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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.
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Michael Sipser Edition 3 Exercise 5 Question 35 (Page No. 242)
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.
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.
asked Oct 20, 2019 in Theory of Computation Lakshman Patel RJIT 99 views
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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 $A$.)
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 $A$.)
asked Oct 17, 2019 in Theory of Computation Lakshman Patel RJIT 57 views
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