## Fitting a regular grammar to strings from a PCFG: how big does it get?

Let $$G=(V, \Sigma, R, S)$$ be a (non regular) probabilistic context-free grammar, and $$u_1, \ldots, u_n$$ a set of $$n$$ strings generated by $$G$$.

For finite $$n$$, it is always possible to find a regular grammar $$\hat G=(\hat V, \Sigma, \hat R, S)$$ which generates the strings $$u_1, \ldots, u_n$$.

Intuitively, as $$n$$ goes to infinity, we expect $$\hat G$$ to get larger: my guess is that the cardinal of $$\hat R$$ (and maybe also the cardinal of $$\hat V$$?) would need to go to infinity.

Are there results which formalize this, e.g. by giving a lower bound on these cardinals as a function of $$n$$?