Lately I have had many works to do at university, therefore I’m sorry for the non regularity of my post’s.
By the way, I also liked to announce that this is my first conscientious post at Planet Haskell.
Why Catamorphisms and Point-Free
Catamorphisms is the way that we can explain in one function how recursive patterns works in some data type.
Here I will use Point-Free notation because what matters here is to show the data flow and the composition of functions.
Point-Free style is used to think in data flow terms, and very useful to program verification, applying formalism to our code.
In point-free style of programming programs are expressed as combinations of simpler functions. This notation is known as write functions without their arguments. Pointwise is the normal form how we write a function.
Couple of examples of point-free notation:
sum = foldr (+) 0 -- point-free
sum l = foldr (+) 0 l -- pointwise
f = (*10).(+2) -- point-free
f n = (n+2)*10 -- pointwise
First of all to define a function, for example , I say:
I will assume that you are familiarized with infix notation, , , and composition function.
In Haskell we have this definition for lists:
data [a] =  | a : [a]
Let’s create the same, but more convenient. Consider the following isomorphic type for lists:
data List a = Empty | Node(a,List a) deriving Show
To represent we wrote .
As you can see, to construct a we have two options, or . Formally we represent the constructor as . And we use to say that our two possibilities are or . We could see as a the following function:
Node :: (a,List a) -> List a
So typologically we have . We use to define that two things occurs in parallel, like tuples do, so we can redefine it:
Now we can say that is isomorphic to .
This is something to say that and keep the same information without any change or any loss.
Catamorphisms as composition of functions
Let , , , be Inductive data types (sets) and , , functions.
We will write using the composition of , , functions. That way we are breaking our problem in small ones. So, in the end we will have the following definition for :
The function that we want is , and that function is over so we have:
cata :: (D -> C) -> List a -> C
Type is . Maybe this isn’t clear yet, let’s start with function
The function is responsible to create the isomorphism between and , so the code could be something like this:
outList :: List a -> Either () (a,List a)
outList Empty = Left ()
outList (Node t) = Right t
In Haskell we represent the type as , as and as .
So, type is .
The function is also known as *gen*, here is where we said the step that pattern do. Imagine that we want to insert all the values of into :
g :: Either () (a,[a]) -> [a]
g (Left()) = 
g (Right(a,h)) = a:h
g = either (const ) (uncurry (:))
We represent as .
Now we can be more specific with our graphic:
Here we have to get a function that transform into . That function, general rec, will be:
recg f g h = f -|- (g >< g) x = ((f . fst) x , (g . snd) x)
With that function we can say exactly what to do with type , , and in domain of .
So we want something like this:
rec g = recG id id g
like that we said that will be the same in the counter domain of . Now we need a function that receive a and give us a …
Yes, that function is ! So, the final graphic became:
Finally we gonna to define the function :
cata g = outList . rec (cata g) . g
where is our *gen* function,
g = either (const ) (uncurry (:))
And our objective:
List2HaskellList = cata $ either (const ) (uncurry (:))
Imagine we have the following data type:
data NList a where
Leaf :: a -> NList a
NNode :: a -> [NList a] -> NList a
, , and became:
out (Leaf a) = Left a
out (NNode a l) = Right (a,l)
Using the previous definitions of and
rec f = id -|- (id >< map f)
cata g = g . rec (cata g) . out
Imagging that has type:
g :: Either a (a,[[a]]) -> [a]
And the graphic for this cata became:
I’ve talked about cata’s without any formalism, the idea was to explain to someone who didn’t know.
I will talk more about catamorphisms and how to calculate programs with them.
In the future I will like to talk about anamorphisms too. And Later on I will talk about point-free over non recursive functions.