Questions tagged [relativization]
36 questions
37
votes
3 answers
Why is Relativization a barrier?
When I was explaining the Baker-Gill-Solovay proof that there exists an oracle with which we can have, $\mathsf{P} = \mathsf{NP}$, and an oracle with which we can have $\mathsf{P} \neq \mathsf{NP}$ to a friend, a question came up as to why such…
Nikhil
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12
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2 answers
An oracle to separate NP from coNP
How to prove that $\mathsf{NP}^A \neq \mathsf{coNP}^A$ ? I am just looking for a such oracle TM $M$ and a recursive language $L(M) = L$ for which this holds.
I know the proof where you show that there is an oracle $A$ such that $\mathsf{P}^A \neq…
stewenson
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11
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1 answer
How can I show that the Cook-Levin theorem does not relativize?
The following is an exercise which I am stuck at ( source: Sanjeev Arora and Boaz Barak; its not homework ) :
Show that there is an oracle $A$ and a language $L \in NP^A$ such that $L$ is not polynomial-time reducible to 3SAT even when the machine…
advocateofnone
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10
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1 answer
Intuition behind Relativization
I take course on Computational Complexity. My problem is I don't understand Relativization method. I tried to find a bit of intuition in many textbooks, unfortunately, so far with no success. I will appreciate if someone could shed the light on this…
com
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9
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2 answers
The meaning of relativization
I don't understand the notion of relativization. I expose with an example. Consider a class $A$ that contains $P$, e.g. $NP$. Why $P^A$ is not necessarily equal to $A$? I can naively think that if one can decide any problem in $A$, one can also…
neophyte
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7
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2 answers
Can two different complexity classes be equal relative to an oracle? Example request
Is there a known example of two complexity classes, $A$ and $B$, such that:
$A \neq B$;
there is an oracle $O$ for which $A^O = B^O$?
I strongly believe that there are such examples, as otherwise $P = PSPACE$ (note that $P^{PSPACE} =…
Guilherme Rito
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7
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1 answer
Why are non-relativizing proofs preferred to relativizing ones?
I apologize, but even after these two other posts: here and here
I'm still having trouble understanding oracle TMs and relativization. This question comes at the issue from a different angle:
Why are non-relativizing proofs considered more valid…
Ari
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7
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1 answer
Do relativized relations between complexity classes tell us anything about the nonrelativized relation?
The existence of relativized relations between complexity classes seems to often be treated as "circumstantial" evidence about the "true" or "real-world" (i.e. nonrelativized) relation between the classes. (For example, this lecture describes one…
tparker
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6
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Confusion about the Time Hierarchy Theorem and relativization
I know that $\mathsf{P}^A = \mathsf{EXP}$
for any $\mathsf{EXPTIME}$-complete language $A$.
Is it true that $\mathsf{DTIME}^A(n^k) = \mathsf{EXP}$
for any fixed $k$ and any $\mathsf{EXPTIME}$-complete oracle $A$?
If not, what do these complexity…
Ari
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5
votes
1 answer
Is $BQP$ in $P^{NP}$?
I read in the introduction of this paper
http://www.scottaaronson.com/papers/uncompute.pdf
that there is a problem $B$ such that $BQP^B \not\subset P^{NP^B}$, and that $B$ is in $BPP$. But, using the fact that $BPP$ is in $BQP$ and that $BQP$ is low…
neophyte
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4
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1 answer
Confusion about $EXP \subseteq P^{EXPCOM}$ claim from Arora and Barak
In Computational Complexity -- A Modern Approach, by Arora and Barak, they have the following claim (Example 3.6).
Let EXPCOM be the following language
$$ \{ \langle M, x, 1^n\rangle \mid M \text{ outputs 1 on $x$ within $2^n$ steps} \} $$
Then…
panto
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4
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1 answer
Can two equal classes be separated wrt an oracle?
Is it known if there are two classes of languages $A$ and $B$ such that:
$A$ and $B$ are defined wrt the exact same type of machine (e.g. 1-tape Deterministic Turing Machines, 2-tape Deterministic Turing Machines, 1-tape Nondeterministic Turing…
Guilherme Rito
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4
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0 answers
Complete problems and relativization barriers for nonuniform complexity classes
Do nonuniform complexity classes like NP/poly have complete problems?
Are there relativization barriers for separations of nonuniform complexity classes?
One way to interpret the second question is by asking whether there are computable languages A…
Thomas Klimpel
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4
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Relativization of NP-completeness
This is actually exercise 3.7 from "Computational Complexity: A Modern Approach".
I need to prove that the NP-Completeness of 3-sat does not relativize, i.e. I need to show that that exists some oracle $A$ such that there exists $L\in \mathrm{NP}^A$…
Ariel
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4
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Why is $NP \subseteq P \implies NP^A \subseteq P^A$ false?
My question is about why does the result of Baker-Gill-Solovay not prove that $P \neq NP$. There have been several questions on this forum about this topic perhaps but I couldn't find my specific question amongst them.
Context: This is what I had in…
kishlaya
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