Lambda Calculus with Explicit Recursion

This paper is concerned with the study of-calculus with explicit recursion, namely of cyclic-graphs. The starting point is to treat a-graph as a system of recursion equations involving-terms, and to manipulate such systems in an unrestricted manner, using equational logic, just as is possible for rst-order term rewriting. Surprisingly, now the connuence property breaks down in an essential way. Connuence can be restored by introducing a restraining mechanism on thèsubstitution' operation. This leads to a family of-graph calculi, which can be seen as an extension of the family of-calculi (-calculi with explicit substitution). While the-calculi treat the let-construct as a rst-class citizen, our calculi support the letrec, a feature that is essential to reason about time and space behavior of functional languages and also about compilation and optimizations of programs.