# Michael Sipser Edition 3 Exercise 5 Question 36 (Page No. 242)

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Say that a $CFG$ is minimal if none of its rules can be removed without changing the language generated. Let $MIN_{CFG} = \{\langle G \rangle \mid \text{G is a minimal CFG}\}$.

1. Show that $MIN_{CFG}$ is $T-$recognizable.
2. Show that $MIN_{CFG}$ is undecidable.

## Related questions

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Say that a variable $A$ in $CFG \:G$ is necessary if it appears in every derivation of some string $w \in G$. Let $NECESSARY_{CFG} = \{\langle G, A\rangle \mid \text{A is a necessary variable in G}\}$. Show that $NECESSARY_{CFG}$ is Turing-recognizable. Show that $NECESSARY_{CFG}$is undecidable.
Show that $EQ_{CFG}$ is co-Turing-recognizable.
Show that if $A$ is Turing-recognizable and $A\leq_{m} \overline{A},$ then $A$ is decidable.
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 decided by any decider $M_{i}$ whose description appears in $A$. (Hint: You may find it helpful to consider an enumerator for $A$.)