Hi,

I am thinking about exploring the space of high-order functions, by constructing (combine) complex fuctions using small and simple functions.

For example, I have a set of simple functions:

$f1(x, y) = x + y$

$f2(x, y) = x*y$

$f3(x) = log(x)$

$f4(x) = exp(x)$

Given 2 variables, x and y, I could create a fairly complex function

$f_{agg}(x, y) = f1( f2(f4(x) ,f3(y)) , f4(y) )$

If we plot the structure of these functions, in this specific case, it is a tree:

                                 f1
             f2                                   f4
f4                       f3

We all know that tree is a special case of a graph.

Now here comes the problem: What if I do not know the stucture of $f_{agg}$ in the first place, and I want to enumerate all possible combinations?

In other words, we have a few nodes , and we want to find all possible graphs that can be created by connecting these nodes.

An obvious constraint is the arity of the function. Unary function takes 1 argument, so it has only 1 inbound edge in a graph, while binary functions have 2, trinary functions have 3.

Another constraint is the domain ( i am not sure if this is the right word), a function $f_{n} (x) = - |x| $ will only give non-positive result, therefore we cannot combine it wiht $f_{p} (x) = log(x) $ .

Can anyone give me some clue how to combine functions in a systematic way? Please leave comments if I haven't explain myself clearly.