Reflecting Stacked Continuations in a Fine-Grained Direct-Style Reduction Theory

The delimited-control operator shift0 has been formally shown to capture the operational semantics of deep handlers for algebraic effects. Its CPS translation generates lambda-terms in which continuation composition is not expressed in terms of nested function calls, as is typical of other delimited-control operators, e.g. shift, but with function applications consuming a sequence of continuations one at a time, as if they formed a stack. We present a novel reduction theory for Moggi's computational lambda-calculus extended with shift0 and a control delimiter dollar, which models the capture of evaluation contexts in a fine-grained manner as an interaction between the let-expressions and the delimiter. We establish a connection between our reduction theory and the existing theories of shif0 and dollar. Moreover, we develop a CPS translation for our calculus along with a direct-style translation that together form a reflection, i.e. the translations preserve reductions and the direct-style translation is a right inverse of the CPS translation. This construction relies on the invariant that CPS root terms are in eta-head-normal form. The results of this work could potentially be used for compiler optimisations and lead to a similar development for algebraic effects.