Two impossibility theorems on behaviour specification of abstract data types (Q1323373)

Two kinds of finite specification of the behaviour of a counter data type are proved impossible. We consider the class of data types (many-sorted algebras) behaving like an encapsulated counter that can be observed only by a test for zero. It is shown that no nonempty subclass of this class can be finitely specified in ``observational first-order logic'', which is a variant of first-order logic in which equality may not be used on encapsulated types. Secondly, it is shown that the class cannot be described exactly by a finite specification in first-order logic.