# Questions tagged [complexity-classes]

Computational complexity classes and their relations

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### Proof that DFS order is P-complete

Suppose we are given an oriented graph G with a selected number of nodes s, where for each node some particular ordering of edges leading from it is specified. If we run a depth-first search algorithm ...
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### What can we do with a generic oracle (as opposed to a random one)?

Let me first recall a few (lengthy but hopefully mostly standard) facts and definitions in order to motivate my question (feel free to skip down to the actual question): Standard definitions: A ...
1 vote
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### Crafting ${NP}^{\#P}$-complete problems

Some related posts: Is $coNP^{\#P}=NP^{\#P}=P^{\#P}$? $\mathsf{NP^{PP}}$ vs $\mathsf{P^{PP}}$ I needed a complete problem for the class ${NP}^{\#P}$ for a reduction to show the hardness of some other ...
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### Assume P != NP, does it imply that one-way functions exist?

I define a function f to be one-way iff for any sufficiently large x computing f(x) bounded ...
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### Deciding finiteness of regular language is NL-complete?

I've been reading the following Habilitation thesis where the author claims (pg. 29): ... First, deciding whether the language of an NFA is finite is in NL ... I'm having trouble seeing why this ... 384 views

1 vote
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### Complexity of analytic functions and integrals

There exist polynomial - time computable functions, log - space computable functions, and NC - functions. Given this: To which class do analytic elementary functions, including trigonometric ones, ...
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### Do fast satisfiability algorithms imply fast algorithms for parity SAT?

$\oplus$SAT is the problem of deciding if the number of satisfying assignments to a CNF formula is odd (and is the standard complete problem for the class $\oplus$P, or Parity-P). Suppose we have a ...
1 vote
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### What complexity class is characterized by having PSPACE verifiers?

Inspired by the 2 definitions (theorems) I am aware of, that are as follows. A language L belongs to QMA if there exists a BQP verifier V. A language L belongs to NP if there exists a P verifier V. ...
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### Is there a well-defined notion of an “R/poly” complexity class?

This would be the complexity class of all problems that are decidable in finite time with a polynomial length advice string that can be arbitrarily hard to compute. But potentially undecidable without ...
1 vote
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### How "Algebrization" is "A New Barrier in Complexity Theory"?

Being an enthusiast in computational complexity theory, I recently came across with this wonderful work Algebrization: A New Barrier in Complexity Theory. My question is about Theorem 5.3 in it (pp. ...
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### Structural Complexity Theory References

I'm a PhD student in mathematics (mostly studying algebraic geometry), but I've always been interested in computational complexity theory. As an undergraduate, I completed an independent reading ...
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### Relationships between Descriptive Complexity and Average Case Complexity

Descriptive complexity gives one a logic or at least a logic to express languages in a complexity class. The PH can be defined as the union of all classes that can be expressed in Second order logic. ...
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### Which 1-player games are EXPTIME-complete? Also, are there any known games that are EXPSPACE-complete?

I noticed a lot of 1-player games have been shown to be NP-Hard, like Pac-Man, The World's Hardest Game, Tetris, etc. For PSPACE-Complete, I noticed that Wikipedia listed these 1-player games: It is ...
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### Reducing the amount of alternations without exponentially increasing the runtime?

Let $\mathsf{AltTime}(g(n), f(n))$ denote the class of languages that are solvable by an alternating machine using $f(n)$ time and $g(n)$ alternations. Is there anything known about the following ...
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### Are there any examples of problems in ZPP not yet in P?

Is there a problem of which we can prove that it is in ZPP but we don’t yet know whether it’s in P or not?
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### Complexity of permanent verification

Consider the problem of permanent verification: $\bullet \$ Given a $n\times n$ matrix $A$ with entries in $\{0,1\}$, and given $k\ge 0$, does $Per(A)=k$? Question: Is it known to be NP-hard? Should ...
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### What the relation between the classes SC and NC?

What the relation between the classes SC (Scott's class) and NC? (Nick's class). Is SC contained in NC? Is NC contained in SC?
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### How do we show directly coNP is in MIP?

I know one can show that by $\mathsf{coNP}\subseteq\mathsf{NEXP}=\mathsf{MIP}$. But here I would like to start with a $\mathsf{coNP}$-complete problem and show there is a two-prover one-round ...
1 vote
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### Explanation of Complexity class $S_2^P$?

I am trying to understand the complexity class $S_2^P$ defined as (https://en.wikipedia.org/wiki/S2P_(complexity)). A language L is in $S_2^P$ if there exists a polynomial-time predicate $P$ such that:...
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### Why is the "balanced vs constant function" problem not a proof that P ≠ BPP?

Recently, I came across the problem of figuring out whether a given binary function $f(x)$ is constant (0 for all values of $x$ or 1 for all values of $x$) or balanced (0 for half of values and 1 for ...
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### Baker–Gill–Solovay Theorem: why $2^n/10$ steps?

Context I'm teaching an introductory complexity theory course right now and although I work in adjacent areas, I'm not an expert on complexity theory myself, so I'm still in the process of working ...
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### Complexity class of optimization problems whose fractional relaxation is polynomial-time solvable

It is known that the problem of integer linear programming is NP-hard, but its fractional relaxation can be solved in polynomial time. The concept of fractional relaxation can be applied to any ...
1 vote
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### Is modular square roots modulo primes in $NC$?

Assume modulus is prime. Is modular square roots then in $NC$? If not then, are there special primes or prime powers or numbers related to these (such as $2^k\pm i$ where $i\in\{-1,0,+1\}$) where it ...
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### Interactive proofs with computation bounded Merlin

Consider usual interactive proofs (Arthur is polynomial-time bounded and can use random bits) where computation power of Merlin is bounded by polynomial-size circuits. For example, every unary NP-...
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### Classes between PH and PSPACE

I am interesting in languages of the following form: $x \in L \Leftrightarrow Q y_1 Q y_2 \ldots Q y_n P(x, y_1, \ldots y_n, x).$ Here every Q is $\forall$ or $\exists$; $n$ is the length of $x$, the ...