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I'm having serious issues fining an unambiguous version of a particular grammar. I've looked online for help for this particular problem and for general methods for finding unambiguous versions of grammars, and though I understand the concept, I can't seem to figure this out. Any help/advice? I keep getting really close then finding something that doesn't fit. Here is the grammar in question:

S -> aS | bS | Sa | Sb | aa

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The language of the grammar $G$ presented here is regular and its corresponding regular expression is $$r = (a \mid b)^*aa(a \mid b)^*.$$ Now, convert the regular expression to a minimal finite automaton and use this question to convert the automaton to an unambiguous regular grammar, which is always possible. In the end you can get something like this:

$$ \begin{align} S &\rightarrow aA \mid bS \\ A &\rightarrow aB \mid bS \mid a \\ B &\rightarrow aB \mid bB \mid \epsilon \end{align} $$


Proof that $\mathscr{L}(r) = \mathscr{L}(G)$: (1) Let $s \in \mathscr{L}(r)$, hence $s = x aa y$, where strings $x,y$ consist of characters $x_i, y_j \in \Sigma, i = [1..m], j = [1..n], m = |x|, n = |y|$. There is a derivation of $s$: $$S \rightarrow x_1S \rightarrow x_1x_2S \rightarrow ... \rightarrow x_1...x_mS = xS \rightarrow xSy_n \rightarrow xSy_{n-1}y_n\rightarrow ... \rightarrow xSy_1...y_n = xSy \rightarrow xaay.$$ Thus $s = xaay \in \mathscr{L}(G)$ and $\mathscr{L}(r) \subseteq \mathscr{L}(G)$.

(2) Let $s \in \mathscr{L}(G)$. This means there exists some derivation $$S \rightarrow \alpha_1 \rightarrow \alpha_2 \rightarrow ... \rightarrow \alpha_p = s.$$ Observe that the only way to make the non-terminal $S$ disappear is to use the rule $S \rightarrow aa$ and it must be the last step of the derivation: $$S \rightarrow \alpha_1 \rightarrow ... \rightarrow \alpha_{p-1} = xSy \rightarrow \alpha_p = s = xaay,$$ where $x,y$ are some strings of terminals. Thus $s = xaay \in \mathscr{L}(r)$ and $\mathscr{L}(G) \subseteq \mathscr{L}(r)$, resulting in $\mathscr{L}(G) = \mathscr{L}(r)$. Qed.

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  • $\begingroup$ This part doesn't seem like it would work... s=xaay Because it wouldn't make it possible for a string like, say, "babaa" because it implies that there has to be some string "y" on the right side of "aa." I've gotten that far before, but can't work out a parse tree that would make "xaay" work. $\endgroup$ – Olivia Feb 11 '16 at 16:11
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    $\begingroup$ Notice that $j = [1..n], n = |y|$. If $y$ is empty, then $|y| = 0$, so $y_1...y_0$ just means the empty string, it's a pretty common convention. The same thing works for $x$. In case of $babaa$, $s = xaay$, where $x = bab$, $y = \epsilon$. $\endgroup$ – Anton Trunov Feb 11 '16 at 16:16

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