Resource Interpretations, Bunched Implications and the alpha lambda-Calculus

We introduce the αλ-calculus, a typed calculus that includes a multiplicative function type -* alongside an additive function type →. It arises proof-theoretically as a calculus of proof terms for the logic of bunched implications of O'Hearn and Pym, and semantically from doubly closed categories, where a single category possesses two closed structures. Typing contexts in αλ are bunches, i.e., trees built from two combining operations, one that admits the structural rules of Weakening and Contraction and another that does not. To illuminate the consequences of αλ's approach to the structural rules we define two resource interpretations, extracted from Reynolds's "sharing reading" of affine λ-calculus. Based on this we show how αλ enables syntactic control of interference and Idealized Algol, imperative languages based on affine and simply-typed λ-calculi, to be smoothly combined in one system.

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