Recursion Schemes

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

Introduction to pattern functors, fix points, anamorphisms, catamorphisms, paramorphisms and hylomorphisms, requiring very little prior knowledge.

Three-part series in which you discover recursion schemes from scratch and implement a small subset of Edward Kmett's library.

Bartosz Milewski explains F-algebras and shows how to use them in the context of catamorphisms.

Series introducing recursion schemes and related concepts in JavaScript, aimed at developers with a minimal functional programming background.

List of various recursion schemes with code samples.

Definition on the Haskell Wiki.

Short definition with code on School of Haskell by Edward Kmett.

Using a hylomorphism to rotate a quadtree by Jared Tobin.

Practical examples of pre- and postpromorphisms by Jared Tobin.

Exploring histo and futu by example by Jared Tobin.

Practical article about histomorphism and the futumorphism.

Map of various recursion schemes and their duals.

Showing that the Visitor pattern implements an f-algebra for use with a catamorphism (in Java).

Introduces the fixpoint combinator, anamorphism, catamorphism, hylomorphism, paramorphism, apomorphism, histomorphism, dynamorphism and futumorphism.

Introduces catamorphism as a generalization of fold.

Advent of Code 2017, Domino challenge

Advent of Code 2018, String comparison challenge

The original paper most of this is based on.

Shows that many basic sorting algorithms exist as a pair, and that these pairs arise naturally out of the duality between folds and unfolds.

Shows how paramorphisms and apomorphisms can be used for more efficient implementations of sorting algorithms.

Design pattern for writing programs that traverse data structures built from rich mutually-recursive data types.

"Recursion Schemes by Example" and "Exotic Tools for Exotic Trades" provide concise definitions as well as practical examples of many recursion schemes.

12 min presentation by Ralf Hinze, Nicolas Wu, and Jeremy Gibbons.

Presented by Tim Williams at the London Haskell meetup.

Presented by Anthony Burzillo at the NYC Haskell User's Group.

Presented by Jean Remi Desjardins at Lambdaconf 2016.

Collection of slide decks about recursion schemes.

High-level talk about structuring programs with coproducts and recursion schemes by Patrick Thomson.

Introduction to recursive fix point data structures and recursion schemes in Haskell and Scala by Greg Pfeil.

Bartosz Milewski's article in talk form, presented at LambdaCon.

Zainab Ali's Introductory talk presented at LambdaWorld.

Overview of different morphisms including a printable PDF.

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

Podcast about Scala that brings up recursion schemes in the second episode.

Recursion schemes come up in Episode 13 with John Wiegley.

for Haskell - The canonical implementation by Edward Kmett.

for Scala using Scalaz - Generalized folds, unfolds, and traversals for fixed point data structures.

for Scala using Cats - Generalized folds, unfolds, and traversals for fixed point data structures.

for Idris, based off Edward Kmett's Haskell library.

for PureScript - Work-in-process port of matryoshka.

for ATS - Demonstration of recursion schemes in ATS.

for Dhall - a library for recursion schemes in Dhall.

for JavaScript/Flow - Schemes for data structures written in the style of flow-static-land.

for Kotlin - a re-envisioning based off Matryoshka using lightweight higher kinded polymorphism.