Questions tagged [relativization]

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Question about the Relativization barrier

Baker, Gill, and Solovay has shown in their famous paper, that there are oracles $A$ and $B$ with $P^A = NP^A$ and $P^B \not= NP^B$. So, one can't solve the $P$ vs. $NP$ Problem with methods like ...
Reiner Czerwinski's user avatar
1 vote
0 answers
32 views

Why results based on padding generally fail to relativize?

I have read in the Algebrization paper that, if we only allow polynomially-long queries to oracles, then, results based on padding will not relativize. For instance: assuming that $A$ is a PSPACE-...
441Juggler's user avatar
3 votes
0 answers
28 views

Understanding a black-box vs white-box simulation and relativization

I am trying to understand the relativization barrier from Baker Gill Solovay (BGS). About this barrier, I have heard that it only applies when using a black-box simulation. Hence, my question is, what ...
441Juggler's user avatar
3 votes
1 answer
105 views

Show that $P^A \neq EXP^A$ for all oracles A

I knew that $P \neq EXP$ by Time Hierarchy Theorem. But, I'd like to show that $P^A \neq EXP^A$ for all oracles A. How can I do it? I knew that If I make A=EXP, PSPACE, NP, and many others, then this ...
user777's user avatar
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1 vote
0 answers
66 views

How do we know that this Karp-Lipton theorem is derived from relativizing arguments?

Luca Trevisan wrote, " The oracle $C$ tells us that we cannot have a relativizing proof that derives the $𝑁𝑃 ⊈ 𝑃/π‘π‘œπ‘™π‘¦$ conclusion from the $𝑃 ≠𝑁𝑃$ assumption, so a theorem such as Karp-...
user777's user avatar
  • 729
2 votes
1 answer
194 views

How to tell if a proof relativizes?

If I have a proof for a separation between two complexity classes (using no oracles) and I want to see if it relativizes, how do I go about doing so? Especially in the case where there already exist ...
DeeDee's user avatar
  • 345
4 votes
1 answer
152 views

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 ...
kishlaya's user avatar
  • 141
1 vote
0 answers
55 views

Which is harder, an NP-complete problem or the Raz-Tal oracle problem?

This is a (hopefully) sharper version of a question that I asked previously. Which of these algorithms is believed to have a longer asymptotic runtime? The optimal algorithm guaranteed to solve some ...
tparker's user avatar
  • 1,034
6 votes
1 answer
95 views

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 ...
tparker's user avatar
  • 1,034
4 votes
1 answer
52 views

Finding a language that is $NP^L$-complete

I'm trying to prove a theorem and as a lemma I would like to identify an $NP^L$-complete language. I was thinking something like a machine that can decide $SAT$ equipped with an oracle for $L$ can ...
Enrico Borba's user avatar
3 votes
1 answer
86 views

The power of relativised proofs

I've been trying to understand why, for instance, even though there are oracles $A$ for which $P^A \neq NP^A$, we still don't know if $P=NP$. As I understand it, it's because it's easy to construct ...
SAS's user avatar
  • 43
1 vote
0 answers
22 views

Any barrier result for this kind of (non-relativising) technique?

We think that non-deterministic machines are more powerful than deterministic machines, by giving an oracle access to $P\subseteq L\subseteq NP$, it seems reasonable to expect there's some $L$ that is ...
Taylor Huang's user avatar
4 votes
1 answer
464 views

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{ ...
panto's user avatar
  • 41
1 vote
1 answer
446 views

Is the existence of an oracle such that $P^O = NP^O$ nontrivial? [duplicate]

This famous paper proves the existence of both oracles $A$ and $B$ such that $\textbf{P}^A = \textbf{NP}^A$ and $\textbf{P}^B \neq \textbf{NP}^B$, therefore proving that any resolution to the P versus ...
tparker's user avatar
  • 1,034
3 votes
1 answer
103 views

Specific example of a problem that shows why **NP** isn't low for itself

Wikipedia says Every class which is low for itself is closed under complement, provided that it is powerful enough to negate the boolean result. This implies that NP isn't low for itself unless NP =...
tparker's user avatar
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1 vote
1 answer
60 views

A' not computable in A

Recall A'= $\{x \mid \phi^A_x(x)$ halts and accepts $\}$ In this article, a proof that A' not computable in A is given: http://www.math.uchicago.edu/~may/VIGRE/VIGRE2006/PAPERS/Flood.pdf He ...
user352102's user avatar
7 votes
2 answers
848 views

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 ...
Guilherme Rito's user avatar
4 votes
1 answer
245 views

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 ...
Guilherme Rito's user avatar
2 votes
1 answer
150 views

Are all proof techniques which only look at black box behaviour of a TM relativizing?

I am currently working on a seminar on $\mathbf{P \stackrel{?}{=} NP}$ and one of the points I want to adress is the Relativization barrier. However, it is hard to find a concrete definition of a "...
Rednaxel59's user avatar
1 vote
1 answer
100 views

Relativized world where $LogCFL^A=PSPACE^A$

I wonder whether there is a known relativization barrier against proving $LogCFL\neq PSPACE$. Hence I'm looking for a language $A$ for which $LogCFL^A=PSPACE^A$. One could try $A:=TQBF$, where $TQBF$ ...
Thomas Klimpel's user avatar
1 vote
0 answers
249 views

Misunderstanding the Baker-Gill-Solovay oracle and obtaining $LOGSPACE^A=PSPACE^A$

Baker, Gill and Solovay [1] gave an oracle $A$ relative to which $P^A=PSPACE^A$. The oracle is the very simple $PSPACE^A$-Complete language $$A = \{\langle M, x, 1^n \rangle | M^A \text{ accepts } x \...
Lieuwe Vinkhuijzen's user avatar
4 votes
0 answers
56 views

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 ...
Thomas Klimpel's user avatar
10 votes
1 answer
3k views

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 ...
advocateofnone's user avatar
4 votes
1 answer
214 views

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 ...
neophyte's user avatar
  • 143
4 votes
0 answers
228 views

Relativization of NP-completeness [duplicate]

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 ...
Ariel's user avatar
  • 13.3k
9 votes
2 answers
1k views

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 ...
neophyte's user avatar
  • 221
3 votes
1 answer
480 views

Is it axiomatic that the Time Hierarchy Theorem holds true in all relativized worlds?

I learned from this post that ${\sf DTIME}^{\text{EXP}}(n^k) \neq \text{EXP}$ for a fixed $k$ for otherwise the Time Hierarchy Theorem would fail in that relativized world. However, is it possible to ...
Ari's user avatar
  • 1,611
1 vote
1 answer
179 views

Why does the principle of locality of computation not relativize?

Although I have trouble understanding oracle TMs, I appreciate that non-relativizing techniques will be needed to resolve P vs. NP (as well as most other open problems in TCS). However, one of the ...
Ari's user avatar
  • 1,611
7 votes
1 answer
2k views

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 ...
Ari's user avatar
  • 1,611
6 votes
1 answer
792 views

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 $\mathsf{EXPTIME}$-...
Ari's user avatar
  • 1,611
10 votes
1 answer
1k views

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 ...
com's user avatar
  • 3,149
12 votes
2 answers
4k views

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 ...
stewenson's user avatar
  • 351
34 votes
3 answers
4k views

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 ...
Nikhil's user avatar
  • 609