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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 \(A_{\mathrm{TM}}\).

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Chapter 6

Prove that there exist two languages \(A\) and \(B\) that are Turing-incomparable- that is, where \(A \not{Z} \mathrm{~T} B\) and \(B \not \mathrm{T} A\).

Chapter 6

In Corollary \(4.18\), we showed that the set of all languages is uncountable. Use this result to prove that languages exist that are not recognizable by an oracle Turing machine with an oracle for \(A_{\mathrm{TM}}\).

Chapter 6

Give a model of the sentence $$ \begin{aligned} \phi_{e q}=& \forall x\left[R_{1}(x, x)\right] \\ & \wedge \forall x, y\left[R_{1}(x, y) \leftrightarrow R_{1}(y, x)\right] \\ & \wedge \forall x, y, z\left[\left(R_{1}(x, y) \wedge R_{1}(y, z)\right) \rightarrow R_{1}(x, z)\right] \end{aligned} $$

Chapter 6

\({ }^{*} 6.17\) Let \(A\) and \(B\) be two disjoint languages. Say that language \(C\) separates \(A\) and \(B\) if \(A \subseteq C\) and \(B \subseteq \bar{C} .\) Describe two disjoint Turing-recognizable languages that aren't separable by any decidable language.

Chapter 6

Describe two different Turing machines, \(M\) and \(N\), where \(M\) outputs \(\langle N\rangle\) and \(N\) outputs \(\langle M\rangle\), when started on any input.

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