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If a $PDA$ can be constructed to check whether a string is not a computation history for a Turing Machine. Like in the proof of $ALL_{CFG}$ is not decidible.

Then we can construct a $PDA$ that accepts all computation histories of a Turing machine $A_{TM}$ on a string $\omega$. This $PDA$ can then be converted into a CFG (Context-Free Grammar). We can use the decider for $E_{CFG}$ to determine whether the language of this CFG is empty.

If $A_{TM}$ does not accept the string $\omega$, there will be no valid computation histories, and thus $L(CFG)$ will be empty. Conversely, if $A_{TM}$ accepts the string $\omega$, there will be valid computation histories, and $L(CFG)$ will not be empty.

My question is: If we can decide whether $L(CFG)$ is empty based on the above construction, does this mean we can decide $A_{TM}$?

If a $PDA$ can be constructed to check whether a string is not a computation history for a Turing Machine. Like in the proof of $ALL_{CFG}$ is not decidable.

Then we can construct a $PDA$ that accepts all computation histories of a Turing machine $A_{TM}$ on a string $\omega$. This $PDA$ can then be converted into a CFG (Context-Free Grammar). We can use the decider for $E_{CFG}$ to determine whether the language of this CFG is empty.

If $A_{TM}$ does not accept the string $\omega$, there will be no valid computation histories, and thus $L(CFG)$ will be empty. Conversely, if $A_{TM}$ accepts the string $\omega$, there will be valid computation histories, and $L(CFG)$ will not be empty.

My question is: If we can decide whether $L(CFG)$ is empty based on the above construction, does this mean we can decide $A_{TM}$?

If a $PDA$ can be constructed to check whether a string is not a computation history for a Turing Machine. Like in the proof of $ALL_{CFG}$

Then we can construct a $PDA$ that accepts all computation histories of a Turing machine $A_{TM}$ on a string $\omega$. This $PDA$ can then be converted into a CFG (Context-Free Grammar). We can use the decider for $E_{CFG}$ to determine whether the language of this CFG is empty.

If $A_{TM}$ does not accept the string $\omega$, there will be no valid computation histories, and thus $L(CFG)$ will be empty. Conversely, if $A_{TM}$ accepts the string $\omega$, there will be valid computation histories, and $L(CFG)$ will not be empty.

My question is: If we can decide whether $L(CFG)$ is empty based on the above construction, does this mean we can decide $A_{TM}$?

If a $PDA$ can be constructed to check whether a string is not a computation history for a Turing Machine. Like in the proof of $ALL_{CFG}$ is not decidible.

Then we can construct a $PDA$ that accepts all computation histories of a Turing machine $A_{TM}$ on a string $\omega$. This $PDA$ can then be converted into a CFG (Context-Free Grammar). We can use the decider for $E_{CFG}$ to determine whether the language of this CFG is empty.

If $A_{TM}$ does not accept the string $\omega$, there will be no valid computation histories, and thus $L(CFG)$ will be empty. Conversely, if $A_{TM}$ accepts the string $\omega$, there will be valid computation histories, and $L(CFG)$ will not be empty.

My question is: If we can decide whether $L(CFG)$ is empty based on the above construction, does this mean we can decide $A_{TM}$?

If a $PDA$ can be constructed to check whether a string is not a computation history for a Turing Machine. Like in the proof of $E_{CFG}$

Then we can construct a $PDA$ that accepts all computation histories of a Turing machine $A_{TM}$ on a string $\omega$. This $PDA$ can then be converted into a CFG (Context-Free Grammar). We can use the decider for $E_{CFG}$ to determine whether the language of this CFG is empty.

If $A_{TM}$ does not accept the string $\omega$, there will be no valid computation histories, and thus $L(CFG)$ will be empty. Conversely, if $A_{TM}$ accepts the string $\omega$, there will be valid computation histories, and $L(CFG)$ will not be empty.

My question is: If we can decide whether $L(CFG)$ is empty based on the above construction, does this mean we can decide $A_{TM}$?

If a $PDA$ can be constructed to check whether a string is not a computation history for a Turing Machine. Like in the proof of $ALL_{CFG}$

Then we can construct a $PDA$ that accepts all computation histories of a Turing machine $A_{TM}$ on a string $\omega$. This $PDA$ can then be converted into a CFG (Context-Free Grammar). We can use the decider for $E_{CFG}$ to determine whether the language of this CFG is empty.

If $A_{TM}$ does not accept the string $\omega$, there will be no valid computation histories, and thus $L(CFG)$ will be empty. Conversely, if $A_{TM}$ accepts the string $\omega$, there will be valid computation histories, and $L(CFG)$ will not be empty.

My question is: If we can decide whether $L(CFG)$ is empty based on the above construction, does this mean we can decide $A_{TM}$?