Awesome Recursion Schemes

A curation of useful resources for learning about and using recursion schemes.

Recursion schemes are simple, composable combinators, that automate the process of traversing and recursing through nested data structures.

Contents

• Introductions
• Articles
• Papers
• Presentations
• Podcasts
• Implementations

Introductions

• Practical Recursion Schemes - Introduction to pattern functors, fix points, anamorphisms, catamorphisms, paramorphisms and hylomorphisms, requiring very little prior knowledge.
• An Introduction to Recursion Schemes - Three-part series in which you discover recursion schemes from scratch and implement a small subset of Edward Kmett's library.
• Understanding Algebras - Bartosz Milewski explains F-algebras and shows how to use them in the context of catamorphisms.
• Recursion Schemes in JavaScript and Flow - Series introducing recursion schemes and related concepts in JavaScript, aimed at developers with a minimal functional programming background.

Articles

• Recursion Schemes: A Field Guide (Redux) - List of various recursion schemes with code samples.
• Catamorphisms - Definition on the Haskell Wiki.
• Catamorphisms - Short definition with code on School of Haskell by Edward Kmett.
• Rotating Squares - Using a hylomorphism to rotate a quadtree by Jared Tobin.
• Recursion Schemes, Part V: Hello, Hylomorphisms
• Promorphisms, Pre and Post - Pratical examples of pre- and postpromorphisms by Jared Tobin.
• Time Traveling Recursion Schemes - Exploring histo and futu by example by Jared Tobin.
• Recursion Schemes, Part IV: Time is of the Essence - Practical article about histomorphism and the futumorphism.
• Cheat Sheet - Map of various recursion schemes and their duals.
• Correcting the Visitor pattern - Showing that the Visitor pattern implements an f-algebra for use with a catamorphism (in Java).

Hylomorphisms in the Wild

Articles by Bartosz Milewski about solving small, practical problems by applying a hylomorphism.

• Stalking a Hylomorphism in the Wild - Advent of Code 2017, Domino challenge
• Open Seasons on Hylomorphisms - Advent of Code 2018, String comparison challenge

Papers

• Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire, 1991, Meijer et al. - The original paper most of this is based on.
• A Duality of Sorts, 2013, Hinze et al. - Shows that many basic sorting algorithms exist as a pair, and that these pairs arise naturally out of the duality between folds and unfolds.
• Sorting with Bialgebras and Distributive Laws, 2012, Hinze et al. - Shows how paramorphisms and apomorphisms can be used for more efficient implementations of sorting algorithms.
• Scrap your boilerplate: a practical design pattern for generic programming, 2003, SPJ et al. - Design pattern for writing programs that traverse data structures built from rich mutually-recursive data types.

Presentations

• Slidedecks by Tim Philip Williams - "Recursion Schemes by Example" and "Exotic Tools for Exotic Trades" provide concise definitions as well as practical examples of many recursion schemes.
• Unifying Structured Recursion Schemes - 12 min presentation by Ralf Hinze, Nicolas Wu, and Jeremy Gibbons.
• Recursion Schemes - Presented by Tim Williams at the London Haskell meetup.
• F-algebras or: How I Learned to Stop Worrying and Love the Type System - Presented by Anthony Burzillo at the NYC Haskell User's Group.
• A Gentle Introduction to Recursion Schemes - Presented by Jean Remi Desjardins at Lambdaconf 2016.
• recursion-scheme-talk - Collection of slide decks about recursion schemes.
• Bracer: Transforming Real-World Languages with Coproducts and Recursion Schemes - High-level talk about structuring programs with coproducts and recursion schemes by Patrick Thomson.
• Recursion: Where Functional Programming Hits Bottom - Introduction to recursive fix point data structures and recursion schemes in Haskell and Scala by Greg Pfeil.
• Programming with algebras - Bartosz Milewski's article in talk form, presented at LambdaCon.
• Peeling the Banana: Recursion Schemes from First Principles - Zainab Ali's Introductory talk presented at LambdaWorld.

Cheat Sheets

• The Hitchhiker's Guide to Morphisms - Overview of different morphisms including a printable PDF.

Podcasts

• Magic Read Along - Casual discussions about category theory that often bring up recursion schemes, including episode 33 which talks about Histomorphisms and Futumorphisms.

Implementations

• recursion-schemes for Haskell - The canonical implementation by Edward Kmett.
• Matryoshka for Scala - Generalized folds, unfolds, and traversals for fixed point data structures.
• recursion_schemes for Idris, based off Edward Kmett's Haskell library.
• purescript-matryoshka for PureScript - Work-in-process port of matryoshka.
• recursion for ATS - Demonstration of recursion schemes in ATS.
• dada for Dhall - a library for recursion schemes in Dhall.
• static-land-recursion-schemes for JavaScript/Flow - Schemes for data structures written in the style of flow-static-land.
• Katalyst for Kotlin - a re-envisioning based off Matryoshka using lightweight higher kinded polymorphism.
• Groupoid Infinity recursion schemes for cubicaltt - a cubical version of recursion schemes.

License

This content is licensed under CC0.