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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)

Abstract

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.

Keywords

recursion semantics completely iterative algebra coalgebra distributive law 

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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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