Effect handlers are a modern and increasingly popular approach to structuring computational effects in functional programming languages. However, while their traditional operational semantics is well-suited to implementation tasks, it is less ideal as a reduction theory. We therefore introduce a fine-grained reduction theory for deep effect handlers, inspired by our existing reduction theory for shift0, along with a standard reduction strategy. We relate this strategy to the traditional, non-local operational semantics via a simulation argument, and show that the reduction theory preserves observational equivalence with respect to the classical semantics of handlers, thus allowing its use as a rewriting theory for handler-equipped programming languages -- this rewriting system mostly coincides with previously studied type-based optimisations. In the process, we establish theoretical properties of our reduction theory, including confluence and standardisation theorems, adapting and extending existing techniques. Finally, we demonstrate the utility of our semantics by providing the first normalisation-by-evaluation algorithm for effect handlers, and prove its soundness and completeness. Additionally, we establish non-expressibility of the lift operator, found in some effect-handler calculi, by the other constructs.

The delimited-control operator shift0 has been formally shown to capture the operational semantics of deep handlers for algebraic effects. Its CPS translation generates λ-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 study of the continuation-composing style (CCS) that describes the image of the CPS translation of Danvy and Filinski’s shift and reset delimited-control operators. In CCS continuations are composable rather than abortive as in the traditional CPS, and, therefore, the structure of terms is considerably more complex. We show that the CPS translation from Moggi’s computational lambda calculus extended with shift and reset has a right inverse and that the two translations form a reflection i.e., a Galois connection in which the target is isomorphic to a subset of the source (the orders are given by the reduction relations). Furthermore, we use this result to show that Plotkin’s call-by-value lambda calculus extended with shift and reset is isomorphic to the image of the CPS translation. This result, in particular, provides a first direct-style transformation for delimited continuations that is an inverse of the CPS transformation up to syntactic identity.