Is the problem of "DFA-TM-INCLUSION" recursively enumerable?

Question

Consider the following problem:

• Input: A Turing Machine M and a DFA D.
• Question: Is $L(D) \subseteq L(M)$?

Of course, this problem is not decidable. Because it is known that judging whether a word belong to a Turing Machine is not decidable. If this problem is decidable, then we can write the word as regular expression and construct DFA from this regular expression, and then check the inclusion relationship between DFA and Turing Machine. So this is not decidable.

However, I feel difficult to prove or disprove whether it is recursively enumerable.

Any ideas are welcome.

Answer 1

No, $\{\langle D,M\rangle\mid D\text{ is a DFA }\land M\text{ is a Turing Machine }\land L(D)\subseteq L(M)\}$ is not recursively enumerable.

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Let $Z=\{0^{n}\mid n\ge0\}$. Call a Turing Machine (TM) $M$ a $Z$-TM if $Z\subseteq L(M)$.

Claim. $\{\langle M\rangle\mid M\text{ is a } Z\text{-TM}\}$ is not recursively enumerable.
Proof (by diagonalization): Towards a contradiction suppose $\{\langle M\rangle\mid M\text{ is a } Z\text{-TM}\}$ is recursively enumerable. That implies there is Turing machine $A$ that enumerates all $Z$-TMs, i.e., given a natural number $n$ as input, $A$ will output (the encoding of ) a $Z$-TM, which is denoted by $C_n$ such that the set of all $C_n$ is the set of all $Z$-TMs.

Let us construct TM $B$ so that on input $w$, $B$ will check first whether $w$ contains only $0$s.

• If not, $B$ accepts. (The behavior of $B$ here does not matter.)
• If yes, $B$ computes $n$, the number of $0$s in $w$. Then $B$ will use $A$ to obtain $C_n$. Then $B$ will simulate $C_n$ on input $0^n$, until $C_n$ accepts. Then $B$ will run one more step and accepts.

$B$ is different from each $C_n$ for all $n$, but $Z\subseteq L(B)$. This means $A$ misses $B$ in its enumeration of all $Z$-TMs. $\quad\checkmark$

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If we can enumerate all pairs of $D$ and $M$ such that $L(D)\subseteq L(M)$, we can further select the pairs such that $L(D)=Z$, since it is decidable whether $L(D)=Z$. Then we can enumerate $Z$-TMs. This contradicts the claim above.

Answer 2

TOTAL is the problem of deciding whether a given Turing machine halts on all inputs. This problem is $\Sigma_2$-complete, and in particular, it is neither r.e. nor co-r.e.

Your problem is also $\Sigma_2$-complete. There is a simple reduction from TOTAL to your problem. In the other direction, given a pair $(M,D)$, you can construct a new Turing machine $M'$ which on input $x$ checks whether $x \in L(D)$. If so, it runs $M$ on $x$, and otherwise, it immediately halts. The new Turing machine $M'$ is total iff $L(D) \subseteq L(M)$.