higherkindness / droste

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

Version Matrix


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"


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._
import cats.implicits._

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(
scala> fib(0)
res0: BigDecimal = 0

scala> fib(1)
res1: BigDecimal = 1

scala> fib(2)
res2: BigDecimal = 1

scala> fib(10)
res3: BigDecimal = 55

scala> 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)
scala> fibAlt(0)
res5: BigDecimal = 0

scala> fibAlt(1)
res6: BigDecimal = 1

scala> fibAlt(2)
res7: BigDecimal = 1

scala> fibAlt(10)
res8: BigDecimal = 55

scala> 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(
scala> sumSquares(0)
res11: BigDecimal = 0

scala> sumSquares(1)
res12: BigDecimal = 1

scala> sumSquares(2)
res13: BigDecimal = 5

scala> sumSquares(10)
res14: BigDecimal = 385

scala> sumSquares(100)
res15: 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) =
    fibAlgebra.gather(Gather.histo) zip
scala> fused(0)
res16: (BigDecimal, BigDecimal) = (0,0)

scala> fused(1)
res17: (BigDecimal, BigDecimal) = (1,1)

scala> fused(2)
res18: (BigDecimal, BigDecimal) = (1,5)

scala> fused(10)
res19: (BigDecimal, BigDecimal) = (55,385)

scala> fused(100)
res20: (BigDecimal, BigDecimal) = (354224848179261915075,338350)

Droste includes athema, a math expression parser/processor, as a more extensive example of recursion schemes.


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

Copyright the maintainers, 2018-present.

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.


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.