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5
votes
2answers
263 views

Baker Gill Solovay $P^B \ne NP^B$ relativization, what class is $B$ in?

A recent question asks whether relativization is well-defined. This question wonders how to put one use of it on firmer ground. In the BGS 1975 proof that there exists a language $B$ such that ...
6
votes
2answers
235 views

Is relativization well-defined?

According to BGS theorem [1], there is an oracle $A$ such that $P^A\neq NP^A$. If the relativization operation $B\mapsto B^A$ was a well-defined function, one would expect that from $B^A\neq C^A$ ...
9
votes
3answers
431 views

What are natural examples of non-relativizable proofs?

As I understand it, a proof that P=NP or P≠NP would need to be non-relativizable (as in recursion theory oracles). Virtually all proofs seem to be relativizable, though. What are good examples of ...
5
votes
0answers
93 views

Relativization of Toda's Theorem

I'm trying to figure out some consequences of the fact that Toda's Theorem relativizes. The (un-relativized) Toda's theorem states that $PH \subset P^{\#P}$ so that for any constant $k$ and any ...
8
votes
1answer
237 views

Techniques for proving that a sentence relativizes

I am interested in how one proves that a sentence relativizes. Of course, proving that a sentence does not relativize is simple, as seen in the Baker-Gill-Solovay result; but how does one prove that ...
3
votes
0answers
49 views

one-way functions vs. secret-coin CRHFs

Is there any paper which can be used to show that there can be no relativizing construction of a secret-coin Collision-Resistant Hash Family from a one-way function and unlike this paper, does not ...
10
votes
1answer
348 views

Relativized world where ${\bf P^A}={\bf NP^A}\not = {\bf PP^A}$

I would like to know if there exists a relativized world where ${\bf P^A}={\bf NP^A}\not = {\bf PP^A}$. I am also interested to know if there exists a relativized world where ${\bf P^B} \not = {\bf ...
8
votes
1answer
291 views

Is $\mathsf{P} = \mathsf{NP}$ relative to a universal predictor?

Consider any language $L$. Define $s(L) \in {\lbrace 0, 1 \rbrace}^\omega$ (an infinite sequence of bits) by the recursive formula $$s(L)_n=\chi_L(s(L)_{<n})$$ Here $\chi_L$ is the characteristic ...
4
votes
0answers
188 views

Can relativization change the direction of separation?

Are any $A$, $B$, and $O$ such that: $O$ is a set (for oracle), $A$ and $B$ are the names of two known complexity classes, $A^X$ and $B^X$ have well-defined accepted meanings, $A=A^\emptyset\subset ...
4
votes
0answers
245 views

A natural result that relativized to a random oracle is true with probability 1/2

There are several well known results regarding random oracles, e.g. $\mathsf{IP}^A \neq \mathsf{PSpace}^A$ for almost all oracles. Are there any known natural examples where a similar statement ...
8
votes
4answers
619 views

Oracle results on P vs BPP

Let $A$ be any EXP complete problem. Then, $P^A = NP^A$. Let $B$ be some oracle that takes into accounts the queries that $M$ (a TM in P) will make, and we can get $P^B \neq NP^B$. Question: Do we ...
10
votes
1answer
216 views

Ruzzo-Simon-Tompa oracle access mechanism

In a paper on relativizing logspace computations, Ladner and Lynch construct an oracle relative to which $\mathsf{NL} \nsubseteq \mathsf{P}$. There are some more pathological examples in this vein in ...
15
votes
1answer
481 views

What's the “real” reason that IP=PSPACE is non-relativizing?

IP=PSPACE is listed as the canonical example of a non-relativizing result, and the proof for this is that there exists an oracle $O$ such that ${\sf coNP}^O \not\subseteq {\sf IP}^O$, while ${\sf ...
1
vote
1answer
187 views

Can an oracle allowing errors be non-relativizing?

I am experimenting with k-SAT. I'm using an oracle that returns the total number of satisfiable truth assignments, which is in #P. The interest here is that this total is returned modulo a natural ...
12
votes
1answer
349 views

Oracle relative to which $\mathsf{BPP}$ is not contained in $Δ_2 \mathsf{P}$

Complexity Zoology by Greg Kuperberg states that there is a language $X$ such that $\mathsf{BPP}^X \nsubseteq \mathsf{\Delta_2 \mathsf{P}}^X$ — in other words, $\mathsf{BPP}^X \nsubseteq ...
26
votes
1answer
839 views

Toy examples for barriers to $P \ne NP$

Are there any toy examples that provide 'essential' insights into understanding the three known barriers to $P = NP$ problem - relativization, natural proofs and algebrization?
2
votes
0answers
87 views

Abilities of Restricted Relativization in Proving Conjectures

Though everyone seems to say that P vs NP cannot be solved using relativization because there are oracles both $A$ and $B$ such that $\text{P}^A = \text{NP}^A$ and $\text{P}^B \neq \text{NP}^B$, why ...
7
votes
1answer
238 views

Separation between existence of crypto primitives

I understand how one can build a crypto primitive from another crypto primitive to some extent. The constructions I know build the later primitive using the former primitive as a black box. My ...
8
votes
4answers
526 views

Does diagonalization captures the essence of class separation ?

I don't remember having seen a class separation not based on diagonalization and relativization results. Diagonalization could still be used to separate remaining known classes, because ...
50
votes
0answers
2k views

more on PH in PP?

A recent question by Huck Bennett asking whether the class PH was contained in the class PP, received somewhat contradictory answers (all true, it seems). On one hand, several oracle results were ...
13
votes
0answers
294 views

Oracle relative to which MA does not have a complete problem?

Babai introduced a hierarchy of complexity classes based on public-coin randomized interactive proof systems, so called Arthur-Merlin games. The game is played by powerful but untrustworthy wizard ...
10
votes
1answer
282 views

Can relativization results be used to prove sentences formally independent?

Is it possible to demonstrate that a sentence must be formally independent based on the fact that it is non-relativizing? In other words, are there examples of sentences in computability/complexity ...
27
votes
1answer
722 views

Are there canonical non-relativizing techniques ?

In a lot of domains, there are canonical technics which everybody working on the field should master. For example, for logspace reductions, the "bit trick" for composition consisting of non ...
8
votes
2answers
314 views

Relativization with Respect to Non-Recursive Oracles

In the paper Relativizations of the P = ? NP Question, Baker et al. showed that there are relativized worlds in which either P = NP or P ≠ NP holds. All oracles in their settings were recursive sets. ...
9
votes
1answer
196 views

Worlds Relative to Which “Invulnerable Generators” Do Not Exist

Invulnerable generators are defined as follows: Let $R$ be an NP relation, and $M$ be a machine which accepts $L(R)$. Informally, a program is an invulnerable generator if, on input $1^n$, it ...
6
votes
1answer
471 views

Oracle relative to which BPP = EXP

An oracle construction relative to which BPP = EXP is usually attributed to Heller (Mathematical System Theory Vol. 17, 1984). Unfortunately I don't have the paper available in my library. Could ...
13
votes
1answer
370 views

Do the proofs that permanent is not in uniform $\mathsf{TC^0}$ relativize?

This is a follow up to this question, and is related to this question of Shiva Kinali. It seems that the proofs in these papers (Allender, Caussinus-McKenzie-Therien-Vollmer, Koiran-Perifel) use ...