7 Search Results for "Marsden, Dan"

Document
DOI: 10.4230/LIPIcs.CALCO.2025.4
Distributive Laws of Monadic Containers

Authors: Chris Purdy and Stefania Damato

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

Abstract
Containers are used to carve out a class of strictly positive data types in terms of shapes and positions. They can be interpreted via a fully-faithful functor into endofunctors on Set. Monadic containers are those containers whose interpretation as a Set functor carries a monad structure. The category of containers is closed under container composition and is a monoidal category, whereas monadic containers do not in general compose. In this paper, we develop a characterisation of distributive laws of monadic containers. Distributive laws were introduced as a sufficient condition for the composition of the underlying functors of two monads to also carry a monad structure. Our development parallels Ahman and Uustalu’s characterisation of distributive laws of directed containers, i.e. containers whose Set functor interpretation carries a comonad structure. Furthermore, by combining our work with theirs, we construct characterisations of mixed distributive laws (i.e. of directed containers over monadic containers and vice versa), thereby completing the "zoo" of container characterisations of (co)monads and their distributive laws. We have found these characterisations amenable to development of existence and uniqueness proofs of distributive laws, particularly in the mechanised setting of Cubical Agda, in which most of the theory of this paper has been formalised.
Cite as

Chris Purdy and Stefania Damato. Distributive Laws of Monadic Containers. In 11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 342, pp. 4:1-4:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{purdy_et_al:LIPIcs.CALCO.2025.4,
  author =	{Purdy, Chris and Damato, Stefania},
  title =	{{Distributive Laws of Monadic Containers}},
  booktitle =	{11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025)},
  pages =	{4:1--4:22},
  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.2025.4},
  URN =		{urn:nbn:de:0030-drops-235633},
  doi =		{10.4230/LIPIcs.CALCO.2025.4},
  annote =	{Keywords: distributive laws, monadic containers, monads, dependent types, cubical agda}
}
Document
DOI: 10.4230/LIPIcs.STACS.2025.10
Monotone Weak Distributive Laws over the Lifted Powerset Monad in Categories of Algebras

Authors: Quentin Aristote

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

Abstract
In both the category of sets and the category of compact Hausdorff spaces, there is a monotone weak distributive law that combines two layers of non-determinism. Noticing the similarity between these two laws, we study whether the latter can be obtained automatically as a weak lifting of the former. This holds partially, but does not generalize to other categories of algebras. We then characterize when exactly monotone weak distributive laws over powerset monads in categories of algebras exist, on the one hand exhibiting a law combining probabilities and non-determinism in compact Hausdorff spaces and showing on the other hand that such laws do not exist in a lot of other cases.
Cite as

Quentin Aristote. Monotone Weak Distributive Laws over the Lifted Powerset Monad in Categories of Algebras. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 10:1-10:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{aristote:LIPIcs.STACS.2025.10,
  author =	{Aristote, Quentin},
  title =	{{Monotone Weak Distributive Laws over the Lifted Powerset Monad in Categories of Algebras}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{10:1--10:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.10},
  URN =		{urn:nbn:de:0030-drops-228356},
  doi =		{10.4230/LIPIcs.STACS.2025.10},
  annote =	{Keywords: weak distributive law, weak extension, weak lifting, iterated distributive law, Yang-Baxter equation, powerset monad, Vietoris monad, Radon monad, Eilenberg-Moore category, regular category, relational extension}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2022.7
Comonadic semantics for hybrid logic

Authors: Samson Abramsky and Dan Marsden

Published in: LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

Abstract
Hybrid logic is a widely-studied extension of basic modal logic, which corresponds to the bounded fragment of first-order logic. We study it from two novel perspectives: (1) We apply the recently introduced paradigm of comonadic semantics, which provides a new set of tools drawing on ideas from categorical semantics which can be applied to finite model theory, descriptive complexity and combinatorics. (2) We give a novel semantic characterization of hybrid logic in terms of invariance under disjoint extensions, a minimal form of locality. A notable feature of this result is that we give a uniform proof, valid for both the finite and infinite cases.
Cite as

Samson Abramsky and Dan Marsden. Comonadic semantics for hybrid logic. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 7:1-7:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{abramsky_et_al:LIPIcs.MFCS.2022.7,
  author =	{Abramsky, Samson and Marsden, Dan},
  title =	{{Comonadic semantics for hybrid logic}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{7:1--7:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.7},
  URN =		{urn:nbn:de:0030-drops-168055},
  doi =		{10.4230/LIPIcs.MFCS.2022.7},
  annote =	{Keywords: comonads, model comparison games, semantic characterizations, hybrid logic, bounded fragment}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2021.115
Arboreal Categories and Resources

Authors: Samson Abramsky and Luca Reggio

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Abstract
We introduce arboreal categories, which have an intrinsic process structure, allowing dynamic notions such as bisimulation and back-and-forth games, and resource notions such as number of rounds of a game, to be defined. These are related to extensional or "static" structures via arboreal covers, which are resource-indexed comonadic adjunctions. These ideas are developed in a very general, axiomatic setting, and applied to relational structures, where the comonadic constructions for pebbling, Ehrenfeucht-Fraïssé and modal bisimulation games recently introduced in [Abramsky et al., 2017; S. Abramsky and N. Shah, 2018; Abramsky and Shah, 2021] are recovered, showing that many of the fundamental notions of finite model theory and descriptive complexity arise from instances of arboreal covers.
Cite as

Samson Abramsky and Luca Reggio. Arboreal Categories and Resources. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 115:1-115:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{abramsky_et_al:LIPIcs.ICALP.2021.115,
  author =	{Abramsky, Samson and Reggio, Luca},
  title =	{{Arboreal Categories and Resources}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{115:1--115:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.115},
  URN =		{urn:nbn:de:0030-drops-141845},
  doi =		{10.4230/LIPIcs.ICALP.2021.115},
  annote =	{Keywords: factorisation system, embedding, comonad, coalgebra, open maps, bisimulation, game, resources, relational structures, finite model theory}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2020.77
Preservation of Equations by Monoidal Monads

Authors: Louis Parlant, Jurriaan Rot, Alexandra Silva, and Bas Westerbaan

Published in: LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Abstract
If a monad T is monoidal, then operations on a set X can be lifted canonically to operations on TX. In this paper we study structural properties under which T preserves equations between those operations. It has already been shown that any monoidal monad preserves linear equations; affine monads preserve drop equations (where some variable appears only on one side, such as x⋅ y = y) and relevant monads preserve dup equations (where some variable is duplicated, such as x ⋅ x = x). We start the paper by showing a converse: if the monad at hand preserves a drop equation, then it must be affine. From this, we show that the problem whether a given (drop) equation is preserved is undecidable. A converse for relevance turns out to be more subtle: preservation of certain dup equations implies a weaker notion which we call n-relevance. Finally, we identify a subclass of equations such that their preservation is equivalent to relevance.
Cite as

Louis Parlant, Jurriaan Rot, Alexandra Silva, and Bas Westerbaan. Preservation of Equations by Monoidal Monads. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 77:1-77:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{parlant_et_al:LIPIcs.MFCS.2020.77,
  author =	{Parlant, Louis and Rot, Jurriaan and Silva, Alexandra and Westerbaan, Bas},
  title =	{{Preservation of Equations by Monoidal Monads}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{77:1--77:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Esparza, Javier and Kr\'{a}l', Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.77},
  URN =		{urn:nbn:de:0030-drops-127460},
  doi =		{10.4230/LIPIcs.MFCS.2020.77},
  annote =	{Keywords: monoidal monads, algebraic theories, preservation of equations}
}
Document
DOI: 10.4230/LIPIcs.CSL.2018.32
Quantitative Foundations for Resource Theories

Authors: Dan Marsden and Maaike Zwart

Published in: LIPIcs, Volume 119, 27th EACSL Annual Conference on Computer Science Logic (CSL 2018)

Abstract
Considering resource usage is a powerful insight in the analysis of many phenomena in the sciences. Much of the current research on these resource theories focuses on the analysis of specific resources such quantum entanglement, purity, randomness or asymmetry. However, the mathematical foundations of resource theories are at a much earlier stage, and there has been no satisfactory account of quantitative aspects such as costs, rates or probabilities. We present a categorical foundation for quantitative resource theories, derived from enriched category theory. Our approach is compositional, with rich algebraic structure facilitating calculations. The resulting theory is parameterized, both in the quantities under consideration, for example costs or probabilities, and in the structural features of the resources such as whether they can be freely copied or deleted. We also achieve a clear separation of concerns between the resource conversions that are freely available, and the costly resources that are typically the object of study. By using an abstract categorical approach, our framework is naturally open to extension. We provide many examples throughout, emphasising the resource theoretic intuitions for each of the mathematical objects under consideration.
Cite as

Dan Marsden and Maaike Zwart. Quantitative Foundations for Resource Theories. In 27th EACSL Annual Conference on Computer Science Logic (CSL 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 119, pp. 32:1-32:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{marsden_et_al:LIPIcs.CSL.2018.32,
  author =	{Marsden, Dan and Zwart, Maaike},
  title =	{{Quantitative Foundations for Resource Theories}},
  booktitle =	{27th EACSL Annual Conference on Computer Science Logic (CSL 2018)},
  pages =	{32:1--32:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-088-0},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{119},
  editor =	{Ghica, Dan R. and Jung, Achim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2018.32},
  URN =		{urn:nbn:de:0030-drops-96996},
  doi =		{10.4230/LIPIcs.CSL.2018.32},
  annote =	{Keywords: Resource Theory, Enriched Category, Profunctor, Monad, Combinatorial Species, Multicategory, Operad, Bimodule}
}
Document
DOI: 10.4230/LIPIcs.CALCO.2017.17
Custom Hypergraph Categories via Generalized Relations

Authors: Dan Marsden and Fabrizio Genovese

Published in: LIPIcs, Volume 72, 7th Conference on Algebra and Coalgebra in Computer Science (CALCO 2017)

Abstract
Process theories combine a graphical language for compositional reasoning with an underlying categorical semantics. They have been successfully applied to fields such as quantum computation, natural language processing, linear dynamical systems and network theory. When investigating a new application, the question arises of how to identify a suitable process theoretic model. We present a conceptually motivated parameterized framework for the construction of models for process theories. Our framework generalizes the notion of binary relation along four axes of variation, the truth values, a choice of algebraic structure, the ambient mathematical universe and the choice of proof relevance or provability. The resulting categories are preorder-enriched and provide analogues of relational converse and taking graphs of maps. Our constructions are functorial in the parameter choices, establishing mathematical connections between different application domains. We illustrate our techniques by constructing many existing models from the literature, and new models that open up ground for further development.
Cite as

Dan Marsden and Fabrizio Genovese. Custom Hypergraph Categories via Generalized Relations. In 7th Conference on Algebra and Coalgebra in Computer Science (CALCO 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 72, pp. 17:1-17:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{marsden_et_al:LIPIcs.CALCO.2017.17,
  author =	{Marsden, Dan and Genovese, Fabrizio},
  title =	{{Custom Hypergraph Categories via Generalized Relations}},
  booktitle =	{7th Conference on Algebra and Coalgebra in Computer Science (CALCO 2017)},
  pages =	{17:1--17:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-033-0},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{72},
  editor =	{Bonchi, Filippo and K\"{o}nig, Barbara},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2017.17},
  URN =		{urn:nbn:de:0030-drops-80494},
  doi =		{10.4230/LIPIcs.CALCO.2017.17},
  annote =	{Keywords: Process Theory, Categorical Compositional Semantics, Generalized Relations, Hypergraph Category, Compact Closed Category}
}
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