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Inversion, Iteration, and the Art of Dual Wielding

Part of the Lecture Notes in Computer Science book series (LNPSE,volume 11497)

Abstract

The humble \(\dagger \) (“dagger”) is used to denote two different operations in category theory: Taking the adjoint of a morphism (in dagger categories) and finding the least fixed point of a functional (in categories enriched in domains). While these two operations are usually considered separately from one another, the emergence of reversible notions of computation shows the need to consider how the two ought to interact.

In the present paper, we wield both of these daggers at once and consider dagger categories enriched in domains. We develop a notion of a monotone dagger structure as a dagger structure that is well behaved with respect to the enrichment, and show that such a structure leads to pleasant inversion properties of the fixed points that arise as a result. Notably, such a structure guarantees the existence of fixed point adjoints, which we show are intimately related to the conjugates arising from a canonical involutive monoidal structure in the enrichment. Finally, we relate the results to applications in the design and semantics of reversible programming languages.

Keywords

  • Reversible computing
  • Dagger categories
  • Iteration categories
  • Domain theory
  • Enriched categories

The author would like to thank Martti Karvonen, Mathys Rennela, Robert Glück, and the anonymous reviewers for their useful comments, corrections, and suggestions; and to acknowledge the support given by COST Action IC1405 Reversible computation: Extending horizons of computing.

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Notes

  1. 1.

    This is not the case in general, as continuous functions are only required to preserve least upper bounds of directed sets, which, by definition, does not include the empty set.

References

  1. Abramsky, S.: Retracing some paths in process algebra. In: Montanari, U., Sassone, V. (eds.) CONCUR 1996. LNCS, vol. 1119, pp. 1–17. Springer, Heidelberg (1996). https://doi.org/10.1007/3-540-61604-7_44

    CrossRef  Google Scholar 

  2. Abramsky, S., Coecke, B.: A categorical semantics of quantum protocol. In: 2004 Proceedings on Logic in Computer Science, pp. 415–425. IEEE (2004)

    Google Scholar 

  3. Abramsky, S., Jung, A.: Domain theory. In: Handbook of Logic in Computer Science, No. 3, pp. 1–168. Clarendon Press (1994)

    Google Scholar 

  4. Adámek, J.: Recursive data types in algebraically \(\omega \)-complete categories. Inf. Comput. 118, 181–190 (1995)

    Google Scholar 

  5. Bainbridge, E.S., Freyd, P.J., Scedrov, A., Scott, P.J.: Functorial polymorphism. Theor. Comput. Sci. 70(1), 35–64 (1990)

    MathSciNet  CrossRef  Google Scholar 

  6. Barr, M.: Algebraically compact functors. J. Pure Appl. Algebra 82(3), 211–231 (1992)

    MathSciNet  CrossRef  Google Scholar 

  7. Beggs, E.J., Majid, S.: Bar categories and star operations. Algebras Represent. Theory 12(2), 103–152 (2009)

    MathSciNet  CrossRef  Google Scholar 

  8. Benton, N., Hyland, M.: Traced premonoidal categories. Theor. Inform. Appl. 37(4), 273–299 (2003)

    MathSciNet  CrossRef  Google Scholar 

  9. Cho, K.: Semantics for a quantum programming language by operator algebras. Master’s thesis, University of Tokyo (2014)

    Google Scholar 

  10. Cho, K., Jacobs, B., Westerbaan, B., Westerbaan, A.: An introduction to effectus theory. arXiv:1512.05813 [cs.LO] (2015)

  11. Cockett, J.R.B., Lack, S.: Restriction categories I: categories of partial maps. Theor. Comput. Sci. 270(1–2), 223–259 (2002)

    MathSciNet  CrossRef  Google Scholar 

  12. Coecke, B., Heunen, C., Kissinger, A.: Categories of quantum and classical channels. Quant. Inf. Process. 15(12), 5179–5209 (2016)

    MathSciNet  CrossRef  Google Scholar 

  13. Egger, J.: Involutive monoidal categories and enriched dagger categories. Seminar talk, University of Oxford (2008)

    Google Scholar 

  14. Ésik, Z.: Fixed point theory. In: Droste, M., Kuich, W., Vogler, H. (eds.) Handbook of Weighted Automata, pp. 29–65. Springer, Berlin (2009). https://doi.org/10.1007/978-3-642-01492-5_2

    CrossRef  Google Scholar 

  15. Ésik, Z.: Equational properties of fixed point operations in cartesian categories: an overview. In: Italiano, G.F., Pighizzini, G., Sannella, D.T. (eds.) MFCS 2015. LNCS, vol. 9234, pp. 18–37. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48057-1_2

    CrossRef  Google Scholar 

  16. Guo, X.: Products, joins, meets, and ranges in restriction categories. Ph.D. thesis, University of Calgary (2012)

    Google Scholar 

  17. Haghverdi, E.: Unique decomposition categories, geometry of interaction and combinatory logic. Math. Struct. Comput. Sci. 10(2), 205–230 (2000)

    MathSciNet  CrossRef  Google Scholar 

  18. Hasegawa, M.: Recursion from cyclic sharing: traced monoidal categories and models of cyclic lambda calculi. In: de Groote, P., Roger Hindley, J. (eds.) TLCA 1997. LNCS, vol. 1210, pp. 196–213. Springer, Heidelberg (1997). https://doi.org/10.1007/3-540-62688-3_37

    CrossRef  MATH  Google Scholar 

  19. Hasegawa, M., Hofmann, M., Plotkin, G.: Finite dimensional vector spaces are complete for traced symmetric monoidal categories. In: Avron, A., Dershowitz, N., Rabinovich, A. (eds.) Pillars of Computer Science. LNCS, vol. 4800, pp. 367–385. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78127-1_20

    CrossRef  Google Scholar 

  20. Heunen, C.: Categorical quantum models and logics. Ph.D. thesis, Radboud University Nijmegen (2009)

    Google Scholar 

  21. Heunen, C.: On the Functor \(\ell ^2\). In: Coecke, B., Ong, L., Panangaden, P. (eds.) Computation, Logic, Games, and Quantum Foundations. The Many Facets of Samson Abramsky. LNCS, vol. 7860, pp. 107–121. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38164-5_8

    CrossRef  Google Scholar 

  22. Heunen, C., Karvonen, M.: Monads on dagger categories. Theory Appl. Categories 31(35), 1016–1043 (2016)

    MathSciNet  MATH  Google Scholar 

  23. Hoshino, N.: A representation theorem for unique decomposition categories. Electron. Notes Theor. Comput. Sci. 286, 213–227 (2012)

    MathSciNet  CrossRef  Google Scholar 

  24. Hyland, M.: Abstract and concrete models for recursion. In: Proceedings of the NATO Advanced Study Institute on Formal Logical Methods for System Security and Correctness, pp. 175–198. IOS Press (2008)

    Google Scholar 

  25. Jacobs, B.: Involutive categories and monoids, with a GNS-correspondence. Found. Phys. 42(7), 874–895 (2012)

    MathSciNet  CrossRef  Google Scholar 

  26. Jacobs, B.: New directions in categorical logic, for classical, probabilistic and quantum logic. Logical Methods Comput. Sci. 11(3), 1–76 (2015)

    MathSciNet  MATH  Google Scholar 

  27. James, R.P., Sabry, A.: Information effects. In: Proceedings, POPL 2012, pp. 73–84. ACM (2012)

    Google Scholar 

  28. James, R.P., Sabry, A.: Theseus: a high level language for reversible computing (2014). Work-in-progress report presented at RC 2014

    Google Scholar 

  29. Joyal, A., Street, R., Verity, D.: Traced monoidal categories. Math. Proc. Cambridge Philos. Soc. 119(3), 447–468 (1996)

    MathSciNet  CrossRef  Google Scholar 

  30. Kaarsgaard, R., Axelsen, H.B., Glück, R.: Join inverse categories and reversible recursion. J. Logical Algebraic Methods Program. 87, 33–50 (2017)

    MathSciNet  CrossRef  Google Scholar 

  31. Karvonen, M.: The way of the dagger. Ph.D. thesis, School of Informatics, University of Edinburgh (2019)

    Google Scholar 

  32. Kastl, J.: Inverse categories. In: Algebraische Modelle, Kategorien und Gruppoide, Studien zur Algebra und ihre Anwendungen, vol. 7, pp. 51–60. Akademie-Verlag (1979)

    Google Scholar 

  33. Kelly, G.M.: Basic Concepts of Enriched Category Theory. London Mathematical Society Lecture Note Series, vol. 64. Cambridge University Press, Cambridge (1982)

    MATH  Google Scholar 

  34. Rennela, M.: Towards a quantum domain theory: order-enrichment and fixpoints in W*-algebras. Electron. Notes Theor. Comput. Sci. 308, 289–307 (2014)

    MathSciNet  CrossRef  Google Scholar 

  35. Selinger, P.: Dagger compact closed categories and completely positive maps. Electron. Notes Theor. Comput. Sci. 170, 139–163 (2007)

    CrossRef  Google Scholar 

  36. Selinger, P.: A survey of graphical languages for monoidal categories. In: Coecke, B. (ed.) New Structures for Physics, pp. 289–355. Springer, Berlin (2011). https://doi.org/10.1007/978-3-642-12821-9_4

    CrossRef  MATH  Google Scholar 

  37. Selinger, P.: Finite dimensional Hilbert spaces are complete for dagger compact closed categories. Logical Methods Comput. Sci. 8, 1–12 (2012)

    MathSciNet  CrossRef  Google Scholar 

  38. Smyth, M.B., Plotkin, G.D.: The category-theoretic solution of recursive domain equations. SIAM J. Comput. 11(4), 761–783 (1982)

    MathSciNet  CrossRef  Google Scholar 

  39. Wadler, P.: Theorems for free! In: Proceedings of the Fourth International Conference on Functional Programming Languages and Computer Architecture, FPCA 1989, pp. 347–359. ACM (1989)

    Google Scholar 

  40. Yokoyama, T., Axelsen, H.B., Glück, R.: Towards a reversible functional language. In: De Vos, A., Wille, R. (eds.) RC 2011. LNCS, vol. 7165, pp. 14–29. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29517-1_2

    CrossRef  MATH  Google Scholar 

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Kaarsgaard, R. (2019). Inversion, Iteration, and the Art of Dual Wielding. In: Thomsen, M., Soeken, M. (eds) Reversible Computation. RC 2019. Lecture Notes in Computer Science(), vol 11497. Springer, Cham. https://doi.org/10.1007/978-3-030-21500-2_3

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  • DOI: https://doi.org/10.1007/978-3-030-21500-2_3

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