Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions regarding oracle machines in computational complexity theory. Oracles can serve as an indicator that a separation between complexity classes is beyond the scope of certain proof techniques.

3
votes
1answer
103 views

Indications that strengthen the conjecture: NEXP ⊊ EXP^NP

I am trying to find indications that strengthen the conjecture of NEXP ⊊ EXP^NP. Clearly NEXP ⊆ EXP^NP, and there are some hints that this inclusion is proper. Some Examples: 1. A paper by Shuichi ...
1
vote
0answers
48 views

An alternative characterization of some NExp-Time Turing machine with oracles

Let me denote by $\Sigma_i^P$ be a class from i-th level of polynomial time hierarchy (see eg. PH). I'm interested in the following type of a Turing Machine $\mathcal{M}$: $\mathcal{M}$ is ...
1
vote
2answers
82 views

Partitioning a square for optimal queries

I have a square plate of size 1x1, full of lots of skittles. I want to eat all of the skittles, but the only way I can get the skittles is through these two oracles: $f(x, y, r)$ tells me how many ...
2
votes
1answer
168 views

P vs. NP in a logic with a random oracle

Choose a random oracle $f : \{0,1\}^\ast \to \{0,1\}$, and define the logic $ZFC^f$ by adding a fresh symbol $g$, an axiom that $g$ has the correct type, and one axiom $g(s) = f(s)$ for each $s \in \{...
1
vote
0answers
53 views

Kolmogorov generic oracle

In Does the Polynomial Hierarchy Collapse if Onto Functions are Invertible?, authors defined a new type of generic oracles named Kolmogorov generic oracles. They proved following results relative to $...
0
votes
0answers
133 views

An oracle that separates BPP from PP?

Is there an oracle that gives $P=BPP\neq NP=PP$?
5
votes
1answer
86 views

Is $UP\not=NP$ with respect to random oracle?

It is shown in An average-case depth hierarchy theorem for Boolean circuits a random oracle makes $PH$ infinite. Is it possible to also show $UP\not=NP\not=\Sigma_2^P\not=\Sigma_3^P\not=\Sigma_4^P\...
0
votes
0answers
17 views

Are $BPP^{BPP^{\oplus P}}$ and $BPP^{NP}$ contained in $BPP^{\oplus P}$?

Does $BPP^{NP}\subseteq\Sigma_k^P\cap P/poly^{NP}\subseteq BPP^{\oplus P}$ hold at some $3\leq k$? Also is $BPP^{BPP^{\oplus P}}\subseteq BPP^{\oplus P}$ known?
1
vote
0answers
51 views

On collapsing the Exponential time hierarchy

Define $\Sigma^E_0 = \Pi^E_0=E$, for every $n>0$, define $\Sigma^E_n=NE^{\Sigma^p_{n-1}}$, for every $n>0$, define $\Pi^E_n=CoNE^{\Sigma^p_{n-1}}$. Define the Exponential time hierarchy by $EH=\...
0
votes
1answer
159 views

How to prove $P^{Halt} = PSPACE^{Halt}$ [closed]

Halt means the halting set. $PSPACE^{Halt}$ is the class of problems that can be solved with polynomial memory (possibly exponential time), given a halting oracle.
6
votes
2answers
150 views

What are examples of complexity classes that have contradictory relativizations but they were proven to be either equal or unequal?

In this article Chang et al. provide a counterexample by giving an oracle $A$ such that $\mathsf{IP}^A \neq \mathsf{PSPACE}^A$. I wanted to know if there are more examples like this.
12
votes
1answer
345 views

Compressing information about the halting problem for oracle Turing machines

The halting problem is well-known to be uncomputable. However, it is possible to exponentially "compress" information about the halting problem, so that decompressing it is computable. More precisely,...
4
votes
2answers
175 views

Oracle comparing $EXP$ with $UP$

Heller (Theorem 6) gave an oracle relative to which $NP=EXP$, and Homer & Selman gave an oracle relative to which $P=UP$ and $\Sigma_2^P=EXP$. Beigel, Buhrman, Fortnow (freely available author's ...
0
votes
0answers
126 views

Grover's for QUBO form?

My preferred formulation of the quadratic unconstrained binary optimization problem (QUBO) is the following: Find $\min(z)=x'Qx$, where $x$ is a $n$-vector of binary variables, $x'$ is its transpose, ...
2
votes
0answers
102 views

Complexity of $FP^{NP[O(n)]}$ with advice string?

$FP^{NP[O(n)]}$ is the functional complexity class with $O(n)$ queries to an $NP$ oracle. Are there any interesting classes $\mathcal C$ such that $$FP^{NP[O(n)]}\subseteq\mathcal C/\log$$ besides $...
7
votes
1answer
2k views

Oracle Construction for Grover's Algorithm

In Mike and Ike's "Quantum Computation and Quantum Information", Grover's algorithm is explained in great detail. However, in the book, and in all explanations I have found online for Grover's ...
9
votes
1answer
223 views

Does there exist an oracle $A$ such that $(P^{\#P})^{A} \neq PSPACE^{A}$?

Background We know that $P^{\#P} \subseteq PSPACE$. In addition, we known from Toda's theorem that $PH \subseteq P^{\#P}$. For more background on $\#P$, see here: https://en.wikipedia....
18
votes
1answer
388 views

What is the minimum complexity oracle that separates PSPACE from the polynomial hierarchy?

Background It is known that there exists an oracle $A$ such that, $PSPACE^A \neq PH^A$. It is even known that the separation holds relative to a random oracle. Informally, one may interpret ...
8
votes
1answer
236 views

Using an oracle to find a vector $b$ for which $Ax=b$ has a solution

There is an oracle built around a hidden $m\times n$ matrix $A$ all of whose entries are 0 or 1, where $m>n$. The oracle takes as input an integer vector $b$ with positive entries, and answers as ...
7
votes
1answer
298 views

Is $\sf{P^{NP \cap coNP}} = \sf{NP \cap coNP}$?

If it is unknown, are there reasons to believe that they might not be equal?
7
votes
0answers
136 views

SETH-like hypothesis for machine with oracle access to some level of PH

I am wondering if hypothesis such as Strong Exponential Time Hypothesis (SETH) have been studied for problems being in a higher level of the polynomial hierarchy when we give the machine access to an ...
4
votes
0answers
224 views

Oracle separating $coNP$ and $NP/poly$

I'd like to prove that, with respect to some adversarial oracle $O$, $coNP^O \not\subseteq NP/poly^O$. I was thinking of using $\textsf{UNSAT}$ for this and to build my oracle as follows: $O$ will "...
8
votes
0answers
142 views

Oracles for sub-exponential circuit complexity of $\Sigma_2 EXP$

Couldn't find this one anywhere... It's an open problem whether $\Sigma_2 EXP$ problems have exponential-size circuit complexity. Is there an oracle relative to which $\Sigma_2 EXP$ has $2^{o(n)}$ ...
5
votes
0answers
118 views

An oracle in $\mathsf{NEXP}$ that separates ZPP from BPP

Does there exist an oracle $A \in \mathsf{NEXP}$ such that $ \mathsf{ZPP}^A \neq \mathsf{BPP}^A$?
-1
votes
1answer
107 views

Given oracle for Max-3SAT compute clauses that cannot be satisfied

We know that Max-3SAT is NP-hard to compute exactly (and also hard to approximate to a particular constant multiplicative factor). However, suppose you are given an oracle for Max-3SAT that tells you ...
9
votes
1answer
247 views

Is there a good notion of non-termination and halting proofs in type theory?

Constructive type theory with its basic interpretation under the curry howard correspondence consists only of total, computable functions. In the literature, some has been said on using "computational ...
8
votes
0answers
53 views

Oracle-Decidability of Algebraic Independence

Consider numbers $x_1,...,x_n\in \mathbb{R}$ given by TMs $M_1,...,M_n$ such that $M_i$ approximates $x_i$ to an arbitrary precision (by allowing it to run longer and longer). I am interested in the ...
6
votes
1answer
122 views

Lower bounds for nonuniform circuits and oracles separating complexity classes

I have read that Furst, Sax, and Sipser came up with their lower bound for nonuniform AC0 while trying to prove an oracle separation. Can someone explain how proving lower bounds for circuits and ...
3
votes
0answers
90 views

Functional oracles

In the traditional oracle Turing machine, the oracle is specified as a decision problem. Roughly speaking, one puts a string in the oracle tape, and asks whether it is true or false. I am wondering ...
8
votes
0answers
137 views

Manuel's trick and oracle separation

Impagliazzo gave a talk last week at Simons Institute on oracle separation. At minute 5:34 he asks whether a one-way permutation can be constructed given oracle access to a random function oracle. ...
30
votes
2answers
757 views

Is there an oracle such that SAT is not infinitely often in sub-exponential time?

Define $io$-$SUBEXP$ to be the class of languages $L$ such that there is a language $L' \in \cap_{\varepsilon > 0} TIME(2^{n^{\varepsilon}})$ and for infinitely many $n$, $L$ and $L'$ agree on all ...
5
votes
0answers
293 views

An oracle relative to which EXP(NP) = BPP

Whether or not $\mathbf{BPP} = \mathbf{EXP}^{\mathbf{NP}}$ is an open problem, although we believe the former is strictly contained in the other. I guess, from the absence of the proof of the ...
18
votes
2answers
560 views

For a random oracle R, does BPP equal the set of computable languages in P^R?

Well, the title pretty much says it all. The interesting question above was asked by commenter Jay on my blog (see here and here). I'm guessing both that the answer is yes and that there's a ...
12
votes
1answer
428 views

Does Kannan's theorem imply that NEXPTIME^NP ⊄ P/poly?

I was reading a paper of Buhrman and Homer “Superpolynomial Circuits, Almost Sparse Oracles and the Exponential Hierarchy”. On the bottom of page 2 they remark that the results of Kannan imply that $...
7
votes
2answers
572 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 $...
12
votes
1answer
591 views

$\mathsf{NP} \cap \mathsf{coNP}$ as oracle

Does $\mathsf{NP^{NP \,\cap\, coNP}=NP}$ hold? Clearly $\mathsf{NP^{NP}\neq NP}$, but it seems to me that $\mathsf{NP\cap coNP}$ is "deterministic" which makes me believe this is true. Is there a ...
8
votes
2answers
372 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 $B^A\neq C^A$ ...
4
votes
0answers
100 views

Reduction from a geometric decision problem to its maximization problem

I am interested in the following NP-complete decision problem: ...
6
votes
1answer
199 views

Oracles which put integer factorization in P

I'm compiling a list of as many problems (decision or function) as I can find such that, if I had an oracle that could solve the problem in P, then integer factorization would also be in P. Here is a ...
11
votes
1answer
488 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 NP^...
2
votes
2answers
831 views

Quantum oracle implementation overhead

I am a physicist getting acquainted with one of the typical constructs for formulation and analysis of quantum algorithms (such as search problems or query complexity models), namely the "oracle ...
5
votes
1answer
456 views

$\mathsf{P}^\mathsf{BPP}$ vs $\mathsf{BPP}$ (Are they known to be equal)

Is it known if $\mathsf{P}^\mathsf{BPP}= \mathsf{BPP}$ ? It's clear that $\mathsf{BPP} \subseteq \mathsf{P}^\mathsf{BPP}$. Now, since $\mathsf{BPP}$ is closed under complementation, union, and ...
2
votes
1answer
72 views

Natural relativized worlds

The oracles that are used in relativized collapses or separations of complexity classes rarely represent $natural$ algorithmic problems. They are typically constructed "artificially" with techniques ...
7
votes
2answers
405 views

What is $DTIME(n^a)^{DTIME(n^b)}$?

This might be embarrassing, but it turned out I don't know what is $DTIME(n^a)^{DTIME(n^b)}$. It is between $DTIME(n^{ab})$ and $DTIME(n^{a(b+1)})$ but where? Update: There are three possible ways to ...
10
votes
1answer
388 views

Is $\mathsf{MA}$ equal to $\mathsf{NP}^\mathsf{RP}$?

I haven't been able to find a statement relating $\mathsf{MA}$ and $\mathsf{NP}^\mathsf{RP}$ in the literature; pointers would be appreciated. I believe they are equal: $\mathsf{MA} \subseteq \...
0
votes
0answers
127 views

Oracle that will provide any computable information about another oracle

Suppose I have an oracle X. Then let Y be an oracle which will answer any computable question about X. In other words, Y takes as input a Turing program which can in turn make calls to X. Y then ...
10
votes
4answers
1k 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 ...
6
votes
2answers
320 views

Oracle complexity of a problem in the Counting Hierarchy

In "On The Complexity of Numerical Analysis" (SIAM J. Comp. Vol. 38, 2009), Allender et al. introduce the problem of PosSLP and show that its complexity lies in the counting hierarchy, and more ...
11
votes
1answer
306 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 ...
1
vote
1answer
107 views

Is predicting (in the limit) computable sequences as hard as a dominating function?

Define a "predicting oracle" to be an oracle that does as described in this question. default (weak) version: Is it the case that, for every predicting oracle $O$, there exists an oracle machine $M$...