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*-Morpisms –
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*-Morpisms
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Monoid? Da war doch was…
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Fortgeschrittene funktionale Programmierung in Haskell
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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] -> 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’ ’ ’
< / p >
< p class = 'mb-3' >
The nice thing is that due to the fact that we haven’
< / 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 => TreeF a [a] -> [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 => [a] -> [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’ ’
< / 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] -> [Integer]
mergeSort =
\ (x :: [Integer]) ->
case x of wild {
[] -> [];
: x1 ds ->
case ds of _ {
[] -> : x1 ([]);
: ipv ipv1 ->
unfoldr
lvl9
(let {
p :: ([Integer], [Integer])
p =
case $wlenAcc wild 0 of ww { __DEFAULT ->
case divInt# ww 2 of ww4 { __DEFAULT ->
case tagToEnum# (< # ww4 0) of _ {
False ->
case $wsplitAt# ww4 wild of _ { (# ww2, ww3 #) -> (ww2, ww3) };
True -> ([], wild)
}
}
} } in
(case p of _ { (x2, ds1) -> mergeSort x2 },
case p of _ { (ds1, y) -> 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 ‘ ’
< / 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 => [a] -> [a] -> [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 < = y = Just (x, (xs, y:ys))
| x > y = Just (y, (x:xs, ys))< / code > < / pre > < / div >
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< div > < a href = 'https://www.youtube.com/watch?v=9EGYSb9vov8' class = 'text-gray-600 hover:underline' target = '_blank' rel = 'noopener' > Unifying Structured Recursion Schemes< / a > aka. < a href = 'Haskell/Code%20Snippets/Morphisms' class = 'text-gray-600 font-bold hover:bg-gray-50' data-wikilink-type = 'WikiLinkNormal' > The Morphism-Zoo< / a > < / div >
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< div > < a href = 'https://www.fpcomplete.com/user/bartosz/understanding-algebras' class = 'text-gray-600 hover:underline' target = '_blank' rel = 'noopener' > Understanding F-Algebras< / a > schöne Erklärung. Man könnte danach anfangen den < a href = 'Haskell/Code%20Snippets/Morphisms' class = 'text-gray-600 font-bold hover:bg-gray-50' data-wikilink-type = 'WikiLinkBranch' > Morphismen-zoo< / a > zu verstehen…< / div >
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