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I'm currently learning for my exams this semester and tried to solve some old exams from the last years.

The question is to show, that L ist undecidable. $L=\{w|T(M_w)\neq\emptyset \land \forall x \in T(M_w): xx\in T(M_w)\land xxx\notin T(M_w)\}$

I showed that the language $L'=\{w | T(M_w)\neq \emptyset\}$ is undecidable due to the Rice Theorem. I created to Turing Machines $M_1$ with $T(M_1)=\emptyset$ and $M_2$ with $T(M_2)=\Sigma^*$ to show that the language L' is not trivial. (since $1\in L$ and $2\notin A$. With out loss of generality i assume those machines have the Gödelindexes 1 and 2)

My problem now is, that I know no way to show, that this result transfers to the language L. I know that the language L must only contain such Gödel Indexes, that for those indexes the following TM must accept infinite words (because in case of $x\in T(M_w)$ there must be $xx\in T(M_w)$... and therefore must be $xxxx\in T(M_w)$ etc.)

I would love to hear suggestions / answers! Thanks in advance

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This is a direct consequence of Rice's theorem. A word $w$ is in $L$ if $T(M_w)$ satisfies the following semantic property:

$T(M_w)$ is non-empty, and for any $x \in T(M_w)$, we have $x^2 \in T(M_w)$ and $x^3 \notin T(M_w)$.

In order to show that this language is undecidable according to Rice's theorem, we need to exhibit two Turing machines $w_1$ and $w_2$: one which doesn't satisfy the property, and another one which does satisfy it. We can take $w_1$ to be some machine such that $T(M_{w_1}) = \emptyset$, and $w_2$ to be some machine such that $T(M_{w_2}) = \{ a^{2^n} : n \geq 0 \}$.

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