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Fantastic Morphisms and Where to Find Them

A Guide to Recursion Schemes

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Mathematics of Program Construction (MPC 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13544))

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Abstract

Structured recursion schemes have been widely used in constructing, optimizing, and reasoning about programs over inductive and coinductive datatypes. Their plain forms, catamorphisms and anamorphisms, are restricted in expressivity. Thus many generalizations have been proposed, which further led to several unifying frameworks of structured recursion schemes. However, the existing work on unifying frameworks typically focuses on the categorical foundation, and thus is perhaps inaccessible to practitioners who are willing to apply recursion schemes in practice but are not versed in category theory. To fill this gap, this expository paper introduces structured recursion schemes from a practical point of view: a variety of recursion schemes are motivated and explained in contexts of concrete programming examples. The categorical duals of these recursion schemes are also explained.

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Notes

  1. 1.

    It requires the DeriveFunctor extension of ghc to derive functors automatically.

  2. 2.

    The recursion scheme requires the ghc extension RankNTypes since the first argument involves a polymorphic function.

  3. 3.

    The name mutumorphism is a bit special in the zoo of recursion schemes: the prefix mutu- is from Latin rather than Greek.

  4. 4.

    Support of gadts is turned on by the extension gadt s in ghc.

  5. 5.

    It requires the RankNTypes extension of ghc.

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Acknowledgements

Particular thanks are due to Jeremy Gibbons for his numerous suggestions and comments. We would also like to thank the reviewers for the efforts in helping us to improve this paper.

We dedicate this paper to the memory of Richard Bird.

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  1. Imperial College London, London, UK

    Zhixuan Yang & Nicolas Wu

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  1. Zhixuan Yang
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Correspondence to Zhixuan Yang .

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  1. School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh, UK

    Ekaterina Komendantskaya

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Yang, Z., Wu, N. (2022). Fantastic Morphisms and Where to Find Them. In: Komendantskaya, E. (eds) Mathematics of Program Construction. MPC 2022. Lecture Notes in Computer Science, vol 13544. Springer, Cham. https://doi.org/10.1007/978-3-031-16912-0_9

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