A symmetric approach to compilation and decompilation

Just as an interpreter for a source language can be turned into a compiler from the source language to a target language, we observe that an interpreter for a target language can be turned into a compiler from the target language to a source language. In both cases, the key issue is the choice of whether to perform an evaluation or to emit code that represents this evaluation. We substantiate this observation with two source interpreters and two target interpreters. We first consider a source language of arithmetic expressions and a target language for a stack machine, and then the λ- calculus and the SECD-machine language. In each case, we prove that the target-to-source compiler is a left inverse of the source-to-target compiler, i.e., that it is a decompiler. In the context of partial evaluation, the binding-time shift of going from a source interpreter to a compiler is classically referred to as a Futamura projection. By symmetry, it seems logical to refer to the binding-time shift of going from a target interpreter to a compiler as a Futamura embedding.