Convergence of Program Transformers in the Metric Space of Trees

In recent years increasing consensus has emerged that program transformers, e.g., partial evaluation and unfold/fold transformations, should terminate; a compiler should stop even if it performs fancy optimizations! A number of techniques to ensure termination of program transformers have been invented, but their correctness proofs are sometimes long and involved. We present a framework for proving termination of program transformers, cast in the metric space of trees. We rst introduce the notion of an abstract program transformer; a number of well-known program transformers can be viewed as instances of this notion. We then formalize what it means that an abstract program transformer terminates and give a general suucient condition for an abstract program transformer to terminate. We also consider some speciic techniques for satisfying the condition. As applications we show that termination of some well-known program transformers either follows directly from the speciic techniques or is easy to establish using the general condition. Our framework facilitates simple termination proofs for program transformers. Also, since our framework is independent of the language being transformed, a single correctness proof can be given in our framework for program transformers using essentially the same technique in the context of diierent languages. Moreover, it is easy to extend termination proofs for program transformers to accommodate changes to these transformers. Finally, the framework may prove useful for designing new termination techniques for program transformers.