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Monoidal Multiplexing

  • Apiwat Chantawibul
  • Paweł Sobociński
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11187)

Abstract

Given a classical algebraic structure—e.g. a monoid or group—with carrier set X, and given a positive integer n, there is a canonical way of obtaining the same structure on carrier set \(X^n\) by defining the required operations “pointwise”. For resource-sensitive algebra (i.e. based on mere symmetric monoidal, not cartesian structure), similar “pointwise” operations are usually defined as a kind of syntactic sugar: for example, given a comonoid structure on X, one obtains a comultiplication on \(X\otimes X\) by tensoring two comultiplications and composing with an appropriate permutation. This is a specific example of a general construction that we identify and refer to as multiplexing. We obtain a general theorem that guarantees that any equation that holds in the base case will hold also for the multiplexed operations, thus generalising the “pointwise” definitions of classical universal algebra.

Keywords

String diagrams Resource sensitivity Symmetric monoidal categories Props 

References

  1. 1.
    Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. In: Proceedings of 19th Annual IEEE Symposium on Logic in Computer Science, LICS 2004, July 2004, Turku, pp. 415–425. IEEE CS Press, Washington, DC (2004).  https://doi.org/10.1109/lics.2004.1319636
  2. 2.
    Baez, J.C., Erbele, J.: Categories in control. arXiv preprint 1405.6881 (2014). https://arxiv.org/abs/1405.6881
  3. 3.
    Baez, J.C., Fong, B.: A compositional framework for passive linear networks. arXiv preprint 1504.05625 (2015). https://arxiv.org/abs/1504.05625
  4. 4.
    Bonchi, F., Gadducci, F., Kissinger, A., Sobociński, P., Zanasi, F.: Rewriting modulo symmetric monoidal structure. In: Proceedings of 31st Annual ACM/IEEE Symposium on Logic and Computer Science, LICS 2016, pp. 710–719. ACM Press, New York (2016).  https://doi.org/10.1145/2933575.2935316
  5. 5.
    Bonchi, F., Gadducci, F., Kissinger, A., Sobociński, P., Zanasi, F.: Confluence of graph rewriting with interfaces. In: Yang, H. (ed.) ESOP 2017. LNCS, vol. 10201, pp. 141–169. Springer, Heidelberg (2017).  https://doi.org/10.1007/978-3-662-54434-1_6CrossRefGoogle Scholar
  6. 6.
    Bonchi, F., Gadducci, F., Kissinger, A., Sobociński, P., Zanasi, F.: Rewriting with Frobenius. In: Proceedings of 33rd Annual ACM/IEEE Symposium on Logic and Computer Science, LICS 2018, July 2018, Oxford, pp. 165–174. ACM Press, New York (2018).  https://doi.org/10.1145/3209108.3209137
  7. 7.
    Bonchi, F., Holland, J., Pavlovic, D., Sobociński, P.: Refinement for signal flow graphs. In: Meyer, R., Nestmann, U. (eds.) Proceedings of 28th International Conference on Concurrency Theory, CONCUR 2017, September 2017, Berlin, Leibniz International Proceedings in Informatics, vol. 85, p. 24. Dagstuhl Publishing, Saarbrücken, Wadern (2017).  https://doi.org/10.4230/lipics.concur.2017.24
  8. 8.
    Bonchi, F., Pavlovic, D., Sobocinski, P.: Functorial semantics for relational theories. arXiv preprint 1711.08699 (2017). https://arxiv.org/abs/1711.08699
  9. 9.
    Bonchi, F., Sobociński, P., Zanasi, F.: Full abstraction for signal flow graphs. In: 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2015, January 2015, Mumbai, pp. 515–526. ACM Press, New York (2015).  https://doi.org/10.1145/2676726.2676993
  10. 10.
    Bonchi, F., Sobociński, P., Zanasi, F.: The calculus of signal flow diagrams I: linear relations on streams. Inf. Comput. 252, 2–29 (2017).  https://doi.org/10.1016/j.ic.2016.03.002MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bonchi, F., Sobociński, P., Zanasi, F.: Deconstructing Lawvere with distributive laws. J. Log. Algebr. Methods Program. 95, 128–146 (2018).  https://doi.org/10.1016/j.jlamp.2017.12.002MathSciNetCrossRefGoogle Scholar
  12. 12.
    Coecke, B., Duncan, R.: Interacting quantum observables. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008. LNCS, vol. 5126, pp. 298–310. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-70583-3_25CrossRefGoogle Scholar
  13. 13.
    Coecke, B., Kissinger, A.: Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning. Cambridge University Press, Cambridge (2017).  https://doi.org/10.1017/9781316219317CrossRefzbMATHGoogle Scholar
  14. 14.
    Fong, B., Rapisarda, P., Sobociński, P.: A categorical approach to open and interconnected dynamical systems. In: Proceedings of 31st Annual ACM/IEEE Symposium on Logic and Computer Science, LICS 2016, July 2016, New York, NY, pp. 495–504. ACM Press, New York (2016).  https://doi.org/10.1145/2933575.2934556
  15. 15.
    Fox, T.: Coalgebras and cartesian categories. Commun. Algebr. 4(7), 665–667 (1976).  https://doi.org/10.1080/00927877608822127MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ghani, N., Hedges, J., Winschel, V., Zahn, P.: Compositional game theory. In: Proceedings of 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018, July 2018, Oxford, pp. 472–481. ACM Press, New York (2018).  https://doi.org/10.1145/3209108.3209165
  17. 17.
    Ghica, D.R.: Diagrammatic reasoning for delay-insensitive asynchronous circuits. In: Coecke, B., Ong, L., Panangaden, P. (eds.) Computation, Logic, Games, and Quantum Foundations. The Many Facets of Samson Abramsky. LNCS, vol. 7860, pp. 52–68. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-38164-5_5CrossRefGoogle Scholar
  18. 18.
    Ghica, D.R., Jung, A.: Categorical semantics of digital circuits. In: Proceedings of 2016 Conference on Formal Methods in Computer-Aided Design, FMCAD 2016, October 2016, Mountain View, CA, pp. 41–48. IEEE CS Press, Washington, DC (2016).  https://doi.org/10.1109/fmcad.2016.7886659
  19. 19.
    Hinze, R.: Kan extensions for program optimisation or: art and dan explain an old trick. In: Gibbons, J., Nogueira, P. (eds.) MPC 2012. LNCS, vol. 7342, pp. 324–362. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-31113-0_16CrossRefzbMATHGoogle Scholar
  20. 20.
    Lack, S.: Composing PROPs. Theory Appl. Categ. 13, 147–163 (2004). http://www.tac.mta.ca/tac/volumes/13/9/13-09abs.html
  21. 21.
    Lawvere, F.W.: Functorial semantics of algebraic theories. Proc. Natl. Acad. Sci. USA 50(5), 869–872 (1963).  https://doi.org/10.1073/pnas.50.5.869MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Mac Lane, S.: Categorical algebra. Bull. Am. Math. Soc. 71, 40–106 (1965).  https://doi.org/10.1090/s0002-9904-1965-11234-4MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Piróg, M., Wu, N.: String diagrams for free monads (functional pearl). In: Proceedings of 21st ACM SIGPLAN International Conference on Functional Programming, ICFP 2016, September 2016, Nara, pp. 490–501. ACM Press, New York (2016).  https://doi.org/10.1145/2951913.2951947
  24. 24.
    Sobociński, P.: Representations of Petri net interactions. In: Gastin, P., Laroussinie, F. (eds.) CONCUR 2010. LNCS, vol. 6269, pp. 554–568. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-15375-4_38CrossRefGoogle Scholar
  25. 25.
    Sobociński, P.: Nets, relations and linking diagrams. In: Heckel, R., Milius, S. (eds.) CALCO 2013. LNCS, vol. 8089, pp. 282–298. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-40206-7_21CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.School of Electronics and Computer ScienceUniversity of SouthamptonSouthamptonUK

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