:

 Computer Science  Logic in Computer Science;
 Computer Science  Programming Languages;
 Mathematics  Category Theory;
 03G99
 EPrint:
 Accepted for publications in the proceedings of the 30th International Conference on Concurrency Theory (CONCUR 2019)