Questions tagged [complexity-classes]

Computational complexity classes and their relations

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10
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0answers
123 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 ...
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1answer
112 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 $C=\{L(\mathcal{M}_i)\mid i=1,2,\ldots\}$. ...
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2answers
269 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 ...
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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 ...
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1answer
137 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?
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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", ...
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1answer
64 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 ...
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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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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 ...
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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 ...
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1answer
160 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 ...
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1answer
177 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 "...
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1answer
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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?
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2answers
230 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 ...
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1answer
446 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 (...
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1answer
81 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 $L$ ?
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1answer
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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 ...
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2answers
285 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\...
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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 ...
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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 ...
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2answers
348 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 ...
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1answer
736 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 ...
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3answers
382 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 ...
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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 ...
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2answers
179 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.
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1answer
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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 ...
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1answer
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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?
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11answers
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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 ...
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2answers
220 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 ...
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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 ...
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1answer
230 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 $DP$-...
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1answer
231 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)
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1answer
233 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
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1answer
456 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}...
3
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1answer
353 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 ...
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On status of Valiant's $NC^2=P^{\#P}$ provability program?

In here it is written 'A most interesting/controversial talk was by Leslie Valiant. He explored paths to try to prove that $NC^2=P^{\#P}\dots$'.... This was a decade back. What is the rationale (at ...
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1answer
165 views

On $\#P\subseteq FP^{\Sigma_{f(n)}^P}$?

Is it known that permanent of a $0/1$ $n\times n$ matrix $M$ is computable in polynomial or randomized polynomial time with access to a ${\Sigma_{(\log n)^c}^P}$ oracle where $0<c$ holds and $\...
3
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1answer
137 views

On $\Delta_i^P$

We know $P\subseteq NP\cap coNP\subseteq\Delta_i^P=P^{\Sigma_{i-1}^P}\subseteq \Sigma_i^P\cap\Pi_i^P=NP^{\Sigma_{i-1}^P}\cap coNP^{\Sigma_{i-1}^P}$. If $P=BPP$ is there a 'higher' randomized class ...
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Co-Partition Problem: Why is this proof for it being in NP wrong? [closed]

So I have this wrong proof that the problem "Co-Partition" is in NP. I know the proof is wrong because I've encountered it in an educational environment and was told that it's not working. I don't, ...
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0answers
110 views

What is the complexity of computation of zero point by Rieman zeta function?

What is the complexity of computation of zero point by Rieman zeta function? That is, given s, whether ζ(s)=0? Is it in P? Any reference is appreciated.
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1answer
305 views

${\bf NP} \not = {\bf E}$ and ${\bf PSPACE} \not = {\bf E}$

We know that ${\bf NP} \subseteq {\bf PH} \subseteq {\bf PSPACE}$. We also know that ${\bf E} \subset {\bf EXP}$, where ${\bf E} = \cup_c DTIME[2^{cn}]$ and ${\bf EXP} = \cup_c DTIME[2^{n^c}]$. It ...
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1answer
156 views

Descriptive model theory classification of Counting hierarchy

Descriptive model theory uses logic to characterize complexity classes How to model Counting Hierarchy PSPACE in descriptive model theory?
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293 views

Is there a computing problem which is in quasi-polynomial time but is (maybe) not in $\beta P$?

Quasi-polynomial time, or QP for short, is a complexity class on deterministic Turing machine. Here is the precise definition:https://complexityzoo.uwaterloo.ca/Complexity_Zoo:Q#qp While βP is a ...
2
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1answer
289 views

On $NP$ and $XP$ classes?

On page 33 venn diagram in http://tcs.rwth-aachen.de/~sanchez/slides/Raleigh2014.pdf it is implied that $XP\subseteq NP$. Below this there is a statement which says $XP\not = NP$ unless $P=NP$. Is ...
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0answers
338 views

Intermediate problems between PSPACE and EXPTIME

Intermediate problems between P and NP are quite famous, and are sometimes considered as complexity classes by themselves. Do you know of any problem that is known to be PSPACE-hard and in EXPTIME, ...
6
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2answers
484 views

Complexity of comparison unary>binary

What is the smallest widely-known complexity class to which $$\left\{\langle i,j\rangle\middle|\begin{array}{@{}l@{\ }l@{}} & i\ \text{is a unary encoding of a positive integer}\ \hat\imath\\\...
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1answer
191 views

Is finding a solution harder than verifying a solution? [closed]

Is there any known problem in Comp science where determinisitically finding a "non-trivial" solution to that problem is asymptotically easier than verifying a solution?
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1answer
454 views

A question of relationships between #P and PSPACE [closed]

Let us assume there is some machine X that converts boolean formula to following form in polynomial time: $$\Phi(x_1, x_2 .. x_m) = r_1(x_{i_1}, x_{j_1}, x_{k_1}) \land r_2(x_{i_2}, x_{j_2}, x_{k_2}) ...
8
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1answer
340 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....
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0answers
221 views

Does the existence of an RP-complete language imply P = RP?

(I'm not sure if this is research-level, but I couldn't find an answer to this question elsewhere) The question of whether there exists an RP-complete language seems to be open, but I guess we believe ...

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