Algebraic Specifications, Higher-order Types and Set-theoretic Models

Abstract

In most algebraic specification frameworks, the type system is restricted to sorts, subsorts, and first‐order function types. This is in marked contrast to the so‐called model‐oriented frameworks, which provide higher‐order types, interpreted set‐theoretically as Cartesian products, function spaces, and power‐sets. This paper presents a simple framework for algebraic specifications with higher‐order types and set‐theoretic models. It may be regarded as the basis for a Horn‐clause approximation to the Z framework, and has the advantage of being amenable to prototyping and automated reasoning. Standard set‐theoretic models are considered, and conditions are given for the existence of initial reducts of such models. Algebraic specifications for various set‐theoretic concepts are considered.