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Let $L=\{R(M) \mid M$ halts when run with $R(M)\}$. a) Show that $\mathrm{L}$ is not recursive. b) Show that $\mathrm{L}$ is recursively enumerable.

   Let $L=\{R(M) \mid M$ halts when run with $R(M)\}$.
a) Show that $\mathrm{L}$ is not recursive.
b) Show that $\mathrm{L}$ is recursively enumerable.

Languages and Machines: An Introduction to the Theory of Computer Science

Languages and Machines: An Introduction to the Theory of Computer Science

Thomas A. Sudkamp 2nd Edition

Chapter 11, Problem 19

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The set $L$ is defined as $L = \{R(M) \mid M \text{ halts when run with } R(M)\}$. Here, $R(M)$ represents the output of a function $R$ applied to a Turing machine $M$. The condition is that $M$ halts when run with its own output $R(M)$ as input. Show more…

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Let $L=\{R(M) \mid M$ halts when run with $R(M)\}$. a) Show that $\mathrm{L}$ is not recursive. b) Show that $\mathrm{L}$ is recursively enumerable.

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