Lambda Calculus and Intuitionistic Linear Logic

The Curry-Howard isomorphism 1 is the basis of typed functional programming. By means of this isomorphism, the intuitionistic proof of a formula can be seen as a functional program, whose type is the formula itself. In this way, the computation process has its logic realization in the proof normalization procedure. Both the implicative fragment of the intuitionistic propositional logic together with the simply typed λ-calculus [3], and the second order propositional logic together with the second order λ-calculus of Girard and Reynold [8, 19] are examples of such an isomorphism. The linear logic, introduced by Girard [9], seems to be particularly interesting from the computational point of view. It is more refined than the classical logic: the use of the structural rules is explicitly controlled through a modal connective, denoted by the unary operator !. In other words, weakening and contraction can be applied only to modal formulas. Since weakening and contraction rules are naturally related to the operations of erasing and copying information, respectively, the linear logic can be seen as a model for a computational environment with an explicit control of the resource management. These features can be effectively studied using a language corresponding to the intuitionistic fragment of the linear logic, through the Curry-Howard isomorphism. Until now, some languages inspired by this isomorphism have been designed. First of all, Lafont [13] defined a calculus of combinators, corresponding to the intuitionistic linear logic (ILL in the following), where combinators were suggested from the categorical interpretation of the logic. Then, he defined a linear abstract machine for the evaluation of his calculus. Abramsky [1] has been the first one proposing a “linear λ-calculus”. His language is inspired to the classical λ-calculus and it is defined using ILL as type assignment for it. The functional language is obtained by “decorating” with terms the rules of the sequent calculus for ILL. Abramsky also proposed an extension of the SECD machine for