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--- | ||
layout: docs | ||
title: "Arrow" | ||
section: "typeclasses" | ||
source: "core/src/main/scala/cats/arrow/Arrow.scala" | ||
scaladoc: "#cats.arrow.Arrow" | ||
--- | ||
# Arrow | ||
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`Arrow` is a type class for modeling something that behaves like functions, such as `Function1`: `A => B`, `Kleisli`: `A => F[B]` (also known as `ReaderT`), `Cokleisli`: `F[A] => B`, etc. So useful, that Haskell provides special syntax (the `proc` notation) for composing and combining Arrows, similar as the `do` notation for sequencing monadic operations. | ||
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To create an `Arrow` instance for a type `F[A, B]`, the following abstract methods need to be implemented: | ||
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```tut:book:silent | ||
trait Arrow[F[_, _]] { | ||
def lift[A, B](f: A => B): F[A, B] | ||
def id[A]: F[A, A] | ||
def compose[A, B, C](f: F[B, C], g: F[A, B]): F[A, C] | ||
def first[A, B, C](fa: F[A, B]): F[(A, C), (B, C)] | ||
} | ||
``` | ||
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Once a type `F` has an `Arrow` instance, it gets for free a number of methods for composing and combining with other `Arrow`s. You will be able to do things like: | ||
- Composing `fab: F[A, B]` and `fbc: F[B, C]` into `fac: F[A, C]` with `fab >>> fbc`. If `F` is `Function1` then `>>>` becomes alias of `andThen`. | ||
- Taking two arrows `fab: F[A, B]` and `fbc: F[B, C]` and combining them into `F[(A, C) => (B, D)]` with `fab *** fbc`. The resulting arrow takes two inputs and processes them with two arrows, one for each input. | ||
- Taking two arrows `fab: F[A, B]` and `fac: F[A, C]` and combining them into `F[A => (B, C)]` with `fab &&& fbc`. The resulting arrow takes an input, duplicates it and processes each copy with a different arrow. | ||
- Taking an arrow `fab: F[A, B]` and turning it into `F[(C, A), (C, B)]` with `fab.second`. The resulting arrow takes two inputs, processes the second input and leaves the first input as it is. | ||
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## Example | ||
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As an example, let's create a fancy version of `Function1` that is capable of maintaining states. Let's call it `FancyFunction`: | ||
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```tut:book:silent | ||
case class FancyFunction[A, B](run: A => (FancyFunction[A, B], B)) | ||
``` | ||
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That is, given an `A`, it not only returns a `B`, but also returns a new `FancyFunction[A, B]`. This sounds similar as the `State` monad (which returns a result and a new `State` from an initial `State`), and indeed, `FancyFunction` can be used to perform stateful transformations. | ||
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Here's an `Arrow` instance for `FancyFunction`: | ||
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```tut:book:silent | ||
import cats.arrow.Arrow | ||
import cats.implicits._ | ||
implicit val arrowInstance: Arrow[FancyFunction] = new Arrow[FancyFunction] { | ||
override def lift[A, B](f: A => B): FancyFunction[A, B] = FancyFunction(lift(f) -> f(_)) | ||
override def first[A, B, C](fa: FancyFunction[A, B]): FancyFunction[(A, C), (B, C)] = FancyFunction {case (a, c) => | ||
val (fa2, b) = fa.run(a) | ||
(first(fa2), (b, c)) | ||
} | ||
override def id[A]: FancyFunction[A, A] = FancyFunction(id -> _) | ||
override def compose[A, B, C](f: FancyFunction[B, C], g: FancyFunction[A, B]): FancyFunction[A, C] = FancyFunction {a => | ||
val (gg, b) = g.run(a) | ||
val (ff, c) = f.run(b) | ||
(gg >>> ff, c) | ||
} | ||
} | ||
``` | ||
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Then we can start to do interesting things with it. For example, we can create a method `accum` that returns a `FancyFunction`, which accumulates values fed to it using the accumulation function `f` and the starting value `b`: | ||
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```tut:book:silent | ||
def accum[A, B](b: B)(f: (A, B) => B): FancyFunction[A, B] = FancyFunction {a => | ||
val b2 = f(a, b) | ||
(accum(b2)(f), b2) | ||
} | ||
``` | ||
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We then define a method `runList` that takes a `List[A]`, feeds the `A` values one by one into the `FancyFunction` and collects the result into a `List[B]`: | ||
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```tut:book:silent | ||
def runList[A, B](ff: FancyFunction[A, B], as: List[A]): List[B] = as match { | ||
case h :: t => | ||
val (ff2, b) = ff.run(h) | ||
b :: runList(ff2, t) | ||
case _ => List() | ||
} | ||
``` | ||
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Now we can try it out: | ||
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```tut:book:silent | ||
runList(accum[Int, Int](0)(_ + _), List(6, 5, 4, 3, 2, 1)) | ||
// res1: List[Int] = List(6, 11, 15, 18, 20, 21) | ||
``` | ||
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As another example, let's make the above `FancyFunction` collect the running average of a list of numbers. To do so it needs to keep track of the count in addition to the sum. In this example we will combine several `FancyFunction`s to get the `FancyFunction` we want. | ||
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We first define arrow `sum` in terms of `accum`, and define arrow `count` by composing `_ => 1` with `sum`: | ||
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```tut:book:silent | ||
import cats.kernel.Monoid | ||
def sum[A: Monoid]: FancyFunction[A, A] = accum(Monoid[A].empty)(_ |+| _) | ||
def count[A]: FancyFunction[A, Int] = Arrow[FancyFunction].lift((_: A) => 1) >>> sum | ||
``` | ||
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Then we define a `combine` function that takes an input and processes two copies of it with two arrows, and finally, the `avg` arrow in terms of the arrows we have so far: | ||
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```tut:book:silent | ||
def combine[F[_, _]: Arrow, A, B, C](fab: F[A, B], fac: F[A, C]): F[A, (B, C)] = | ||
Arrow[F].lift((a: A) => (a, a)) >>> (fab *** fac) | ||
def avg: FancyFunction[Int, Double] = | ||
combine(sum[Int], count[Int]) >>> Arrow[FancyFunction].lift{case (x, y) => x.toDouble / y} | ||
``` | ||
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And try it out: | ||
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```tut:book:silent | ||
runList(avg, List(1, 10, 100, 1000)) | ||
// res2: List[Int] = List(1.0, 5.5, 37.0, 277.75) | ||
``` |