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Coalgebraic description of generalised binary methods†

Published online by Cambridge University Press:  01 August 2007

FURIO HONSELL ,
MARINA LENISA  and
REKHA REDAMALLA
FURIO HONSELL
Affiliation:
Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, 33100 Udine, Italy
MARINA LENISA
Affiliation:
Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, 33100 Udine, Italy
REKHA REDAMALLA
Affiliation:
Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, 33100 Udine, Italy B.M. Birla Science Centre, Adarsh Nagar, Hyderabad, 500 063 A.P., India
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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 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.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 17 , Issue 4 , August 2007 , pp. 647 - 673
Copyright
Copyright © Cambridge University Press 2007

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