Suppose you are given a deterministic Turing machine and you are guaranteed it runs in polynomial time. What's the computational complexity of determining whether the language accepted by the machine is empty?


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It's equivalent (under Turing reduction) to deciding whether the language of any given TM is empty. That is, it is co-RE-complete. This can be shown using a standard padding argument.

Here are the details. Let $E_P$ denote the problem in question (deciding whether $L(M)=\emptyset$ for a given TM $M$, with the promise that $M$ runs in polynomial time).

Lemma 1. $E_P$ is co-RE-complete.

Proof. Let $E_{TM} = \{\langle M \rangle : L(M) = \emptyset\}$ denote the problem of deciding whether the language of a given TM is empty. Note that $E_{TM}$ is co-RE-complete. (The complement of $E_{TM}$ is RE-hard by Rice's theorem, and in RE by a standard dovetailing argument.)

Clearly $E_P$ reduces to $E_{TM}$ (the reduction, given $\langle M\rangle$, can simply output $\langle M \rangle$).

To finish, we show the converse, that $E_{TM}$ reduces to $E_P$. Given (the encoding of) an arbitrary TM $M$, the reduction will output the encoding of a TM $M'$ such that $M'$ is guaranteed to run in polytime, and $L(M') = \emptyset$ iff $L(M)=\emptyset$.

The idea is that $L(M')$ will be the set of pairs $(w, 1^i)$ such that $M(w)$ accepts within $i$ steps. Given an input $(w, 1^i)$, the machine $M'$ simply simulates $M$ on input $w$ for $i$ steps, then accepts if $M$ does (within $i$ steps). The simulation is done efficiently so that the time it takes is polynomial in the input size, $|w| + i$. (This is not hard.) By construction, $L(M')$ is as desired, so $L(M')$ is empty iff $L(M)$ is empty.

Finally, given the encoding of $M$, the encoding of $M'$ is computable. $~~~~\Box$


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