A Term Calculus for Intuitionistic Linear Logic

In this paper we consider the problem of deriving a term assignment system for Girard's Intuitionistic Linear Logic for both the sequent calculus and natural deduction proof systems. Our system di ers from previous calculi (e.g. that of Abramsky [1]) and has two important properties which they lack. These are the substitution property (the set of valid deductions is closed under substitution) and subject reduction (reduction on terms is well-typed). We also consider term reduction arising from cut-elimination in the sequent calculus and normalisation in natural deduction. We explore the relationship between these and consider their computational content. 1 Intuitionistic Linear Logic Girard's Intuitionistic Linear Logic [3] is a re nement of Intuitionistic Logic where formulae must be used exactly once. Given this restriction the familiar logical connectives become divided into multiplicative and additive versions. Within this paper, we shall only consider the multiplicatives. Intuitionistic Linear Logic can be most easily presented within the sequent calculus. The linearity constraint is achieved by removing the Weakening and Contraction rules. To regain the expressive power of Intuitionistic Logic, we introduce a new logical operator, !, which allows a formula to be used as many times as required (including zero). The fragment we shall consider is given in Fig. 1. We use capital Greek letters ; for sequences of formulae and A;B for single formulae. The system has multiplicative conjunction or tensor, , linear implication, , and a logical operator, !. The Exchange rule simply allows the permutation of assumptions. In what follows we shall consider this rule to be implicit, whence the convention that ; denote multisets (and not sequences). The `! rules' have been given names by other authors. ! L 1 is called Weakening , ! L 2 Contraction, ! L 3 Dereliction and (! R ) Promotion 3 . We shall use these terms throughout this paper. In the Promotion rule, ! means that every formula in the set is modal, in other words, if is the set fA 1 ; . . . ; A n g, then ! denotes the set f!A 1 ; . . . ; !A n g. We shall defer the question of a term assignment system until Section 3. 2 Linear Natural Deduction In the natural deduction system, originally due to Gentzen [11], but expounded by Prawitz [9], a deduction is a derivation of a proposition from a nite set of assumption 3 Girard, Scedrov and Scott [5] prefer to call this rule Storage.