Two impossibility theorems on behaviour specification of abstract data types

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.An extended abstract of a part of this paper appeared as: Schoett, O.: An observational subset of first-order logic cannot speacify the behaviour of a counter, in: Choffrut, C., Jantzen, M. (eds) STACS 91. 8th Annual Symposium on Theoretical Aspects of Computer Science (Lect. Notes comput. Sci., vol. 480, pp. 499–510) Berlin Heidelberg New York: Springer 1991