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Chapter 9: Q3E (page 389)

Prove that NTIME(n)桅PSPACE.

Short Answer

Expert verified

NTIME (n) is strict subset of PSPACE (n).

At most t(n) tape cells on each branch can be used by any Turning machine that operates in time t(n) on each computation branch. So, it can be stated that NTIME 桅NSPACE (n).

Step by step solution

01

Considering the Savitch’s theorem

Now, consider the Savitch鈥檚 theorem which says that: 鈥淟et f:N鈫�R

be a function with fn鈮�nthen NSPACE(f(n)) SPACE((f(n))2). Therefore according to Savitch鈥檚 theorem NTIME (n)鈯� PSPACE(n2).

02

Considering the space hierarchy theorem

Now, consider the space hierarchy theorem which says that 鈥渋f g is space-constructible (1n 1g(n) can be computed in space o(g(n))), f(n)=o(g(n)) then SPACE(f (n)) 桅SPACE(g(n)).

Therefore, according to space hierarchy theorem it can be said that SPACE(n2) 桅SPACE(n3). The result follows because SPACE(n3)鈯� PSPACE.

Hence NTIME (n)桅PSPACE(n).

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Most popular questions from this chapter

If R is a regular expression, let R{m,n} represent the expression

Rm鈭�Rm+1鈭�...鈭�Rn

Show how to implement the R{m,n} operator, using the ordinary exponentiation operator, but without 鈥溌� 路 路 鈥�.

Prove that if A鈭�P, thenPA=P

Problem 8.13 showedthat ALBAis PSPACE-complete.

a) Do we know whetherALBA鈭�NL?Explain your answer.

b) Do we know whetherALBA鈭�P ?Explain your answer.

Question: LetEREXI^

Prove that ifNEXPTIME鈮�EXPTIME, then P鈮�NP. You may find the function pad, defined in problem 9.13, to be helpful.
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