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A type-theoretic foundation of delimited continuations

  • Published:
Higher-Order and Symbolic Computation

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.

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Author information

Authors and Affiliations

  1. University of Oregon, Eugene, OR, 97403, USA

    Zena M. Ariola

  2. INRIA-Futurs, Orsay, France

    Hugo Herbelin

  3. Indiana University, Bloomington, USA

    Amr Sabry

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  1. Zena M. Ariola
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  2. Hugo Herbelin
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  3. Amr Sabry
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Correspondence to Hugo Herbelin.

Additional information

This paper is an extended version of the conference article “A Type-Theoretic Foundation of Continuations and Prompts.” (Ariola et al., 2004).

Z.M. Ariola supported by National Science Foundation grant number CCR-0204389.

A. Sabry supported by National Science Foundation grant number CCR-0204389, by a Visiting Researcher position at Microsoft Research, Cambridge, U.K., and by a Visiting Professor position at the University of Genova, Italy.

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Ariola, Z.M., Herbelin, H. & Sabry, A. A type-theoretic foundation of delimited continuations. Higher-Order Symb Comput 22, 233–273 (2009). https://doi.org/10.1007/s10990-007-9006-0

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