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DOI: 10.4230/LIPIcs.MFCS.2025.68

Quantitative Monoidal Algebra: Axiomatising Distance with String Diagrams

Authors: Gabriele Lobbia, Wojciech Różowski, Ralph Sarkis, and Fabio Zanasi

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract

String diagrammatic calculi have become increasingly popular in fields such as quantum theory, circuit theory, probabilistic programming, and machine learning, where they enable resource-sensitive and compositional algebraic analysis. Traditionally, the equations of diagrammatic calculi only axiomatise exact semantic equality. However, reasoning in these domains often involves approximations rather than strict equivalences. In this work, we develop a quantitative framework for diagrammatic calculi, where one may axiomatise notions of distance between string diagrams. Unlike similar approaches, such as the quantitative theories introduced by Mardare et al., this requires us to work in a monoidal rather than a cartesian setting. We define a suitable notion of monoidal theory, the syntactic category it freely generates, and its models, where the concept of distance is established via enrichment over a quantale. To illustrate the framework, we provide examples from probabilistic and linear systems analysis.

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Gabriele Lobbia, Wojciech Różowski, Ralph Sarkis, and Fabio Zanasi. Quantitative Monoidal Algebra: Axiomatising Distance with String Diagrams. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 68:1-68:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{lobbia_et_al:LIPIcs.MFCS.2025.68,
  author =	{Lobbia, Gabriele and R\'{o}\.{z}owski, Wojciech and Sarkis, Ralph and Zanasi, Fabio},
  title =	{{Quantitative Monoidal Algebra: Axiomatising Distance with String Diagrams}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{68:1--68:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.68},
  URN =		{urn:nbn:de:0030-drops-241759},
  doi =		{10.4230/LIPIcs.MFCS.2025.68},
  annote =	{Keywords: string diagram, symmetric monoidal category, quantitative algebraic theory, quantale, metric}
}

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DOI: 10.4230/LIPIcs.CALCO.2025.5

String Diagrams for Graded Monoidal Theories, with an Application to Imprecise Probability

Authors: Ralph Sarkis and Fabio Zanasi

Published in: LIPIcs, Volume 342, 11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025)

Abstract

We introduce string diagrams for graded symmetric monoidal categories. Our approach includes a definition of graded monoidal theory and the corresponding freely generated syntactic category. Also, we show how an axiomatic presentation for the graded theory may be modularly obtained from one for the grading theory and one for the base category. The Para construction on monoidal actegories is a motivating example for our framework. As a case study, we show how to axiomatise a variant of the graded category ImP, recently introduced by Liell-Cock and Staton to model imprecise probability [Liell-Cock and Staton, 2025]. This culminates in a representation, as string diagrams with grading wires, of programs with primitives for nondeterministic and probabilistic choices and conditioning.

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Ralph Sarkis and Fabio Zanasi. String Diagrams for Graded Monoidal Theories, with an Application to Imprecise Probability. In 11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 342, pp. 5:1-5:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{sarkis_et_al:LIPIcs.CALCO.2025.5,
  author =	{Sarkis, Ralph and Zanasi, Fabio},
  title =	{{String Diagrams for Graded Monoidal Theories, with an Application to Imprecise Probability}},
  booktitle =	{11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025)},
  pages =	{5:1--5:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-383-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{342},
  editor =	{C\^{i}rstea, Corina and Knapp, Alexander},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2025.5},
  URN =		{urn:nbn:de:0030-drops-235641},
  doi =		{10.4230/LIPIcs.CALCO.2025.5},
  annote =	{Keywords: string diagrams, graded categories, probability, nondeterminism}
}

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Quantitative Monoidal Algebra: Axiomatising Distance with String Diagrams

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String Diagrams for Graded Monoidal Theories, with an Application to Imprecise Probability

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