CIA structures and the semantics of recursion

  • Authors:

  • Stefan Milius;Lawrence S. Moss;Daniel Schwencke

  • Affiliations:

  • Institut für Theoretische Informatik, Technische Universität Braunschweig, Germany;Department of Mathematics, Indiana University, Bloomington, IN;Institut für Theoretische Informatik, Technische Universität Braunschweig, Germany

  • Venue:
  • FOSSACS'10 Proceedings of the 13th international conference on Foundations of Software Science and Computational Structures
  • Year:
  • 2010

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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 lawλ. 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.