Bialgebraic Semantics for String Diagrams

Authors Filippo Bonchi, Robin Piedeleu, Pawel Sobocinski, Fabio Zanasi

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Author Details

Filippo Bonchi
  • University of Pisa, Italy
Robin Piedeleu
  • University College London, UK
Pawel Sobocinski
  • University of Southampton, UK
Fabio Zanasi
  • University College London, UK


RP and FZ acknowledge support from EPSRC grant EP/R020604/1.

Cite AsGet BibTex

Filippo Bonchi, Robin Piedeleu, Pawel Sobocinski, and Fabio Zanasi. Bialgebraic Semantics for String Diagrams. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 37:1-37:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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 for 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Categorical semantics
  • String Diagram
  • Structural Operational Semantics
  • Bialgebraic semantics


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