Affine Monads and Side-Effect-Freeness

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).

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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