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Converting a HOAS term GADT into a de Bruijn term GADTTwo approaches to representing variables in a typed term language implemented as a GADT are (1) to use higherorder abstract syntax (HOAS) or to use de Bruijn indices.. Both approaches are convenient for manipulating terms as they require no explicit alpha conversion and avoid name capture during substitution. Nevertheless, there are trade offs between the two. In particular, the HOAS representation doesn't support comparing variable names and the de Bruijn representation is inconvenient for humans to read and write. For a more detailed discussion, see for example Conor McBride and James McKinna's I am not a number: I am a free variable, where they discuss a mixed representation using de Bruijn indices for bound variables and variables of the metalanguage for free variables. The tension between the HOAS and de Bruijn representation also has relevance for the design and implementation of embedded domain specific languages (EDSLs) — aka internal languages. When an internal includes higherorder functions, it is usually most convenient for the user to use the function abstraction mechanisms of the host language. However, to execute or compile the internal language, de Bruijn notation is often more convenient; at least, if optimisations are performed. An obvious way to relief the tension between the representations is to use HOAS in the surface representation of the internal language and to convert that to the de Bruijn representation before optimisation and execution. However, the conversion is not entirely straight forward for a strongly typed internal language with typed intermediate representations and typepreserving transformations between those representations. The difficulties were already mentioned by LouisJulien Guillemette and Stefan Monnier in their Haskell'07 paper A TypePreserving Closure Conversion in Haskell (which is concerned with typepreserving compilation in general and doesn't specifically address internal languages). However, their description of the conversion (in Section 5 of the paper) is sketchy and intermingled with concerns specific to their application. The following Haskell code demonstrates the typepreserving conversion from a HOAS to a de Bruijn representation for a simple term language (essentially the lambda calculus with constants of arbitrary type). The two important aspects of the method are two:
The implementation consists of four Haskell modules:
(This code has been tested with GHC 6.11.) Update [9Jul9] Robert Atkey, Sam Lindley & Jeremy Yallop have a forthcoming paper entitled Unembedding DomainSpecific Languages in the upcoming Haskell Symposium 2009 that addresses the exact same translation, but on the basis of a somewhat different representation of HOAS terms. 
• Copyright 2009 Manuel M T Chakravarty • Last modified: 3 July 2009 • 