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Understanding Algebraic Effect Handlers via Delimited Control Operators

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Trends in Functional Programming (TFP 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13401))

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Abstract

Algebraic effects and handlers are a powerful and convenient abstraction for user-defined effects. We aim to understand effect handlers through the lens of control operators, a similar but more well-studied tool for expressing effects. In this paper, we establish two program transformations and a type system for effect handlers, all by reusing the existing results about control operators and their relationship to effect handlers.

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Notes

  1. 1.

    The dollar operator was originally proposed by Kiselyov and Shan [20]. In their formalization, the construct takes the form \(N \, \$ \, M\), where M is the main computation and N is an arbitrary expression representing an ending continuation. We use the bracket notation of Forster et al., restricting N to be an abstraction \(\lambda x.\, N\). This makes it easier to compare dollar with effect handlers.

  2. 2.

    To see what a meta-context is, consider the following program:

    $$\begin{aligned} {\langle 2 * \langle 1 + \mathcal {S}_0 k_1.\, \mathcal {S}_0 k_2.\, k_2 \ 3 \mid x.\texttt {return}\ x \rangle \mid x.\texttt {return}\ x \rangle \leadsto ^*6} \end{aligned}$$

    The first shift0 operator captures the delimited context \(\langle 1 + [] \mid x.\texttt {return}\ x \rangle \), whereas the second one captures the meta-context \(\langle 2 * [] \mid x.\texttt {return}\ x \rangle \) surrounding the innermost dollar operator.

  3. 3.

    The macro translation is originally defined as:

    $$\begin{aligned} (\lambda k.\, \langle (\lambda x.\, \mathcal {S}_0 z.\, k \ x) \ \llbracket M \rrbracket _m \rangle ) \ \llbracket N \rrbracket _m \end{aligned}$$

    We adapted the translation by sequencing the application, and by incorporating the fact that N is always a \(\lambda \)-abstraction.

  4. 4.

    If we wish to apply these translations to a typed language, we would need to slightly modify the definition. The reason is that, the translation of regular handlers yields a single pattern variable x representing the arguments of the ret and op operations, which have different types in general.

  5. 5.

    An anonymous reviewer suggests a different translation that does not use labels:

    figure d

    where \(M \ V\) is a shorthand for \(\texttt {let}\ x = M \ \texttt {in}\ x \ V\) and 0 can be replaced by any constant. The translation implements the return clause by creating thunks (to exit from a handler) and passing around a dummy argument (to trigger computation). This eliminates the need for labels, but in our view, it also makes the translation slightly harder to understand.

  6. 6.

    By “unoptimized translation”, we mean the first-order translation in Fig. 5 of Hillerström et al. [17].

  7. 7.

    The CPS translation of the original dollar operator \(N \, \$ \, M\) is defined as \(\lambda k.\, \llbracket N \rrbracket _c \ (\lambda f.\, \llbracket M \rrbracket _c \ f \ k)\).

  8. 8.

    The original translation of Hillerström et al. [17] packages the two arguments to h into a tuple and associates the tuple with an operation label.

  9. 9.

    In the original type system of Forster et al. [13], effects are attached to the typing judgment instead of being part of a computation type.

  10. 10.

    The effect representation does not allow answer-type modification. A more general (and involved) type system supporting answer-type modification is given by Materzok and Biernacki [30].

  11. 11.

    The interpreter is available at https://github.com/YouyouCong/tfp22.

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Acknowledgments

We gratefully thank the anonymous reviewers for their constructive feedback on the presentation and technical development. This work was supported by JSPS KAKENHI under Grant No. JP18H03218, No. JP19K24339, and No. JP22H03563.

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Authors and Affiliations

  1. Tokyo Institute of Technology, Tokyo, Japan

    Youyou Cong

  2. Ochanomizu University, Tokyo, Japan

    Kenichi Asai

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  1. Utrecht University, Utrecht, The Netherlands

    Wouter Swierstra

  2. Imperial College, London, UK

    Nicolas Wu

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Cong, Y., Asai, K. (2022). Understanding Algebraic Effect Handlers via Delimited Control Operators. In: Swierstra, W., Wu, N. (eds) Trends in Functional Programming. TFP 2022. Lecture Notes in Computer Science, vol 13401. Springer, Cham. https://doi.org/10.1007/978-3-031-21314-4_4

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