# 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
- Cheat Sheets
- Podcasts
- Implementations

## Introductions

- Awesome Recursion Schemes - A curation of useful resources for learning about and using recursion schemes.
- 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 - Practical 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).
- Recursion Schemes in Scala - Introduces the fixpoint combinator, anamorphism, catamorphism, hylomorphism, paramorphism, apomorphism, histomorphism, dynamorphism and futumorphism.
- What’s in a Fold: The Basic Catamorphism in recursion-schemes - Introduces catamorphism as a generalization of fold.

### 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.
- Scala Love - Podcast about Scala that brings up recursion schemes in the second episode.
- The Haskell Cast - Recursion schemes come up in Episode 13 with John Wiegley.

## Implementations

- recursion-schemes for Haskell - The canonical implementation by Edward Kmett.
- Matryoshka for Scala using Scalaz - Generalized folds, unfolds, and traversals for fixed point data structures.
- andyscott/droste for Scala using Cats - 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.

## License

This content is licensed under CC0.