Shape of Homework Assignments

My current thinking about the research for the whole automatic generation of homework assignments is about descriptions from which a problem can be constructed. The best I have so far is that type signatures should be linked to elements of stories, where the types can be specific or general (polymorphic). An abstract type (like a type constructor) can be made more specific, or kept abstract, and should be linked to one or more stories. These stories might be composable if the functions that describe them are composable

For example- give the subject "lists" we have functions related to lists- (a->b) -> [a] -> [b] for example, which is the familiar map. This type can have a, b, neither, or both specified as a concrete type, and may have either or both of its arguments fixed. Any of these combinations of things should give rise to some story about lists where we want a list of something based on the list we have or are given.

An example of composition is a function a->b and and a function b->c. We hope that if each function has some set of stories behind it, then there is some set of stories that are the composition a->c. For example- we have a price and want to know how much is will cost to buy n of something and we have a coupon that takes %10 off a price. The composition of those stories is- we want to know the price of buying n of something with a 10% off coupon and the type signatures are (Double->Double)o(Double->Double) making a Double->Double.
Interestingly the layer of specificity seems to be related to the difficulty of the problem. This even works in other domains- in physics it is considered more difficult to give a homework problem where we need to solve an equation symbolically then numerically, where needing to solve symbolically corresponds to not specifying a value for some input variable.

I wonder about the ability to describe problems with arbitrary depth- lists can hold lists can hold lists, etc. At each level we can "add" the lists, or we can add up all elements of each list (assuming the lists hold some monoid or something). It makes me wonder if shape analysis, something I've just recently become interested in, may help. We could say- lists of length n to m as part of the type if we can be explicit about shapes.

It seems like to work around all sorts of nonsense problems that may appear, we would have to guide our search for specifying an abstract type by some real world knowledge. This is where the whole concept map, knowledge domain thing comes in to this project. I'm not sure exactly how this would happen.

I will have to flesh out exactly how these types are arrived at, and how the stories they specify can be distinguished (as I suspect the underlying stories will direct the search for specificity as well as our ability to compose).