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why is this Problem$$L = \{ \langle M\rangle \mid L(M) \text{ is undecidable}\}$$ undecidable?

I thought if we know $L(M)$ the turingmaschine accepts all $x \in L(M)$, so $L(M)$ is in every case decidable and $L=\emptyset$ especially finite and decidable.

I know we cannot find an algorithm but every TM has a language and we even if we don't it, we know it is decidable.

Can someone help me where I do have a misunderstanding?

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  • $\begingroup$ Can you prove your claim step by step very slowly? By "very slowly", I mean nothing beyond definitions and known theorems and syllogism in every step. $\endgroup$ – Apass.Jack Jan 23 at 18:46
  • $\begingroup$ I cannot proof my claim, it is just an idea. I claim L(M) always decidable if we know L(M), because as I said it contains all words the TM M accepts. So L is empty in my consideration. So I know this is false, but why? Where is my fallacy? $\endgroup$ – Marc Jan 23 at 19:00
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You're very close!

The key is the difference between recognizable and decidable.

A language $R$ is recognizable if we can build a TM $T$ such that $L(T) = R$. In other words, there's a Turing machine that accepts exactly those strings in the language—and for strings not in the language, it might halt in a non-accepting state (i.e. it might reject), or it might run forever without halting (i.e. it might diverge).

A language $D$ is decidable if we can build a TM $T$ such that $L(T) = D$, and $T$ rejects (halts in a non-accepting state) on strings not in $D$.

It turns out that a whole lot of languages are recognizable but not decidable. The language $H_0 = \{ \langle M \rangle | M \textrm{ halts on empty input} \}$ is undecidable (this is the Empty-Input Halting Problem), but $H_0$ can be recognized by the following procedure (call it $RH$):

Simulate $M$ on empty input. If it ever halts, accept.

Now we have a Turing machine whose language, $L(RH) = H_0$, is undecidable! Which means the property in question is non-trivial, so by Rice's Theorem that property is undecidable.

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