Q2E Prove that 聽TIME(2n)鈯圱IME(22n... [FREE SOLUTION]

91影视

Introduction to Theory of Computation

Michael Sipser

Computer Science

3rd Edition 路 458 pages

ISBN: 9781133187790

Chapter 9: Q2E (page 389)

Prove that $TIME (2^{n}) 鈯� TIME ({2^{2}}^{n})$

Short Answer

Expert verified

The above problem can be solved using Turing Machine, and the time hierarchy theorem which shows that the function $2 n$ is time constructible.

Step by step solution

Using the Time Hierarchy Theorem

The $TIME (2^{n}) 鈯� TIME ({2^{2}}^{n})$ holds because $2^{n} 鈮� 2^{2 n}$ .

According to the time Hierarchy theorem, if $f (n) \log f (n) = O (g (n))$ and f, g are time-constructible functions then, $D TIME (f (n)) 鈯� D TIME (g (n))$

Thus, as discussed in time-hierarchy theorem the above containment

$TIME (2^{n}) 鈯� TIME ({2^{2}}^{n})$ is proper

Using Turing Machine to prove the problem

As we can write the number 1 followed by $2 n$ 0s in $O (2^{2 n})$ time by making the function $2^{2 n}$ time constructible

Hence by the guarantees of the Time Hierarchy theorem, it can be decided whether a language A exists in $O (2^{2 n})$ time but not in $o (\frac{2^{2 n}}{\log 2^{2 n}}) = o (\frac{2^{2 n}}{2 n})$

time.

So, $A 鈭� TIME ({2^{2}}^{n})$ but $A 鈭� TIME (2^{n})$

Hence, we can say that. $TIME (2^{n}) 鈯� TIME ({2^{2}}^{n})$

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91影视

Prove that $TIME (2^{n}) 鈯� TIME ({2^{2}}^{n})$

Short Answer

Step by step solution

Using the Time Hierarchy Theorem

Using Turing Machine to prove the problem

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91影视

Prove that TIME(2n)鈯�TIME(22n)

Short Answer

Step by step solution

Using the Time Hierarchy Theorem

Using Turing Machine to prove the problem

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Computer Science Textbooks

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Game Design in Computer Science

Study anywhere. Anytime. Across all devices.

Prove that $TIME (2^{n}) 鈯� TIME ({2^{2}}^{n})$