Embedding by Normalisation

This paper presents the insight that practical embedding techniques, commonly used for implementing Domain-Specific Languages, correspond to theoretical Normalisation-By-Evaluation (NBE) techniques, commonly used for deriving canonical form of terms with respect to an equational theory. NBE constitutes of four components: a syntactic domain, a semantic domain, and a pair of translations between the two. Embedding also often constitutes of four components: an object language, a host language, encoding of object terms in the host, and extraction of object code from the host. The correspondence is deep in that all four components in embedding and NBE correspond to each other. Based on this correspondence, this paper introduces Embedding-By-Normalisation (EBN) as a principled approach to study and structure embedding. The correspondence is useful in that solutions from NBE can be borrowed to solve problems in embedding. In particular, based on NBE techniques, such as Type-Directed Partial Evaluation, this paper presents a solution to the problem of extracting object code from embedded programs involving sum types, such as conditional expressions, and primitives, such as literals and operations on them.