Prove that [latex]TIME (2^{n}) = TIME (2^{n+1})[/latex].

The time complexity classes are defined in terms of the big-O notation, so constant factors have no effect. The function [latex]2^{n+1}[/latex] is O[latex](2^{n})[/latex] and thus [latex]A \in TIME (2^{n})[/latex] iff [latex]A \in TIME (2^{n+})[/latex].

Question 9.1: Prove that TIME(2^n) = TIME(2^n+1).

Introduction to the Theory of Computation

Prove that $TIME (2^{n}) = TIME (2^{n+1})$ .

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Define pad as in Problem 9.13. a. Prove that for every A and natural number k, A ∈ P iff pad(A, n^k) ∈ P. b. Prove that P ≠ SPACE(n). ...

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(a) Let A be any language and k ∈ N. If A ∈ P, the...

Question: 9.7

Give regular expressions with exponentiation that generate the following languages over the alphabet {0,1}. a. All strings of length 500 b. All strings of length 500 or less c. All strings of length 500 or more d. All strings of length different than 500 ...

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(a)

\Sigma ^{500}

; (b)

(\Sig...

Question: 9.3

Prove that NTIME(n) ⊊ PSPACE. ...

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NTIME(n)\subseteq NSPACE(n)

because...

Question: 9.2

Prove that TIME(2^n) ⊊ ( TIME(2^2n). ...

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The containment

TIME (2^{n}) ⊆ TIME (2^{2n+...

Question 9.1: Prove that TIME(2^n) = TIME(2^n+1).

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