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Chapter 9: Q11E (page 389)

Show that the language MAX-CLIQUEfrom problem 7.48 isin.PSAT

Short Answer

Expert verified

It is known that every language in DP is polynomial time reducible to Z and DP is also in NP. Also,MAX-CLIQUE isNP-complete. Hence, using this we can solve the above problem.

Step by step solution

01

Checking for the given facts from the problem 7.48 

In a graph Gclique is maximum of the size kif and only if it has kclique size and does not havek+1clique size.

The languageCLIQUE is in NPand also inco-NP

02

Continuing with the solution

It is known that every language in DP is polynomial time reducible to Z and DP is also in NP. Also,MAX-CLIQUE isNP-complete. Hence, using this we can solve the above problem.

Since,NP鈭�PSATand,coNP鈭�PSATisMAXCLIQUEin.PSAT

And PSAT鈭�NPSATwhich implies that MAXCLIQUE is in.NPSAT

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Most popular questions from this chapter

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

e. All strings that contain exactly 500 1s

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

i. All strings that contain two 0s that have at least 500 symbols between them

Consider the following function pad:鈭�*脳螡鈫�鈭�*#*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鈫�N, define the language pad(A, f) aspad(A,f)={pad(s,f(m))| where s鈭�Aand 鈥榤鈥� is the length of 鈥榮鈥� }. Prove that if A鈭�TIME(n6), then pad(A,n2)鈭�TIME(n3).

Describe the error in the following fallacious 鈥減roof鈥� that.P鈮�NPAssume that P=NP and obtain a contradiction. If P=NP, then SAT? P and so, for some k,SAT?TIME(nk)Because every language in NP is polynomial time reducible to SAT, you haveNP鈯�TIME(nk)

. Therefore,.P鈯�TIME(nk)Butby the time hierarchy theorem, TIME(n k+1) contains a language that isn鈥檛 in ,TIME(nk) which contradicts.P鈯�TIME(nk)Therefore, P鈮�NP.

Prove that NTIME(n)桅PSPACE.

Question: Read the definition of a 2DFA (two-headed finite automation) given in Problem 5.26. Prove that P contains a language that is not recognisable by a 2DFA.
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