Maximal sharing in the Lambda calculus with letrec

Increasing sharing in programs is desirable to compactify the code, and to avoid duplication of reduction work at run-time, thereby speeding up execution. We show how a maximal degree of sharing can be obtained for programs expressed as terms in the lambda calculus with letrec. We introduce a notion of 'maximal compactness' for λletrec-terms among all terms with the same infinite unfolding. Instead of defined purely syntactically, this notion is based on a graph semantics. λletrec-terms are interpreted as first-order term graphs so that unfolding equivalence between terms is preserved and reflected through bisimilarity of the term graph interpretations. Compactness of the term graphs can then be compared via functional bisimulation. We describe practical and efficient methods for the following two problems: transforming a λletrec-term into a maximally compact form; and deciding whether two λletrec-terms are unfolding-equivalent. The transformation of a λletrec-terms L into maximally compact form L0 proceeds in three steps: (i) translate L into its term graph G = [[L]] ; (ii) compute the maximally shared form of G as its bisimulation collapse G0 ; (iii) read back a λletrec-term L0 from the term graph G0 with the property [[L0]] = G0. Then L0 represents a maximally shared term graph, and it has the same unfolding as L. The procedure for deciding whether two given λletrec-terms L1 and L2 are unfolding-equivalent computes their term graph interpretations [[L1]] and [[L2]], and checks whether these are bisimilar. For illustration, we also provide a readily usable implementation.