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Contradiction proof for inequality of P and NP?
Announcing the arrival of Valued Associate #679: Cesar Manara
Unicorn Meta Zoo #1: Why another podcast?Proof for P-complete is not closed under intersectionProof of sum of powerset?Contradiction between best-case running time of insertion sort and $nlog n$ lower bound?bounded length CoNP proofLogarithmic Randomness is Necessary for PCP TheoremTrouble seeing the contradiction in diagonalization proofIs it always possible to have one part of the reduction?Is this language NP Hard?Testing algorithm for a modified sieve of EratosthenesFinding a complexity by solving inequality
$begingroup$
I'm trying to argue that N is not equal NP using hierarchy theorems. This is my argument, but when I showed it to our teacher and after deduction, he said that this is problematic where I can't find a compelling reason to accept.
We start off by assuming that $P=NP$. Then it yields that $SAT in P$ which itself then follows that $SAT in TIME(n^k)$. As stands, we are able to do reduce every language in $NP$ to $SAT$. Therefore, $NP subseteq TIME(n^k)$. On the contrary, the time hierarchy theorem states that there should be a language $A in TIME(n^k+1)$, that's not in $TIME(n^k)$. This would lead us to conclude that $A$ is in $P$, while not in $NP$, which is a contradiction to our first assumption. So, we came to the conclusion that $P neq NP$.
Is there something wrong with my proof? I was struggling for hours before asking this, though!
complexity-theory time-complexity
asked 1 hour ago
inverted_indexinverted_index
1384
$endgroup$
add a comment |
$begingroup$
I'm trying to argue that N is not equal NP using hierarchy theorems. This is my argument, but when I showed it to our teacher and after deduction, he said that this is problematic where I can't find a compelling reason to accept.
We start off by assuming that $P=NP$. Then it yields that $SAT in P$ which itself then follows that $SAT in TIME(n^k)$. As stands, we are able to do reduce every language in $NP$ to $SAT$. Therefore, $NP subseteq TIME(n^k)$. On the contrary, the time hierarchy theorem states that there should be a language $A in TIME(n^k+1)$, that's not in $TIME(n^k)$. This would lead us to conclude that $A$ is in $P$, while not in $NP$, which is a contradiction to our first assumption. So, we came to the conclusion that $P neq NP$.
Is there something wrong with my proof? I was struggling for hours before asking this, though!
complexity-theory time-complexity
asked 1 hour ago
inverted_indexinverted_index
1384
$endgroup$
add a comment |
$begingroup$
I'm trying to argue that N is not equal NP using hierarchy theorems. This is my argument, but when I showed it to our teacher and after deduction, he said that this is problematic where I can't find a compelling reason to accept.
We start off by assuming that $P=NP$. Then it yields that $SAT in P$ which itself then follows that $SAT in TIME(n^k)$. As stands, we are able to do reduce every language in $NP$ to $SAT$. Therefore, $NP subseteq TIME(n^k)$. On the contrary, the time hierarchy theorem states that there should be a language $A in TIME(n^k+1)$, that's not in $TIME(n^k)$. This would lead us to conclude that $A$ is in $P$, while not in $NP$, which is a contradiction to our first assumption. So, we came to the conclusion that $P neq NP$.
Is there something wrong with my proof? I was struggling for hours before asking this, though!
complexity-theory time-complexity
asked 1 hour ago
inverted_indexinverted_index
1384
$endgroup$
I'm trying to argue that N is not equal NP using hierarchy theorems. This is my argument, but when I showed it to our teacher and after deduction, he said that this is problematic where I can't find a compelling reason to accept.
We start off by assuming that $P=NP$. Then it yields that $SAT in P$ which itself then follows that $SAT in TIME(n^k)$. As stands, we are able to do reduce every language in $NP$ to $SAT$. Therefore, $NP subseteq TIME(n^k)$. On the contrary, the time hierarchy theorem states that there should be a language $A in TIME(n^k+1)$, that's not in $TIME(n^k)$. This would lead us to conclude that $A$ is in $P$, while not in $NP$, which is a contradiction to our first assumption. So, we came to the conclusion that $P neq NP$.
Is there something wrong with my proof? I was struggling for hours before asking this, though!
complexity-theory time-complexity
complexity-theory time-complexity
asked 1 hour ago
inverted_indexinverted_index
1384
asked 1 hour ago
inverted_indexinverted_index
1384
asked 1 hour ago
inverted_indexinverted_index
1384
asked 1 hour ago
inverted_indexinverted_index
1384
asked 1 hour ago
inverted_indexinverted_index
1384
1384
add a comment |
add a comment |
1 Answer
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$begingroup$
Then it yields that $SAT in P$ which itself then follows that $SAT in TIME(n^k)$.
Sure.
As stands, we are able to do reduce every language in $NP$ to $SAT$. Therefore, $NP subseteq TIME(n^k)$.
No. Polynomial time reductions aren't free. We can say it takes $O(n^r(L))$ time to reduce language $L$ to $SAT$, where $r(L)$ is the exponent in the polynomial time reduction used. This is where your argument falls apart. There is no finite $k$ such that for all $L in NP$ we have $r(L) < k$. At least this does not follow from $P = NP$ and would be a much stronger statement.
And this stronger statement does indeed conflict with the time hierarchy theorem, which tells us that $P$ can not collapse into $TIME(n^k)$, let alone all of $NP$.
answered 1 hour ago
orlporlp
6,1251826
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$begingroup$
Then it yields that $SAT in P$ which itself then follows that $SAT in TIME(n^k)$.
Sure.
As stands, we are able to do reduce every language in $NP$ to $SAT$. Therefore, $NP subseteq TIME(n^k)$.
No. Polynomial time reductions aren't free. We can say it takes $O(n^r(L))$ time to reduce language $L$ to $SAT$, where $r(L)$ is the exponent in the polynomial time reduction used. This is where your argument falls apart. There is no finite $k$ such that for all $L in NP$ we have $r(L) < k$. At least this does not follow from $P = NP$ and would be a much stronger statement.
And this stronger statement does indeed conflict with the time hierarchy theorem, which tells us that $P$ can not collapse into $TIME(n^k)$, let alone all of $NP$.
answered 1 hour ago
orlporlp
6,1251826
$endgroup$
add a comment |
$begingroup$
Then it yields that $SAT in P$ which itself then follows that $SAT in TIME(n^k)$.
Sure.
As stands, we are able to do reduce every language in $NP$ to $SAT$. Therefore, $NP subseteq TIME(n^k)$.
No. Polynomial time reductions aren't free. We can say it takes $O(n^r(L))$ time to reduce language $L$ to $SAT$, where $r(L)$ is the exponent in the polynomial time reduction used. This is where your argument falls apart. There is no finite $k$ such that for all $L in NP$ we have $r(L) < k$. At least this does not follow from $P = NP$ and would be a much stronger statement.
And this stronger statement does indeed conflict with the time hierarchy theorem, which tells us that $P$ can not collapse into $TIME(n^k)$, let alone all of $NP$.
answered 1 hour ago
orlporlp
6,1251826
$endgroup$
add a comment |
$begingroup$
Then it yields that $SAT in P$ which itself then follows that $SAT in TIME(n^k)$.
Sure.
As stands, we are able to do reduce every language in $NP$ to $SAT$. Therefore, $NP subseteq TIME(n^k)$.
No. Polynomial time reductions aren't free. We can say it takes $O(n^r(L))$ time to reduce language $L$ to $SAT$, where $r(L)$ is the exponent in the polynomial time reduction used. This is where your argument falls apart. There is no finite $k$ such that for all $L in NP$ we have $r(L) < k$. At least this does not follow from $P = NP$ and would be a much stronger statement.
And this stronger statement does indeed conflict with the time hierarchy theorem, which tells us that $P$ can not collapse into $TIME(n^k)$, let alone all of $NP$.
answered 1 hour ago
orlporlp
6,1251826
$endgroup$
Then it yields that $SAT in P$ which itself then follows that $SAT in TIME(n^k)$.
Sure.
As stands, we are able to do reduce every language in $NP$ to $SAT$. Therefore, $NP subseteq TIME(n^k)$.
No. Polynomial time reductions aren't free. We can say it takes $O(n^r(L))$ time to reduce language $L$ to $SAT$, where $r(L)$ is the exponent in the polynomial time reduction used. This is where your argument falls apart. There is no finite $k$ such that for all $L in NP$ we have $r(L) < k$. At least this does not follow from $P = NP$ and would be a much stronger statement.
And this stronger statement does indeed conflict with the time hierarchy theorem, which tells us that $P$ can not collapse into $TIME(n^k)$, let alone all of $NP$.
answered 1 hour ago
orlporlp
6,1251826
edited 17 mins ago
answered 1 hour ago
orlporlp
6,1251826
answered 1 hour ago
orlporlp
6,1251826
answered 1 hour ago
orlporlp
6,1251826
6,1251826
add a comment |
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