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 $n$?