Towards a Theory of Abstract Data Types: A Discussion on Problems and Tools

This paper aims to show that, in order to capture a quite relevant feature such as the recursiveness of abstract data types, Model Theory works better than Category Theory. First, various eategorial notions such as "initiality", "finality", "monoinitiality", "epifinality", "weak monoinitiality" and "weak epifinality' are analyzed, from a model theoretic point of view, as regards the "abstractness" and the "cardinality" of the models they determine. In particular~ countability is seen as a necessary condition to get recursive data types~ and it is shown that only "initiality", ~'monoinitiality"~ "epifinality" and "weak epifinality ~ allow to select countable models. An extensive analysis is then devoted to the problem of the recursiveness of abstract data types: we provide a formal definition of recursiveness and show that it neither collapses, nor it is incompatible with the "abstractness" requirement. We also show" that none of the above quoted categorial notions captures recursiveness. Finally, we consider our own definition of abstract data type, based on model-theoretic notions; we analyze this definition in the frame of the proposed formalization of recursiveness, and illustrate the sense according to which it captures recursiveness.