Coalgebraic description of generalised binary methods

Abstract

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 classconstructors, we model classes asbialgebras. 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 byfreezingthe types of the extra class parameters to constant types. If all parameter types arefinitary, thebisimilarity equivalenceinduced on objects by this model yields thegreatest congruencewith respect to method application.In the second solution, we treat binary methods asgraphsinstead of functions, thus turning contravariant occurrences in the functor into covariant ones.We show the existence offinal coalgebrasin both cases.