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Quotienting the Delay Monad by Weak Bisimilarity

  • James Chapman
  • Tarmo Uustalu
  • Niccolò Veltri
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9399)

Abstract

The delay datatype was introduced by Capretta [3] as a means to deal with partial functions (as in computability theory) in Martin-Löf type theory. It is a monad and it constitutes a constructive alternative to the maybe monad. It is often desirable to consider two delayed computations equal, if they terminate with equal values, whenever one of them terminates. The equivalence relation underlying this identification is called weak bisimilarity. In type theory, one commonly replaces quotients with setoids. In this approach, the delay monad quotiented by weak bisimilarity is still a monad. In this paper, we consider Hofmann’s alternative approach [6] of extending type theory with inductive-like quotient types. In this setting, it is difficult to define the intended monad multiplication for the quotiented datatype. We give a solution where we postulate some principles, crucially proposition extensionality and the (semi-classical) axiom of countable choice. We have fully formalized our results in the Agda dependently typed programming language.

Notes

Acknowledgement

We thank Thorsten Altenkirch, Andrej Bauer, Bas Spitters and our anonymous referees for comments.

This research was supported by the ERDF funded Estonian CoE project EXCS and ICT national programme project “Coinduction”, the Estonian Science Foundation grants No. 9219 and 9475 and the Estonian Ministry of Education and Research institutional research grant IUT33-13.

References

  1. 1.
    Adámek, J., Milius, S., Velebil, J.: Elgot algebras. Log. Methods Comput. Sci. 2(5:4), 1–31 (2006)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Benton, N., Kennedy, A., Varming, C.: Some domain theory and denotational semantics in Coq. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) TPHOLs 2009. LNCS, vol. 5674, pp. 115–130. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  3. 3.
    Capretta, V.: General recursion via coinductive types. Log. Methods Comput. Sci. 1(2:1), 1–28 (2005)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Chicli, L., Pottier, L., Simpson, D.: Mathematical quotients and quotient types in Coq. In: Geuvers, H., Wiedijk, F. (eds.) TYPES 2002. LNCS, vol. 2646, pp. 95–107. Springer, Heidelberg (2003) CrossRefGoogle Scholar
  5. 5.
    Cockett, R., Díaz-Boïls, J., Gallagher, J., Hrubes, P.: Timed sets, complexity, and computability. In: Berger, U., Mislove, M. (eds.) Proceedings of 28th Conference on the Mathematical Foundations of Program Semantics, MFPS XXVIII. Electron. Notes in Theor. Comput. Sci., vol. 286, pp. 117–137. Elsevier, Amsterdam (2012)Google Scholar
  6. 6.
    Hofmann, M.: Extensional Constructs in Intensional Type Theory. CPHS/BCS Distinguished Dissertations. Springer, London (1997)CrossRefGoogle Scholar
  7. 7.
    Hughes, J.: Generalising monads to arrows. Sci. Comput. Program. 37(1–3), 67–111 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Jacobs, B., Heunen, C., Hasuo, I.: Categorical semantics for arrows. J. Funct. Program. 19(3–4), 403–438 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kraus, N., Escardó, M., Coquand, T., Altenkirch, T.: Notions of anonymous existence in Martin-Löf type theory. Manuscript (2014)Google Scholar
  10. 10.
    Maietti, M.E.: About effective quotients in constructive type theory. In: Altenkirch, T., Naraschewski, W., Reus, B. (eds.) TYPES 1998. LNCS, vol. 1657, pp. 166–178. Springer, Heidelberg (1999) CrossRefGoogle Scholar
  11. 11.
    Martin-Löf, P.: 100 years of Zermelo’s axiom of choice: what was the problem with it? Comput. J. 49(3), 345–350 (2006)CrossRefGoogle Scholar
  12. 12.
    Moggi, E.: Notions of computation and monads. Inf. Comput. 93(1), 55–92 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Norell, U.: Dependently typed programming in Agda. In: Koopman, P., Plasmeijer, R., Swierstra, D. (eds.) AFP 2008. LNCS, vol. 5832, pp. 230–266. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  14. 14.
    Troelstra, A.S., Van Dalen, D.: Constructivism in Mathematics: An Introduction, v. I. Studies in Logic and the Foundations of Mathematics, vol. 121. North-Holland, Amsterdam (1988)Google Scholar
  15. 15.
    The Univalent Foundations Program: Homotopy Type Theory: Univalent Foundations of Mathematics. Institute for Advanced Study, Princeton, NY (2013). http://homotopytypetheory.org/book

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institute of CyberneticsTallinn University of TechnologyTallinnEstonia

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