Cyclic lambda calculi

Abstract

We precisely characterize a class of cyclic lambda-graphs, and then give a sound and complete axiomatization of the terms that represent a given graph. The equational axiom system is an extension of lambda calculus with the letrec construct. In contrast to current theories, which impose restrictions on where the rewriting can take place, our theory is very liberal, e.g., it allows rewriting under lambda-abstractions and on cycles. As shown previously, the reduction theory is non-confluent. We thus introduce an approximate notion of confluence. Using this notion we define the infinite normal form or Lévy-Longo tree of a cyclic term. We show that the infinite normal form defines a congruence on the set of terms. We relate our cyclic lambda calculus to the traditional lambda calculus and to the infinitary lambda calculus. Since most implementations of non-strict functional languages rely on sharing to avoid repeating computations, we develop a variant of our calculus that enforces the sharing of computations and show that the two calculi are observationally equivalent. For reasoning about strict languages we develop a call-by-value variant of the sharing calculus. We state the difference between strict and non-strict computations in terms of different garbage collection rules. We relate the call-by-value calculus to Moggi's computational lambda calculus and to Hasegawa's calculus.KeywordsBlack HoleNormal FormGarbage CollectionLambda CalculusComplete AxiomatizationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.