On the Power of Model Theory in Specifying Abstract Data Types and in Capturing Their Recursiveness

In this paper a comparative analysis of some algebraic and model-theoretic tools to specify abstract data types is presented: our aim is 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. To do so, we analyze various notions such as “initiality”, “finality”, “monoinitiality”, “epifinality”, “weak monoinitiality” and “weak epifinality”, both from the point of view of “abstractness” and of “cardinality”, in a general model theoretical frame. For the second aspect, it is shown that only “initiality”, “monoinitiality”, “epifinality” and “weak epifinality” allow us to select countable models (for theories with a countable language), a necessary condition to get recursive data types, while this is not the case for “finality” and “weak monoinitiality”. 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 typically model-theoretic notions, and illustrate the sense according to which it captures recursiveness.