Linearizing Intuitionistic Implication

An embedding of the implicational propositional intuitionistic logic (iil) into the nonmodal fragment of intuitionistic linear logic (imall) is given. The embedding preserves cut-free proofs in a proof system that is a variant of iil. The embedding is eecient and provides an alternative proof of the pspace-hardness of imall. It exploits several proof-theoretic properties of intuitionistic implication that analyze the use of resources in iil proofs. Linear logic is a reenement of classical and intuitionistic logic that provides an intrinsic and natural accounting of resources. In Girard's words 12], \linear logic is a logic behind logic." A convenient way to present linear logic is by modifying the traditional Gentzen-style sequent calculus axiomatization of classical logic (see, e.g., 15, 22]). The modiication may be brieey described in three steps. The rst step is to remove two structural rules, contraction and weakening , which manipulate the use of hypotheses and conclusions in classical proofs. For expository purposes let us concentrate on the treatment of hypotheses. The contraction rule states that if a property follows from two