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Tape Diagrams for Monoidal Monads

Authors Filippo Bonchi , Cipriano Junior Cioffo , Alessandro Di Giorgio , Elena Di Lavore

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ACM Subject Classification
  • Theory of computation → Logic
Keywords
  • rig categories
  • string diagrams
  • monads
  • probabilistic control

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Abstract

Tape diagrams provide a graphical representation for arrows of rig categories, namely categories equipped with two monoidal structures, ⊕ and ⊗, where ⊗ distributes over ⊕. However, their applicability is limited to categories where ⊕ is a biproduct, i.e., both a categorical product and a coproduct. In this work, we extend tape diagrams to deal with Kleisli categories of symmetric monoidal monads, presented by algebraic theories.

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Filippo Bonchi, Cipriano Junior Cioffo, Alessandro Di Giorgio, and Elena Di Lavore. Tape Diagrams for Monoidal Monads. In 11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 342, pp. 11:1-11:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CALCO.2025.11

Author Details

Filippo Bonchi
  • University of Pisa, Italy
Cipriano Junior Cioffo
  • University of Pisa, Italy
Alessandro Di Giorgio
  • Tallinn University of Technology, Estonia
Elena Di Lavore
  • University of Oxford, UK

Funding

This research was partly funded by the Advanced Research + Invention Agency (ARIA) Safeguarded AI Programme.
  • Bonchi, Filippo: Bonchi is supported by the Ministero dell'Università e della Ricerca of Italy grant PRIN 2022 PNRR No. P2022HXNSC - RAP (Resource Awareness in Programming). This study was carried out within the National Centre on HPC, Big Data and Quantum Computing - SPOKE 10 (Quantum Computing) and received funding from the European Union Next-GenerationEU - National Recovery and Resilience Plan (NRRP) – MISSION 4 COMPONENT 2, INVESTMENT N. 1.4 – CUP N. I53C22000690001.
  • Di Giorgio, Alessandro: Supported by the EU grant No. 101087529.

References

  1. John Baez and Jason Erbele. Categories in control. Theory and Applications of Categories, 30:836-881, 2015. URL: http://arxiv.org/abs/1405.6881.
  2. Guillaume Boisseau and Robin Piedeleu. Graphical Piecewise-Linear Algebra. In Patricia Bouyer and Lutz Schröder, editors, Foundations of Software Science and Computation Structures, Lecture Notes in Computer Science, pages 101-119, Cham, 2022. Springer International Publishing. URL: https://doi.org/10.1007/978-3-030-99253-8_6.
  3. Filippo Bonchi, Cipriano Junior Cioffo, Alessandro Di Giorgio, and Elena Di Lavore. Tape diagrams for monoidal monads, 2025. URL: https://doi.org/10.48550/arXiv.2503.22819.
  4. Filippo Bonchi, Alessandro Di Giorgio, and Elena Di Lavore. A diagrammatic algebra for program logics. In FoSSaCS 2025 (to appear), 2025. URL: https://arxiv.org/abs/2410.03561.
  5. Filippo Bonchi, Alessandro Di Giorgio, and Alessio Santamaria. Deconstructing the calculus of relations with tape diagrams. Proceedings of the ACM on Programming Languages, 7(POPL):1864-1894, 2023. URL: https://doi.org/10.1145/3571257.
  6. Filippo Bonchi, Fabio Gadducci, Aleks Kissinger, Pawel Sobocinski, and Fabio Zanasi. String diagram rewrite theory I: rewriting with frobenius structure. J. ACM, 69(2):14:1-14:58, 2022. URL: https://doi.org/10.1145/3502719.
  7. Filippo Bonchi, Joshua Holland, Robin Piedeleu, Paweł Sobociński, and Fabio Zanasi. Diagrammatic algebra: From linear to concurrent systems. Proceedings of the ACM on Programming Languages, 3(POPL):25:1-25:28, January 2019. URL: https://doi.org/10.1145/3290338.
  8. Filippo Bonchi, Pawel Sobocinski, and Fabio Zanasi. Full Abstraction for Signal Flow Graphs. In Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL '15, pages 515-526, New York, NY, USA, January 2015. Association for Computing Machinery. URL: https://doi.org/10.1145/2676726.2676993.
  9. Francis Borceux. Handbook of categorical algebra. 2, volume 51 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1994. Categories and structures. Google Scholar
  10. Roberto Bruni and Fabio Gadducci. Some algebraic laws for spans (and their connections with multi-relations). In RelMiS 2001. Elsevier, 2001. Google Scholar
  11. Aurelio Carboni, Stephen Lack, and Robert FC Walters. Introduction to extensive and distributive categories. Journal of Pure and Applied Algebra, 84(2):145-158, 1993. Google Scholar
  12. Kenta Cho and Bart Jacobs. Disintegration and bayesian inversion via string diagrams. Mathematical Structures in Computer Science, 29(7):938-971, 2019. URL: https://doi.org/10.1017/S0960129518000488.
  13. Kenta Cho, Bart Jacobs, Bas Westerbaan, and Abraham Westerbaan. An introduction to effectus theory. ArXiv, abs/1512.05813, 2015. URL: https://arxiv.org/abs/1512.05813.
  14. Cipriano Junior Cioffo, Fabio Gadducci, and Davide Trotta. A taxonomy of categories for relations, 2025. URL: https://arxiv.org/abs/2502.10323.
  15. Bob Coecke and Ross Duncan. Interacting quantum observables. In Proceedings of the 35th international colloquium on Automata, Languages and Programming (ICALP), Part II, pages 298-310, 2008. URL: https://doi.org/10.1007/978-3-540-70583-3_25.
  16. Bob Coecke and Ross Duncan. Interacting quantum observables: categorical algebra and diagrammatics. New Journal of Physics, 13(4):043016, 2011. Google Scholar
  17. Bob Coecke and Aleks Kissinger. Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning. Cambridge University Press, 2017. URL: https://doi.org/10.1017/9781316219317.
  18. Bob Coecke, Dusko Pavlovic, and Jamie Vicary. A new description of orthogonal bases. Math. Struct. Comp. Sci., 23(3):557-567, 2012. Google Scholar
  19. Cole Comfort, Antonin Delpeuch, and Jules Hedges. Sheet diagrams for bimonoidal categories. arXiv preprint arXiv:2010.13361, 2020. Google Scholar
  20. Andrea Corradini and Fabio Gadducci. An algebraic presentation of term graphs, via GS-monoidal categories. Applied Categorical Structures, 7(4):299-331, 1999. URL: https://doi.org/10.1023/A:1008647417502.
  21. Ernst-Erich Doberkat. Eilenberg-moore algebras for stochastic relations. Information and Computation, 204(12):1756-1781, 2006. URL: https://doi.org/10.1016/J.IC.2006.09.001.
  22. Ross Duncan. Generalised proof-nets for compact categories with biproducts. CoRR, abs/0903.5154, 2009. URL: https://arxiv.org/abs/0903.5154.
  23. Samuel Eilenberg and G Max Kelly. Closed categories. In Proceedings of the Conference on Categorical Algebra, pages 421-562. Springer, 1966. Google Scholar
  24. Brendan Fong, Paolo Rapisarda, and Paweł Sobociński. A categorical approach to open and interconnected dynamical systems. In LICS 2016, 2016. Google Scholar
  25. Brendan Fong and David Spivak. String diagrams for regular logic (extended abstract). In John Baez and Bob Coecke, editors, Applied Category Theory 2019, volume 323 of Electronic Proceedings in Theoretical Computer Science, pages 196-229. Open Publishing Association, September 2020. URL: https://doi.org/10.4204/eptcs.323.14.
  26. Thomas Fox. Coalgebras and cartesian categories. Communications in Algebra, 4(7):665-667, 1976. URL: https://doi.org/10.1080/00927877608822127.
  27. Tobias Fritz. A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics. Advances in Mathematics, 370:107239, 2020. URL: https://arxiv.org/abs/1908.07021.
  28. Dan R. Ghica and Achim Jung. Categorical semantics of digital circuits. In 2016 Formal Methods in Computer-Aided Design (FMCAD), pages 41-48, 2016. URL: https://doi.org/10.1109/FMCAD.2016.7886659.
  29. 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, pages 454-464, 2020. Google Scholar
  30. Esfandiar Haghverdi. A categorical approach to linear logic, geometry of proofs and full completeness. University of Ottawa (Canada), 2000. Google Scholar
  31. John Harding. Orthomodularity in dagger biproduct categories. Preprint, 2008. Google Scholar
  32. Ichiro Hasuo, Bart Jacobs, and Ana Sokolova. Generic trace semantics via coinduction. Logical Methods in Computer Science, 3, 2007. URL: https://doi.org/10.2168/LMCS-3(4:11)2007.
  33. 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 Electron. Notes Theor. Comput. Sci., pages 437-458. Elsevier Sci. B. V., Amsterdam, 2007. URL: https://doi.org/10.1016/j.entcs.2007.02.019.
  34. Bart Jacobs. Coalgebraic trace semantics for combined possibilitistic and probabilistic systems. Electronic Notes in Theoretical Computer Science, 203(5):131-152, 2008. URL: https://doi.org/10.1016/J.ENTCS.2008.05.023.
  35. Bart Jacobs. Convexity, duality and effects. In IFIP International Conference on Theoretical Computer Science, pages 1-19. Springer, 2010. URL: https://doi.org/10.1007/978-3-642-15240-5_1.
  36. Bart Jacobs. From coalgebraic to monoidal traces. Electronic Notes in Theoretical Computer Science, 264(2):125-140, 2010. URL: https://doi.org/10.1016/J.ENTCS.2010.07.017.
  37. Bart Jacobs. From probability monads to commutative effectuses. Journal of Logical and Algebraic Methods in Programming, 94:200-237, 2018. URL: https://doi.org/10.1016/j.jlamp.2016.11.006.
  38. Roshan P James and Amr Sabry. Information effects. ACM SIGPLAN Notices, 47(1):73-84, 2012. URL: https://doi.org/10.1145/2103656.2103667.
  39. Niles Johnson and Donald Yau. Bimonoidal categories, e_n-monoidal categories, and algebraic k-theory. arXiv preprint arXiv:2107.10526, 2021. Google Scholar
  40. André Joyal and Ross Street. The geometry of tensor calculus, I. Advances in Mathematics, 88(1):55-112, July 1991. URL: https://doi.org/10.1016/0001-8708(91)90003-P.
  41. Anders Kock. Monads on symmetric monoidal closed categories. Archiv der Mathematik, 21:1-10, 1970. Google Scholar
  42. Anders Kock. Strong functors and monoidal monads. Archiv der Mathematik, 23:113-120, 1972. Google Scholar
  43. Anna Labella. Categories with sums and right distributive tensor product. J. Pure Appl. Algebra, 178(3):273-296, 2003. URL: https://doi.org/10.1016/S0022-4049(02)00169-X.
  44. Stephen Lack. Non-canonical isomorphisms. J. Pure Appl. Algebra, 216(3):593-597, 2012. URL: https://doi.org/10.1016/j.jpaa.2011.07.012.
  45. Miguel L. Laplaza. Coherence for distributivity. In G. M. Kelly, M. Laplaza, G. Lewis, and Saunders Mac Lane, editors, Coherence in Categories, Lecture Notes in Mathematics, pages 29-65, Berlin, Heidelberg, 1972. Springer. URL: https://doi.org/10.1007/BFb0059555.
  46. Jack Liell-Cock and Sam Staton. Compositional imprecise probability: A solution from graded monads and markov categories. Proc. ACM Program. Lang., 9(POPL):1596-1626, 2025. URL: https://doi.org/10.1145/3704890.
  47. Saunders Mac Lane. Categories for the Working Mathematician, volume 5 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1978. URL: //www.springer.com/gb/book/9780387984032.
  48. Koko Muroya, Steven W. T. Cheung, and Dan R. Ghica. The geometry of computation-graph abstraction. In Anuj Dawar and Erich Grädel, editors, Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018, Oxford, UK, July 09-12, 2018, pages 749-758. ACM, 2018. URL: https://doi.org/10.1145/3209108.3209127.
  49. Robin Piedeleu, Mateo Torres-Ruiz, Alexandra Silva, and Fabio Zanasi. A complete axiomatisation of equivalence for discrete probabilistic programming. In ESOP 2025 (to appear), 2025. URL: https://arxiv.org/abs/2408.14701.
  50. Robin Piedeleu and Fabio Zanasi. A String Diagrammatic Axiomatisation of Finite-State Automata. In Stefan Kiefer and Christine Tasson, editors, Foundations of Software Science and Computation Structures, Lecture Notes in Computer Science, pages 469-489, Cham, 2021. Springer International Publishing. URL: https://doi.org/10.1007/978-3-030-71995-1_24.
  51. Ralph Sarkis and Fabio Zanasi. String diagrams for graded monoidal theories with an application to imprecise probability. CoRR, abs/2501.18404, 2025. URL: https://doi.org/10.48550/arXiv.2501.18404.
  52. Peter Selinger. A survey of graphical languages for monoidal categories. In New structures for physics, pages 289-355. Springer, 2010. Google Scholar
  53. Zbigniew Semadeni. Monads and their eilenberg-moore algebras in functional analysis. (No Title), 1973. Google Scholar
  54. Sam Staton. Algebraic effects, linearity, and quantum programming languages. ACM SIGPLAN Notices, 50(1):395-406, 2015. URL: https://doi.org/10.1145/2676726.2676999.
  55. Tobias Stollenwerk and Stuart Hadfield. Diagrammatic analysis for parameterized quantum circuits. Bulletin of the American Physical Society, 2022. Google Scholar
  56. Tadeusz Swirszcz. Monadic functors and convexity. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys, 22(3):35, 1974. Google Scholar
  57. Alexis Toumi, Richie Yeung, and Giovanni de Felice. Diagrammatic differentiation for quantum machine learning. In Chris Heunen and Miriam Backens, editors, Proceedings 18th International Conference on Quantum Physics and Logic, QPL 2021, Gdansk, Poland, and online, 7-11 June 2021, volume 343 of EPTCS, pages 132-144, 2021. URL: https://doi.org/10.4204/EPTCS.343.7.
  58. Alejandro Villoria, Henning Basold, and Alfons Laarman. Enriching diagrams with algebraic operations. In International Conference on Foundations of Software Science and Computation Structures, pages 121-143. Springer, 2024. URL: https://doi.org/10.1007/978-3-031-57228-9_7.
  59. Chen Zhao and Xiao-Shan Gao. Analyzing the barren plateau phenomenon in training quantum neural networks with the zx-calculus. Quantum, 5:466, 2021. URL: https://doi.org/10.22331/Q-2021-06-04-466.
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