We investigate the relation between the set-theoretical description of coinduction based on Tarski Fixpoint Theorem, and the categorical description of coinduction based on coalgebras. In particular, we examine set-theoretic generalizations of the coinduction proof principle, in the spirit of Milner's bisimulation "up-to", and we discuss categorical counterparts for these. Moreover, we investigate the connection between these and the equivalences induced by T -coiterative functions. These are morphisms into final coalgebras, satisfying the T -coiteration scheme, which is a generalization of both the coiteration and the corecursion scheme. We generalize Rutten's transformation from coalgebraic bisimulations to set-theoretic bisimulations, in order to cover also the case of bisimulations "up-to". A list of examples of set-theoretic coinductive specifications which appear not to be easily expressible in coalgebraic terms are discussed. Introduction Coinductive definitions and ...

C'era una volta un re seduto in canap`e, che disse alla regina raccontami una storia. La regina cominci`o: "C'era una volta un re seduto in canap`e

We study two observational equivalences of Fickle programs. Fickle is a class-based object oriented imperative language...

Fickle is a class-based object oriented imperative language, which extends Java with object re-classification. In this paper, we introduce a natural observational equivalence over Fickle programs. This is a contextual equivalence over main methods with respect to a given sequence of class definitions, i.e. a program. In order to study it, we utilize the formal computational model for OO-programming based on coalgebras, which has recently emerged, whereby objects are taken to be equal when the actions of methods on them yield the same observations and equivalent next states. However, in order to deal with imperative features, we need to extend the original approach of H.Reichel and B.Jacobs in various ways. In particular, we introduce a coalgebraic description of objects (states of a class), which induces a coinductive behavioural equivalence on programs. For simplicity, we focus on Fickle objects whose methods do not take more than one object parameter as argument. Completeness results as well as problematic issues arising from binary methods are also discussed.

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