CIA Structures and the Semantics of Recursion

  • Stefan Milius
  • Lawrence S. Moss
  • Daniel Schwencke
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6014)


Final coalgebras for a functor serve as semantic domains for state based systems of various types. For example, formal languages, streams, non-well-founded sets and behaviors of CCS processes form final coalgebras. We present a uniform account of the semantics of recursive definitions in final coalgebras by combining two ideas: (1) final coalgebras are also initial completely iterative algebras (cia); (2) additional algebraic operations on final coalgebras may be presented in terms of a distributive law λ. We first show that a distributive law leads to new extended cia structures on the final coalgebra. Then we formalize recursive function definitions involving operations given by λ as recursive program schemes for λ, and we prove that unique solutions exist in the extended cias. We illustrate our results by the four concrete final coalgebras mentioned above, e. g., a finite stream circuit defines a unique stream function and we show how to define new process combinators from given ones by sos rules involving recursion.


recursion semantics completely iterative algebra coalgebra distributive law 


  1. 1.
    Aceto, L., Fokking, W., Verhoef, C.: Structural Operational Semantics. In: Handbook of Process Algebra. Elsevier, Amsterdam (2001)Google Scholar
  2. 2.
    Aczel, P.: Non-Well-Founded Sets. CLSI Lecture Notes, vol. 14. CLSI Publications, Stanford (1988)zbMATHGoogle Scholar
  3. 3.
    Aczel, P., Adámek, J., Milius, S., Velebil, J.: Infinite trees and completely iterative theories: A coalgebraic view. Theoret. Comput. Sci. 300, 1–45 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Adámek, J.: Free algebras and automata realizations in the language of categories. Comment. Math. Univ. Carolin. 15, 589–602 (1974)zbMATHGoogle Scholar
  5. 5.
    Adámek, J.: Introduction to coalgebra. Theory Appl. Categ. 14, 157–199 (2005)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Adámek, J., Milius, S., Velebil, J.: On coalgebras based on classes. Theoret. Comput. Sci. 316, 3–23 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Adámek, J., Milius, S., Velebil, J.: Elgot algebras. Log. Methods Comput. Sci. 2(5:4), 31 (2006)Google Scholar
  8. 8.
    Bartels, F.: Generalized coinduction. Math. Structures Comput. Sci. 13(2), 321–348 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Bartels, F.: On generalized coinduction and probabilistic specification formats. PhD thesis, Vrije Universiteit Amsterdam (2004)Google Scholar
  10. 10.
    Barwise, J., Moss, L.S.: Vicious circles. CLSI Publications, Stanford (1996)zbMATHGoogle Scholar
  11. 11.
    Capretta, V., Uustalu, T., Vene, V.: Recursive coalgebras from comonads. Inform. and Comput. 204, 437–468 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Jacobs, B.: A bialgebraic review of deterministic automata, regular expressions and languages. In: Futatsugi, K., Jouannaud, J.-P., Meseguer, J. (eds.) Algebra, Meaning, and Computation. LNCS, vol. 4060, pp. 375–404. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  13. 13.
    Jacobs, B.: Distributive laws for the coinductive solution of recursive equations. Inform. and Comput. 204(4), 561–587 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Lambek, J.: A fixpoint theorem for complete categories. Math. Z. 103, 151–161 (1968)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Lenisa, M., Power, A.J., Watanabe, H.: Distributivity for endofunctors, pointed and co-pointed endofunctors, monads and comonads. In: Reichel, H. (ed.) Proc. Coalgebraic Methods in Computer Science. Electron. Notes Theor. Comput. Sci., vol. 33. Elsevier, Amsterdam (2000)Google Scholar
  16. 16.
    Lenisa, M., Power, A.J., Watanabe, H.: Category theory for operational semantics. Theoret. Comput. Sci. 327, 135–154 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    MacLane, S.: Categories for the working mathematician, 2nd edn. Springer, Heidelberg (1998)Google Scholar
  18. 18.
    Milius, S.: Completely iterative algebras and completely iterative monads. Inform. and Comput. 196, 1–41 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Milius, S., Moss, L.S.: The category theoretic solution of recursive program schemes. Theoret. Comput. Sci. 366, 3–59 (2006) (fundamental study)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Milius, S., Moss, L.S.: Equational properties of recursive program scheme solutions. Cah. Topol. Gèom. Diffèr. Catèg. 50, 23–66 (2009)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Milner, R.: Communication and Concurrency. International Series in Computer Science. Prentice Hall, Englewood Cliffs (1989)zbMATHGoogle Scholar
  22. 22.
    Plotkin, G.D., Turi, D.: Towards a mathematical operational semantics. In: Proc. Logic in Computer Science, LICS (1997)Google Scholar
  23. 23.
    Rutten, J.J.M.M.: Universal coalgebra: a theory of systems. Theoret. Comput. Sci. 249(1), 3–80 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Rutten, J.J.M.M.: A coinductive calculus of streams. Math. Structures Comput. Sci. 15(1), 93–147 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    Schwencke, D.: Coequational logic for accessible functors. Accepted for publication in Inform. and Comput. (2009)Google Scholar
  26. 26.
    Uustalu, T., Vene, V., Pardo, A.: Recursion schemes from comonads. Nordic J. Comput. 8(3), 366–390 (2001)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Worrell, J.: On the final sequence of a finitary set functor. Theoret. Comput. Sci. 338, 184–199 (2005)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Stefan Milius
    • 1
  • Lawrence S. Moss
    • 2
  • Daniel Schwencke
    • 1
  1. 1.Institut für Theoretische InformatikTechnische Universität BraunschweigGermany
  2. 2.Department of MathematicsIndiana UniversityBloomingtonUSA

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