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.
- 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 Scholar
Digital Library
- 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 Scholar
Cross Ref
- 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 Scholar
- 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 Scholar
- 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 Scholar
Cross Ref
- 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 Scholar
Digital Library
- 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 Scholar
- 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 Scholar
- 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 Scholar
Digital Library
- 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 Scholar
Cross Ref
- 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 Scholar
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- 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 Scholar
Cross Ref
- 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 Scholar
Cross Ref
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- 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 Scholar
Cross Ref
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- 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 Scholar
Cross Ref
- 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 Scholar
Cross Ref
- 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 Scholar
Cross Ref
- 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 Scholar
- 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 Scholar
- 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 Scholar
- 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 Scholar
Digital Library
- 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 Scholar
Cross Ref
- 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 Scholar
Cross Ref
- Peter Freyd and Andre Scedrov. 1990. Categories, Allegories (North-Holland Mathematical Library, Vol. 39). Elsevier B.V. isbn:978-0-444-70368-2
Google Scholar
- Tobias Fritz. 2009. A Presentation of the Category of Stochastic Matrices. https://doi.org/10.48550/arXiv.0902.2554 arxiv:0902.2554.
Google Scholar
- 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 Scholar
Cross Ref
- John Harding. 2008. Orthomodularity in dagger biproduct categories. Preprint, https://www.researchgate.net/publication/228354796_Orthomodularity_in_dagger_biproduct_categories
Google Scholar
- 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 Scholar
Cross Ref
- 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 Scholar
Digital Library
- 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 Scholar
Cross Ref
- 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 Scholar
Cross Ref
- 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 Scholar
Digital Library
- Niles Johnson and Donald Yau. 2022. Bimonoidal Categories, E_n -Monoidal Categories, and Algebraic K -Theory. https://nilesjohnson.net/En-monoidal.html
Google Scholar
- 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 Scholar
Cross Ref
- 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 Scholar
- 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 Scholar
Cross Ref
- 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 Scholar
- 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 Scholar
Cross Ref
- 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 Scholar
Cross Ref
- 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 Scholar
Cross Ref
- 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 Scholar
- 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 Scholar
Digital Library
- Donald Monk. 1964. On representable relation algebras. Michigan Mathematical Journal, 11, 3 (1964), 207 – 210. https://doi.org/10.1307/mmj/1028999131
Google Scholar
Cross Ref
- 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 Scholar
Digital Library
- 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 Scholar
- 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 Scholar
Digital Library
- 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 Scholar
Cross Ref
- Valentin N Redko. 1964. On defining relations for the algebra of regular events. Ukrainskii Matematicheskii Zhurnal, 16 (1964), 120–126.
Google Scholar
- 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 Scholar
Digital Library
- Peter Selinger. 1998. A Note on Bainbridge’s Power Set Construction. BRICS, Basic Research in Computer Science, Aarhus, Denmark.
Google Scholar
- 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 Scholar
Cross Ref
- 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 Scholar
Cross Ref
- 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 Scholar
Digital Library
- Tobias Stollenwerk and Stuart Hadfield. 2022. Diagrammatic Analysis for Parameterized Quantum Circuits. arxiv:2204.01307. arxiv:2204.01307
Google Scholar
- 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 Scholar
Cross Ref
- 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 Scholar
Cross Ref
- 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 Scholar
- 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 Scholar
Cross Ref
Index Terms
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