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Introduction to the Theory of Computation

Sipser Michael

Chapter 6: Chapter 6

Problem 18

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

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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 $A_{\mathrm{TM}}$.

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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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Chapter 6: Chapter 6

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

Short Answer

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

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

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More chapters from the book ‘Introduction to the Theory of Computation’

Chapter 0

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 7

Chapter 8

Chapter 9

Chapter 10

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