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\(\lambda \) to SKI, Semantically

Declarative Pearl

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Functional and Logic Programming (FLOPS 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10818))

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Abstract

We present a technique for compiling lambda-calculus expressions into SKI combinators. Unlike the well-known bracket abstraction based on (syntactic) term re-writing, our algorithm relies on a specially chosen, compositional semantic model of generally open lambda terms. The meaning of a closed lambda term is the corresponding SKI combination. For simply-typed as well as unityped terms, the meaning derivation mirrors the typing derivation. One may also view the algorithm as an algebra, or a non-standard evaluator for lambda-terms (i.e., denotational semantics).

The algorithm is implemented as a tagless-final compiler for (uni)typed lambda-calculus embedded as a DSL into OCaml. Its type preservation is clear even to OCaml. The correctness of both the algorithm and of its implementation becomes clear.

Our algorithm is easily amenable to optimizations. In particular, its output and the running time can both be made linear in the size (i.e., the number of all constructors) of the input De Bruijn-indexed term.

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Notes

  1. 1.

    By ‘weakening’ we mean a (structural) inference rule stating that adding more premises to hypotheses of a valid logical deduction preserves the validity.

  2. 2.

    As we will see in Sect. 6, it is enough to keep the length of \(\varGamma \), that is, the number of free variables in a term.

  3. 3.

    It is natural to wish a denotation of an open term be non-divergent: if \(\tau _n,\ldots ,\tau _1\models d\) then d, until applied to n other terms, should have only a finite number of reductions, if any at all. The wish is already granted: in the present simply-typed calculus, all terms are (strongly) normalizing. We have to wait until Sect. 6 to say something non-trivial about termination.

  4. 4.

    The optimal solution is described in Sect. 6.

  5. 5.

    http://okmij.org/ftp/tagless-final/course/optimizations.html.

  6. 6.

    It is worth pointing out one, comprehensive web page: http://www.cantab.net/users/antoni.diller/brackets/intro.html.

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Acknowledgments

I thank Yukiyoshi Kameyama for his challenge to write the SK conversion in the tagless-final style, and helpful discussions. I am very grateful to Doaitse Swierstra, Fritz Henglein and Noam Zeilberger for many helpful comments and discussions. Numerous suggestions by anonymous reviewers have greatly helped improve the presentation.

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Authors and Affiliations

  1. Tohoku University, Sendai, Japan

    Oleg Kiselyov

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  1. Oleg Kiselyov
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Correspondence to Oleg Kiselyov .

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Editors and Affiliations

  1. Roskilde University, Roskilde, Denmark

    John P. Gallagher

  2. Karlsruhe University of Applied Sciences, Karlsruhe, Germany

    Martin Sulzmann

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Kiselyov, O. (2018). \(\lambda \) to SKI, Semantically. In: Gallagher, J., Sulzmann, M. (eds) Functional and Logic Programming. FLOPS 2018. Lecture Notes in Computer Science(), vol 10818. Springer, Cham. https://doi.org/10.1007/978-3-319-90686-7_3

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  • DOI: https://doi.org/10.1007/978-3-319-90686-7_3

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-90685-0

  • Online ISBN: 978-3-319-90686-7

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