Algebraic implementation of abstract data types

Starting with a review of the theory of algebraic specifications in the sense of the ADJ-group a new theory for algebraic implementations of abstract data types is presented.While main concepts of this new theory were given already at several conferences this paper provides the full theory of algebraic implementations developed in Berlin except of complexity considerations which are given in a separate paper. The new concept of algebraic implementations includes implementations for algorithms in specific programming languages and on the other hand it meets also the requirements for stepwise refinement of structured programs and software systems as introduced by Dijkstra and Wirth. On the syntactical level an algebraic implementation corresponds to a system of recursive programs while the semantical level is defined by algebaic constructions, called synthesis, restriction and identification. Moreover the concept allows composition of implementations and a rigorous study of correctness. The main results of the paper are different kinds of correctness criteria which are applied to a number of illustrating examples including the implementation of sets by hash-tables. Algebraic implementations of larger systems like a histogram or a parts system are given in separate case studies which, however, are not included in this paper.