ACE: an automatic complexity evaluator

Reviewer: D. John Cooke

The ACE system provides a partial procedure for finding the worst-case complexity of FP programs. The notion of complexity used is essentially the (order of magnitude of the) number of recursive calls necessary for the program evaluation; as such, the complexity is expressed as a composite of standard functions such as exponentials and logarithms. Having such a measure facilitates comparison of different equivalent programs such as might be obtained by a program transformation system. This is paradoxical because ACE uses the FP algebra to perform transformations and simplifications of Cf, the complexity function for the program under investigation ( Cf is derived by a straightforward syntax-directed translation from the given FP program, f). Subsequently, recursion induction is used to extract a nonrecursive function. Other topics that the reader will encounter include fixpoint theory, folding, computational induction, and the splitting of complex conditionals into a set of simpler forms that each involve a single condition (following McCarthy). Unfortunately, as complexity is inexorably related to termination and termination is undecidable, the system does not always successfully yield a complexity function for a given program. Adding more transformations, more information on `standard' functions, and better approximating functions extends the set of programs on which ACE will succeed. The addition of more rules within the system, however, makes the rule selection process harder and the system slower—but that's another problem. The paper flows well; it consists of two introductory sections, two technical sections, a sizable example (example 3 of section 5 includes illustrations of most of the technical points mentioned earlier), and a conclusion. There is little comparable material in the literature, but all related background is adequately cross-referenced for those who need to brush up on their theory. The ACE system, emanating from ESPRIT project 302 (funded by the EEC), is a very neat application of established theory that is used to harness the emerging technology of program transformation. The use of the FP algebra is central to the methodology, but extensions to other similar languages are clearly possible by using FP as an internal/intermediate form. There are a few bothersome typographical errors and other minor irritations (most notably `verify' for `satisfy,' lower case `o' for the function composition symbol 3:9D, and :3WCfor the atom that is the empty sequence rather than &fgr; as used by Backus in his original FP paper) but all in all this is a very readable paper describing interesting and useful work.