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Michael Sipser Edition 3 Exercise 2 Question 21 (Page No. 156)
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Let $\Sigma = \{a,b\}.$ Give a $CFG$ generating the language of strings with twice as many $a’s$ as $b’s.$ Prove that your grammar is correct$.$
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number of a's=2*number of b's
S->aaSb | baSa| aSba | SS | ε
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Michael Sipser Edition 3 Exercise 2 Question 19 (Page No. 156)
Let $CFG$ $G$ be the following grammar$.$ $S\rightarrow aSb \mid bY \mid Y a$ $Y\rightarrow bY \mid aY \mid \epsilon$ Give a simple description of $L(G)$ in English$.$ Use that description to give a $CFG$ for $\overline{L(G)},$ the complement of $L(G).$
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Michael Sipser Edition 3 Exercise 2 Question 15 (Page No. 156)
Give a counterexample to show that the following construction fails to prove that the class of context-free languages is closed under star. Let $A$ be a $\text{CFL}$ that is generated by the $\text{CFG}$ ... new rule $S\rightarrow SS$ and call the resulting grammar $G'.$This grammar is supposed to generate $A^{*}.$
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Michael Sipser Edition 3 Exercise 2 Question 22 (Page No. 156)
Let $C = \{x\#y \mid x, y\in\{0,1\}^{*}$ and $x\neq y\}.$ Show that $C$ is a context-free language$.$
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Michael Sipser Edition 3 Exercise 2 Question 14 (Page No. 156)
Convert the following $\text{CFG}$ into an equivalent $\text{CFG}$ in Chomsky normal form,using the procedure given in $\text{Theorem 2.9.}$ $A\rightarrow BAB \mid B \mid \epsilon$ $B\rightarrow 00 \mid \epsilon$
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