Unfolding Semantics of the Untyped λ-Calculus with lectrec-Calculus with letrec

Abstract

We investigate the relationship between finite terms in lambda-letrec, the lambda calculus with letrec, and the infinite lambda terms they express. We say that a lambda-letrec term expresses a lambda term if the latter can be obtained as an infinite unfolding of the former. Unfolding is the process of substituting occurrences of function variables by the right-hand side of their definition. We consider the following questions: (i) How can we characterise those infinite lambda terms that are lambda-letrec-expressible? (ii) given two lambda-letrec terms, how can we determine whether they have the same unfolding? (iii) given a lambda-letrec term, can we find a more compact version of the term with the same unfolding? To tackle these questions we introduce and study the following formalisms: - a rewriting system for unfolding lambda-letrec terms into lambda terms - a rewriting system for `observing' lambda terms by dissecting their term structure - higher-order and first-order graph formalisms together with translations between them as well as translations from and to lambda-letrec We identify a first-order term graph formalism on which bisimulation preservesand reflects the unfolding semantics of lambda-letrec and which is closed under functional bisimulation. From this we derive efficient methods to determine whether two terms are equivalent under infinite unfolding and to compute the maximally shared form of a given lambda-letrec term.