Show that the language $L=\{w|w$ has odd length and the middle symbol is a $0\}$ is Context-Free and construct a PDA that accepts it

Question

Were w is any string composed over the alphabet $\Sigma = \{0,1\}$.

For the first part of the exercise I've tried decomposing the problem into three different ones, mainly the first one is for the string before the 0 the second one for the 0 itself and the third for the rest of the string after the 0. This is what I came up with: $$ \begin{align} S &\rightarrow S_{1}S_{2}S_{3} \\ S_{1} &\rightarrow 0S_{1}|1S_{1}|\varepsilon \\ S_{2} &\rightarrow 0 \\ S_{3} &\rightarrow 0S_{3}|1S_{3}|\varepsilon \end{align} $$

But I'm not sure if the presence of the $\epsilon$ means that input strings such as $01$ or $10$ could be accepted, or if the $\epsilon$ at the end just ensure the recursion of the grammar.

For the second part I have no idea about a possible solution.

Any help would be most appreciated. Thanks in advance

Please put your "answer" as part of your question. You can show what you did up to this point and where you got stuck but don't use the answer section for that purpose. — ExpressionCoder, Jul 04 '23 at 13:11
@Michel as it clearly shown I didn't use any space from the answer section. — Lorenzo, Jul 04 '23 at 13:35

score 0 · Accepted Answer · answered Jul 03 '23 at 16:49

In your solution there is no guarantee that the words generated by $S_1$ and $S_2$ will have the same length. The trick is to generate both sides simultaneously: $$S \to 0S0 \mid0S1 \mid1S0 \mid1S1 \mid 0.$$

As for the second part, there are mechanical procedures to convert a CFG into a PDA.

Show that the language $L=\{w|w$ has odd length and the middle symbol is a $0\}$ is Context-Free and construct a PDA that accepts it

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