Embedding is a language development technique that implements the object language as a library in a host language. There are many advantages of the approach, including being lightweight and the ability to inherit features of the host language. A notable example is the technique of Higher-Order Abstract Syntax (HOAS), which makes crucial use of higher-order functions to represent abstract syntax trees with binders. Despite its popularity, HOAS has its limitations. We observe that HOAS struggles with semantic domains that cannot be naturally expressed as functions, particularly when open expressions are involved. Prominent examples of this include incremental computation and reversible/bidirectional languages.

In this paper, we pin-point the challenge faced by HOAS as a mismatch between the semantic domain of host and object language functions, and propose a solution. The solution is based on the technique of unembedding, which converts from the finally-tagless representation to de Bruijn-indexed terms with strong correctness guarantees. We show that this approach is able to extend the applicability of HOAS while preserving its elegance. We provide a generic strategy for Embedding by Unembedding, and then demonstrate its effectiveness with two substantial case studies in the domains of incremental computation and bidirectional transformations. The resulting embedded implementations are comparable in features to the state-of-the-art language implementations in the respective areas.