# Questions tagged [circuit-complexity]

Circuit complexity is the study of resource-bounded circuits and the functions computed by such circuits.

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### How to prove “obvious” facts?

The title is somewhat "arrogant": say, most of us treat $P\neq NP$ as an "obvious" fact, albeit no proof is in sight. But my question is at a much, much lower level, is about a fact which "should be" ...
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### Wide and shallow circuits for $\mathrm P$

The $\mathrm{NC} \stackrel?= \mathrm P$ question is not as famous as the $\mathrm P$ versus $\mathrm{NP}$ problem, but still a deep and interesting question. It is generally accepted that there are ...
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### Boolean functions computed approximately by k-CNFs and l-DNFs and the depth of trees

A well-known fact is that if a function f is computable by a k-CNF and an l-DNF then it can be computed by a tree of depth at most kl. Long time ago I heard the problem what happens if the CNF and DNF ...
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### Complexity of a problem over acyclic context-free grammars

Let $G$ be an acyclic, context-free grammar over a fixed alphabet $\Sigma=\{a_1,\dots,a_k\}$ with the restriction (without loss of generality) that $|w|=2$ for each rule $A\to w$ in the grammar. ...
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### $\mathsf{TC^0}$-completeness and reductions

AFAIU, we don't know any problem which is complete for $\mathsf{TC^0}$ w.r.t. many-one $\mathsf{AC^0}$ reductions ($\leq^\mathsf{AC^0}_m$). On the other hand, proving that they don't exist would ...
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### Does ${\bf CFLPAD}={\bf PPAD}$?

What happens if we define ${\bf PPAD}$ such that instead of a polytime Turing-machine/polysize circuit, a (non-)deterministic finite/push-down automaton encodes the problem? I asked a similar ...
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### Reference for a circuit lower bound for slightly superexponential time

It is known that $EXP$ doesn't have circuits of size $n^k$. On the other hand proving $10 n$ lower bound on circuit size for $E$, $NE$ or even $E^{NP}$ is a known open problem. My question is ...
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### Parameterized complexity of deciding if a string can be computed by circuits of size $k\log(n)$

In the following, we will describe what seems to be a parameterized version of the minimum circuit size problem (MCSP). Before we get started, we need the following concepts: For every natural ...
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### Is there a counting complexity class for succint problems?

Encoding NP-complete problems succintly often makes them NEXP-complete. I am wondering if counting the number of solutions to such a problem with a succint encoding would be any harder than solving ...
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### Complexity of Polynomial Division

Let $P(x)$ be a univariate polynomial with integer coefficients where both coefficients and degrees are in binary and let $q(x)$ be another polynomial also with integer coefficients where the degrees ...
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### Given a Boolean formula, does there exist a small circuit that computes a satisfying assignment?

The 3-SAT problem can be defined as follows: 3-SAT Input: A 3-CNF formula $\phi$ of size $m$ with $n$ variables. Question: Does there exist a variable assignment that satisfies $\phi$? ...
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### Learning function with a few low-order Fourier coefficients, from uniformly random samples

Let $f:\{-1,+1\}^n \to \{-1,+1\}$ be a boolean function where all of the energy of the Fourier transform of $f$ is concentrated in a small number of low-order coefficients, say $k$ coefficients each ...
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### Oracles for sub-exponential circuit complexity of $\Sigma_2 EXP$

Couldn't find this one anywhere... It's an open problem whether $\Sigma_2 EXP$ problems have exponential-size circuit complexity. Is there an oracle relative to which $\Sigma_2 EXP$ has $2^{o(n)}$ ...
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### Computing the Fourier-Walsh coefficients of an arithmetic circuit

Given an $f$-fan in arithmetic circuit of depth $d$ over $GF(q)$ (for some prime $q$) and the set of variables $(x_1,\ldots,x_n)$, what is the state of the art upper bound for calculating the Fourier-...
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### What are problems in SC we don't know to be in NC?

NC and SC are, roughly, the class of problems decidable by shallow circuits of polynomial size and the class of problems decidable by narrow circuits of polynomial size, respectively. They are both ...
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### Is computational complexity of neural networks so old-fashioned?

I find that works on computational power and complexity of neural nets are all from 1980s/1990s or even earlier. Surveys and books are also from that time. Personally, I find problems in this field ...