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Michael Sipser Edition 3 Exercise 4 Question 21 (Page No. 212)
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Let $S = \{\langle M \rangle \mid \text{M is a DFA that accepts}\: \text{ $w^{R}$ whenever it accepts $w$}\}$. Show that $S$ is decidable.
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Michael Sipser Edition 3 Exercise 4 Question 31 (Page No. 212)
Say that a variable $A$ in $CFL\: G$ is usable if it appears in some derivation of some string $w \in G$. Given a $CFG\: G$ and a variable $A$, consider the problem of testing whether $A$ is usable. Formulate this problem as a language and show that it is decidable.
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Michael Sipser Edition 3 Exercise 4 Question 30 (Page No. 212)
Let $A$ be a Turing-recognizable language consisting of descriptions of Turing machines, $\{ \langle M_{1}\rangle,\langle M_{2}\rangle,\dots\}$, where every $M_{i}$ is a decider. Prove that some decidable language $D$ is not ... $A$. (Hint: You may find it helpful to consider an enumerator for $A$.)
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Michael Sipser Edition 3 Exercise 4 Question 29 (Page No. 212)
Let $C_{CFG} = \{\langle G, k \rangle \mid \text{ G is a CFG and L(G) contains exactly $k$ strings where $k \geq 0$ or $k = \infty$}\}$. Show that $C_{CFG}$ is decidable.
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Michael Sipser Edition 3 Exercise 4 Question 28 (Page No. 212)
Let $C = \{ \langle G, x \rangle \mid \text{G is a CFG $x$ is a substring of some $y \in L(G)$}\}$. Show that $C$ is decidable. (Hint: An elegant solution to this problem uses the decider for $E_{CFG}$.)
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