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Affine Monads and Side-Effect-Freeness

  • Bart Jacobs
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9608)

Abstract

The notions of side-effect-freeness and commutativity are typical for probabilistic models, as subclass of quantum models. This paper connects these notions to properties in the theory of monads. A new property of a monad (‘strongly affine’) is introduced. It is shown that for such strongly affine monads predicates are in bijective correspondence with side-effect-free instruments. Also it is shown that these instruments are commutative, in a suitable sense, for monads which are commutative (monoidal).

Notes

Acknowledgements

Thanks are due to Kenta Cho and Fabio Zanasi for helpful discussions on the topic of the paper, and to the anonymous referees for suggesting several improvements.

References

  1. 1.
    Abramsky, S., Brandenburger, A.: The sheaf-theoretic structure of non-locality and contextuality. New J. Phys. 13, 113036 (2011)CrossRefGoogle Scholar
  2. 2.
    Adámek, J., Velebil, J.: Analytic functors and weak pullbacks. Theor. Appl. Categ. 21(11), 191–209 (2008)MathSciNetMATHGoogle Scholar
  3. 3.
    Adams, R.: QPEL: quantum program and effect language. In: Coecke, B., Hasuo, I., Panangaden, P. (eds.) Electrical Proceedings in Theoretical Computer Science on Quantum Physics and Logic (QPL) 2014, no. 172, pp. 133–153 (2014)Google Scholar
  4. 4.
    Adams, R., Jacobs, B.: A type theory for probabilistic and Bayesian reasoning (2015). arXiv.org/abs/1511.09230
  5. 5.
    Cho, K.: Total and partial computation in categorical quantum foundations. In: Heunen, C., Selinger, P., Vicary, J. (eds.) Electrical Proceedings in Theoretical Computer Science on Quantum Physics and Logic (QPL) 2015, no. 195, pp. 116–135 (2015)Google Scholar
  6. 6.
    Cho, K., Jacobs, B., Westerbaan, A., Westerbaan, B.: An introduction to effectus theory (2015). arXiv.org/abs/1512.05813
  7. 7.
    Coecke, B., Heunen, C., Kissinger, A.: Categories of quantum and classical channels. Quantum Inf. Process. 1–31 (2014)Google Scholar
  8. 8.
    Fong, B.: Causal theories: a categorical perspective on Bayesian networks. Master’s thesis, Univ. of Oxford (2012). arXiv.org/abs/1301.6201
  9. 9.
    Furber, R., Jacobs, B.: Towards a categorical account of conditional probability. In: Heunen, C., Selinger, P., Vicary, J. (eds.) Electronic Proceedings in Theoretical Computer Science of Quantum Physics and Logic (QPL) 2015, no. 195, pp. 179–195 (2015)Google Scholar
  10. 10.
    Giry, M.: A categorical approach to probability theory. In: Banaschewski, B. (ed.) Categorical Aspects of Topology and Analysis. Lecture Notes in Mathematics, vol. 915, pp. 68–85. Springer, Berlin (1982)CrossRefGoogle Scholar
  11. 11.
    Jacobs, B.: Semantics of weakening and contraction. Ann. Pure Appl. Logic 69(1), 73–106 (1994)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Jacobs, B.: Measurable spaces and their effect logic. In: Logic in Computer Science. IEEE, Computer Science Press (2013)Google Scholar
  13. 13.
    Jacobs, B.: New directions in categorical logic, for classical, probabilistic and quantum logic. Logical Methods in Comp. Sci. 11(3), 1–76 (2015)MathSciNetMATHGoogle Scholar
  14. 14.
    Jacobs, B.: Effectuses from monads. In: MFPS 2016 (2016, to appear)Google Scholar
  15. 15.
    Jacobs, B., Mandemaker, J.: The expectation monad in quantum foundations. In: Jacobs, B., Selinger, P., Spitters, B. (eds.) Elecronic Proccedings in Theoretical Computer Science of Quantum Physics and Logic (QPL) 2011, no. 95, pp. 143–182 (2012)Google Scholar
  16. 16.
    Jacobs, B., Westerbaan, B., Westerbaan, B.: States of convex sets. In: Pitts, A. (ed.) FOSSACS 2015. LNCS, vol. 9034, pp. 87–101. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  17. 17.
    Jacobs, B., Zanasi, F.: A predicate/state transformer semantics for Bayesian learning. In: MFPS 2016 (2016, to appear)Google Scholar
  18. 18.
    Klin, Bartek: Structural operational semantics for weighted transition systems. In: Palsberg, Jens (ed.) Semantics and Algebraic Specification. LNCS, vol. 5700, pp. 121–139. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  19. 19.
    Kock, A.: Monads on symmetric monoidal closed categories. Arch. Math. XXI, 1–10 (1970)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Kock, A.: Bilinearity and cartesian closed monads. Math. Scand. 29, 161–174 (1971)MathSciNetMATHGoogle Scholar
  21. 21.
    Lindner, H.: Affine parts of monads. Arch. Math. XXXIII, 437–443 (1979)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Panangaden, P.: Labelled Markov Processes. Imperial College Press, London (2009)CrossRefMATHGoogle Scholar
  23. 23.
    Trnková, V.: Some properties of set functors. Comment. Math. Univ. Carolinae 10, 323–352 (1969)MathSciNetMATHGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2016

Authors and Affiliations

  1. 1.Institute for Computing and Information SciencesRadboud UniversiteitNijmegenThe Netherlands

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