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$L =\big\{\langle T\rangle \mid T\text{ is a Turing machine that recognizes }\{00, 01\}\big\}$. Prove $L$ is undecidable.

I am really having difficulties even understanding the reduction to use here.

I'm not asking for free lunch, just a push in the right direction.

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1 Answer 1

up vote 1 down vote accepted

That follows from Rice's theorem:

Rice's Theorem.

Let S be a set of languages that is nontrivial, meaning

  • there exists a Turing machine that recognizes a language in S
  • there exists a Turing machine that recognizes a language not in S

Then, it is undecidable to determine whether the language decided by an arbitrary Turing machine lies in S.

[Copied from Wikipedia]

The language $\{00,01\}$ is nontrivial, thus $L$ is not decidable.

Hint: It is also easy to prove that directly, not using Rice's theorem. Let $M$ be a Turing machine. Consider Turing machine $M'$ defined as

if $x\in\{00,\,01\}$ then

  • run $M$ on empty tape

  • accept (if $M$ halts)

otherwise, reject

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Unfortunately, I need to be more precise than this. It has to be constructed. We have to reduce some undecidable language to L to prove by contradiction that L is undecidable, which is where I am confused: what undecidable language do I use? –  user1405177 Dec 5 '12 at 1:59
Rice's theorem is precise. The proof is very rigorous. I added a hint how to prove that $L$ is not decidable directly without using Rice's theorem. (BTW, please, use tag “homework” if it's a homework problem.) –  Yury Dec 5 '12 at 2:08

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