Questions tagged [relativization]

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11
votes
4answers
744 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 non-...
12
votes
3answers
744 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 ...
8
votes
2answers
418 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 $B^A\neq C^A$ ...
7
votes
2answers
688 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 $...
63
votes
1answer
3k 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 ...
28
votes
1answer
1k views

Are there canonical non-relativizing techniques?

In a lot of domains, there are canonical techniques which everybody working in the field should master. For example, for logspace reductions, the "bit trick" for composition consisting of not ...
8
votes
2answers
373 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. ...
18
votes
1answer
868 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 coNP}...
15
votes
1answer
504 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 ...
4
votes
0answers
223 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 B^...
13
votes
0answers
358 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 ...
8
votes
1answer
355 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 ...
7
votes
1answer
332 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 ...
5
votes
1answer
317 views

Relativized world where $L^A=NP^A$

I wonder1 whether there is a known relativization barrier against proving $L\neq NP$. Hence I'm looking for a language $A$ for which $L^A=NP^A$. My first idea was to try $A:=SAT$, but then I thought ...
2
votes
1answer
117 views

What is conjunctive truth table reduction?

What are conjunctive/disjunctive truth table reductions and how do they compare with other reductions?
12
votes
1answer
438 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 \mathsf{P}^{\...
8
votes
1answer
371 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 ...
1
vote
1answer
200 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 ...