Executable higher-order algebraic specifications

Conventional algebraic specifications are first-order. Using higher-order equations in combination with first-order ones raises several fundamental model-theoretic and proof-theoretic questions. The model theory of higher-order equations is well understood (see [20] for a survey of algebraic specifications). The proof theory of higher-order equations is equally well understood, it requires higher- order matching, and higher-order rewriting therefore providing with a simple execution model. Higher-order variables may be instantiated by functions described by λ-expressions, bringing in λ-calculus, whose execution model is again rewriting (β-redexes). Hence rewriting is at the heart of all three execution models, which makes their combination quite simple on the operational side. The main question reviewed in this paper is whether the Church-Rosser and termination properties of these three execution models are preserved within their combination. We will see that the answer is to a large extent positive.