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A Sound and Complete Bisimulation for Contextual Equivalence in \(\lambda \)-Calculus with Call/cc

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Programming Languages and Systems (APLAS 2016)

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Abstract

We develop a sound and complete proof method of contextual equivalence in \(\lambda \)-calculus with the abortive control operator call/cc (as opposed to delimited control operators like \(\mathtt {shift}\) and \(\mathtt {reset}\)), and prove the non-trivial equivalence between \(\lambda f.\,f()\) and \(\lambda f.\,f();f()\) for example, both for the first time to our knowledge. Although our method is based on environmental bisimulations (Sumii et al. 2004-), it makes an essential and general change to their metatheory, which is not only necessary for handling call/cc but is also applicable in other languages with no control operator.

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Notes

  1. 1.

    http://www.smlnj.org/doc/SMLofNJ/pages/cont.html

  2. 2.

    http://pauillac.inria.fr/~xleroy/software.html#callcc

  3. 3.

    In fact, we finished most of our proofs before discovering his unpublished manuscript [23], often reinventing essential ideas that are specific to each example.

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Acknowledgments

We thank Oleg Kiselyov for informing us of the problem of contextual equivalence between M and M; M, and the anonymous reviewers for useful comments. This research and the authors are partially supported by KAKENHI JP16K12409, JP15H02681, JP25540001, and JP22300005.

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  1. Tohoku University, Sendai, Japan

    Taichi Yachi & Eijiro Sumii

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  1. Taichi Yachi
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  2. Eijiro Sumii
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Correspondence to Taichi Yachi or Eijiro Sumii .

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Editors and Affiliations

  1. Kyoto University , Kyoto, Japan

    Atsushi Igarashi

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Yachi, T., Sumii, E. (2016). A Sound and Complete Bisimulation for Contextual Equivalence in \(\lambda \)-Calculus with Call/cc . In: Igarashi, A. (eds) Programming Languages and Systems. APLAS 2016. Lecture Notes in Computer Science(), vol 10017. Springer, Cham. https://doi.org/10.1007/978-3-319-47958-3_10

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  • DOI: https://doi.org/10.1007/978-3-319-47958-3_10

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