Show that [latex]A_{TM}[/latex] is not mapping reducible to [latex]E_{TM}[/latex]. In other words, show that no computable function reduces [latex]A_{TM}[/latex] to [latex]E_{TM}[/latex]. (Hint: Use a proof by contradiction, and facts you already know about [latex]A_{TM}[/latex] and [latex]E_{TM}[/latex].)

Suppose for a contradiction that [latex]A_{TM}\leq m E_{TM}[/latex] via reduction f. It follows from the definition of mapping reducibility that [latex]\overline{A_{TM}} \leq m \overline{E_{TM}}[/latex] via the same reduction function f. However, [latex]\overline{E_{TM}}[/latex] is Turing-recognizable (see the solution to Exercise 4.5) and [latex]\overline{A_{TM}}[/latex] is not Turing-recognizable, contradicting Theorem 5.28.

Question 5.5: Show that ATM is not mapping reducible to ETM. In other word...

Introduction to the Theory of Computation

Show that $A_{TM}$ is not mapping reducible to $E_{TM}$ . In other words, show that no computable function reduces $A_{TM}$ to $E_{TM}$ . (Hint: Use a proof by contradiction, and facts you already know about $A_{TM}$ and $E_{TM}$ .)

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Use Rice’s theorem, which appears in Problem 5.28, to prove the undecidability of each of the following languages. a. INFINITETM = {〈M〉 | M is a TM and L(M) is an infinite language}. ...

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(a)

INFINITE_{TM}

is a language of ...

Question: 5.28

Rice’s theorem. Let P be any nontrivial property of the language of a Turing machine. Prove that the problem of determining whether a given Turing machine’s language has property P is undecidable. In more formal terms, let P be a language consisting of Turing machine descriptions where P fulfills ...

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Assume for the sake of contradiction that P is a d...

Question: 5.11

Consider the problem of determining whether a two-tape Turing machine ever writes a nonblank symbol on its second tape during the course of its computation on any input string. Formulate this problem as a language and show that it is undecidable. ...

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Let C = { 〈M〉 | M is a two-tape TM that writes a...

Question: 5.8

In the proof of Theorem 5.15, we modified the Turing machine M so that it never tries to move its head off the left-hand end of the tape. Suppose that we did not make this modification to M. Modify the PCP construction to handle this case. ...

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You need to handle the case where the head is at t...

Question: 5.7

Show that if A is Turing-recognizable and A ≤m Ā, then A is decidable. ...

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Suppose that

A\leq _{m} \overline{A}[/latex...

Question: 5.6

Show that ≤m is a transitive relation. ...

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Suppose

A \leq _{m} B

and

B ...

Question: 5.10

Consider the problem of determining whether a two-tape Turing machine ever writes a nonblank symbol on its second tape when it is run on input w. Formulate this problem as a language and show that it is undecidable. ...

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Let B = {〈M,w〉 | M is a two-tape TM that writes ...

Question 5.5: Show that ATM is not mapping reducible to ETM. In other word...

Related Answered Questions

Use Rice’s theorem, which appears in Problem 5.28, to prove the undecidability of each of the following languages. a. INFINITETM = {〈M〉 | M is a TM and L(M) is an infinite language}. ...

Verified Answer:

Verified Answer:

Consider the problem of determining whether a two-tape Turing machine ever writes a nonblank symbol on its second tape during the course of its computation on any input string. Formulate this problem as a language and show that it is undecidable. ...

Verified Answer:

In the proof of Theorem 5.15, we modified the Turing machine M so that it never tries to move its head off the left-hand end of the tape. Suppose that we did not make this modification to M. Modify the PCP construction to handle this case. ...

Verified Answer:

Show that if A is Turing-recognizable and A ≤m Ā, then A is decidable. ...

Verified Answer:

Show that ≤m is a transitive relation. ...

Verified Answer:

Consider the problem of determining whether a two-tape Turing machine ever writes a nonblank symbol on its second tape when it is run on input w. Formulate this problem as a language and show that it is undecidable. ...

Verified Answer: