In this post I look at some of the traversal patterns' functional implementations using scalaz. In the paper on applicative functors, McBride and Paterson defines traverse as an applicative mapping operation ..

traverse :: Applicative f => (a -> f b) -> [a] -> f [b]

Gibbons et. al. uses this abstraction to study various traversal structures in the presence of *effects*. The paper starts with a C# code snippet that uses the syntax sugar of foreach to traverse over a collection of elements .

public static int loop<MyObj> (IEnumerable<MyObj> coll){

int n = 0;

foreach (MyObj obj in coll){

n = n+1;

obj.touch();

}

return n;

}

In the above loop method, we do two things simultaneously :-

- mapping - doing some operation touch() on the elements of coll with the expectation that we get the modified collection at the end of the loop
- accumulating - counting the elements, which is a stateful operation for each iteration and which is independent of the operation which we do on the elements

Over the last weekend I was exploring how much of these effectful functional traversals can be done using scalaz, the closest to Haskell you can get with Scala. Section 4.2 of the original paper talks about two definite patterns of effectful traversal. Both of these patterns combine mapping and accumulation (like the C# code above) but separates the concerns skillfully using functional techniques. Let's see how much of that we can manage with scalaz functors.

**Pattern #1**

The first pattern of traversal accumulates elements *effectfully*, but modifies the elements of the collection *purely* and *independently of this accumulation*. Here's the scalaz implementation of collect (see the original paper for the haskell implementation) .

def collect[T[_]:Traverse, A, B, S](f: A => B, t: T[A], g: S => S) =

t.traverse[({type λ[x] = State[S,x]})#λ, B](a => state((s: S) => (g(s), f(a))))

To the uninitiated, the type annotation in traverse looks ugly - it's there because scalac cannot infer partial application of type constructors, a problem which will be rectified once Adriaan fixes issue 2712 on the Scala Trac.

Traverse is one of the functors in scalaz similar to the model of Data.Traversable in Haskell.

trait Traverse[T[_]] extends Functor[T] {

def traverse[F[_] : Applicative, A, B](f: A => F[B], t: T[A]): F[T[B]]

import Scalaz._

override def fmap[A, B](k: T[A], f: A => B) = traverse[Identity, A, B](f(_), k)

}

and scalaz defines implementations of the Traverse typeclass for a host of classes on which you can invoke traverse.

The above implmentation uses the State monad to handle effectful computations. For an introduction to the State monad in scalaz, have a look at this post from Tony Morris.

Note, f is the pure function that maps on the elements of the collection, g is the function that does the effectful accumulation through the State monad. Using collect, here's a version of the C# loop method that we did at the beginning ..

val loop = collect((a: Int) => 2 * a, List(10, 20, 30, 40), (i: Int) => i + 1)

loop(0) should equal((4, List(20, 40, 60, 80)))

Now we have the effectful iteration without using any mutable variables.

**Pattern #2**

The second pattern of traversal modifies elements *purely* but dependent on some *state* that evolves independently of the elements. Gibbons et. al. calls this abstraction disperse, whose scalaz implementation can be as follows ..

def disperse[T[_]: Traverse, A, S, B](t: T[A], s: A => State[S, B]) =

t.traverse[({type λ[x] = State[S,x]})#λ, B](s)

Note how the elements of the collection are being modified through the State monad. Using disperse, we can write a labeling function that labels every element with its position in order of traversal ..

def label[T[_]: Traverse, A](t: T[A]) =label(List(10, 20, 30, 40)) should equal(List(0, 1, 2, 3))

disperse(t, ((a: A) => state((i: Int) => (i+1, i)))) ! 0

disperse can also be used to implement the wordCount example that ships with scalaz distribution. Actually it counts the number of characters and lines in a stream.

def charLineCount[T[_]:Traverse](t: T[Char]) =charLineCount("the cat in the hat\n sat on the mat\n".toList).last should equal((35, 2))

disperse(t, ((a: Char) => state((counts: (Int, Int)) =>

((counts._1 + 1, counts._2 + (if (a == '\n') 1 else 0)), (counts._1, counts._2))))) ! (1,1)

*From http://debasishg.blogspot.com/2011/01/iteration-in-scala-effectful-yet.html *

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