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.

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.
@JosephJnk/recursion-schemes-in-javascript-and-flow-with-static-land-recursision-schemes-97cf10599fb7">Recursion Schemes in JavaScript and Flow -
Series introducing recursion schemes and related concepts in JavaScript,
aimed at developers with a minimal functional programming background.

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.

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

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.

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.

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

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.

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.

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