String Diagram Rewriting Modulo Commutative (Co)Monoid Structure

Authors Aleksandar Milosavljević, Robin Piedeleu , Fabio Zanasi

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Aleksandar Milosavljević
  • University College London, UK
Robin Piedeleu
  • University College London, UK
Fabio Zanasi
  • University College London, UK
  • University of Bologna, Italy


We thank Tobias Fritz for helpful discussion and the anonymous reviewers of CALCO for their suggestions.

Cite AsGet BibTex

Aleksandar Milosavljević, Robin Piedeleu, and Fabio Zanasi. String Diagram Rewriting Modulo Commutative (Co)Monoid Structure. In 10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 270, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


String diagrams constitute an intuitive and expressive graphical syntax that has found application in a very diverse range of fields including concurrency theory, quantum computing, control theory, machine learning, linguistics, and digital circuits. Rewriting theory for string diagrams relies on a combinatorial interpretation as double-pushout rewriting of certain hypergraphs. As previously studied, there is a "tension" in this interpretation: in order to make it sound and complete, we either need to add structure on string diagrams (in particular, Frobenius algebra structure) or pose restrictions on double-pushout rewriting (resulting in "convex" rewriting). From the string diagram viewpoint, imposing a full Frobenius structure may not always be natural or desirable in applications, which motivates our study of a weaker requirement: commutative monoid structure. In this work we characterise string diagram rewriting modulo commutative monoid equations, via a sound and complete interpretation in a suitable notion of double-pushout rewriting of hypergraphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Rewrite systems
  • String diagrams
  • Double-pushout rewriting
  • Commutative monoid


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