Show that every grammar for an inherently ambiguous CFL has infinitely many ambiguities Prove that if a CFL $L$ is inherently ambiguous, then for any grammar $G$ with $L(G) = L$, there are infinitely many strings in $L$ that have (at least) 2 different derivations in $G$.


Here's a sketch of a proof I have in mind:


  Suppose $G = (V, \Sigma, P, S)$ is a grammar satisfying $L(G) = L$.  Then, since $L$ is inherently ambiguous, it must be that $G$ is ambiguous.  Hence, there exists a string, call it $z$, with two different derivations in $G$.  Now, if the right-hand side of every production in G is of length $\leq d$, and if $|z| \geq d^{|V| + 1}$, then by Ogden's Lemma, if
  $d^{|V| + 1}$ or more symbols of $z$ are marked arbitrarily, there exists a decomposition $z = uvwxy$ such that there exists a variable $A$ in G such that
  
  $S \Longrightarrow^* uAy $ 
  
  $ A \Longrightarrow^* vAx$
  
  $A \Longrightarrow^* w$   
  
  Hence, every derivation of $z$ must at least contain the sentential forms $S \Longrightarrow^* uAy \Longrightarrow^* uvAxy \Longrightarrow^* uvwxy = z$
  
  In particular, we may indefinitely "pump" the sentential form $uvAxy$ in both derivations via repeated application of $ A \Longrightarrow^* vAx$.  Hence, the resulting two derivations must at least contain the sentential forms
  
  $S \Longrightarrow^* uAy \Longrightarrow^* uvAxy \Longrightarrow^* uv^iAx^iy \Longrightarrow^* uv^iwx^iy \in L$
  
  Both derivations; however, are still different, and thus we have that strings of the form $uv^iwx^iy \in L$, with $i \geq 2$ have 2 distinct derivations.  Hence, there are infinitely many strings in $L$ with 2 different derivations.


Now, I do see some problems with this sketch.  For one, we cannot guarantee that $|z| \geq d^{|V| + 1}$.  Second, even if $|z| \geq d^{|V| + 1}$, I'm not sure that we can guarantee that both derivations of z must contain the sentential forms $S \Longrightarrow^* uAy \Longrightarrow^* uvAxy \Longrightarrow^* uvwxy = z$.


  Given $k = |V|$, suppose that every string of length $\geq d^{k+1}$ is unambiguous in G.  Let $z$ be an arbitrary string in $L$, with $|z| \geq d^{k+1}$.  Then by Ogden's lemma, there exists a decomposition $z = uvwxy$ and a derivation of $uv^2wx^2y$, namely:
  
  $S \Longrightarrow^* uAy \Longrightarrow^* uvAxy \Longrightarrow^* uv^2Ax^2y \Longrightarrow^* uv^2wx^2y$.  Clearly, $|uv^2wx^2y| \geq d^{|V| + 1}$, and thus the provided derivation for the string is unique.  But now, mark only the characters in $uv^2wx^2y$ which appear in $u, w, \text{ or } y$. The resulting derivation for $uv^2wx^2y$, obtained using Ogden's Lemma, must differ from the previous derivation for $uv^2wx^2y$, contradiction.  Hence, there exists an ambiguous string of length $\geq d^{|V| + 1}$, and we may apply the argument above.
Intuitively, the reason the claim is true is that if a grammar $G$ had only finitely many ambiguous strings, then you could somehow resolve the ambiguity and come up with an equivalent unambiguous grammar $G'$, thus contradicting the inherent ambiguity. The way you would resolve the ambiguity is something like this: you would generate all small strings (say at most $\ell$, the length of the maximal ambiguous word in $G$) explicitly (and unambiguously), and modify the grammar so that all other strings that it generates are longer; for example, you could have a different subgrammar for every prefix of length $\ell+1$ which simulates $G$.

Of course, this is only one argument, there might be completely different ones.