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Verifying Selective CPS Transformation for Shift and Reset

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Trends in Functional Programming (TFP 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12053))

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Abstract

A selective CPS transformation enables us to execute a program with delimited control operators, \(\texttt {shift}\) and \(\texttt {reset}\), in a standard functional language without support for control operators. The selective CPS transformation dispatches not only on the structure of the input term but also its purity: it transforms only those parts that actually involve control effects. As such, the selective CPS transformation consists of many rules, each for one possible combination of the purity of subterms, making its verification tedious and error-prone. In this paper, we first formalize a monomorphic version of the selective CPS transformation in the Agda proof assistant. We use intrinsically typed term and context representations together with parameterized higher-order abstract syntax (PHOAS) to represent binding structures. We then prove the correctness of the transformation, i.e., the equality of terms is preserved by the CPS transformation. Through the formalization, we confirmed that all the rules of the selective CPS transformation in the previous work are correct, but found that one lemma on the behavior of \(\texttt {shift}\) was not precise.

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Notes

  1. 1.

    PHOAS prevents us from case splitting on the body of an abstraction and a \(\texttt {shift}\) construct until they are applied. It is not a problem for formalizing and verifying CPS transformations.

  2. 2.

    We do not consider let-polymorphism in this paper. See [19] for the formalization of a (non-selective) CPS transformation of lambda calculus (without control operators) extended with let-polymorphism.

  3. 3.

    Following Kameyama and Hasegawa [11], the word ‘pure’ in a pure frame (and a pure context to be introduced soon) is used to mean “no surrounding reset constructs”, not whether control effects are used or not.

  4. 4.

    On the other hand, it looks difficult to prove the progress property in this formalization, because the relational definition of substitution prohibits us from extracting the result of substitution (reduct) from the higher-order representation of a redex. This is not a problem since we do not need the progress property.

  5. 5.

    We conjecture that the same lemma holds not only for a \(\texttt {shift}\) construct but also for an arbitrary expression. We have not formalized the general case yet, however.

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Acknowledgements

We would like to thank Youyou Cong and anonymous reviewers for valuable comments and feedbacks. This work was partly supported by JSPS KAKENHI under Grant No. JP18H03218.

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Authors and Affiliations

  1. Ochanomizu University, Tokyo, Japan

    Chiaki Ishio & Kenichi Asai

Authors
  1. Chiaki Ishio
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  2. Kenichi Asai
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Correspondence to Chiaki Ishio or Kenichi Asai .

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Editors and Affiliations

  1. University of British Columbia, Vancouver, BC, Canada

    William J. Bowman

  2. University of British Columbia, Vancouver, BC, Canada

    Ronald Garcia

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Ishio, C., Asai, K. (2020). Verifying Selective CPS Transformation for Shift and Reset. In: Bowman, W., Garcia, R. (eds) Trends in Functional Programming. TFP 2019. Lecture Notes in Computer Science(), vol 12053. Springer, Cham. https://doi.org/10.1007/978-3-030-47147-7_3

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  • DOI: https://doi.org/10.1007/978-3-030-47147-7_3

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-030-47147-7

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