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*-Morpisms – Home
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Einrichtung Android-Smartphones
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*-Morpisms
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<a class='hover:underline truncate' title='Monoid? Da war doch was…' href='Haskell/Code%20Snippets/Monoid'>
Monoid? Da war doch was…
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<a class='hover:underline truncate' title='Fortgeschrittene funktionale Programmierung in Haskell' href='Haskell/FFPiH'>
Fortgeschrittene funktionale Programmierung in Haskell
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</svg>
<a class='hover:underline truncate' title='Lenses' href='Haskell/Lenses'>
Lenses
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</svg>
<a class='hover:underline truncate' title='Talks und Posts zu Haskell' href='Haskell/Advantages'>
Talks und Posts zu Haskell
</a>
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<h1 class='flex items-end justify-center mb-4 p-3 bg-red-100 text-5xl font-extrabold text-black rounded'>
<a class='z-40 tracking-tighter '>
*-Morpisms
</a>
</h1>
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<p class='mb-3'>
<strong>Backup eines Blogposts eines Kommilitonen:</strong>
</p>
<p class='mb-3'>
This weekend I spend some time on Morphisms.
</p>
<p class='mb-3'>
Knowing that this might sound daunting to many dabbling Haskellers (like I am), I decided to write a real short MergeSort hylomorphism quickstarter.
</p>
<hr class='mb-3' />
<p class='mb-3'>
For those who need a refresher: MergeSort works by creating a balanced binary tree from the input list and directly collapsing it back into itself while treating the children as sorted lists and merging these with an O(n) algorithm.
</p>
<hr class='mb-3' />
<p class='mb-3'>
First the usual prelude:
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>{-# LANGUAGE DeriveFunctor #-}
{-# LANGUAGE TypeFamilies #-}
import Data.Functor.Foldable
import Data.List (splitAt, unfoldr)</code></pre></div>
<hr class='mb-3' />
<p class='mb-3'>
We will use a binary tree like this. Note that there is no explicit recursion used, but <code class='py-0.5 px-0.5 bg-gray-100'>NodeF</code> has two <em>holes</em>. These will eventually filled later.
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>data TreeF c f = EmptyF | LeafF c | NodeF f f
deriving (Eq, Show, Functor)</code></pre></div>
<hr class='mb-3' />
<p class='mb-3'>
Aside: We could use this as a <em>normal</em> binary tree by wrapping it in <code class='py-0.5 px-0.5 bg-gray-100'>Fix</code>: <code class='py-0.5 px-0.5 bg-gray-100'>type Tree a = Fix (TreeF a)</code> But this would require us to write our tree like <code class='py-0.5 px-0.5 bg-gray-100'>Fix (NodeF (Fix (LeafF 'l')) (Fix (LeafF 'r')))</code> which would get tedious fast. Luckily Edward build a much better way to do this into <em>recursion-schemes</em>. I will touch on this later.
</p>
<hr class='mb-3' />
<p class='mb-3'>
Without further ado we start to write a Coalgebra, which in my book is just a scary name for “function that is used to construct datastructures”.
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>unflatten :: [a] -&gt; TreeF a [a]
unflatten ( []) = EmptyF
unflatten (x:[]) = LeafF x
unflatten ( xs) = NodeF l r where (l,r) = splitAt (length xs `div` 2) xs</code></pre></div>
<p class='mb-3'>
From the type signature it’s immediately obvious, that we take a list of ’a’s and use it to create a part of our tree.
</p>
<p class='mb-3'>
The nice thing is that due to the fact that we haven’t commited to a type in our tree nodes we can just put lists in there.
</p>
<hr class='mb-3' />
<p class='mb-3'>
Aside: At this point we could use this Coalgebra to construct (unsorted) binary trees from lists:
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>example1 = ana unflatten [1,3] == Fix (NodeF (Fix (LeafF 1)) (Fix (LeafF 3)))</code></pre></div>
<hr class='mb-3' />
<p class='mb-3'>
On to our sorting, tree-collapsing Algebra. Which again is just a creepy word for “function that is used to deconstruct datastructures”.
</p>
<p class='mb-3'>
The function <code class='py-0.5 px-0.5 bg-gray-100'>mergeList</code> is defined below and just merges two sorted lists into one sorted list in O(n), I would probably take this from the <code class='py-0.5 px-0.5 bg-gray-100'>ordlist</code> package if I were to implement this <em>for real</em>.
</p>
<p class='mb-3'>
Again we see that we can just construct our sorted output list from a <code class='py-0.5 px-0.5 bg-gray-100'>TreeF</code> that apparently contains just lists.
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>flatten :: Ord a =&gt; TreeF a [a] -&gt; [a]
flatten EmptyF = []
flatten (LeafF c) = [c]
flatten (NodeF l r) = mergeLists l r </code></pre></div>
<hr class='mb-3' />
<p class='mb-3'>
Aside: We could use a Coalgebra to deconstruct trees:
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>example2 = cata flatten (Fix (NodeF (Fix (LeafF 3)) (Fix (LeafF 1)))) == [1,3]</code></pre></div>
<hr class='mb-3' />
<p class='mb-3'>
Now we just combine the Coalgebra and the Algebra with one from the functions from Edwards <code class='py-0.5 px-0.5 bg-gray-100'>recursion-schemes</code> library:
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>mergeSort :: Ord a =&gt; [a] -&gt; [a]
mergeSort = hylo flatten unflatten
example3 = mergeSort [5,2,7,9,1,4] == [1,2,4,5,7,9] </code></pre></div>
<hr class='mb-3' />
<p class='mb-3'>
What have we gained?
</p>
<p class='mb-3'>
We have implemented a MergeSort variant in 9 lines of code, not counting the <code class='py-0.5 px-0.5 bg-gray-100'>mergeLists</code> function below. Not bad, but <a href='http://en.literateprograms.org/Merge_sort_(Haskell)' class='text-red-600 hover:underline' target='_blank' rel='noopener'>this implementation</a> is not much longer.
</p>
<p class='mb-3'>
On the other hand the morphism based implementation cleanly describes what happens during construction and deconstruction of our intermediate structure.
</p>
<p class='mb-3'>
My guess is that, as soon as the algortihms get more complex, this will really make a difference.
</p>
<hr class='mb-3' />
<p class='mb-3'>
At this point I wasn’t sure if this was useful or remotely applicable. Telling someone “I spend a whole weekend learning about Hylomorphism” isn’t something the cool developer kids do.
</p>
<p class='mb-3'>
It appeared to me that maybe I should have a look at the Core to see what the compiler finally comes up with (edited for brevity):
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'> mergeSort :: [Integer] -&gt; [Integer]
mergeSort =
\ (x :: [Integer]) -&gt;
case x of wild {
[] -&gt; [];
: x1 ds -&gt;
case ds of _ {
[] -&gt; : x1 ([]);
: ipv ipv1 -&gt;
unfoldr
lvl9
(let {
p :: ([Integer], [Integer])
p =
case $wlenAcc wild 0 of ww { __DEFAULT -&gt;
case divInt# ww 2 of ww4 { __DEFAULT -&gt;
case tagToEnum# (&lt;# ww4 0) of _ {
False -&gt;
case $wsplitAt# ww4 wild of _ { (# ww2, ww3 #) -&gt; (ww2, ww3) };
True -&gt; ([], wild)
}
}
} } in
(case p of _ { (x2, ds1) -&gt; mergeSort x2 },
case p of _ { (ds1, y) -&gt; mergeSort y }))
}
}
end Rec }</code></pre></div>
<p class='mb-3'>
While I am not really competent in reading Core and this is actually the first time I bothered to try, it is immediately obvious that there is no trace of any intermediate tree structure.
</p>
<p class='mb-3'>
This is when it struck me. I was dazzled and amazed. And am still. Although we are writing our algorithm as if we are working on a real tree structure the library and the compiler are able to just remove the whole intermediate step.
</p>
<hr class='mb-3' />
<p class='mb-3'>
Aftermath:
</p>
<p class='mb-3'>
In the beginning I promised a way to work on non-functor data structures. Actually that was how I began to work with the <code class='py-0.5 px-0.5 bg-gray-100'>recursion-schemes</code> library.
</p>
<p class='mb-3'>
We are able to create a ‘normal’ version of our tree from above:
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>data Tree c = Empty | Leaf c | Node (Tree c) (Tree c)
deriving (Eq, Show)</code></pre></div>
<p class='mb-3'>
But we can not use this directly with our (Co-)Algebras. Luckily Edward build a little bit of type magic into the library:
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>type instance Base (Tree c) = (TreeF c)
instance Unfoldable (Tree c) where
embed EmptyF = Empty
embed (LeafF c) = Leaf c
embed (NodeF l r) = Node l r
instance Foldable (Tree c) where
project Empty = EmptyF
project (Leaf c) = LeafF c
project (Node l r) = NodeF l r</code></pre></div>
<p class='mb-3'>
Without going into detail by doing this we establish a relationship between <code class='py-0.5 px-0.5 bg-gray-100'>Tree</code> and <code class='py-0.5 px-0.5 bg-gray-100'>TreeF</code> and teach the compiler how to translate between these types.
</p>
<p class='mb-3'>
Now we can use our Alebra on our non functor type:
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>example4 = cata flatten (Node (Leaf 'l') (Leaf 'r')) == "lr"</code></pre></div>
<p class='mb-3'>
The great thing about this is that, looking at the Core output again, there is no traces of the <code class='py-0.5 px-0.5 bg-gray-100'>TreeF</code> structure to be found. As far as I can tell, the algorithm is working directly on our <code class='py-0.5 px-0.5 bg-gray-100'>Tree</code> type.
</p>
<hr class='mb-3' />
<p class='mb-3'>
Literature:
</p>
<ul class='my-3 ml-6 space-y-1 list-disc'>
<li>
<a href='https://www.fpcomplete.com/user/bartosz/understanding-algebras' class='text-red-600 hover:underline' target='_blank' rel='noopener'>Understanding F-Algebras</a>
</li>
<li>
<a href='http://www.timphilipwilliams.com/slides.html' class='text-red-600 hover:underline' target='_blank' rel='noopener'>Recursion Schemes by Example</a>
</li>
<li>
<a href='http://comonad.com/reader/2009/recursion-schemes/' class='text-red-600 hover:underline' target='_blank' rel='noopener'>Recursion Schemes: A Field Guide</a>
</li>
<li>
<a href='http://stackoverflow.com/questions/6941904/recursion-schemes-for-dummies' class='text-red-600 hover:underline' target='_blank' rel='noopener'>This StackOverflow question</a>
</li>
</ul>
<hr class='mb-3' />
<p class='mb-3'>
Appendix:
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>mergeLists :: Ord a =&gt; [a] -&gt; [a] -&gt; [a]
mergeLists = curry $ unfoldr c where
c ([], []) = Nothing
c ([], y:ys) = Just (y, ([], ys))
c (x:xs, []) = Just (x, (xs, []))
c (x:xs, y:ys) | x &lt;= y = Just (x, (xs, y:ys))
| x &gt; y = Just (y, (x:xs, ys))</code></pre></div>
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