Compositional semantics for composable continuations: from abortive to delimited control

Parigot's λμ-calculus, a system for computational reasoning about classical proofs, serves as a foundation for control operations embodied by operators like Scheme's callcc. We demonstrate that the call-by-value theory of the λμ-calculus contains a latent theory of delimited control, and that a known variant of λμ which unshackles the syntax yields a calculus of composable continuations from the existing constructs and rules for classical control. To relate to the various formulations of control effects, and to continuation-passing style, we use a form of compositional program transformations which preserves the underlying structure of equational theories, contexts, and substitution. Finally, we generalize the call-by-name and call-by-value theories of the λμ-calculus by giving a single parametric theory that encompasses both, allowing us to generate a call-by-need instance that defines a calculus of classical and delimited control with lazy evaluation and sharing.