Say you have a (context-free) grammar, and you wish to mathematically model the magnitude of the ambiguity possible under this grammar, across the space of all possible** input strings.
Practically, one can feed in a large number of inputs strings to a parser of the grammar, and get a distribution over that set of inputs. This may be useful if one assumes the set of inputs to somehow represent a more general case, but for the sake of exercise, let us arbitrarily assume, that this approach does not generalize, or that we cannot a-priori come up with a sufficiently representative set of strings.
Can you suggest a mathematical model for the magnitude of ambiguity allowed by a given grammar? differently phrased, how would you creatively model the combinatorial magnitude of possible parses?
** this is of course ill-defined, but it takes some creativity to come up with a definition for "reasonable" input strings, or think of a measure for the length or complexity of a string, that can serve as a useful parameter in the model.