Give CFGs for the following languages

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Alphabet is $\{0,1\}$.

a) $\{w \ \vert \ w \text{ contains at least three 1s} \}$

b) $\{w \ \vert \ w \text{ stars and ends with the same symbol} \}$

c) $\{w \ \vert \ w \text{ has odd length} \}$

d) $\{w \ \vert \ w \text{ has odd length and middle symbol is 0} \}$

e) $\{w \ \vert \ w \text{ is a palindrome} \}$

f) $\emptyset$

Solution:

a) $A \rightarrow X1X1X1X $, $X \rightarrow XX \ \vert \ 0 \ \vert \ 1 \ \vert \ \varepsilon$

b) $A \rightarrow 0X0 \ \vert \ 1X1 \ $, $X \rightarrow 0 \ \vert \ 1 \ \vert \ \varepsilon \ \vert \ XX$

c) $A \rightarrow X, X \rightarrow XXX \ \vert \ 0 \ \vert \ 1$

d) $A \rightarrow B, B \ \rightarrow 0 \ \vert \ XBX, X \rightarrow 1 \ \vert \ 0$

e) $A \rightarrow B , B \rightarrow 0 \ \vert \ 1 \ \vert \ \varepsilon \ \vert \ 0B0 \ \vert \ 1B1$

f)