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Certifying CPS Transformation of Let-Polymorphic Calculus Using PHOAS

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Programming Languages and Systems (APLAS 2018)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 11275))

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Abstract

This paper formalizes the correctness of a one-pass CPS transformation for the lambda calculus extended with let-polymorphism. We prove in Agda that equality is preserved through the CPS transformation. Parameterized higher-order abstract syntax is used to represent binders both at the term level and the type level. Unlike the previous work based on denotational semantics, we use small-step operational semantics to formalize the equality. Thanks to the small-step formalization, we can establish the correctness without any hypothesis on the well-formedness of input terms. The resulting formalization is simple enough to serve as a basis for more complex CPS transformations such as selective one for a calculus with delimited control operators.

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Notes

  1. 1.

    We follow the notation employed by Chlipala [6].

  2. 2.

    The two cases, \(\lfloor {e}\rfloor _{\mathsf {S}}\) and \(\lfloor {e}\rfloor _{\mathsf {D}}\), correspond to \([\![{e}]\!]\) and \([\![{e}]\!]'\) in Danvy and Filinski [10].

  3. 3.

    Values and terms correspond to serious expressions and trivial expressions in Danvy’s notation, respectively. Besides the introduction of let terms, our notation differs from Danvy’s in that we allow a term of the form \({c}\,@^{\mathsf {K}}\,{k}\) where c is not necessarily a continuation variable. We need the new form during the correctness proof.

  4. 4.

    As for continuations and terms, we keep the type after the CPS transformation. Since the answer type is always \(\mathsf {Nat}\), we could elide it and write the type of continuations and terms as \(\lnot \tau \) and \(\bot \), respectively.

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Acknowledgements

We would like to thank the reviewers for useful and constructive comments. This work was partly supported by JSPS KAKENHI under Grant No. JP18H03218.

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Authors and Affiliations

  1. Ochanomizu University, Tokyo, Japan

    Urara Yamada & Kenichi Asai

Authors
  1. Urara Yamada
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  2. Kenichi Asai
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Correspondence to Urara Yamada .

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  1. Korea Advanced Institute of Science and Technology, Daejeon, Korea (Republic of)

    Sukyoung Ryu

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Yamada, U., Asai, K. (2018). Certifying CPS Transformation of Let-Polymorphic Calculus Using PHOAS. In: Ryu, S. (eds) Programming Languages and Systems. APLAS 2018. Lecture Notes in Computer Science(), vol 11275. Springer, Cham. https://doi.org/10.1007/978-3-030-02768-1_20

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  • DOI: https://doi.org/10.1007/978-3-030-02768-1_20

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-02767-4

  • Online ISBN: 978-3-030-02768-1

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