Total correctness by local improvement in program transformation

The goal of program transformation is to improve efficiency while preserving meaning. One of the best known transformation techniques is Burstall and Darlington's unfold-fold method. Unfortunately the unfold-fold method itself guarantees neither improvement in efficiency nor total-correctness. The correctness problem for unfold-fold is an instance of a strictly more general problem: transformation by locally equivalence-preserving steps does not necessarily preserve (global) equivalence. This paper presents a condition for the total correctness of transformations on recursive programs, which, for the first time, deals with higher-order functional languages (both strict and non-strict) including lazy data structures. The main technical result is an improvement theorem which says that if the local transformation steps are guided by certain optimisation concerns (a fairly natural condition for a transformation, then correctness of the transformation follows. The improvement theorem makes essential use of a formalised improvement-theory; as a rather pleasing corollary it also guarantees that the transformed program is a formal improvement over the original. The theorem has immediate practical consequences: • It is a powerful tool for proving the correctness of existing transformation methods for higher-order functional programs, without having to ignore crucial factors such as memoization or folding. We have applied the theorem to obtain a particularly simple proof of correctness for a higher-order variant of deforestation. • It yields a simple syntactic method for guiding and constraining the unfold/fold method in the general case so that total correctness (and improvement) is always guaranteed.