String Diagrammatic Trace Theory

Authors Matthew Earnshaw , Paweł Sobociński

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Author Details

Matthew Earnshaw
  • Department of Software Science, Tallinn University of Technology, Estonia
Paweł Sobociński
  • Department of Software Science, Tallinn University of Technology, Estonia


We thank Chad Nester, Mario Román, and Niels Voorneveld for discussions.

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Matthew Earnshaw and Paweł Sobociński. String Diagrammatic Trace Theory. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 43:1-43:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We extend the theory of formal languages in monoidal categories to the multi-sorted, symmetric case, and show how this theory permits a graphical treatment of topics in concurrency. In particular, we show that Mazurkiewicz trace languages are precisely symmetric monoidal languages over monoidal distributed alphabets. We introduce symmetric monoidal automata, which define the class of regular symmetric monoidal languages. Furthermore, we prove that Zielonka’s asynchronous automata coincide with symmetric monoidal automata over monoidal distributed alphabets. Finally, we apply the string diagrams for symmetric premonoidal categories to derive serializations of traces.

Subject Classification

ACM Subject Classification
  • Theory of computation → Concurrency
  • Theory of computation → Formal languages and automata theory
  • Theory of computation → Categorical semantics
  • symmetric monoidal categories
  • Mazurkiewicz traces
  • asynchronous automata


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