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Michael Sipser Edition 3 Exercise 2 Question 24 (Page No. 157)
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Let $E=\{a^{i}b^{j}\mid i\neq j$ $\text{and}$ $2i\neq j\}.$ Show that $E$ is a context-free language$.$
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S->aaAb | aBbbb | C | D
A->aAb | aA | a
B->aBb | a
C->aA | a
D->bB | b
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Michael Sipser Edition 3 Exercise 2 Question 35 (Page No. 157)
Let $G$ be a $CFG$ in Chomsky normal form that contains $b$ variables$.$ Show that if $G$ generates some string with a derivation having at least $2^{b}$ steps$, L(G)$ is infinite$.$
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Michael Sipser Edition 3 Exercise 2 Question 34 (Page No. 157)
Let $G = (V, \Sigma, R, S)$ be the following grammar. $V = \{S, T, U\}; \Sigma = \{0, \#\};$ and $R$ is the set of rules$:$ $S\rightarrow TT\mid U$ $T\rightarrow 0T\mid T0\mid \#$ ... existence of a pumping length $p$ for $B.$ What is the minimum value of $p$ that works in the pumping lemma$?$ Justify your answer$.$
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Michael Sipser Edition 3 Exercise 2 Question 32 (Page No. 157)
Let $\Sigma = \{1, 2, 3, 4\}$ and $C = \{w\in\Sigma^{*}\mid$ $\text{in }w,\text{ the number of }1\text{'s equals the number of }2\text{'s, and the number of } 3\text{'s equals the number of }4\text{'s}\}.$ Show that $C$ is not context free.
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Michael Sipser Edition 3 Exercise 2 Question 25 (Page No. 157)
For any language $A,$ let $SUFFIX(A) = \{v\mid uv \in A$ $\text{for some string u\}}.$ Show that the class of context-free languages is closed under the $\text{SUFFIX operation.}$
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