Unembedding domain-specific languages

Abstract

Higher-order abstract syntax provides a convenient way of embedding domain-specific languages, but is awkward to analyse and manipulate directly.We explore the boundaries of higher-order abstract syntax. Our key tool is the unembedding of embedded terms as de Bruijn terms, enabling intensional analysis. As part of our solution we present techniques for separating the definition of an embedded program from its interpretation, giving modular extensions of the embedded language, and different ways to encode the types of the embedded language.

A fresh look at programming with names and binders

“…For instance, our implementation of normalization by evaluation ( §6) illustrates how to move back and forth between "syntactic" name abstractions and "semantic" name abstractions in the style of higherorder abstract syntax. Atkey and co-authors [2,3] investigate how to move back and forth between higher-order abstract syntax and de Bruijn indices. The translation out of higher-order abstract syntax produces well-scoped de Bruijn indices, but the proof of this fact is meta-theoretic.…”

Optimising purely functional GPU programs

“…In particular, it converts the source representation based on higher-order abstract syntax (HOAS) to a type-safe internal representation based on de Bruijn indices. Although developed independently, this conversion is like the unembedding of Atkey et al [1]. Unembedding and sharing recovery necessarily need to go hand in hand for the following reasons.…”

Generic conversions of abstract syntax representations

“…We will introduce the parametric higher-order abstract syntax representation proposed by Atkey et al [2], which builds on Washburn's and Weirich's idea. Their representation is based on generalized Church encodings for term formers of the syntax of a language syntax comparable to Church encodings of inductive datatypes.…”

“…Because of this property the introduced variable has a de Bruijn index in ∆. We would like to create this index by using a proof of the sub-context relation between Γ and ∆, but as Atkey et al [2] point out, Agda's type system unfortunately does not provide us with enough information to obtain such a proof. Atkey [1] provides a meta-theoretical proof of this property for an encoding in System F, however, his proof relies on the parametricity of the universal quantification in C λ .…”

“…In fact, people do use both approaches simultaneously and convert between them when necessary [2,3,5]. However, to the best of our knowledge, a datatype-generic treatment of such a conversion has not been addressed before.…”