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Suppose that A and B are two oracles. One of them is an oracle for TQBF, but if you don’t know which. Give an algorithm that has access to both A and B, and that is guaranteed to solve TQBF in polynomial time.

Short Answer

Expert verified

An oracle machine is a type of Turing Machine having many different tapes and these tapes are called oracle tapes. The different states may be representedqstates .

Step by step solution

01

 Using Baker Gill Solovay Theorem

TQBF is referred as True Qualified Boolean Formula. To show TQBF in polynomial problem having access to Turing machine A and B can be solved using Baker Gill Solovay Theorem.

For our convenience the problem is broken into two parts.

For Oracle A:

  • Let, TQBF has access to A then,A=TQBF .
  • The above problem can be solved by showing PA=NPB.
  • We know thatTQBF is PSPACE problem,
  • Therefore,PSPACEPTQBF andPSPACETQBFPSPACE
  • Hence, by the above results we can say thatPA=NPB
02

Solving the part two of the problem

For Oracle B:

  • It is required to show that PANPBfor oracle B.
  • Now, define a language L as:LB={0nlw0,1}where,B(w)=1
  • For any oracle B the defined language L isNPB
  • If X=0nwith|w|=|x| then, the machine’s output will be 1.
  • Here we can say that TQBFNLwhich shows that PSPACENL
  • Baker Gill theorem states thatNLϕPSPACE
  • Now, using Baker Gill and hierarchy Theorem, it can be said that PANPB

Thus, from the results of the two parts it is proven that TQBF is in polynomial time for both oracle A and B.

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Most popular questions from this chapter

Prove that ifNEXPTIMEEXPTIME, then PNP. You may find the function pad, defined in problem 9.13, to be helpful.

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:NN, define the language pad(A, f) aspad(A,f)={pad(s,f(m))| where sAand ‘m’ is the length of ‘s’ }. Prove that if ATIME(n6), then pad(A,n2)TIME(n3).

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Problem 9.24

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majorityn(x1,,xn)=0xi<n2;1xin2.

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a. O(n2) size circuits.

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

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Give regular expressions with exponentiation that generate the following languages over the alphabet {0,1}.

a. All strings of length 500

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f. All strings that contain at least 500 1s

g. All strings that contain at most 500 1s

h. All strings of length 500 or more that contain a 0 in the 500th position

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