. Various formalizations of the concept of "refinement step" as used in the formal development of programs from algebraic specifications are presented and compared. 1 Introduction Algebraic specification aims to provide a formal basis to support the systematic development of correct programs from specifications by means of verified refinement steps. Obviously, a central piece of the puzzle is how best to formalize concepts like "specification", "program" and "refinement step". Answers are required that are simple, elegant and general and which enjoy useful properties, while at the same time taking proper account of the needs of practice. Here I will concentrate on the last of these concepts, but first I need to deal with the other two. For "program", I take the usual approach of algebraic specification whereby programs are modelled as many-sorted algebras consisting of a collection of sets of data values together with functions over those sets. This level of abstraction is commens...

Abstract. Realizability theory can produce code interfaces for the data structure corresponding to a mathematical theory. Our tool, called RZ, serves as a bridge between constructive mathematics and programming by translating specifications in constructive logic into annotated interface code in Objective Caml. The system supports a rich input language allowing descriptions of complex mathematical structures. RZ does not extract code from proofs, but allows any implementation method, from handwritten code to code extracted from proofs by other tools. 1