David Sands
Abstract
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 article 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 nonstrict) including lazy data structures. The main technical result is an improvement theorem which says that if the local transformation steps are guided by certain optimization concerns (a fairly natural condition for a transformation), then correctness of the transformation follows. The improvement theorem makes essential use of a formalized 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, and 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.
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Index Terms
- Total correctness by local improvement in the transformation of functional programs
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Reviews
Jan Hendrik Jongejan
Sands discusses program transformations (such as the unfold/fold method [1]). More specifically, this paper is about the (total) correctness of such transformations. It is intended for researchers in functional programming languages. The author presents a generic functional language with higher-order functions, lazy data structures, and strict as well as nonstrict application. The language is defined by its syntax and an operational semantics for its evaluation. Transformation is defined in terms of substitution and equivalence of expressions. In general, these transformations do not guarantee correctness, although they are based on locally equivalence-preserving steps. A restriction, namely improvement, is needed to guarantee total correctness. Improvement is defined in terms of the cost of evaluating an expression. If e1 costs the same as or more than e2 , then e1 is improved by e2 . The main theorem of the paper states that a transformation guarantees total correctness if all transformation steps are improvements: if fx=e and e is improved by e ? (where the free variables of e ? are included in the vector x of formal parameters of f ) then f is improved by g , where gx=e ? f/g . The proof of the theorem is technically involved, needing several auxiliary definitions and lemmas. However, this part of the paper may be safely skipped if the reader is more interested in the applications. One application is the correctness proof of a simple mechanizable transformation [2], which aims to eliminate calls to the concatenate function: the transformation rules are proven to be improvements. Another application is the development of a totally correct, improved unfold/fold transformation. There is a substantive section on related work and a large list of references, although the author states that not much work has been done on correctness of transformations. <__?__Pub Caret>Since the paper is rather long, some readers might choose to read only the first two sections, which depict the problem and present an overview of the solution. I found reading the paper a stimulating experience, so I recommend it heartily.
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