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Behavioural Metrics and Quantitative Logics (Dagstuhl Seminar 24432)

Authors: Barbara König, Radu Mardare, Prakash Panangaden, Jurriaan Rot, and Florence Clerc

Published in: Dagstuhl Reports, Volume 14, Issue 10 (2025)

Abstract

This report documents the program and the outcome of Dagstuhl Seminar "Behavioural Metrics and Quantitative Logics" (24432). Behavioural metrics and quantitative logics specify quantitative aspects of systems. A metric measures how far apart two systems are in their behaviour while a quantititative logic evaluates the degree to which a state satisfies a formula. They are often intimately connected via a Hennessy-Milner theorem stating that the distance induced by a quantitative logic coincides with behavioural distance. There are various applications in model-checking, differential privacy, hybrid systems and learning. Several challenges in this area have been identified: studying suitable metrics and their corresponding logics, generalizing to the setting of coalgebras by parameterizing the branching type of the system under consideration, developing methods of quantitative algebraic reasoning, finding efficient methods for computing behavioural methods. This Dagstuhl Seminar provided a forum to researchers working in this area, to discuss the state-of-the-art and further developments, and in particular address applications in various domains.

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Barbara König, Radu Mardare, Prakash Panangaden, Jurriaan Rot, and Florence Clerc. Behavioural Metrics and Quantitative Logics (Dagstuhl Seminar 24432). In Dagstuhl Reports, Volume 14, Issue 10, pp. 58-75, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@Article{konig_et_al:DagRep.14.10.58,
  author =	{K\"{o}nig, Barbara and Mardare, Radu and Panangaden, Prakash and Rot, Jurriaan and Clerc, Florence},
  title =	{{Behavioural Metrics and Quantitative Logics (Dagstuhl Seminar 24432)}},
  pages =	{58--75},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2025},
  volume =	{14},
  number =	{10},
  editor =	{K\"{o}nig, Barbara and Mardare, Radu and Panangaden, Prakash and Rot, Jurriaan and Clerc, Florence},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.14.10.58},
  URN =		{urn:nbn:de:0030-drops-230222},
  doi =		{10.4230/DagRep.14.10.58},
  annote =	{Keywords: Behavioural metrics, quantitative equational reasoning, quantitative logics}
}

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DOI: 10.4230/LIPIcs.RTA.2015.89

Presenting a Category Modulo a Rewriting System

Authors: Florence Clerc and Samuel Mimram

Published in: LIPIcs, Volume 36, 26th International Conference on Rewriting Techniques and Applications (RTA 2015)

Abstract

Presentations of categories are a well-known algebraic tool to provide descriptions of categories by the means of generators, for objects and morphisms, and relations on morphisms. We generalize here this notion, in order to consider situations where the objects are considered modulo an equivalence relation (in the spirit of rewriting modulo), which is described by equational generators. When those form a convergent (abstract) rewriting system on objects, there are three very natural constructions that can be used to define the category which is described by the presentation: one is based on restricting to objects which are normal forms, one consists in turning equational generators into identities (i.e. considering a quotient category), and one consists in formally adding inverses to equational generators (i.e. localizing the category). We show that, under suitable coherence conditions on the presentation, the three constructions coincide, thus generalizing celebrated results on presentations of groups. We illustrate our techniques on a non-trivial example, and hint at a generalization for 2-categories.

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Florence Clerc and Samuel Mimram. Presenting a Category Modulo a Rewriting System. In 26th International Conference on Rewriting Techniques and Applications (RTA 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 36, pp. 89-105, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{clerc_et_al:LIPIcs.RTA.2015.89,
  author =	{Clerc, Florence and Mimram, Samuel},
  title =	{{Presenting a Category Modulo a Rewriting System}},
  booktitle =	{26th International Conference on Rewriting Techniques and Applications (RTA 2015)},
  pages =	{89--105},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-85-9},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{36},
  editor =	{Fern\'{a}ndez, Maribel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2015.89},
  URN =		{urn:nbn:de:0030-drops-51916},
  doi =		{10.4230/LIPIcs.RTA.2015.89},
  annote =	{Keywords: presentation of a category, quotient category, localization, residuation}
}

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Behavioural Metrics and Quantitative Logics (Dagstuhl Seminar 24432)

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Presenting a Category Modulo a Rewriting System

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