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Coalgebraic description of generalised binary methods


We extend the coalgebraic account of specification and refinement of objects and classes in object-oriented programming given by Reichel and Jacobs to (generalised) binary methods. These are methods that take more than one parameter of a class type. Class types include products, sums and powerset type constructors. To allow for class constructors, we model classes as bialgebras. We study and compare two solutions for modelling generalised binary methods, which use purely covariant functors.

In the first solution, which applies when we already have a class implementation, we reduce the behaviour of a generalised binary method to that of a bunch of unary methods. These are obtained by freezing the types of the extra class parameters to constant types. If all parameter types are finitary, the bisimilarity equivalence induced on objects by this model yields the greatest congruence with respect to method application.

In the second solution, we treat binary methods as graphs instead of functions, thus turning contravariant occurrences in the functor into covariant ones.

We show the existence of final coalgebras in both cases.

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S. Abramsky and L. Ong (1993) Full Abstraction in the Lazy Lambda Calculus. Information and Computation 105 (2) 159267.

A. Corradini , R. Heckel and U. Montanari (2002) Compositional SOS and beyond: A coalgebraic view of open systems. Theoretical Computer Science 280 163192.

J. Goguen and G. Malcolm (2000) A Hidden Agenda. Theoretical Computer Science 245 55101.

R. Hennicker and A. Kurz (1999) (Ω, Ξ)-Logic: On the algebraic extension of coalgebraic specifications. CMCS'1999. Electronic Notes in Theoretical Computer Science 19

C. Hermida and B. Jacobs (1998) Structural induction and coinduction in a fibrational setting. Information and Computation 145 (2) 107152.

F. Honsell and M. Lenisa (1999) Coinductive Characterizations of Applicative Structures. Mathematical Structures in Computer Science 9 403435.

B. Jacobs (1996) Objects and Classes, co-algebraically. In: B. Freitag . (eds.) Object-Orientation with Parallelism and Book Persistence, Kluwer Academic Publishers83103.

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Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
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