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Computational complexity classes and their relations

3 votes

Problems in $\text{PSPACE} \cap \text{Co-NP-Hard}$

Like the Boolean Formula Isomorphism problem, the Group Equations Isomorphism problem is $\mathsf{coNP}$-hard and in $\mathsf{\Sigma_2P}$, for any fixed non-abelian group. See The Complexity of Equiva …
argentpepper's user avatar
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1 vote
0 answers
227 views

Is $\mathsf{NP}$ in $\mathsf{NNC}^1$?

Theorem 2.2 in "Nondeterministic circuits, space complexity and quasigroups", by Wolf, 1994 (a technical report version is available here without fee), proves that NP = NNC, where NNC is the class of …
argentpepper's user avatar
  • 2,291
1 vote

In what class are randomized algorithms that err with exactly 25% chance?

This is a partial answer; maybe it will inspire someone else to provide a better one. $\newcommand{\EBPP}{\mathsf{EBPP}}$ $\newcommand{\CP}{\mathsf{C}_=\mathsf{P}}$ Your class $\EBPP$ is a special ca …
argentpepper's user avatar
  • 2,291
20 votes
0 answers
488 views

Interesting PCP characterization of classes smaller than P?

The PCP theorem, $\mathsf{NP} = \mathsf{PCP}(\mathsf{log}\, n, 1)$, involves probabilistically checkable proofs with polynomial time verifiers, so the smallest class that can be characterized in this …
argentpepper's user avatar
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10 votes
1 answer
216 views

Can $\log^k n$ alternations be simulated in $\mathsf{NC}^k$?

Let $\mathsf{ATISP}(f(n), g(n))$ be the class of languages decided by alternating Turing machines that halt in time $f(n)$ using space $g(n)$. Let $\mathsf{AALTSP}(f(n), g(n))$ be the class of languag …
argentpepper's user avatar
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26 votes
1 answer
807 views

What is $\mathsf{NP}$ restricted to linear size witnesses?

This is related to the question Is the Witness Size of Membership for Every NP Language Already Known? Some natural $\mathsf{NP}$(-complete) problems have linear length witnesses: a satisfying assign …
argentpepper's user avatar
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28 votes

What are the consequences of $\mathsf{L}^2 \subseteq \mathsf{P}$?

$ \newcommand{\DSPACE}{\mathsf{DSPACE}} \newcommand{\L}{\mathsf{L}} \newcommand{\P}{\mathsf{P}} \newcommand{\DTIME}{\mathsf{DTIME}} $ $\L^2 \subseteq \P$ would refute the Exponential Time Hypothesis. …
argentpepper's user avatar
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49 votes
4 answers
2k views

What are the consequences of $\mathsf{L}^2 \subseteq \mathsf{P}$?

We know that $\mathsf{L} \subseteq \mathsf{NL} \subseteq \mathsf{P}$ and that $\mathsf{L} \subseteq \mathsf{NL} \subseteq \mathsf{L}^2 \subseteq $ $\mathsf{polyL}$, where $\mathsf{L}^2 = \mathsf{DSPAC …
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