. 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 L'evy-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 ...

S extends the -calculus with recursive bindings, barriers, and updatable memory cells with synchronized operations. The calculus can express both deterministic and nondeterministic computations. It is designed to be useful for reasoning about compiler optimizations and thus allows reductions anywhere, even inside 's. Despite the presence of side effects, the calculus retains fine-grained, implicit parallelism and non-strict functions: there is no global, sequentializing store. Barriers, for sequencing, capture a robust notion of termination. Although S was developed as a foundation for the parallel functional languages pH and Id, we believe that barriers give it wider applicability---to sequential, explicitly parallel and concurrent languages. In this paper we describe the S -calculus and its properties, based on a notion of observable information in a term. We also describe reduction strategies to compute maximal observable information even in the presence of unbounded nondeterminis...