higherkindness/droste

recursion schemes for cats; to iterate is human, to recurse, divine

Source code at GitHub https://github.com/higherkindness/droste

{
  "createdAt": "2018-01-12T08:17:00Z",
  "defaultBranch": "main",
  "description": "recursion schemes for cats; to iterate is human, to recurse, divine",
  "fullName": "higherkindness/droste",
  "homepage": "",
  "language": "Scala",
  "name": "droste",
  "pushedAt": "2025-09-08T05:22:05Z",
  "stargazersCount": 393,
  "topics": [
    "functional-programming",
    "recursion-schemes",
    "scala"
  ],
  "updatedAt": "2025-09-21T21:40:58Z",
  "url": "https://github.com/higherkindness/droste"
}

Droste

Droste is a recursion library for Scala.

SBT installation

Select a tagged release version (x.y.z) and then add the following to your SBT build:

libraryDependencies += "io.higherkindness" %% "droste-core" % "x.y.z"

Usage

Droste makes it easy to assemble morphisms. For example, calculating Fibonacci values can be done with a histomorphism if we model natural numbers as a chain of Option. We can easily unfold with an anamorphism and then fold to our result with a histomorphism.

import higherkindness.droste._
import higherkindness.droste.data._

val natCoalgebra: Coalgebra[Option, BigDecimal] =
  Coalgebra(n => if (n > 0) Some(n - 1) else None)

val fibAlgebra: CVAlgebra[Option, BigDecimal] = CVAlgebra {
  case Some(r1 :< Some(r2 :< _)) => r1 + r2
  case Some(_ :< None)           => 1
  case None                      => 0
}

val fib: BigDecimal => BigDecimal = scheme.ghylo(
  fibAlgebra.gather(Gather.histo),
  natCoalgebra.scatter(Scatter.ana))

fib(0)
// res0: BigDecimal = 0
fib(1)
// res1: BigDecimal = 1
fib(2)
// res2: BigDecimal = 1
fib(10)
// res3: BigDecimal = 55
fib(100)
// res4: BigDecimal = 354224848179261915075

An anamorphism followed by a histomorphism is also known as a dynamorphism. Recursion scheme animals like dyna are available in the zoo:

val fibAlt: BigDecimal => BigDecimal =
  scheme.zoo.dyna(fibAlgebra, natCoalgebra)

fibAlt(0)
// res5: BigDecimal = 0
fibAlt(1)
// res6: BigDecimal = 1
fibAlt(2)
// res7: BigDecimal = 1
fibAlt(10)
// res8: BigDecimal = 55
fibAlt(100)
// res9: BigDecimal = 354224848179261915075

What if we want to do two things at once? Let’s calculate a Fibonacci value and the sum of all squares.

val fromNatAlgebra: Algebra[Option, BigDecimal] = Algebra {
  case Some(n) => n + 1
  case None    => 0
}

// note: n is the fromNatAlgebra helper value from the previous level of recursion
val sumSquaresAlgebra: RAlgebra[BigDecimal, Option, BigDecimal] = RAlgebra {
  case Some((n, value)) => value + (n + 1) * (n + 1)
  case None             => 0
}

val sumSquares: BigDecimal => BigDecimal = scheme.ghylo(
  sumSquaresAlgebra.gather(Gather.zygo(fromNatAlgebra)),
  natCoalgebra.scatter(Scatter.ana))

sumSquares(0)
// res10: BigDecimal = 0
sumSquares(1)
// res11: BigDecimal = 1
sumSquares(2)
// res12: BigDecimal = 5
sumSquares(10)
// res13: BigDecimal = 385
sumSquares(100)
// res14: BigDecimal = 338350

Now we can zip the two algebras into one so that we calculate both results in one pass.

val fused: BigDecimal => (BigDecimal, BigDecimal) =
  scheme.ghylo(
    fibAlgebra.gather(Gather.histo) zip
    sumSquaresAlgebra.gather(Gather.zygo(fromNatAlgebra)),
    natCoalgebra.scatter(Scatter.ana))

fused(0)
// res15: (BigDecimal, BigDecimal) = (0, 0)
fused(1)
// res16: (BigDecimal, BigDecimal) = (1, 1)
fused(2)
// res17: (BigDecimal, BigDecimal) = (1, 5)
fused(10)
// res18: (BigDecimal, BigDecimal) = (55, 385)
fused(100)
// res19: (BigDecimal, BigDecimal) = (354224848179261915075, 338350)

Droste includes [athema]!(athema), a math expression parser/processor, as a more extensive example of recursion schemes.

Credits

A substantial amount of Droste’s code is a derivation— or an alternative encoding— of patterns pioneered by others. Droste has benefited from the excellent work in many other recursion libraries, blog posts, academic papers, etc. Notably, Droste has benefited from:

Thank you to everyone involved. Additionally, thanks to Greg Pfeil (@sellout) for answering my random questions over the last few years while I’ve been slowly learning (and using recursion) schemes.

Copyright and License

All code is available to you under the Apache License, Version 2.0, available at http://www.apache.org/licenses/LICENSE-2.0.

Logos provided by the very excellent @impurepics.

Disclamer

Please be advised that I have no idea what I am doing. Nevertheless, this project is already being used for real work with real data in real life.