Questions tagged [complexity-classes]

Computational complexity classes and their relations

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Is mathematical proof itself NP-hard?

At the 8:00 mark of this video, he claims that proving things is itself an NP problem. I'm looking for more insight into this. Could someone help explain this concept to me and also provide a link to ...
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Are all RegExp solvable in O(n)?

I'm wondering if all features, that are often part of modern RegEx engines, are solvable in O(n). I'm talking about features like repeating patterns ([abc]+);\1 ...
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Diagonalization arguments for QMA type proof systems

Diagonalization is a very common technique to find oracle separations. For example, it can be used to separate $\cal{P}$ and $\cal{NP}$, with the essential idea being that of constructing an oracle in ...
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Why can MIP be restricted to just two provers?

In several places I see it referred to that the MIP class can be assumed to be two interactive provers that don't communicate with each other, rather than any polynomial number of provers. Why are ...
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Is there an assumption that implies $P=ZPP$ which is not known to imply $P=BPP$?

There are assumptions that are known to imply that $P = BPP$. For example, if there exists a function in $E = DTIME(2^{O(n)})$ that has circuit complexity $2^{\Omega(n)}$, then $P = BPP$ [1]. Clearly, ...
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Complexity class of efficient streaming algorithms

Consider the class of problems $\mathsf{StreamL}$ which can be solved in logarithmic space reading the input in a single pass from left to right. In other words: $L \in \mathsf{StreamL}$ if there ...
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Is there an analogue of QMA where Merlin gives Arthur unitaries rather than states?

$\def\braket#1#2{\langle#1|#2\rangle}\def\bra#1{\langle#1|}\def\ket#1{|#1\rangle}$Is there an analogue to $\mathsf{QMA}$ where Merlin provides to Arthur single-use access to a unitary operator $U$? By ...
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270 views

The theoretical complexity of Go - The state of the art

What are the latest advances in theoretical complexity of Go? I know some early works about the complexity of Go: "Go is polynomial-space hard" proved that Go is PSPACE-hard. "Ladders are PSPACE-...
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Is PP invariant under changing its cut-off from 1/2 to another number?

Suppose I have a fixed family of quantum circuits $\{C_i\}$ for which determining whether the maximum output acceptance probabilities are $p\geq 1/2$ or $p< 1/2$ is PP-hard. Now suppose I have the ...
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108 views

Order of quantifiers in the definition of NP-completeness: does the reduction allow arbitrary polynomials? [closed]

Arora and Barak define NP-completeness as the following: "We say that a language $A \subseteq \{0, 1\}^∗$ is polynomial-time Karp reducible to a language $B \subseteq \{0, 1\}^∗$ denoted by $A \leq_p ...
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137 views

Is $L \subset 1NL$ when $L \neq NL$?

A log-space Turing machine has a read-only input tape, a write-only output tape and uses at most $O(\log n)$ space in its read-write work tapes. The classes $L$ and $NL$ contain those languages which ...
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82 views

Are there complexity theory consequences of the collapse NEXP=EXP^NP?

It is clear that $NEXP\subseteq EXP^{NP}$, as a TM with exponential run time can simply query the NP oracle with an exponentially long query. However, it's not clear that the reverse $EXP^{NP}\...
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Sorting using comparisons that are not simple mappings of simple comparisons

The Python language has a sort(x) function that sorts a list based on the intrinsic comparison operator associated with the type of the elements of its input list x. One can also provide a cmp ...
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2answers
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Problem in deterministic time $n^p$ and not lower

I'm looking for any language $L$ candiate to be in $DTIME(n^p) -DTIME(n^{p-1})$ (it takes at least $n^{p-1}$ steps to determine if an input is in L with a 2-tape $TM$, but L is polynomially solvable). ...
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158 views

Containment problem of an acyclic NFA in an NFA

Let $A$ and $B$ be NFAs, such that $A$ is acyclic. In the general case, deciding whether $L(A)\subseteq L(B)$ is $PSPACE$-hard. However, since $A$ is acyclic, we know that for every $w \in L(A)$, it ...
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Randomized Reduction for Maximization Problem

I have two maximization problems $P_1$ and $P_2$ where the decision version $L_1 = \{(x, t) : \operatorname{Val}_1(x)\ge t\}$ of $P_1$ is $\mathsf{NP}$-complete. Let $f:P_1\to P_2$ be a randomized ...
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What is the difference between ${\bf P ^{\bf FewP}}$ and ${\bf UP ^{\bf FewP}}$?

What is the difference between ${\bf P ^{\bf FewP}}$ and ${\bf UP ^{\bf FewP}}$ ? In other words, what is the language that can be decided by ${\bf UP ^{\bf FewP}}$, but cannot be decided by ${\bf P ^{...
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BPP fragment of a PSPACE complete problem

Consider a PSPACE-complete problem (e.g., TQBF). Is there a sub-problem in BPP, that is not known to be in P? Is there a general technique of finding such sub-problems? Are any of them "natural" (i.e....
5
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1answer
214 views

If NP in BPP then NP equals RP

I am looking for a reference to the fact that if NP is included in BPP then NP is equal to RP. See for instance these links: https://cs.stackexchange.com/q/80509 http://www.inf.ed.ac.uk/teaching/...
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134 views

The decision procedure of theory of closed real field is in NP-hard?

The decision procedure of theory of closed real field refers to https://en.wikipedia.org/wiki/Decidability_of_first-order_theories_of_the_real_numbers
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Terminology: FNP, with P replaced by NP?

Consider these two classes of search problems: Search problems with poly-sized solutions s.t. verifying solutions is in P. Search problems with poly-sized solutions s.t. verifying solutions is in NP. ...
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72 views

Does Descriptive Complexity techniques have the naturalisation barrier?

I wished to know if the proof attempts at separation of complexity classes via the methods outlined by Descriptive Complexity theorists naturalise? By naturalise I'm talking about the Idea of Natural ...
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100 views

Term for a set that is not immune

At the outer bounds of computational complexity classes are those defined through computability theory (AKA recursion theory). This is where we get the well known complexity classes such as R, RE, and ...
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298 views

Which complexity class does this problem belong to?

Consider the following problem $\mathcal{P}$. Instance: A Boolean formula $F$ of $n$ Boolean variables ($x_1,...,x_n$) and $m$ Boolean parameters ($b_1,...,b_m$) where $0 \leq m \leq n$. Problem: ...
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139 views

Co-NP definition

The class Co-NP is defined as all the languages $L$ such that $\overline{L} \in NP$. An example that appears in the book of Arora and Barak is of $\overline{SAT}$, which is defined as $\overline{SAT} =...
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75 views

Are there common names for the subtiers of PTIME?

We all know P, or PTIME, I think, as a common name for the class of polynomial-time problems. Are there common names for the first few levels inside P; that is, for constant-time, linear-time, ...
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165 views

On complexity class $\mathsf{\Pi_2 L}$

I suggest the following definition of $\mathsf{\Pi_2 L}$ (similarly to the certificate definition of $\mathsf{NL}$): A language $L$ belongs to $\mathsf{\Pi_2 L}$ iff there exists a deterministic ...
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On Courcelle's question about Monadic second-order logic with cardinality predicates

I have found the following question at openproblemgarden.org: The problem concerns the extension of Monadic Second Order Logic (over a binary relation representing the edge relation) with the ...
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A canonical complete problem for EXP and NEXP in terms of formulae

3SAT is a complete problem for NP. TQBF is a complete problem for PSPACE. Is there direct way to define canonical complete problems for EXP and NEXP in terms of boolean formulae? I have only seen ...
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308 views

Is unary $\Pi_2$-SUBSETSUM coNP-complete?

Consider the following problem: for given integers $a_1, \ldots, a_{2n}$ and $A$ that are given in unary representation define is it true that for every $S \subseteq \{1, ..., 2n \}$ such that $|...
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748 views

PPAD and Quantum

Today in New York and all over the world Christos Papadimitriou's birthday is celebrated. This is a good opportunity to ask about the relations between Christos' complexity class PPAD (and his other ...
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134 views

Consequences/existence of problems without any “optimal” algorithm

Let $P$ be some kind of "problem" such as addition or graph coloring, that has an input size $n$. Let $S_P$ denote the set of algorithms $A_1, A_2, \dots$ which deterministically solve $P$. Based off ...
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932 views

A game on several graphs

Consider the following game on a directed weighted graph $G$ with a chip at some node. All nodes of $G$ are marked by A or B. There are two players Alice and Bob. The goal of Alice (Bob) is to ...
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115 views

what does NP ⊆ DTIME(…) mean?

Recently I've seen inside theory of a paper. This time complexity, DTIME, is completely new for me. Can somebody explain it? Also, the paper shows that the misinformation containment problem cannot ...
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111 views

Separation of AM and SZK

Are any results on the separation of AM from SZK known (e.g. relativized separation, or a separation assuming one-way functions exist, etc.)?
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161 views

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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169 views

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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$XP_{\text{uniform}}=FPT$ and update to $EPTAS$ section in complexity zoo?

Complexity zoo in https://complexityzoo.uwaterloo.ca/Complexity_Zoo:E#eptas has the following: $FPT = XPuniform\implies EPTAS = PTAS$. Fundamentals of Parametrized complexity on page $534$ has ...
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Is the following problem in $coNP$?

Given an $n\times n$ matrix $M$ with $\mathbb Z$ entries is 'does an $\frac n2\times\frac n2$ minor of $M$ vanish?' in $\bf{coNP}$? At least one $\frac n2\times\frac n2$ minor non-vanish implies rank ...
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Why is $BPP^{NP}$ in polynomial hierarchy? [closed]

Why is $BPP^{NP}$ in the polynomial hierarchy? I know that $BPP$ is contained in $NP^{NP}$, so $BPP$ is inside $PH$. However, how does that imply $BPP^{NP}$ is inside the polynomial hierarchy?
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263 views

Is there a relation between BBH (black box hypothesis) and SETH (strong exponential time hypothesis)?

Is there a relation between BBH (black box hypothesis) and SETH (strong exponential time hypothesis)?
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220 views

String theory based computations

I was reading Arora and Barak's book on computational complexity and in the section on 'criticism on Turing machine model and the class P' along with quantum computer it also mentions possibilities of ...
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Canonical complete problem for $\mathrm{FP}^{\Sigma^p_2}$

Given a $\Sigma^p_2$-complete oracle (i.e., $\Sigma_2 \mathrm{SAT}$), I have a problem that requires to call this oracle polynomially many times and returns an integer. Essentially, this is a function ...
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Can we define a meaningful concept of exptime reductions (as opposed to polytime reductions) for classes like NEXP or NEEXP?

Typically we are only interested in polytime reductions as we are usually interested in showing a reduction from one NP-problem to another. However, if we consider larger complexity classes such as ...
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205 views

Computational Complexity of the Frobenius Problem

The Frobenius problem takes as input $n$ positive integers $a_1,\ldots,a_n$ with $\gcd(a_1,\ldots,a_n)=1$ and asks for the largest integer $F$ that cannot be written in the form $F=a_1x_1+a_2x_2+\...
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110 views

When is a problem specified on a TM contained in non-uniform classes such as P/poly? [closed]

In this paper by Gottesman and Irani: https://arxiv.org/abs/0905.2419 , they prove NEXP-hardness of tiling an $N\times N$ grid. They do so by encoding a TM in the tiles making up the grid. However, ...
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245 views

Example problem that is not in $2^{o(n)}$ but could be solved in $O(2^{cn})$ for any $c > 0$ (suggested by wording of ETH)

In the wikipedia article on Time Complexity it is written that: The exponential time hypothesis (ETH) is that 3SAT, the satisfiability problem of Boolean formulas in conjunctive normal form with, ...
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93 views

Are there any known languages in the intersection of NP and co-NP but not in P? [closed]

We currently don't know the relationship between NP and co-NP, but would it be possible to show whether the intersection is equal to P? I can't think of any languages in both NP and co-NP, but not in ...
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190 views

Natural candidates for NP-E and E-NP

It has been known since the early 70's that ${\bf NP}$ and ${\bf E}=DTIME(2^{O(n)})$ are not equal (because ${\bf E}$ is not closed under polynomial-time many-one reductions, in contrast to ${\bf NP}$...
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454 views

Is there a P-complete language X such that succinct-X is in P?

I came across a paper called "A Note on Succinct Representation of Graphs". It seems that in the discussion section they claim that for any problem $X$ that is $\mathrm{P}$-hard under projections, $\...

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