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 tagged [complexity-classes]

Computational complexity classes and their relations

1
vote
1answer
72 views

PromiseBQP and expectation values of operators

This question is regarding The Equivalence of Searching and Sampling by Aaronson. In page 4 he makes the following statement, ... a difficult and unsolved meta-question is whether PromiseBPP = ...
5
votes
2answers
710 views

Isn't it trivial to represent any classical Physics problem in a Spin-Glass format which is NP-Complete?

In the late 80's there were several efforts to use Spin-Glass models to formulate other computational problems such as: Protein Folding and Neural Networks. Wouldn't it be straight forward to reduce ...
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 $...
8
votes
1answer
368 views

Complexity of modal logic IK5

What is the complexity of local satisfiability problem for modal logic $\mathit{IK5}$? Herein we denote by $IK5$ the modal logic over euclidean frames extended with inverse modality. Could you provide ...
0
votes
0answers
65 views

Using idea of entropy (maybe Shannon entropy or other continuous entropy) to the topic of functional analysis

I am an electrical engineer without a detailed background in theoretical computer science. I am posting here since I hypothesize that the concept of entropy or other branches of information theory (as ...
1
vote
0answers
87 views

Does the NP-hardness of finding any valid solution imply NPO-hardness?

Suppose we have an NP optimization problem $A$ such that the problem of finding any valid solution, regardless of whether it is good or bad, is NP-hard (as it happens in NPO-complete problems such as ...
-5
votes
1answer
122 views

Should GCT focus on $PSPACE\not\subseteq P/poly$?

GCT tries to show $P$ is not $NP$ by showing $NP$ is not in $P/poly$. Could it be useful in showing $\Sigma_{i+1}\not\subseteq P^{\Sigma_i}/Poly$ at every $i>0$? Suppose if it turns out that $\...
10
votes
1answer
131 views

On sparse complete sets and P vs L

Mahaney's Theorem tells us that if there is a sparse $NP$-complete set under polynomial-time many-one reductions, then $P = NP$. (See "Sparse complete sets for NP: Solution of a conjecture of Berman ...
3
votes
0answers
75 views

How to charactorize computational complexity based on finding solution to algebraic equations? [closed]

The Matiyasevich/MRDP Theorem relates two notions:one from computability theory, the other from number theory, thus Turing Machine and algorithms finding integral solution to algebraic equations can ...
1
vote
1answer
193 views

Relation between transcendental numbers and computational complexity?

Regarding the relation, there is the Hartmanis-Stearns conjecture, but beyond Turing Machine of the realtime output, there is no further conjecture or theorem. Obviously irrational algebraic number ...
7
votes
1answer
125 views

Presburger arithmetic: is it known to be in $EXPSPACE \setminus EXP$?

My understanding of the Presburger arithmetic decision problem is that it requires doubly-exponential time, but only singly-exponential space, meaning that it is in $EXPSPACE \setminus EXP$. ...
2
votes
0answers
89 views

Random self reducibility and NP

I was reading the Wikipedia page https://en.m.wikipedia.org/wiki/Random_self-reducibility and it states: "If an NP-complete problem is non-adaptively random self-reducible the polynomial hierarchy ...
2
votes
0answers
61 views

On reduction between two classes?

https://link.springer.com/article/10.1007/s00153-013-0351-x gives seven reductions $m,c,d,p,btt(1),\ell,tt$. What does norm $1$ mean in $btt(1)$? Is there illustrative examples that help understand ...
2
votes
1answer
103 views

What is conjunctive truth table reduction?

What are conjunctive/disjunctive truth table reductions and how do they compare with other reductions?
10
votes
0answers
104 views

Problems in $\mathsf{NC^{2}}$ that are not known to be in $\mathsf{AC^{1}}$ or $\mathsf{DET}$

Do we know of any problems in $\mathsf{NC^{2}}$ that are not known to be in $\mathsf{AC^{1}}$ or $\mathsf{DET}$? Context: based on Josh's answer to this question, it could be possible that all ...
1
vote
1answer
58 views

Recursively presenting or even enumerating all P-hard languages

A class of languages $C$ is recursively presentable if there is an effective enumeration of Turing machines $\mathcal{M}_1,\mathcal{M}_2,\ldots$ such that $C=\{L(\mathcal{M}_i)\mid i=1,2,\ldots\}$. ...
4
votes
2answers
240 views

On $NP$, $\oplus P$ and $PP$?

We know $\oplus P^{\oplus P}=\oplus P$, $PP^{\oplus P}\subseteq P^{PP}$ and $NP\subseteq PP$. Is $\oplus P^{PP}=PP$? Why is it difficult to show $NP^{NP}\subseteq PP$? What is the smallest known ...
2
votes
0answers
35 views

A class of languages admitted by a class of grammars equivalent to $\mathbf{PR}$?

Is there a class of languages $L(G)$ admitted by a class of phrase structure grammars $G$ equivalent to $\mathbf{PR}$? (the class of primitive recursive languages = $\mathbf{LOOP}$)? In greater ...
-1
votes
1answer
130 views

Two queries related to Toda

Is the Isolation lemma crucial for $PH\subseteq BPP^{\oplus P}$ theorem and would avoiding the Isolation lemma say anything more that is not known?
3
votes
0answers
104 views

Approximation class of finding decision trees with minimal depth

We are given some sets $S_1, \cdots , S_n$ and two disjoint sets $A$ and $B$. A decision tree is a binary tree where each node asks "$x \in S_i? $" for some $i$, taking the left branch means "yes", ...
-4
votes
1answer
59 views

Proving NP-complete problem

Suppose the following problem: Given an undirected graph G=(V,E), is it possible to choose a subset V' of vertex set V, such that deleting it removes all triangles (cycles of length 3), where |V'| is ...
1
vote
0answers
56 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=\...
4
votes
0answers
284 views

What happens when PSPACE contains NEXP?

The complexity class NEXP is defined as the set of all languages that an arbitrary nondeterministic exponential time Turing Machine accepts (i.e. NTIME($2^{p(n)}$) for $p()$ a polynomial). In the ...
2
votes
0answers
48 views

Given an algebraic variaties of n multivarieties polynomial equations, is there any algorithm to decide whether there is n-cube inscribing to it?

Given an algebraic variaties of n multivarieties polynomial equations, is there any algorithm to decide whether there is n-cube inscribing to it? And if there is, what is the computational ...
1
vote
1answer
147 views

What is the computational complexity of solutions over $\mathbb{Q}$ of polynomial equation with coeffiecents over $\mathbb{Z}$

What is the complexity of the following problem? (e.g. best-known running time, space, best upper bound in terms of complexity classes, etc.) Input: A multivariate polynomial $f$ with coefficients ...
3
votes
1answer
103 views

Almost-P and related definitions

I'm pretty sure this has a trivial answer but it's always faster to ask the community :-) I understand that, relative to a random oracle, P=BPP. But this is sometimes phrased via the shorthand "...
-1
votes
1answer
76 views

What is the relation between P-immune languages and NP-complete languages? [closed]

Can a NP-complete language be P-immune? Why can't existence of P-immune languages separate NP from P?
4
votes
2answers
214 views

When studying the computational complexity of functions $\{0, 1\}^\ast \to \{0, 1\}^\ast$, is it enough to restrict to $\{0, 1\}^\ast \to \{0, 1\}$?

I started reading Avi Wigderson's paper $\mathcal{P}$, $\mathcal{NP}$ and Mathematics – a Computational Complexity Perspective (link). (Notation: $\{0, 1\}^\ast$ is the set of all finite binary ...
1
vote
1answer
206 views

Is there any known Poly-APX-complete minimimization problem?

All Poly-APX-complete problems I know are maximization problems, e.g. Max Clique, Max Independent Set, Max One for some set of contraints, and even choosing the attributes of a product to maximize (...
-1
votes
1answer
71 views

Is there an inherently ambiguous language which can not be recognized by Deterministic LBA?

Is there inherently ambiguous language which can not be recognized by Deterministic LBA? For example, $L=\{wv: w,v=(x|y)^*, w=w^R,v=v^R\}$, is there any deterministc LBA that recognizes $L$ ?
2
votes
1answer
83 views

Is sliding blocks linear space complete?

Sliding blocks is PSPACE complete even in its simplest form involving 1x2 and 2x1 blocks (without rotation or fractional positions) in a rectangular area, with goal being to move a designated block to ...
11
votes
2answers
223 views

Example of something that’s different for generic and random oracles?

Let $G$ be a generic oracle in the sense of Cohen / Baire category. Let $R$ be a random oracle. Are there complexity classes A and B with $$\mathrm{A}^G=\mathrm{B}^G\quad\text{and}\quad\mathrm{A}^R\...
7
votes
0answers
153 views

Complexity Class Equalities on the Edge of Inconsistency

What are some of the most extreme potential equalities between computational complexity classes (especially if there is a barrier to refuting them)? These may give us an opportunity to prove better ...
4
votes
0answers
160 views

Is there a language in NSPACE(O(n)) and (very likely) not in DSPACE(O(n))?

Actually I found that the set of context-sensitive Languages, $\mathbf{CSL}$ ($\mathbf{=NSPACE}(O(n))$ $= \mathbf{LBA}$ accepted languages) are not so widely discussed as $\mathbf{REG}$ (regular ...
1
vote
2answers
142 views

Is Asymptotic PTAS $\subseteq$ APX?

The definition of asymptotic polynomial-time approximation scheme (Asymptotic PTAS) is defined as follows: A minimization problem $\Pi$ is Asymptotic PTAS if for all $\epsilon$ there exists an ...
6
votes
1answer
707 views

Power of randomness vs. power of indefinite computation

I am writing a paragraph on the power of randomness, part of which I am trying to ground in theory of computation (I am no expert/researcher in this field). First off, I am aware that for ...
4
votes
3answers
307 views

Classes between $\textbf{PSPACE}$ and $\textbf{EXP}$

1) What classes contain $\textbf{PSPACE}$, are contained in $\textbf{EXP}$ and (presumably) are not equal to $\textbf{PSPACE}$ nor to $\textbf{EXP}$? A possible class satisfying this requirement: the ...
9
votes
0answers
100 views

Logarithmic levels of the polynomial hierarchy (below PSPACE)

We generally define $PH = \cup_i\Sigma_i^p$ (or various equivalent forms.) In the same notation we can also define $PSPACE = \cup_c\Sigma_{n^c}^p$--that is, like the polynomial hierarchy, but with a ...
6
votes
2answers
154 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.
22
votes
1answer
1k views

Tardos Function Counterexample to Blum's $P\neq NP$ Claim

In this thread, Norbet Blum's attempted $P \neq NP$ proof is succinctly disproved by noting that the Tardos function is a counterexample to Theorem 6. Theorem 6: Let $f \in \mathcal{B}_n$ be any ...
6
votes
1answer
179 views

Cases of Linear programming known to be in $NC$?

Linear programming is $P$-complete. However are there special situations where we know an $NC$ algorithm?
232
votes
11answers
114k views

Is Norbert Blum's 2017 proof that $P \ne NP$ correct?

Norbert Blum recently posted a 38-page proof that $P \ne NP$. Is it correct? Also on topic: where else (on the internet) is its correctness being discussed? Note: the focus of this question text has ...
4
votes
2answers
181 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 ...
4
votes
0answers
85 views

A succinct version of permanent that is $EXP$-complete

Succinct version of permanent is $NEXP$-hard (https://eccc.weizmann.ac.il/report/2012/086/) and so unlikely to be $EXP$-complete. Permanent mod $2$ is in $\oplus L$ and so succinct version is ...
9
votes
1answer
193 views

variant of Critical SAT

The language Critical SAT is defined as the set of $CNF$ boolean formulas $f$ such that $f \in UNSAT$ but removing any clause from $f$ makes it satisfiable. It is known that Critical SAT is $DP$-...
-2
votes
1answer
150 views

Closure properties of $L$ (DLOGSPACE)? [closed]

What are the closure properties of $L$ (DLOGSPACE)? I'm not only intrested in these properties (if of course $S$ and $T$ are in $L$) : $S \cap T$ $S^*$ (kleene-star) $S.T$ (concat)
2
votes
1answer
172 views

What does $\#P\subseteq FP^{PPAD}$ imply?

We know $\#P\subseteq {PPAD}\implies PH\subseteq P^{{PPAD}}\subseteq P^{{NP}}$ and the polynomial hierarchy collapses ($FP^{PPAD}=PPAD$ following Emil Jerabek's comment). Can $\#P\subseteq {PPAD}...
12
votes
1answer
343 views

Time Hierarchies in DSPACE(O(s(n)))

The time hierarchy theorem states that turing machines can solve more problems if they have (enough) more time. Does it hold in some way if the space is limited asymptotically? How does $\textrm{DTISP}...
1
vote
0answers
78 views

A uniform computability model to define time and space complexity (even in the sublinear case) [closed]

To define time complexity the Turing machine model with only one tape (for input, work and output) is used. This TMM is also used to define the $s(n)$-space complexity for $s(n) \ge n$. But if $s(n)$ ...
3
votes
1answer
205 views

Dp completeness of a problem

Given a Boolean formula $\varphi$ over the variables $\{x_1...x_n\}$ , an assignment $T_0$ for $\varphi$ and an integer $k$, I am interested in the following question: Does $k$ is the minimal number ...