Questions tagged [circuit-complexity]

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

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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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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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Monotone circuit complexity of matroids?

Call a monotone boolean function $f$ a matroid function if its minterms are bases of some matroid. I am interested in monotone circuit complexity of such functions, even when we "tie hands" of these ...
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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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Fundamental assumptions in complexity analysis

I am a software engineer and I need a bit of clarification. The practical performance of algorithms is usually compared against models where arithmetic and dereferencing are instantaneous, such as RAM....
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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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Is nonuniform $\mathsf{TC^0}$ equal to the composition closure of $\mathsf{AC^0}$ and Majority?

D.A.M. Barrington, N. Immerman and H. Straubing show in their 1990 paper "On Uniformity Within $\mathsf{NC^1}$" that the uniform $\mathsf{TC^0}$ is equal to $\mathsf{FOM}$ ($\mathsf{FO}$ plus ...
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Complexity Class Equalities on the Edge of Inconsistency

What are some of the most extreme potential equalities between computational complexity classes (especially if there is a barrier to refuting them)? These may give us an opportunity to prove better ...
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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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Number of circuits computing a given function

Let's say we have some function that can be computed by a minimal circuit of size $m$ (using some metric, say, the number of gates). Other than this minimal circuit, there will be many other circuits ...
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Has what I am calling “helpfulness” here been studied?

We say that a Boolean function $f : \{0, 1\}^n \rightarrow \{0, 1\}$ is helpful for another Boolean function $g$ if $f(x)$ can be computed with a smaller circuit given $g(x)$ as an extra input bit. I'...
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Are there generalizations for Chow's theorem?

The Chow's theorem as it stands holds only for a single linear threshold gate. That these gates are uniquely determined by their first $n+1$ Fourier coefficients. Are there other circuits for which ...
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Status of constant-depth isomorphism in $AC^0[\oplus]$

In [1], Agrawal proved the following: For any complexity class $\mathcal{C}$ closed under $NC^1$ reductions, problems complete for $\mathcal{C}$ under $FO$ reductions are isomorphic via an ...
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Using epsilon biased sets for circuit lower bounds

I have seen instances of how the technique of epsilon biased sets can be used to construct hard functions against a circuit class - like how in the recent paper of Kane-Williams this was used to ...
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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 ...