Separating Weakening and Contraction in a Linear Lambda Calculus

We present a separated-linear lambda calculus of resource consumption based on a refinement of linear logic which allows separate control of weakening and contraction. The calculus satisfies subject reduction and confluence, and inherits previous results on the relationship of Girard’s two translations from minimal intuitionistic logic to linear logic with call-by-name and call-by-value. We construct a hybrid translation from Girard’s two which is sound and complete for mapping types and reduction sequences from call-by-need into separatedlinear . This treatment of call-by-need is more satisfying than in previous work, allowing a contrasting of all three reduction strategies in the manner (for example) that the CPS translations allow for call-by-name and call-by-value. HOW can we explain the differences between parameter-passing styles? With the continuation-passing style (CPS) transforms [24, 25], one makes the flow of control explicit. Each parameter-passing style is associated with a different CPS transform, which reveals the control-flow decisions implicit in each mechanism. In fact these implicit control decisions explain the difference between the mechanisms quite well, and evaluation of the transformed terms actually becomes independent of any particular parameter-passing mechanism. In this paper, we will also compare different calling mechanisms by mapping them into a common system, but rather than focusing on the flow of control as with the CPS transforms, we will contrast the mechanisms in terms of structural typing operations, which reveal how programs use resources, namely their parameters. In particular, we will use linear systems, which allow structural operations only in conjunction with a new modal connective. Moreover, rather than considering linear systems with a single intuitionistic mode, as in previous linear calculi, we will construct as a target of the translations a separated-linear lambda calculus motivated by Jacobs’ model theory [17], where the two key structural operations of weakening and contraction are enabled by distinct modal connectives, rather than the same single connective. This presentation assumes familiarity with the lambda calculus and its simple types [5], intuitionistic logic [27] and the basics of linear logic [13].