Understanding the Expressive Power of Unhygienic Substitution in Metaprogramming via Combinatory Logic

Recent work on combinatory logic demonstrates a compositional translation from lambda calculus that gives meaning to open terms. As the meaning of open terms is a key difficulty in the study of metaprogramming, we investigate whether this idea can be extended to metaprogramming systems with hygienic and unhygienic substitution, specifically $\lambda_{e}^{\rightarrow \Box}$ and $\lambda_{\mathcal{S}}$. We conclude that there is quantitative evidence to support the claim that metaprogramming with unhygienic substitution is more powerful than metaprogramming with hygienic substitution.