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/ ===== 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?
