Can a Turing machine decide the language $L_\emptyset$ of machines with empty language?

Let $$L_\emptyset = \{\langle M\rangle \mid M \text{ is a Turing Machine and }L(M)=\emptyset\}.$$

Is there a Turing machine R that decides (I don't mean recognizes) the language $L_\emptyset$?

It seems that the same technique used to show that $\{A \mid A \text{ is a DFA and } L(A)=\emptyset\}$ should work here as well.

$A$ is undecidable because of Rice's Theorem, which states that non trivial properties of partial functions are not decidable.

• There is a TM which doesn't accept any string. (Which directly goes to the reject state).

• There is a TM which accepts every string. (Which directly goes to the accept state).

This means that the functions computed by elements of $A$ have a non trivial property. Hence $A$ is not decidable.

$E$ is decidable only under the assumption that DFAs are encoded in a special way like state transition table or etc. (we cannot decide whether a TM only accepts regular languages, because of Rice's Theorem!). In this case Rice's Theorem is not applicable because the particular encoding of an element is required to decide on $E$. So we don't just decide on partial functions.

(That is to say if the problem were, deciding whether a particular TM is a DFA -- or DFA computable -- and the language accepted by it is empty, $E$ would be undecidable through Rice's Theorem. Notice that in this case $A = E$.)

StackExchange Computer Science Q#10897, answer score: 10