From shift and reset to polarized linear logic

Griffin [22] pointed out that, just as the pure λ-calculus corresponds to intuitionistic logic, a λ-calculus with firstclass continuations corresponds to classical logic. We study how first-class delimited continuations [13], in the form of Danvy and Filinski’s shift and reset operators [10, 11], can also be logically interpreted. First, we refine Danvy and Filinski’s type system for shift and reset to distinguish between pure and impure functions. This refinement not only paves the way for answer type polymorphism, which makes more terms typable, but also helps us invert the continuation-passing-style (CPS) transform: any pure λ-term with an appropriate type is βη-equivalent to the CPS transform of some shift-reset expression. We conclude that the λ-calculus with shift and reset and the pure λ-calculus have the same logical interpretation, namely good old intuitionistic logic. Second, we mix delimited continuations with undelimited ones. Informed by the preceding conclusion, we translate the λ-calculus with shift and reset into a polarized variant of linear logic [34] that integrates classical and intuitionistic reasoning. Extending previous work on the λμ-calculus [36, 37, 40], this unifying intermediate language expresses computations with and without control effects, on delimited and undelimited continuations, in call-by-value and call-byname settings.