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I've been thinking about this one for a while:

Consider the language of TMs which do not recognize themselves: $L_{s}=\{ \langle M\rangle ~|~ M \text{ does not accept } \langle M\rangle \}$. If $N$ is a TM where $L(N) \subseteq L_s$, then is $\langle N\rangle \in L_s$?

Intuitively, I'm pretty sure it is true. I pictured $N$ as a compiler, and imagined some machine $P$ that $N$ compiled. It seemed that if $P$ was not able to bootstrap, then $N$ wouldn't be able to either. But I'm having trouble thinking about this problem more formally.

Am I on the right track? A hint would be really helpful.

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You have to argue formally. Intuition can be helpful in pointing the way, but you need a formal argument in the end.

Hint: Suppose that $\langle N \rangle \notin L_s$. Then $N$ accepts $\langle N \rangle$, and so $\langle N \rangle \in L(N) \subseteq L_s$, a contradiction.

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  • $\begingroup$ I completely agree, I just thought sharing my intuition might help people answering. Thank you very much for the hint, it "clicked" for me. $\endgroup$ – Craig Spurr Nov 10 '13 at 7:00

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