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Problem 18

Show that $$\overline{E Q_{\mathrm{TM}}}$$ is recognizable by a Turing machine with an oracle for $$A_{\mathrm{TM}}$$.

Expert verified
We can construct a Turing machine, $$O$$, using an oracle for $$A_{\mathrm{TM}}$$ to recognize $$\overline{E Q_{\mathrm{TM}}}$$, the complement of the equivalence problem for Turing machines. $$O$$ operates on input $$\langle M_1, M_2 \rangle$$ and queries the oracle for $$A_{\mathrm{TM}}$$ for each string $$w$$ to check if $$M_1$$ and $$M_2$$ have different behavior on $$w$$. If the oracle returns a positive result for one of the Turing machines and a negative result for the other, it accepts the input, indicating that $$O$$ recognizes $$\overline{E Q_{\mathrm{TM}}}$$. Therefore, $$\overline{E Q_{\mathrm{TM}}}$$ is recognizable by a Turing machine with an oracle for $$A_{\mathrm{TM}}$$.
See the step by step solution

Step 1: Design the Turing machine recognizing $$\overline{E Q_{\mathrm{TM}}}$$

Let's design a Turing machine, $$O$$, which uses an oracle for $$A_{\mathrm{TM}}$$ to recognize $$\overline{E Q_{\mathrm{TM}}}$$. $$O$$ operates as follows on input $$\langle M_1, M_2 \rangle$$: 1. For any string $$w$$: a. Query the $$A_{\mathrm{TM}}$$ oracle if $$M_1$$ accepts $$w$$. If yes, move to the next step. b. Query the $$A_{\mathrm{TM}}$$ oracle if $$M_2$$ accepts $$w$$. If yes, move to the next step. c. If either of the oracle queries returns positive for one Turing machine and negative for the other, $$\langle M_1, M_2 \rangle$$ is in $$\overline{E Q_{\mathrm{TM}}}$$ and we can accept the input. d. Move to the next string $$w$$ and repeat the process. Notice that this Turing machine never explicitly rejects any input. Instead, it continues to search for a string on which the two Turing machines act differently. This indicates that the Turing machine $$O$$ recognizes the language $$\overline{E Q_{\mathrm{TM}}}$$.

Step 2: Use oracle for $$A_{\mathrm{TM}}$$

In the construction of the Turing machine $$O$$, we used an oracle for $$A_{\mathrm{TM}}$$ as a subroutine. This oracle allows us to determine if a specific Turing machine accepts a specific input string. Since $$O$$ can recognize $$\overline{E Q_{\mathrm{TM}}}$$ by relying on this oracle, we can say that $$\overline{E Q_{\mathrm{TM}}}$$ is recognizable by a Turing machine with an oracle for $$A_{\mathrm{TM}}$$. To summarize, we have shown that it is possible to construct a Turing machine that uses an oracle for $$A_{\mathrm{TM}}$$ and recognizes the complement of the equivalence problem for Turing machines. This shows that $$\overline{E Q_{\mathrm{TM}}}$$ is recognizable by a Turing machine with an oracle for $$A_{\mathrm{TM}}$$.

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