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Consider the following CFG

$S \rightarrow \epsilon\ |\ aSbS\ |\ bSaS$

How can we prove formally that an equivalent $LL(1)$ grammar does not exist. I feel that intuitively an equivalent $LL(1)$ grammar doesn't exist, but I'm unable to prove this formally.

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The language for the grammar in the question is the set of all words with equal number of $a$'s and $b$'s.

I am afraid your intuition is incorrect. Here is an $LL(1)$ grammar for it.

$S \to aAbS \mid bBaS \mid \epsilon$
$A \to aAb \mid \epsilon$
$B \to bBa \mid\epsilon$

Here is some statistics about the above $LL(1)$ grammar. $$\begin{array} {|c|c|c|c|} \hline \text{nonterminal} &\text{first set} &\text{follow set} &\text{nullable} &\text{endable}\\\hline S &a\ b &\emptyset &\text{yes} &\text{yes}\\\hline A &a &b &\text{yes} &\text{no}\\\hline B &b &a &\text{yes} &\text{no}\\\hline \end{array}$$

Here is the parsing table. $$\begin{array} {|c|c|c|c|}\hline &a & b &\$ \\\hline S &S\to aAbS &S\to bBaS &S\to \epsilon\\\hline A &A\to aAb &A\to \epsilon\quad\ & \\\hline B &B\to \epsilon\quad\ &B\to bBa & \\\hline \end{array}$$


Exercise. Show that both the grammar in the question and the grammar in this answer generate the language of words with equal number of $a$'s and $b$'s.

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  • $\begingroup$ Thanks a lot! I realize my intuition was wrong. $\endgroup$ – user104014 Jan 31 at 7:54

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