Proving the Correctness of Unfold/Fold Program Transformations Using Bisimulation

Abstract

Abstract. This paper shows that a bisimulation approach can be used to prove the correctness of unfold/fold program transformation algorithms. As an illustration, we show how our approach can be use to prove the correctness of positive supercompilation (due to Sørensen et al). Traditional program equivalence proofs show the original and transformed programs are contextually equivalent, i.e., have the same termination behaviour in all closed contexts. Contextual equivalence can, however, be difficult to establish directly. Gordon and Howe use an alternative approach: to represent a program's behaviour by a labelled transition system whose bisimilarity relation is a congruence that coincides with contextual equivalence. Labelled transition systems are well-suited to represent global program behaviour. On the other hand, unfold/fold program transformations use generalization and folding, and neither is easy to describe contextually, due to use of non-local information. We show that weak bisimulation on labelled transition systems gives an elegant framework to prove contextual equivalence of original and transformed programs. One reason is that folds can be seen in the context of corresponding unfolds.

Distillation with labelled transition systems

“…(αi σ = α i ∧ ti ρ σ∪σ t i ) Figure 7. Embedding Relation for LTSs The main difference between distillation and positive supercompilation [9,21] is that generalization and folding are performed with respect to LTSs in distillation, while they are performed with respect to expressions in positive supercompilation.…”

“…However, the correctness of this algorithm was not perfectly clear; it needed a more solid theoretical foundation. In more recent work by both authors [9], a theoretical framework using labelled transition systems was defined that can be used to prove the correctness of unfold/fold program transformations using bisimilarity, and the correctness of positive supercompilation was proved within this framework. In this paper, we define the distillation algorithm in the same framework and prove its correctness.…”