Compatibility of parameter passing and implementation of parameterized data types

The basis for this paper are the concepts of parameterization and implementation of abstract data types which have been developed in the theory of algebraic specifications with initial algebra semantics. In this paper we combine both concepts defining implementations of parameterized data types and studying the compatibility of parameter passing and implementation of parameterized data types. In our main result we show that parameter passing commutes with implementation. This is an important step in order to apply the theory of algebraic specifications to development and stepwise refinement of software systems. We illustrate our notion and results by a small example implementing binary trees over arbitrary data by corresponding strings with brackets. Finally we consider the problem of 2-dimensional compatibility of parameter passing and implementation and discuss the kind of compatibility results which have been shown by other authors in the case of loose and final algebra semantics.