[Haskell-cafe] Visualizing Typed Functions

[Duane Johnson]

I'm not sure what the policy is on posting with graphical attachments,  
but I'll give this a tentative try.  In case this does not come  
through with images, I've also posted this on my blog at

	http://blog.inquirylabs.com/2009/05/07/visualizing-typed-functions/

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As I've worked in Haskell over the past several months, I've noticed  
that my mind has kind of implicitly modeled functional programming as  
a visual/mechanical representation of concepts.  I'd like to make the  
ideas more explicit, however, so I'm writing this post and cooking up  
some graphics.  I wonder if others have similar "internal  
representations" of functions, and if perhaps others might suggest  
improvements over my model.

The basic idea is that a function is a semi-circle with inputs on one  
side (left) and the output at the right.  Colors represent types.   
Here is a function that takes one input and produces one output (of  
the same type):



Haskell would type this as "a -> a".  Next, we have a function that  
takes one type and returns a different type... e.g. a hashing function  
might take a string and return an integer:



(Please ignore the vertical stripe, it's an artifact of my poor  
Inkscape skills).  We can also create functions with more than one  
input, of course, so here is a binary function.  It might be the '+'  
operator, or some other binary operator, e.g. "a -> a -> a":



Haskell and many other languages also have the concept of higher-order  
functions, i.e. functions that take functions as inputs.  Here is a  
binary function that takes a binary function as input, along with  
another input, to produce an output of the same type:



I imagine polymorphic types as "banded colors" so that the bands  
contain another type (color).  Lists are polymorphic types.  Here is  
what the "head" function might look like ("head" returns the first  
element in a list):



Finally, here is a composite of all of the above visualizations.  The  
"map" function in Haskell is a function that takes a function and a  
list, and applies that function to each element in the list, returning  
a new list with the "mapped" elements:





The type signature for the "map" function is "(a -> b) -> [a] ->  
[b]".  I wonder what a Monad would look like...

With these functions visualized, one could make a kind of "drag and  
drop" interface for Haskell programming, although that isn't really my  
intention.  I admit this is a little convoluted even for the purpose  
of visualization, but at least it's a starting place.  Does anyone  
know of another system or better representation?

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