Your mind has four purposes: to make you like the world, and to make the world like you.

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I was reading the following synopsis of a science article today:

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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 function that takes a binary function as input, along with another input, to produce an output of the same type:

There exist polymorphic types in many languages as well.  In Haskell, for example, the list is a polymorphic type–it is a linked list of any type you want.  I imagine these polymorphic types as “banded colors” so that the bands contain another type (color).  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 in Haskell is “(a -> b) -> [a] -> [b]“.

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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The semester is over, my final project was a success (at least in that I passed the class) and it’s time now to release the game I made for Graphics 455: Silkworm!

See and download the full gallery on posterous

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