Advertisement

Complete Iterativity for Algebras with Effects

  • Stefan Milius
  • Thorsten Palm
  • Daniel Schwencke
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5728)

Abstract

Completely iterative algebras (cias) are those algebras in which recursive equations have unique solutions. In this paper we study complete iterativity for algebras with computational effects (described by a monad). First, we prove that for every analytic endofunctor on Set there exists a canonical distributive law over any commutative monad M, hence a lifting of that endofunctor to the Kleisli category of M. Then, for an arbitrary distributive law λ of an endofunctor H on Set over a monad M we introduce λ-cias. The cias for the corresponding lifting of H (called Kleisli-cias) form a full subcategory of the category of λ-cias. For various monads of interest we prove that free Kleisli-cias coincide with free λ-cias, and these free algebras are given by free algebras for H. Finally, for three concrete examples of monads we prove that Kleisli-cias and λ-cias coincide and give a characterisation of those algebras.

Keywords

iterative algebra monad distributive law initial algebra terminal coalgebra 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adámek, J., Milius, S., Velebil, J.: Iterative algebras at work. Math. Structures Comput. Sci. 16, 1085–1131 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    America, P., Rutten, J.J.M.M.: Solving reflexive domain equations in a category of complete metric spaces. J. Comput. System Sci. 39(3), 343–375 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Elgot, C.: Monadic computation and iterative algebraic theories. In: Rose, H.E., Shepherdson, J.C. (eds.) Logic Colloquium 1973, pp. 175–230. North-Holland, Amsterdam (1975)Google Scholar
  4. 4.
    Hasuo, I., Jacobs, B., Sokolova, A.: Generic trace semantics via coinduction. Log. Methods Comput. Sci. 3(4:11), 1–36 (2007)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Joyal, A.: Une théorie combinatoire des séries formelles. Adv. Math. 42, 1–82 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Joyal, A.: Foncteurs analytiques et espèces de structures. In: Labelle, G., Leroux, P. (eds.) Combinatoire énumérative. Lecture Notes in Math., vol. 1234, pp. 126–159 (1986)Google Scholar
  7. 7.
    Joyal, A., Street, R.: Braided tensor categories. Adv. Math. 102, 20–78 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kock, A.: Monads on symmetric monoidal closed categories. Arch. Math. (Basel) 21, 1–10 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kock, A.: Strong functors and monoidal monads. Arch. Math. (Basel) 23, 113–120 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lambek, J.: A fixpoint theorem for complete categories. Math. Z. 103(2), 151–161 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Milius, S.: Completely iterative algebras and completely iterative monads. Inform. and Comput. 196, 1–41 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Milius, S., Moss, L.S.: The category-theoretic solution of recursive program schemes. Theoret. Comput. Sci. 366, 3–59 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Milius, S., Palm, T., Schwencke, D.: Complete iterativity for algebras with effects, http://www.stefan-milius.eu
  14. 14.
    Moggi, E.: Notions of computation and monads. Inform. and Comput. 93(1), 55–92 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Nelson, E.: Iterative algebras. Theoret. Comput. Sci. 25, 67–94 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Tiuryn, J.: Unique fixed points vs. least fixed points. Theoret. Comput. Sci. 12, 229–254 (1980)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Stefan Milius
    • 1
  • Thorsten Palm
    • 1
  • Daniel Schwencke
    • 1
  1. 1.Institut für Theoretische InformatikTechnische Universität BraunschweigBraunschweigGermany

Personalised recommendations