Proving that a context-free grammar is unambiguous

Question

I have to find an unambiguous context-free grammar that generates the following language. $$L= \{ w \in \{a,b\}^+ : |w|_a = |w|_b\}$$ I think I have found the context-free one, it should be this one. $$S \to aS'bS'|bS'aS'$$ $$S' \to aS'bS' | bS'aS' | \lambda$$ Now I am stuck with proving whether it is unambiguous or not. I think it is but how should I do that?

Answer 1

Consider the word $abab$. It has two derivation trees (rendered using syntree):

Hence your grammar is ambiguous. In order to make an unambiguous grammar, we have to force the parse to be unique. We identify the word with a path in which $a$ corresponds to $\nearrow$ and $b$ to $\searrow$. We're looking for a non-empty path that ends at level ground.

If the path starts with $\nearrow$, then we can look at the first time at which it returns to level ground, and decompose the path accordingly: $\nearrow x \searrow y$, where $x$ is a path that never goes below level ground, and $y$ is an arbitrary path that ends at level ground. Hence we need a way to capture paths that end at level ground and never go below level ground. Such a path can also be decomposed uniquely as $\nearrow x \searrow y$, where this time both $x$ and $y$ never go below level ground.

The considerations above lead to the following unambiguous grammar: $$ \begin{align*} &S \to aAbS' \mid bBaS' \\ &S' \to aAbS' \mid bBaS' \mid \lambda \\ &A \to aAbA \mid \lambda \\ &B \to bBaB \mid \lambda \end{align*} $$

As a somewhat degenerate example, the derivation tree for $abab$ is:

The proof that this grammar is unambiguous goes along the description above, though the formal proof by induction is somewhat challenging.