Coalgebraic Semantics of an Imperative Class Based Language

We study two observational equivalences of Fickle programs. Fickle is a class-based object oriented imperative language, which extends Java with object re-classification. The first is a contextual equivalence of expressions with respect to a given program. We provide an adequate coalgebraic semantics for it, which is compositional w.r.t. the operators of the language. The second observational equivalence is defined on programs implementing the same specification, given as an abstract class. We introduce a coalgebraic description of classes which gives a sound coinduction principle for this latter equivalence. To this end we need to extend the original coalgebraic approach of H.Reichel and B.Jacobs to deal with binary methods, i.e. methods which take more than one instance of the hosting class as argument. This coalgebraic description induces in particular a coinductive observational equivalence on objects of a program, where objects (states of a class) are taken to be equal when the action of methods on them yield the same observations and equivalent next states.