Project Euclid

Storage operators have been introduced by J. L. Krivine in [5] they are closed $\lambda$-terms which, for a data type, allow one to simulate a "call by value" while using the "call by name" strategy. In this paper, we introduce the directed $\lambda$-calculus and show that it has the usual properties of the ordinary $\lambda$-calculus. With this calculus we get an equivalent--and simple--definition of the storage operators that allows to show some of their properties: $\bullet$ the stability of the set of storage operators under the $\beta$-equivalence (Theorem 5.1.1); $\bullet$ the undecidability (and semidecidability) of the problem "is a closed $\lambda$-term $t$ a storage operator for a finite set of closed normal $\lambda$-terms?" (Theorems 5.2.2 and 5.2.3); $\bullet$ the existence of storage operators for every finite set of closed normal $\lambda$-terms (Theorem 5.4.3); $\bullet$ the computation time of the "storage operation" (Theorem 5.5.2).