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Given the grammar:

$S \to AS\mid \varepsilon$
$A \to A1\mid 0A1 \mid \varepsilon$

Generate a new unambiguous grammar that generates the same language as the grammar above.

I have no idea how to make it unambiguous, I have proven already that the given language has two leftmost derivations for the string 01101.

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Your grammar generates the language $L(A)^*$, where $$ L(A) = \{0^i 1^j : j \geq i\}. $$ You can generate $L(A)$ unambiguously in many ways. Two options are:

  1. Use $0^i 1^j = 0^i 1^i 1^{j-i}$.
  2. Use $0^i 1^j = 0^i 1^{j-i} 1^i$.

Given that, the current productions for $S$ will already generate $L(A)^*$ unambiguously.

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