# Questions tagged [complexity-classes]

Computational complexity classes and their relations

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### complexity class of a function - linear combinations and reductions (Fermionant, immanant, $GL_n$ representations)

The fermionant is a matrix function from physics, which is indexed by a positive integer $k$: \begin{align} \operatorname{Ferm}_k(A) = \sum_{\lambda} d_{\lambda}^{(k)} \operatorname{Imm}_{\lambda^T}(A)...
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### What is the best simulation of majority utilizing $\bmod\{2,3,\dots,p\}$ gates?

It is known $AC^0$ cannot get majority function. Is there a literature on simulation of $MAJ$ function utilizing $AC^0[2,3,\dots,p]$ gates for a finite fixed set of primes $2$ to $p=O(1)$? What is ...
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### Language in $PSPACE$ and not necessarily in $P$ if $P=PP$?

If $P=PP$ then the counting hierarchy collapses to $CH=P$. Because so many complexity classes are contained in $CH$, this causes most classes to now be contained in $P$. My question is whether this is ...
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### On Determinant modulo $2^k$ complexity

Determinant of integer matrix modulo $2$ is complete for the class $\oplus L$. Is determinant modulo $2^k$ computable in $\oplus L$ at any fixed $k$? How about if $k=o(n)$ where matrix is $n\times n$?
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### The graph of problem reductions

A classical approach to study the complexity of a problem $P$ is to efficiently reduce a well known problem $P'$ to $P$, thus showing that $P$ is at least as difficult as $P'$. The TCS literature ...
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### Where is $MA$ more relevant than $\exists BPP$?

(EDITED) A previous version of this question asked about a complexity class I called $MA^*$, which has been recognized by users to be $\exists BPP$. The difference between $MA$ and $\exists BPP$ is ...
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### Is anything known about NC$^1$ with NP oracle

A few things are known about the class $\textsf{L}$ provided with an $\textsf{NP}$ oracle ($\textsf{L}^\textsf{NP} = \Theta_2^\textsf{P}$ has attracted a bit of attention, for instance ) On the ...
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### Logarithmic queries to $\Sigma_i^P$ oracle and the Boolean hiearchies

If I understood correctly (the complexity zoo, wikipedia, and some of the cited articles), the class $\textsf{P}^{\textsf{NP}[\log]}$, also known as $\Theta_2^{\textsf{P}}$, sits at the top of the ...
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### $\mathsf{ACC}^0$ and $\mathsf{TC}^0$ with $\mathsf{Cuniform}$-$\oplus\mathsf{L}$ or $\mathsf{Cuniform}$-$\mathsf{NC}^1$ oracle?

$\mathsf{TC}^0$ is a small class with $\oplus\mathsf{L}$ containing it. Following inclusions are known: \mathsf{Cuniform}\mbox{ -}\mathsf{ACC}^0\subseteq\mathsf{Cuniform}\mbox{ -}\mathsf{TC}^0\...
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### Is there any relation between $\mathsf{PPP}\subseteq\mathsf{TFNP}$ and $\mathsf{UP}$? [closed]

A complete problem for $\mathsf{PPP}$ is the pigeon problem which is 'Given a Boolean circuit $C$ having the same number $n$ of input bits as output bits, find either ...
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### Is there a containment between $\mathsf{ZPP}$ and $\mathsf{UP}$?

Do we know if $\mathsf{ZPP}\subseteq\mathsf{UP}$ known or is there oracles against the hypothesis?
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### NP complete problem help

I'm currently trying to find a reduction to this problem: Given a set S of n points (in the plane) in general position, is there a set of at least k triangles (formed using only points in S as ...
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### What do we know about provably correct programs?

The ever increasing complexity of computer programs and the increasingly crucial position computers have in our society leaves me wondering why we still don't collectively use programming languages in ...
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### In what class are randomized algorithms that err with exactly 25% chance?

Suppose I consider the following variant of BPP, which let us call E(xact)BPP: A language is in EBPP if there is a polynomial time randomized TG that accepts every word of the language with exactly 3/...
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### Is it to solid to conclude APX-complete after showing a problem cannot be approximated better than 1.5 and also develop a 2-approximation algorithm

I know the canonical way to show APX-Complete is to give an L-reduction from an already-known APX-complete problem. If I have used gap-preserving reduction to show a problem cannot be approximated ...
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### Can a NEXP machine simulate invalid queries to a promise problem oracle?

Let $A=(A_{YES},A_{NO})$ be some promise problem (such as xSAT, the Local Hamiltonian problem, etc). Suppose we want to show that a P machine with access to a the oracle A can always have its output ...
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### What graphs on $\mathbb{N}$ can be encoded as regular languages?

Suppose I represent the natural number 0 by "x", and use the symbol "s" for successor so that I get the following encoding of $\alpha : \mathbb{N} \rightarrow V$ of natural numbers ...
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### Unambiguous Problems and Classes over Reals

Are there unambiguous analogues of $NP_{R}$ (using the BSS model, in all discussion)complete problems, and any results known about them? For instance, the canonical $NP_{R}$ complete problem $4FEAS$ (...
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### Has parameterized complexity led to better algorithms?

I know that for the vertex cover problem, if we know that the parameter $k$ (which is the number of vertices in the solution) is small, then we can expect to solve it feasibly in practice. So far, ...