Derivation trees to show a given grammar is ambiguous

Question

Given the grammar with productions: \begin{align} S \rightarrow aSb \mid SS \mid \lambda\\ \end{align} I would like to show that it is ambiguous. As I understand it, if you can show that some string can be produced with these rules through more than one leftmost or rightmost derivation, then the grammar is ambiguous.

As such, I tried to get two ways that derive the string "ab" using leftmost derivations: \begin{align} S \Rightarrow aSb \Rightarrow a\lambda b \Rightarrow ab\\ S \Rightarrow SS \Rightarrow aSbS \Rightarrow a\lambda b\lambda \Rightarrow ab \end{align}

Because there are two different leftmost derivations of the same string "ab", the grammar must be ambiguous.

However, my professor indicated to me that my answer is wrong, and I did not use all the production rules. I thought I had, though. He told me I could instead derive the string "aabb" to prove the ambiguity, but I don't know what would make that fundamentally different.

Can anyone help explain why this relatively simple string "ab" doesn't work, at least as I attempted to prove it? Thanks!

Answer 1

I believe your professor is wrong. You exhibited two leftmost derivations, which is all that is required to show a grammar is ambiguous. Although, one small thing: $S \Rightarrow SS \Rightarrow aSbS \Rightarrow a\lambda b\lambda \Rightarrow ab$ should have another rule as you have followed two rules at once, $S \Rightarrow SS \Rightarrow aSbS \Rightarrow abS \Rightarrow ab$ (in other words, you eliminated two $S$ variables at once). Also, as you have mentioned, you actually did use every production rule.

Answer 2

if the ambiguity was the problem, then your solution is completely correct, showing 1 example, where derivation isn't unique is enough to prove that grammar is ambiguous in this case.