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DOI: 10.4230/LIPIcs.LICS.2026.45

Monads and Distributive Laws in Substructural Contexts

Authors: Soichiro Fujii, Yun Chen Tsai, Yoàv Montacute, and Ichiro Hasuo

Published in: LIPIcs, Volume 380, 41st Annual Symposium on Logic in Computer Science (LICS 2026)

Abstract

We present a categorical theory of monads and distributive laws in substructural contexts. In the study of distributive laws, the roles of (the absence of) structural rules for variable contexts have been recognized; our theory formalizes these substructural situations using Tronin’s verbal categories W, in a uniform and presentation-independent manner. We introduce the classes of W-operadic monads (those defined via the structural rules in W) and of W-commutative monads (those invariant under the structural rules in W). We give a canonical construction of a distributive law ST → TS of monads on Set; it is applicable when S is W-operadic and T is W-commutative (under mild conditions). This accounts for many known and new distributive laws. Even when S fails to be W-operadic, we can refine S and force W-operadicity; this captures Varacca and Winskel’s construction of indexed valuations.

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Soichiro Fujii, Yun Chen Tsai, Yoàv Montacute, and Ichiro Hasuo. Monads and Distributive Laws in Substructural Contexts. In 41st Annual Symposium on Logic in Computer Science (LICS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 380, pp. 45:1-45:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{fujii_et_al:LIPIcs.LICS.2026.45,
  author =	{Fujii, Soichiro and Tsai, Yun Chen and Montacute, Yo\`{a}v and Hasuo, Ichiro},
  title =	{{Monads and Distributive Laws in Substructural Contexts}},
  booktitle =	{41st Annual Symposium on Logic in Computer Science (LICS 2026)},
  pages =	{45:1--45:28},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-434-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{380},
  editor =	{Faggian, Claudia and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.LICS.2026.45},
  URN =		{urn:nbn:de:0030-drops-268324},
  doi =		{10.4230/LIPIcs.LICS.2026.45},
  annote =	{Keywords: Monad, distributive law, operad, category theory, effect}
}

Document

DOI: 10.4230/LIPIcs.CSL.2022.19

Dynamic Cantor Derivative Logic

Authors: David Fernández-Duque and Yoàv Montacute

Published in: LIPIcs, Volume 216, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022)

Abstract

Topological semantics for modal logic based on the Cantor derivative operator gives rise to derivative logics, also referred to as d-logics. Unlike logics based on the topological closure operator, d-logics have not previously been studied in the framework of dynamical systems, which are pairs (X,f) consisting of a topological space X equipped with a continuous function f: X → X. We introduce the logics wK4C, K4C and GLC and show that they all have the finite Kripke model property and are sound and complete with respect to the d-semantics in this dynamical setting. In particular, we prove that wK4C is the d-logic of all dynamic topological systems, K4C is the d-logic of all T_D dynamic topological systems, and GLC is the d-logic of all dynamic topological systems based on a scattered space. We also prove a general result for the case where f is a homeomorphism, which in particular yields soundness and completeness for the corresponding systems wK4H, K4H and GLH. The main contribution of this work is the foundation of a general proof method for finite model property and completeness of dynamic topological d-logics. Furthermore, our result for GLC constitutes the first step towards a proof of completeness for the trimodal topo-temporal language with respect to a finite axiomatisation - something known to be impossible over the class of all spaces.

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David Fernández-Duque and Yoàv Montacute. Dynamic Cantor Derivative Logic. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 19:1-19:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{fernandezduque_et_al:LIPIcs.CSL.2022.19,
  author =	{Fern\'{a}ndez-Duque, David and Montacute, Yo\`{a}v},
  title =	{{Dynamic Cantor Derivative Logic}},
  booktitle =	{30th EACSL Annual Conference on Computer Science Logic (CSL 2022)},
  pages =	{19:1--19:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-218-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{216},
  editor =	{Manea, Florin and Simpson, Alex},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2022.19},
  URN =		{urn:nbn:de:0030-drops-157397},
  doi =		{10.4230/LIPIcs.CSL.2022.19},
  annote =	{Keywords: dynamic topological logic, Cantor derivative, temporal logic, modal logic}
}

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Monads and Distributive Laws in Substructural Contexts

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Dynamic Cantor Derivative Logic

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