Questions tagged [oracles]

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.

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Turing meta-oracle

Let H(P) be some program that given P('s source code) computes whether or not P terminates, i.e. solves the halting problem. H only needs to terminate if P terminates. (This disallows solutions like ...
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Results comparing BQP and NEXP

Are there oracle results with $$P=NP\neq BQP=NEXP\mbox{ and }P=NP\neq BQP\neq NEXP?$$ Also is there a $PCP$ characterization of $BQP$ like $$PCP(O(poly(n)),1)=PCP(O(poly(n)),O(poly(n)))=NEXP?$$
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Possibility of hierarchy with $UP$ class?

I am not sure if this is a cheap query. However I am unable to find this myself. So I am posting here. The standard complexity class is built with $NP$ and $coNP$ and leads up to $PSPACE$. The ...
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Oracle separation between PH and PSPACE

I am having difficulty understanding the concept and intuition behind this proof. The proof deals with constructing an oracle $A$ relative to which $PH$ is separated from $PSPACE$. I have several ...
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1answer
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Lower bound on alternations needed in $BQP$ versus $PH$ result?

What is the fastest $f(n)$ the relatively new result of oracle separation of $\mathsf{BQP}$ from $\mathsf{PH}$ provides such that ${\#\mathsf{SAT}}\not\subseteq\mathsf{FP}^{\mathsf{PH}[O(f(n))]}$ ...
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Why is the “general notion of a reduction […] inherent to the notion of self-reducibility”?

While reading "Computational Complexity: A Conceptual Perspective" by Oded Goldreich, I have come across the following passage, which I simply cannot get my head around: Note that the general ...
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Can relativization technique be applied to natural NP-complete languages?

Levin [1] defined distNP is the distributional problem (L,D), where L ∈ NP, and D is an ensemble of efficiently samplable distributions over problem instances. We say that a distNP problem (L,D) is ...
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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 ...
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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 ...
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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 ...
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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 \{...
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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 $...
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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\...
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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=\...
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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.
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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.
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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,...
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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 ...
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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 $...
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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 ...
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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....
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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 ...
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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 ...
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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?
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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 ...
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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 "...
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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 $2^{o(n)}$ ...
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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$?
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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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 ...
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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 $...
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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 $...
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$\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 ...
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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 $B^A\neq C^A$ ...
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Reduction from a geometric decision problem to its maximization problem

I am interested in the following NP-complete decision problem: ...
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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 ...
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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^...
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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 ...
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$\mathsf{P}^\mathsf{BPP}$ 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 ...
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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 ...
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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 ...
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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 \...
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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 ...