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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}
}

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DOI: 10.4230/LIPIcs.CONCUR.2025.33

Chance and Mass Interpretations of Probabilities in Markov Decision Processes

Authors: Yun Chen Tsai, Kittiphon Phalakarn, S. Akshay, and Ichiro Hasuo

Published in: LIPIcs, Volume 348, 36th International Conference on Concurrency Theory (CONCUR 2025)

Abstract

Markov decision processes (MDPs) are a popular model for decision-making in the presence of uncertainty. The conventional view of MDPs in verification treats them as state transformers with probabilities defined over sequences of states and with schedulers making random choices. An alternative view, especially well-suited for modeling dynamical systems, defines MDPs as distribution transformers with schedulers distributing probability masses. Our main contribution is a unified semantical framework that accommodates these two views and two new ones. These four semantics of MDPs arise naturally through identifying different sources of randomness in an MDP (namely schedulers, configurations, and transitions) and providing different ways of interpreting these probabilities (called the chance and mass interpretations). These semantics are systematically unified through a mathematical construct called chance-mass (CM) classifier. As another main contribution, we study a reachability problem in each of the two new semantics, demonstrating their hardness and providing two algorithms for solving them.

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Yun Chen Tsai, Kittiphon Phalakarn, S. Akshay, and Ichiro Hasuo. Chance and Mass Interpretations of Probabilities in Markov Decision Processes. In 36th International Conference on Concurrency Theory (CONCUR 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 348, pp. 33:1-33:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{tsai_et_al:LIPIcs.CONCUR.2025.33,
  author =	{Tsai, Yun Chen and Phalakarn, Kittiphon and Akshay, S. and Hasuo, Ichiro},
  title =	{{Chance and Mass Interpretations of Probabilities in Markov Decision Processes}},
  booktitle =	{36th International Conference on Concurrency Theory (CONCUR 2025)},
  pages =	{33:1--33:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-389-8},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{348},
  editor =	{Bouyer, Patricia and van de Pol, Jaco},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2025.33},
  URN =		{urn:nbn:de:0030-drops-239838},
  doi =		{10.4230/LIPIcs.CONCUR.2025.33},
  annote =	{Keywords: MDP, distribution transformer, antichain, template-based synthesis}
}

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Documents authored by Tsai, Yun Chen

Monads and Distributive Laws in Substructural Contexts

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Chance and Mass Interpretations of Probabilities in Markov Decision Processes

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