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Chapter 9: Q14P (page 390)

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

Short Answer

Expert verified

IfNEXPTIME≠EXPTIME, thenP≠NPproved by taking its contrapositive i.e., If P=NPthen NEXPTIME≠EXPTIME,

Step by step solution

01

Assuming  and proving the contrapositive

IfL∈NTIME(22°) then the languageLpad={<x,12x°>:x∈L} is in NP. This process of adding a string of symbols to each string in the language is known as padding.

So, if P=NPthenLpad is in P but if Lpad is in P then L is in EXPTIME.

02

Using the polynomial-time machine

In order to conclude whether an input x is in L, it just pads the input and decides whether it is inLpad (Using polynomial-time machine for)

Hence, we can say that if P=NP is assumed then NEXPTIME =EXPTIME and so it can also be said that if NEXPTIME≠EXPTIME, then P≠NP.

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Most popular questions from this chapter

Prove that TIME(2n)⊈TIME(22n)

Consider the following function pad:∑*×Ν→∑*#*that is defined as follows. Let PAD (s, l) = s#3, where j = max (0,l - m) and mis the length of s. Thus, pad (s, l)simply adds enough copies of the new symbol # to the end of s so that the length of the result is at least l. For any language A and function f:N→N, define the language pad(A, f) aspad(A,f)={pad(s,f(m))| where s∈Aand ‘m’ is the length of ‘s’ }. Prove that if A∈TIME(n6), then pad(A,n2)∈TIME(n3).

Define the functionas in problem9.24. Show that it may be computed withO(n)size circuits.

Problem 9.24

Define the functionmajorityn:{0,1}n→{0,1}as

majorityn(x1,…,xn)=0∑xi<n2;1∑xi⩾n2.

Thus, the majorityn function returns the majority vote of the inputs. Show thatmajorityn can be computed with:

a. O(n2) size circuits.

b. localid="1663252609202" O(nlogn) size circuits.

Problem 8.13 showedthat ALBAis PSPACE-complete.

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

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

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

Rm∪Rm+1∪...∪Rn

Show how to implement the R{m,n} operator, using the ordinary exponentiation operator, but without “· · · ”.
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