Abstract. There is a correspondence between classical logic and programming language calculi with first-class continuations. With the addition of control delimiters, the continuations become composable and the calculi become more expressive. We present a fine-grained analysis of control delimiters and formalise that their addition corresponds to the addition of a single dynamically-scoped variable modelling the special top-level continuation. From a type perspective, the dynamically-scoped variable requires effect annotations. In the presence of control, the dynamically-scoped variable can be interpreted in a purely functional way by applying a store-passing style. At the type level, the effect annotations are mapped within standard classical logic extended with the dual of implication, namely subtraction. A continuation-passing-style transformation of lambda-calculus with control and subtraction is defined. Combining the translations provides a decomposition of standard CPS transformations for delimited continuations. Incidentally, we also give a direct normalisation proof of the simply-typed lambda-calculus with control and subtraction.

Abstract. Functional and delimited continuations are more expressive than traditional abortive continuations and they apparently seem to require a framework beyond traditional continuation or monadic semantics. We show that this is not the case: standard continuation semantics is sufficient to explain directly the common control operators for delimited continuations. This implies a monadic framework for typed and encapsulated functional and delimited continuations which we design and implement as a Haskell library.