Finer Control of Weakening and Contraction : Towards aSeparated - Linear

P lotkin demonstrated how two diierent parameter-passing mechanisms could be explained by two diierent translations from the-calculus into the calculus of the continuation passing style 12]. Each of his transformations makes the control operations of each particular mechanism explicit in the transformed term, so that reduction of the transformed term by either mechanism produces equivalent results. In this presentation, we will also compare diierent calling mechanisms by mapping them into a common system, but rather than focusing on an explicit ow of control such as through the continuation-passing style, we will contrast the mechanisms in terms of the substructural operations suggested by Girard's linear logic 5]. Moreover, rather than considering linear systems with a single intutitionistic mode, we will construct as a target of the translations a separated-linear lambda calculus, where the two key substructural operations of weakening and contraction are enabled by distinct modal connectives. Typed lambda calculi generally have a Curry-Howard correspondence 4, 7]: a close relationship between their type systems and formal logical frameworks, usually some variety of minimal intuitionistic logic. In such logics, structural inference rules play an important, if often overlooked, role: weakening allows assumptions to be discarded, while contraction allows duplication. Moreover , these rules are the only facility for duplicating and discarding assumptions. So at the level of the calculi, the corresponding typing rules are the mechanisms by which we introduce the copying or discarding of terms. A term is discarded when it is substituted for a variable which appears only on the left side of a typing judgement, which can occur only as the result of applying a weakening rule; a term is duplicated when it is substituted for a variable which appears more than once on the right side of a typing judgement, which can occur only by a contraction rule. In the substructural lambda calculi I 3], A and L 8], the use of (respectively) weakening, contraction and both are simply banned. These systems are not suucient here: while we do want to make the use of such rules explicit, we do not want to prohibit them altogether. Instead, we will take as target calculi systems whose type systems are related to logics where the use of the structural rules must be explicitly enabled. In previous work 10], we explored translations into a system based on the linear logic of Girard 5]. In linear logic the ability to weaken or …