DROPS

Document

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.

Cite as

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)

Copy BibTex To Clipboard

@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}
}

Search Results

Documents authored by Lobbia, Gabriele

Quantitative Monoidal Algebra: Axiomatising Distance with String Diagrams

Abstract

Cite as

Thanks for your feedback!

Could not send message