Advertisement

Abstract

The paper investigates non-deterministic, probabilistic and quantum walks, from the perspective of coalgebras and monads. Non-deterministic and probabilistic walks are coalgebras of a monad (powerset and distribution), in an obvious manner. It is shown that also quantum walks are coalgebras of a new monad, involving additional control structure. This new monad is also used to describe Turing machines coalgebraically, namely as controlled ‘walks’ on a tape.

References

  1. 1.
    Bartels, F., Sokolova, A., de Vink, E.: A hierarchy of probabilistic system types. Theor. Comp. Sci. 327(1-2), 3–22 (2004)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Coumans, D., Jacobs, B.: Scalars, monads and categories (2010), http://arxiv.org/abs/1003.0585
  3. 3.
    Jacobs, B.: Semantics of weakening and contraction. Ann. Pure & Appl. Logic 69(1), 73–106 (1994)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Jacobs, B.: Dagger categories of tame relations (2011), http://arxiv.org/abs/1101.1077
  5. 5.
    Kempe, J.: Quantum random walks – an introductory overview. Contemporary Physics 44, 307–327 (2003)CrossRefGoogle Scholar
  6. 6.
    Kock, A.: Bilinearity and cartesian closed monads. Math. Scand. 29, 161–174 (1971)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kock, A.: Closed categories generated by commutative monads. Journ. Austr. Math. Soc. XII, 405–424 (1971)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Liang, S., Hudak, P., Jones, M.: Monad transformers and modular interpreters. In: Principles of Programming Languages, pp. 333–343. ACM Press, New York (1995)Google Scholar
  9. 9.
    Pavlović, D., Mislove, M., Worrell, J.: Testing semantics: Connecting processes and process logics. In: Johnson, M., Vene, V. (eds.) AMAST 2006. LNCS, vol. 4019, pp. 308–322. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  10. 10.
    Perdrix, S.: Partial observation of quantum Turing machine and weaker well-formedness condition. In: Proceedings of: Quantum Physics and Logic and Development of Computational Models (2008), http://web.comlab.ox.ac.uk/people/simon.perdrix/publi/weakerQTM.pdf
  11. 11.
    Rutten, J.: Universal coalgebra: a theory of systems. Theor. Comp. Sci. 249, 3–80 (2000)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Venegas-Andraca, S.: Quantum Walks for Computer Scientists. Morgan & Claypool, San Francisco (2008)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Bart Jacobs
    • 1
  1. 1.Institute for Computing and Information Sciences (iCIS)Radboud University NijmegenThe Netherlands

Personalised recommendations