skip to main content
research-article
Open Access

Deconstructing the Calculus of Relations with Tape Diagrams

Published:11 January 2023Publication History
Skip Abstract Section

Abstract

Rig categories with finite biproducts are categories with two monoidal products, where one is a biproduct and the other distributes over it. In this work we present tape diagrams, a sound and complete diagrammatic language for these categories, that can be intuitively thought as string diagrams of string diagrams. We test the effectiveness of our approach against the positive fragment of Tarski's calculus of relations.

References

  1. Matteo Acclavio. 2019. Proof Diagrams for Multiplicative Linear Logic: Syntax and Semantics. Journal of Automated Reasoning, 63, 4 (2019), Dec., 911–939. issn:1573-0670 https://doi.org/10.1007/s10817-018-9466-4 Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. H. Andréka and D. A. Bredikhin. 1995. The Equational Theory of Union-Free Algebras of Relations. Algebra Universalis, 33, 4 (1995), Dec., 516–532. issn:1420-8911 https://doi.org/10.1007/BF01225472 Google ScholarGoogle ScholarCross RefCross Ref
  3. M Backens. 2015. Completeness and the ZX-calculus. Ph. D. Dissertation. University of Oxford. https://ora.ox.ac.uk/objects/uuid:0120239e-b504-4376-973d-d720a095f02e Google ScholarGoogle Scholar
  4. John Baez and Jason Erbele. 2015. Categories In Control. Theory and Applications of Categories, 30 (2015), 836–881. http://www.tac.mta.ca/tac/volumes/30/24/30-24abs.html Google ScholarGoogle Scholar
  5. Edwin S. Bainbridge. 1976. Feedback and Generalized Logic. Information and Control, 31, 1 (1976), May, 75–96. issn:0019-9958 https://doi.org/10.1016/S0019-9958(76)90390-9 Google ScholarGoogle ScholarCross RefCross Ref
  6. Paolo Baldan and Fabio Gadducci. 2019. Petri nets are dioids: a new algebraic foundation for non-deterministic net theory. Acta Informatica, 56, 1 (2019), 61–92. https://doi.org/10.1007/s00236-018-0314-0 Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. Bruce Bartlett, Christopher L. Douglas, Christopher J. Schommer-Pries, and Jamie Vicary. 2015. Modular Categories as Representations of the 3-Dimensional Bordism 2-Category. https://doi.org/10.48550/arXiv.1509.06811 arxiv:1509.06811. Google ScholarGoogle Scholar
  8. Corrado Böhm and Giuseppe Jacopini. 1979. Flow Diagrams, Turing Machines and Languages with Only Two Formation Rules. In Classics in Software Engineering. Yourdon Press, USA. 11–25. isbn:978-0-917072-14-7 https://dl.acm.org/doi/abs/10.5555/1241515.1241517 Google ScholarGoogle Scholar
  9. Guillaume Boisseau and Robin Piedeleu. 2022. Graphical Piecewise-Linear Algebra. In Foundations of Software Science and Computation Structures, Patricia Bouyer and Lutz Schröder (Eds.) (Lecture Notes in Computer Science). Springer International Publishing, Cham. 101–119. isbn:978-3-030-99253-8 https://doi.org/10.1007/978-3-030-99253-8_6 Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. Benedikt Bollig, Alain Finkel, and Amrita Suresh. 2020. Bounded Reachability Problems Are Decidable in FIFO Machines. In 31st International Conference on Concurrency Theory (CONCUR 2020), Igor Konnov and Laura Kovács (Eds.) (Leibniz International Proceedings in Informatics (LIPIcs), Vol. 171). Schloss Dagstuhl– Leibniz-Zentrum für Informatik, Dagstuhl, Germany. 49:1–49:17. isbn:978-3-95977-160-3 issn:1868-8969 https://doi.org/10.4230/LIPIcs.CONCUR.2020.49 Google ScholarGoogle ScholarCross RefCross Ref
  11. Filippo Bonchi, Alessandro Di Giorgio, and Alessio Santamaria. 2022. Deconstructing the Calculus of Relations with Tape Diagrams. https://doi.org/10.48550/arXiv.2210.09950 arxiv:2210.09950. Google ScholarGoogle Scholar
  12. Filippo Bonchi, Fabio Gadducci, Aleks Kissinger, Pawel Sobocinski, and Fabio Zanasi. 2022. String Diagram Rewrite Theory I: Rewriting with Frobenius Structure. J. ACM, 69, 2 (2022), 14:1–14:58. https://doi.org/10.1145/3502719 Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. Filippo Bonchi, Joshua Holland, Robin Piedeleu, Paweł Sobociński, and Fabio Zanasi. 2019. Diagrammatic Algebra: From Linear to Concurrent Systems. Proceedings of the ACM on Programming Languages, 3, POPL (2019), Jan., 25:1–25:28. https://doi.org/10.1145/3290338 Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. Filippo Bonchi, Robin Piedeleu, Pawel Sobociński, and Fabio Zanasi. 2019. Graphical Affine Algebra. In 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). 1–12. https://doi.org/10.1109/LICS.2019.8785877 Google ScholarGoogle ScholarCross RefCross Ref
  15. Filippo Bonchi, Jens Seeber, and Pawel Sobocinski. 2018. Graphical Conjunctive Queries. In 27th EACSL Annual Conference on Computer Science Logic (CSL 2018), Dan Ghica and Achim Jung (Eds.) (Leibniz International Proceedings in Informatics (LIPIcs), Vol. 119). Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany. 13:1–13:23. isbn:978-3-95977-088-0 issn:1868-8969 https://doi.org/10.4230/LIPIcs.CSL.2018.13 Google ScholarGoogle ScholarCross RefCross Ref
  16. Filippo Bonchi, Pawel Sobocinski, and Fabio Zanasi. 2015. Full Abstraction for Signal Flow Graphs. In Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL ’15). Association for Computing Machinery, New York, NY, USA. 515–526. isbn:978-1-4503-3300-9 https://doi.org/10.1145/2676726.2676993 Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. Paul Brunet and Damien Pous. 2015. Petri Automata for Kleene Allegories. In 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science. 68–79. issn:1043-6871 https://doi.org/10.1109/LICS.2015.17 Google ScholarGoogle ScholarDigital LibraryDigital Library
  18. Roberto Bruni, Ivan Lanese, and Ugo Montanari. 2006. A Basic Algebra of Stateless Connectors. Theoretical Computer Science, 366, 1 (2006), Nov., 98–120. issn:0304-3975 https://doi.org/10.1016/j.tcs.2006.07.005 Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. A. Carboni and R. F. C. Walters. 1987. Cartesian Bicategories I. Journal of Pure and Applied Algebra, 49, 1 (1987), Nov., 11–32. issn:0022-4049 https://doi.org/10.1016/0022-4049(87)90121-6 Google ScholarGoogle ScholarCross RefCross Ref
  20. Ashok K. Chandra and Philip M. Merlin. 1977. Optimal Implementation of Conjunctive Queries in Relational Data Bases. In Proceedings of the Ninth Annual ACM Symposium on Theory of Computing (STOC ’77). Association for Computing Machinery, New York, NY, USA. 77–90. isbn:978-1-4503-7409-5 https://doi.org/10.1145/800105.803397 Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. Bob Coecke and Ross Duncan. 2008. Interacting Quantum Observables. In Automata, Languages and Programming, Luca Aceto, Ivan Damgård, Leslie Ann Goldberg, Magnús M. Halldórsson, Anna Ingólfsdóttir, and Igor Walukiewicz (Eds.) (Lecture Notes in Computer Science). Springer, Berlin, Heidelberg. 298–310. isbn:978-3-540-70583-3 https://doi.org/10.1007/978-3-540-70583-3_25 Google ScholarGoogle ScholarDigital LibraryDigital Library
  22. Bob Coecke and Ross Duncan. 2011. Interacting Quantum Observables: Categorical Algebra and Diagrammatics. New Journal of Physics, 13, 4 (2011), April, 043016. issn:1367-2630 https://doi.org/10.1088/1367-2630/13/4/043016 Google ScholarGoogle ScholarCross RefCross Ref
  23. Bob Coecke and Aleks Kissinger. 2017. Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning. Cambridge University Press. https://doi.org/10.1017/9781316219317 Google ScholarGoogle ScholarCross RefCross Ref
  24. Bob Coecke, John Selby, and Sean Tull. 2018. Two Roads to Classicality. 266 (2018), Feb., 104–118. https://doi.org/10.4204/EPTCS.266.7 Google ScholarGoogle ScholarCross RefCross Ref
  25. Cole Comfort, Antonin Delpeuch, and Jules Hedges. 2020. Sheet Diagrams for Bimonoidal Categories. https://doi.org/10.48550/arXiv.2010.13361 arxiv:2010.13361. Google ScholarGoogle Scholar
  26. Ross Duncan. 2009. Generalised Proof-Nets for Compact Categories with Biproducts. In Semantics of Quantum Computation, Simon Gay and Ian Mackie (Eds.). Cambridge University Press. Google ScholarGoogle Scholar
  27. Leonardo Bigolli Pisani vulgo Fibonacci. 2020. Liber Abbaci / edidit Enrico Giusti coadiuvante Paolo D’Alessandro (Olschki ed.) (Biblioteca di « Nuncius», Vol. 79). Firenze, Italy. isbn:978-88-222-6658-3 Google ScholarGoogle Scholar
  28. Brendan Fong, Paweł Sobociński, and Paolo Rapisarda. 2016. A Categorical Approach to Open and Interconnected Dynamical Systems. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS ’16). Association for Computing Machinery, New York, NY, USA. 495–504. isbn:978-1-4503-4391-6 https://doi.org/10.1145/2933575.2934556 Google ScholarGoogle ScholarDigital LibraryDigital Library
  29. Brendan Fong and David Spivak. 2020. String Diagrams for Regular Logic (Extended Abstract). In Applied Category Theory 2019, John Baez and Bob Coecke (Eds.) (Electronic Proceedings in Theoretical Computer Science, Vol. 323). Open Publishing Association, 196–229. issn:2075-2180 https://doi.org/10.4204/eptcs.323.14 Google ScholarGoogle ScholarCross RefCross Ref
  30. T. Fox. 1976. Coalgebras and Cartesian Categories. Communications in Algebra, 4, 7 (1976), 665–667. issn:0092-7872 https://doi.org/10.1080/00927877608822127 Google ScholarGoogle ScholarCross RefCross Ref
  31. Peter Freyd and Andre Scedrov. 1990. Categories, Allegories (North-Holland Mathematical Library, Vol. 39). Elsevier B.V. isbn:978-0-444-70368-2 Google ScholarGoogle Scholar
  32. Tobias Fritz. 2009. A Presentation of the Category of Stochastic Matrices. https://doi.org/10.48550/arXiv.0902.2554 arxiv:0902.2554. Google ScholarGoogle Scholar
  33. Dan R. Ghica and Achim Jung. 2016. Categorical semantics of digital circuits. In 2016 Formal Methods in Computer-Aided Design (FMCAD). 41–48. https://doi.org/10.1109/FMCAD.2016.7886659 Google ScholarGoogle ScholarCross RefCross Ref
  34. John Harding. 2008. Orthomodularity in dagger biproduct categories. Preprint, https://www.researchgate.net/publication/228354796_Orthomodularity_in_dagger_biproduct_categories Google ScholarGoogle Scholar
  35. Masahito Hasegawa, Martin Hofmann, and Gordon Plotkin. 2008. Finite Dimensional Vector Spaces Are Complete for Traced Symmetric Monoidal Categories. In Pillars of Computer Science: Essays Dedicated to Boris (Boaz) Trakhtenbrot on the Occasion of His 85th Birthday, Arnon Avron, Nachum Dershowitz, and Alexander Rabinovich (Eds.). Springer, Berlin, Heidelberg. 367–385. isbn:978-3-540-78127-1 https://doi.org/10.1007/978-3-540-78127-1_20 Google ScholarGoogle ScholarCross RefCross Ref
  36. Charles A. R. Hoare. 1969. An Axiomatic Basis for Computer Programming. Commun. ACM, 12, 10 (1969), Oct., 576–580. issn:0001-0782 https://doi.org/10.1145/363235.363259 Google ScholarGoogle ScholarDigital LibraryDigital Library
  37. Tony Hoare, Bernhard Möller, Georg Struth, and Ian Wehrman. 2011. Concurrent Kleene Algebra and Its Foundations. The Journal of Logic and Algebraic Programming, 80, 6 (2011), Aug., 266–296. issn:1567-8326 https://doi.org/10.1016/j.jlap.2011.04.005 Google ScholarGoogle ScholarCross RefCross Ref
  38. Ian Hodkinson and Szabolcs Mikulás. 2000. Axiomatizability of Reducts of Algebras of Relations. Algebra Universalis, 43, 2 (2000), Aug., 127–156. issn:1420-8911 https://doi.org/10.1007/s000120050150 Google ScholarGoogle ScholarCross RefCross Ref
  39. Roshan P. James and Amr Sabry. 2012. Information Effects. ACM SIGPLAN Notices, 47, 1 (2012), Jan., 73–84. issn:0362-1340 https://doi.org/10.1145/2103621.2103667 Google ScholarGoogle ScholarDigital LibraryDigital Library
  40. Niles Johnson and Donald Yau. 2022. Bimonoidal Categories, E_n -Monoidal Categories, and Algebraic K -Theory. https://nilesjohnson.net/En-monoidal.html Google ScholarGoogle Scholar
  41. André Joyal and Ross Street. 1991. The Geometry of Tensor Calculus, I. Advances in Mathematics, 88, 1 (1991), July, 55–112. issn:0001-8708 https://doi.org/10.1016/0001-8708(91)90003-P Google ScholarGoogle ScholarCross RefCross Ref
  42. David Kaiser. 2009. Drawing Theories Apart: The Dispersion of Feynman Diagrams in Postwar Physics. University of Chicago Press. isbn:978-0-226-42265-7 https://doi.org/10.7208/9780226422657 Google ScholarGoogle Scholar
  43. Tobias Kappé, Paul Brunet, Alexandra Silva, and Fabio Zanasi. 2018. Concurrent Kleene Algebra: Free Model and Completeness. In Programming Languages and Systems, Amal Ahmed (Ed.) (Lecture Notes in Computer Science). Springer International Publishing, Cham. 856–882. isbn:978-3-319-89884-1 https://doi.org/10.1007/978-3-319-89884-1_30 Google ScholarGoogle ScholarCross RefCross Ref
  44. Stephen Lack. 2004. Composing PROPs. Theory and Application of Categories, 13, 9 (2004), 147–163. http://www.tac.mta.ca/tac/volumes/13/9/13-09abs.html Google ScholarGoogle Scholar
  45. Yves Lafont. 2003. Towards an Algebraic Theory of Boolean Circuits. Journal of Pure and Applied Algebra, 184, 2 (2003), Nov., 257–310. issn:0022-4049 https://doi.org/10.1016/S0022-4049(03)00069-0 Google ScholarGoogle ScholarCross RefCross Ref
  46. Miguel L. Laplaza. 1972. Coherence for Distributivity. In Coherence in Categories, G. M. Kelly, M. Laplaza, G. Lewis, and Saunders Mac Lane (Eds.) (Lecture Notes in Mathematics). Springer, Berlin, Heidelberg. 29–65. isbn:978-3-540-37958-4 https://doi.org/10.1007/BFb0059555 Google ScholarGoogle ScholarCross RefCross Ref
  47. Saunders Mac Lane. 1965. Categorical Algebra. Bull. Amer. Math. Soc., 71, 1 (1965), 40–106. issn:0002-9904, 1936-881X https://doi.org/10.1090/S0002-9904-1965-11234-4 Google ScholarGoogle ScholarCross RefCross Ref
  48. S. Mac Lane. 1978. Categories for the Working Mathematician (second ed.) (Graduate Texts in Mathematics, Vol. 5). Springer-Verlag, New York. isbn:978-0-387-98403-2 https://www.springer.com/gb/book/9780387984032 Google ScholarGoogle Scholar
  49. Paul-André Melliès. 2006. Functorial Boxes in String Diagrams. In Computer Science Logic, Zoltán Ésik (Ed.) (Lecture Notes in Computer Science). Springer, Berlin, Heidelberg. 1–30. isbn:978-3-540-45459-5 https://doi.org/10.1007/11874683_1 Google ScholarGoogle ScholarDigital LibraryDigital Library
  50. Donald Monk. 1964. On representable relation algebras. Michigan Mathematical Journal, 11, 3 (1964), 207 – 210. https://doi.org/10.1307/mmj/1028999131 Google ScholarGoogle ScholarCross RefCross Ref
  51. Koko Muroya, Steven W. T. Cheung, and Dan R. Ghica. 2018. The Geometry of Computation-Graph Abstraction. In Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018, Oxford, UK, July 09-12, 2018, Anuj Dawar and Erich Grädel (Eds.). ACM, 749–758. https://doi.org/10.1145/3209108.3209127 Google ScholarGoogle ScholarDigital LibraryDigital Library
  52. Roger Penrose. 1971. Applications of Negative Dimensional Tensors. In Combinatorial Mathematics and Its Applications, D. J. A. Welsh (Ed.). Academic Press. isbn:0-12-743350-3 Google ScholarGoogle Scholar
  53. Robin Piedeleu and Fabio Zanasi. 2021. A String Diagrammatic Axiomatisation of Finite-State Automata. In Foundations of Software Science and Computation Structures, Stefan Kiefer and Christine Tasson (Eds.) (Lecture Notes in Computer Science). Springer International Publishing, Cham. 469–489. isbn:978-3-030-71995-1 https://doi.org/10.1007/978-3-030-71995-1_24 Google ScholarGoogle ScholarDigital LibraryDigital Library
  54. Damien Pous. 2018. On the Positive Calculus of Relations with Transitive Closure. In 35th Symposium on Theoretical Aspects of Computer Science, STACS 2018, February 28 to March 3, 2018, Caen, France, Rolf Niedermeier and Brigitte Vallée (Eds.) (LIPIcs, Vol. 96). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 3:1–3:16. https://doi.org/10.4230/LIPIcs.STACS.2018.3 Google ScholarGoogle ScholarCross RefCross Ref
  55. Valentin N Redko. 1964. On defining relations for the algebra of regular events. Ukrainskii Matematicheskii Zhurnal, 16 (1964), 120–126. Google ScholarGoogle Scholar
  56. Yehoshua Sagiv and Mihalis Yannakakis. 1980. Equivalences Among Relational Expressions with the Union and Difference Operators. J. ACM, 27, 4 (1980), Oct., 633–655. issn:0004-5411 https://doi.org/10.1145/322217.322221 Google ScholarGoogle ScholarDigital LibraryDigital Library
  57. Peter Selinger. 1998. A Note on Bainbridge’s Power Set Construction. BRICS, Basic Research in Computer Science, Aarhus, Denmark. Google ScholarGoogle Scholar
  58. Peter Selinger. 2010. A survey of graphical languages for monoidal categories. In New structures for physics. Springer, 289–355. https://doi.org/10.1007/978-3-642-12821-9_4 Google ScholarGoogle ScholarCross RefCross Ref
  59. Peter Selinger. 2012. Finite Dimensional Hilbert Spaces Are Complete for Dagger Compact Closed Categories. Logical Methods in Computer Science, Volume 8, Issue 3 (2012), Aug., https://doi.org/10.2168/LMCS-8(3:6)2012 Google ScholarGoogle ScholarCross RefCross Ref
  60. Sam Staton. 2015. Algebraic Effects, Linearity, and Quantum Programming Languages. ACM SIGPLAN Notices, 50, 1 (2015), Jan., 395–406. issn:0362-1340 https://doi.org/10.1145/2775051.2676999 Google ScholarGoogle ScholarDigital LibraryDigital Library
  61. Tobias Stollenwerk and Stuart Hadfield. 2022. Diagrammatic Analysis for Parameterized Quantum Circuits. arxiv:2204.01307. arxiv:2204.01307 Google ScholarGoogle Scholar
  62. Alfred Tarski. 1941. On the Calculus of Relations. The Journal of Symbolic Logic, 6, 3 (1941), Sept., 73–89. issn:0022-4812, 1943-5886 https://doi.org/10.2307/2268577 Google ScholarGoogle ScholarCross RefCross Ref
  63. Alexis Toumi, Richie Yeung, and Giovanni de Felice. 2021. Diagrammatic Differentiation for Quantum Machine Learning. In Proceedings 18th International Conference on Quantum Physics and Logic, QPL 2021, Gdansk, Poland, and online, 7-11 June 2021, Chris Heunen and Miriam Backens (Eds.) (EPTCS, Vol. 343). 132–144. https://doi.org/10.4204/EPTCS.343.7 Google ScholarGoogle ScholarCross RefCross Ref
  64. Fabio Zanasi. 2015. Interacting Hopf Algebras- the Theory of Linear Systems. Ph. D. Dissertation. Ecole normale supérieure de Lyon - ENS LYON. https://tel.archives-ouvertes.fr/tel-01218015 Google ScholarGoogle Scholar
  65. Chen Zhao and Xiao-Shan Gao. 2021. Analyzing the Barren Plateau Phenomenon in Training Quantum Neural Networks with the ZX-calculus. Quantum, 5 (2021), June, 466. https://doi.org/10.22331/q-2021-06-04-466 Google ScholarGoogle ScholarCross RefCross Ref

Index Terms

  1. Deconstructing the Calculus of Relations with Tape Diagrams

      Recommendations

      Comments

      Login options

      Check if you have access through your login credentials or your institution to get full access on this article.

      Sign in

      Full Access

      • Published in

        cover image Proceedings of the ACM on Programming Languages
        Proceedings of the ACM on Programming Languages  Volume 7, Issue POPL
        January 2023
        2196 pages
        EISSN:2475-1421
        DOI:10.1145/3554308
        • Editor:
        Issue’s Table of Contents

        Copyright © 2023 Owner/Author

        Publisher

        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 11 January 2023
        Published in pacmpl Volume 7, Issue POPL

        Permissions

        Request permissions about this article.

        Request Permissions

        Check for updates

        Qualifiers

        • research-article
      • Article Metrics

        • Downloads (Last 12 months)310
        • Downloads (Last 6 weeks)67

        Other Metrics

      PDF Format

      View or Download as a PDF file.

      PDF

      eReader

      View online with eReader.

      eReader
      About Cookies On This Site

      We use cookies to ensure that we give you the best experience on our website.

      Learn more

      Got it!