Document Open Access Logo

Monads and Distributive Laws in Substructural Contexts

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

PDF

File

Thumbnail PDF
PDF
LIPIcs.LICS.2026.45.pdf
  • Filesize: 2.36 MB
  • 28 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Categorical semantics
  • Theory of computation → Probabilistic computation
  • Theory of computation → Program semantics
Keywords
  • Monad
  • distributive law
  • operad
  • category theory
  • effect

Metrics

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.

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.LICS.2026.45

Author Details

Soichiro Fujii
  • National Institute of Informatics, Tokyo, Japan
Yun Chen Tsai
  • National Institute of Informatics, Tokyo, Japan
  • SOKENDAI (The Graduate University for Advanced Studies), Kanagawa, Japan
Yoàv Montacute
  • National Institute of Informatics, Tokyo, Japan
Ichiro Hasuo
  • National Institute of Informatics, Tokyo, Japan
  • SOKENDAI (The Graduate University for Advanced Studies), Kanagawa, Japan
  • Imiron Co., Ltd., Tokyo, Japan

Funding

All authors are supported by ASPIRE Grant No. JPMJAP2301, JST, Japan.
  • Montacute, Yoàv: Supported by ACT-X, Grant No. JPMJAX24CR, JST, Japan.

Acknowledgements

We thank the anonymous referees for their helpful comments.

References

  1. 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), volume 327 of Leibniz International Proceedings in Informatics (LIPIcs), pages 10:1-10:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. URL: https://doi.org/10.4230/LIPIcs.STACS.2025.10.
  2. Falk Bartels. On Generalised Coinduction and Probabilistic Specification Formats: Distributive laws in coalgebraic modelling. PhD thesis, Vrije Universiteit Amsterdam, 2004. URL: https://hdl.handle.net/1871.1/56f40c24-8cb4-4359-beb3-c28c3263838e.
  3. Jon Beck. Distributive laws. In Seminar on Triples and Categorical Homology Theory, volume 80 of Lecture Notes in Mathematics, pages 119-140. Springer Berlin Heidelberg, 1969. URL: https://doi.org/10.1007/BFb0083084.
  4. Filippo Bonchi and Alessio Santamaria. Convexity via weak distributive laws. Logical Methods in Computer Science, 18(4):8:1-8:55, 2022. URL: https://doi.org/10.46298/LMCS-18(4:8)2022.
  5. Marta Bunge. On the relationship between composite and tensor product triples. Journal of Pure and Applied Algebra, 13(2):139-156, 1978. URL: https://doi.org/10.1016/0022-4049(78)90004-X.
  6. Titouan Carette, Louis Lemonnier, and Vladimir Zamdzhiev. Central submonads and notions of computation: soundness, completeness and internal languages. In Proceedings of the 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2023), pages 1-13. IEEE, 2023. URL: https://doi.org/10.1109/LICS56636.2023.10175687.
  7. Chris Chen, Annabelle McIver, and Carroll Morgan. Probabilistic datatypes. In Theoretical Aspects of Computing - ICTAC 2024, volume 15373 of Lecture Notes in Computer Science, pages 3-16. Springer Nature Switzerland, 2025. URL: https://doi.org/10.1007/978-3-031-77019-7_1.
  8. Chris Chen, Annabelle McIver, and Carroll Morgan. Forward and backward simulations for partially observable probability. In Theoretical Aspects of Computing - ICTAC 2025, volume 16237 of Lecture Notes in Computer Science, pages 279-297. Springer Nature Switzerland, 2026. URL: https://doi.org/10.1007/978-3-032-11176-0_17.
  9. Alfred Hoblitzelle Clifford and Gordon Bamford Preston. The Algebraic Theory of Semigroups, Volume I, volume 7 of Mathematical Surveys. American Mathematical Society, 1961. URL: https://doi.org/10.1090/surv/007.1/01.
  10. Pierre-Louis Curien. Operads, clones, and distributive laws. In Operads and universal algebra. Proceedings of the summer school and international conference, Tianjin, China, July 5-9, 2010, volume 9 of Nankai Series in Pure, Applied Mathematics and Theoretical Physics, pages 25-49. World Scientific, 2012. URL: https://doi.org/10.1142/9789814365123_0002.
  11. Fredrik Dahlqvist, Louis Parlant, and Alexandra Silva. Layer by layer - combining monads. In Theoretical Aspects of Computing - ICTAC 2018, volume 11187 of Lecture Notes in Computer Science, pages 153-172. Springer International Publishing, 2018. URL: https://doi.org/10.1007/978-3-030-02508-3_9.
  12. Valeria de Paiva. The dialectica categories. Technical Report UCAM-CL-TR-213, University of Cambridge, Computer Laboratory, 1991. URL: https://doi.org/10.48456/tr-213.
  13. Marcelo Fiore, Gordon Plotkin, and Daniele Turi. Abstract syntax and variable binding. In Proceedings of the Fourteenth Annual IEEE Symposium on Logic in Computer Science (LICS 1999), pages 193-202. IEEE Computer Society, 1999. URL: https://doi.org/10.1109/LICS.1999.782615.
  14. Soichiro Fujii, Yun Chen Tsai, Yoàv Montacute, and Ichiro Hasuo. Monads and distributive laws in substructural contexts (extended version), 2026. URL: https://arxiv.org/abs/2605.13533v1.
  15. Richard Garner. The Vietoris monad and weak distributive laws. Applied Categorical Structures, 28(2):339-354, 2020. URL: https://doi.org/10.1007/s10485-019-09582-w.
  16. Richard Garner. Cartesian closed varieties I: the classification theorem. Algebra Universalis, 85(4):38, 2024. URL: https://doi.org/10.1007/s00012-024-00869-1.
  17. Alexandre Goy and Daniela Petrişan. Combining probabilistic and non-deterministic choice via weak distributive laws. In Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2020), pages 454-464. ACM, 2020. URL: https://doi.org/10.1145/3373718.3394795.
  18. Martin Hyland, Gordon Plotkin, and John Power. Combining effects: sum and tensor. Theoretical Computer Science, 357(1):70-99, 2006. URL: https://doi.org/10.1016/j.tcs.2006.03.013.
  19. Martin Hyland and John Power. The category theoretic understanding of universal algebra: Lawvere theories and monads. In Computation, Meaning, and Logic: Articles dedicated to Gordon Plotkin, volume 172 of Electronic Notes in Theoretical Computer Science, pages 437-458. Elsevier, 2007. URL: https://doi.org/10.1016/j.entcs.2007.02.019.
  20. Bart Jacobs. Semantics of weakening and contraction. Annals of Pure and Applied Logic, 69(1):73-106, 1994. URL: https://doi.org/10.1016/0168-0072(94)90020-5.
  21. Bart Jacobs. Trace semantics for coalgebras. In Proceedings of the Seventh Workshop on Coalgebraic Methods in Computer Science (CMCS 2004), volume 106 of Electronic Notes in Theoretical Computer Science, pages 167-184. Elsevier, 2004. URL: https://doi.org/10.1016/j.entcs.2004.02.031.
  22. Bart Jacobs. From multisets over distributions to distributions over multisets. In Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2021), pages 1-13. IEEE, 2021. URL: https://doi.org/10.1109/LICS52264.2021.9470678.
  23. Peter Johnstone. Collapsed toposes and cartesian closed varieties. Journal of Algebra, 129(2):446-480, 1990. URL: https://doi.org/10.1016/0021-8693(90)90230-L.
  24. André Joyal. Foncteurs analytiques et espèces de structures. In Combinatoire énumérative, volume 1234 of Lecture Notes in Mathematics, pages 126-159. Springer Berlin Heidelberg, 1986. URL: https://doi.org/10.1007/BFb0072514.
  25. Gregory Maxwell Kelly. On the operads of J.P. May. Reprints in Theory and Applications of Categories, 13:1-13, 2005. URL: http://www.tac.mta.ca/tac/reprints/articles/13/tr13abs.html.
  26. David King and Philip Wadler. Combining monads. In Functional Programming, Glasgow 1992, pages 134-143. Springer London, 1993. URL: https://doi.org/10.1007/978-1-4471-3215-8_12.
  27. Bartek Klin and Julian Salamanca. Iterated covariant powerset is not a monad. In Proceedings of the Thirty-Fourth Conference on the Mathematical Foundations of Programming Semantics (MFPS XXXIV), volume 341 of Electronic Notes in Theoretical Computer Science, pages 261-276. Elsevier, 2018. URL: https://doi.org/10.1016/j.entcs.2018.11.013.
  28. Anders Kock. Monads on symmetric monoidal closed categories. Archiv der Mathematik, 21:1-10, 1970. URL: https://doi.org/10.1007/BF01220868.
  29. Anders Kock. Bilinearity and Cartesian closed monads. Mathematica Scandinavica, 29:161-174, 1972. URL: https://doi.org/10.7146/math.scand.a-11042.
  30. Anders Kock. Strong functors and monoidal monads. Archiv der Mathematik, 23:113-120, 1972. URL: https://doi.org/10.1007/BF01304852.
  31. Anders Kock. Commutative monads as a theory of distributions. Theory and Applications of Categories, 26:97-131, 2012. URL: http://www.tac.mta.ca/tac/volumes/26/4/26-04abs.html.
  32. Dexter Kozen and Alexandra Silva. Multisets and distributions. In Logics and Type Systems in Theory and Practice - Essays Dedicated to Herman Geuvers on The Occasion of His 60th Birthday, volume 14560 of Lecture Notes in Computer Science, pages 168-187. Springer, 2024. URL: https://doi.org/10.1007/978-3-031-61716-4_11.
  33. Tom Leinster. Higher Operads, Higher Categories, volume 298 of London Mathematical Society Lecture Note Series. Cambridge University Press, 2004. URL: https://doi.org/10.1017/CBO9780511525896.
  34. Fabian Lenke, Stefan Milius, and Henning Urbat. A unified treatment of substitution for presheaves, nominal sets, renaming sets, and so on, 2026. URL: https://arxiv.org/abs/2602.11907v1.
  35. Jack Liell-Cock and Sam Staton. Compositional imprecise probability: A solution from graded monads and Markov categories. Proceedings of the ACM on Programming Languages, 9(POPL):54:1-54:31, 2025. URL: https://doi.org/10.1145/3704890.
  36. Harald Lindner. Affine parts of monads. Archiv der Mathematik, 33:437-443, 1980. URL: https://doi.org/10.1007/BF01222782.
  37. Saunders Mac Lane. Categories for the Working Mathematician, volume 5 of Graduate Texts in Mathematics. Springer, 2nd edition, 1998. URL: https://doi.org/10.1007/978-1-4757-4721-8.
  38. Ernie Manes and Philip Mulry. Monad compositions I: General constructions and recursive distributive laws. Theory and Applications of Categories, 18:172-208, 2007. URL: http://www.tac.mta.ca/tac/volumes/18/7/18-07abs.html.
  39. Cristina Matache and Sam Staton. A sound and complete logic for algebraic effects. In Foundations of Software Science and Computation Structures (FOSSACS 2019), volume 11425 of Lecture Notes in Computer Science, pages 382-399. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-17127-8_22.
  40. Dylan McDermott and Tarmo Uustalu. What makes a strong monad? In Proceedings of the Ninth Workshop on Mathematically Structured Functional Programming (MSFP 2022), volume 360 of Electronic Proceedings in Theoretical Computer Science, pages 113-133. Open Publishing Association, 2022. URL: https://doi.org/10.4204/EPTCS.360.6.
  41. Matteo Mio, Ralph Sarkis, and Valeria Vignudelli. Combining nondeterminism, probability, and termination: Equational and metric reasoning. In Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2021), pages 1-14. IEEE, 2021. URL: https://doi.org/10.1109/LICS52264.2021.9470717.
  42. Matteo Mio and Valeria Vignudelli. Monads and quantitative equational theories for nondeterminism and probability. In 31st International Conference on Concurrency Theory (CONCUR 2020), volume 171 of Leibniz International Proceedings in Informatics (LIPIcs), pages 28:1-28:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.CONCUR.2020.28.
  43. Eugenio Moggi. Notions of computation and monads. Information and Computation, 93(1):55-92, 1991. URL: https://doi.org/10.1016/0890-5401(91)90052-4.
  44. Shawn Ong, Stephanie Ma, and Dexter Kozen. Probabilistic Kleene algebra with angelic nondeterminism. Proceedings of the ACM on Programming Languages, 9(PLDI):897-919, 2025. URL: https://doi.org/10.1145/3729286.
  45. Louis Parlant. Monad Composition via Preservation of Algebras. PhD thesis, University College London, 2020. URL: https://discovery.ucl.ac.uk/id/eprint/10112228.
  46. 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), volume 170 of Leibniz International Proceedings in Informatics (LIPIcs), pages 77:1-77:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.MFCS.2020.77.
  47. Maciej Piróg and Sam Staton. Backtracking with cut via a distributive law and left-zero monoids. Journal of Functional Programming, 27:e17, 2017. URL: https://doi.org/10.1017/S0956796817000077.
  48. Gordon Plotkin and Matija Pretnar. Handling algebraic effects. Logical Methods in Computer Science, 9(4):23, 2013. URL: https://doi.org/10.2168/LMCS-9(4:23)2013.
  49. Aloïs Rosset, Maaike Zwart, Helle Hvid Hansen, and Jörg Endrullis. Correspondence between composite theories and distributive laws. In Coalgebraic Methods in Computer Science (CMCS 2024), volume 14617 of Lecture Notes in Computer Science, pages 194-215. Springer, 2024. URL: https://doi.org/10.1007/978-3-031-66438-0_10.
  50. Michael Shulman. Categorical logic from a categorical point of view. Draft, version of July 28, 2016. URL: https://mikeshulman.github.io/catlog/catlog.pdf.
  51. Alex Simpson and Niels Voorneveld. Behavioural equivalence via modalities for algebraic effects. ACM Transactions on Programming Languages and Systems, 42(1):4:1-4:45, 2020. URL: https://doi.org/10.1145/3363518.
  52. Ross Street. The formal theory of monads. Journal of Pure and Applied Algebra, 2(2):149-168, 1972. URL: https://doi.org/10.1016/0022-4049(72)90019-9.
  53. Stanisław Szawiel and Marek Zawadowski. Monads of regular theories. Applied Categorical Structures, 23(3):215-262, 2015. URL: https://doi.org/10.1007/s10485-013-9331-x.
  54. Miki Tanaka and John Power. Pseudo-distributive laws and axiomatics for variable binding. Higher-Order and Symbolic Computation, 19(2-3):305-337, 2006. URL: https://doi.org/10.1007/s10990-006-8750-x.
  55. Miki Tanaka and John Power. A unified category-theoretic semantics for binding signatures in substructural logics. Journal of Logic and Computation, 16(1):5-25, 2006. URL: https://doi.org/10.1093/logcom/exi070.
  56. Regina Tix, Klaus Keimel, and Gordon Plotkin. Semantic domains for combining probability and non-determinism. Electronic Notes in Theoretical Computer Science, 222:3-99, 2009. URL: https://doi.org/10.1016/j.entcs.2009.01.002.
  57. Serge Tronin. Abstract clones and operads. Siberian Mathematical Journal, 43(4):746-755, 2002. URL: https://doi.org/10.1023/A:1016340806503.
  58. Serge Tronin. Operads and varieties of algebras defined by polylinear identities. Siberian Mathematical Journal, 47(3):670-694, 2006. URL: https://doi.org/10.1007/s11202-006-0067-9.
  59. Serge Tronin. On algebras over multicategories. Russian Mathematics, 60(2):52-61, 2016. URL: https://doi.org/10.3103/S1066369X16020092.
  60. Serge Tronin and Oleg Kopp. Matrix linear operads. Russian Mathematics, 44(6):50-59, 2000. Google Scholar
  61. Yun Chen Tsai, Kittiphon Phalakarn, Sundararaman Akshay, and Ichiro Hasuo. Chance and mass interpretations of probabilities in Markov decision processes. In 36th International Conference on Concurrency Theory (CONCUR 2025), volume 348 of Leibniz International Proceedings in Informatics (LIPIcs), pages 33:1-33:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. URL: https://doi.org/10.4230/LIPIcs.CONCUR.2025.33.
  62. Daniele Varacca and Glynn Winskel. Distributing probability over non-determinism. Mathematical Structures in Computer Science, 16(1):87-113, 2006. URL: https://doi.org/10.1017/S0960129505005074.
  63. Mark Weber. Generic morphisms, parametric representations and weakly cartesian monads. Theory and Applications of Categories, 13:191-234, 2004. URL: http://www.tac.mta.ca/tac/volumes/13/14/13-14abs.html.
  64. Noam Zilberstein, Dexter Kozen, Alexandra Silva, and Joseph Tassarotti. A demonic outcome logic for randomized nondeterminism. Proceedings of the ACM on Programming Languages, 9(POPL):539-568, 2025. URL: https://doi.org/10.1145/3704855.
  65. Maaike Zwart. On the Non-Compositionality of Monads via Distributive Laws. PhD thesis, Department of Computer Science, University of Oxford, September 2020. URL: https://www.cs.ox.ac.uk/publications/publication14285-abstract.html.
  66. Maaike Zwart and Dan Marsden. No-go theorems for distributive laws. Logical Methods in Computer Science, 18(1):13:1-13:61, 2022. URL: https://doi.org/10.46298/lmcs-18(1:13)2022.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact