# Questions tagged [circuit-complexity]

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

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### Sparsity Bounds for Probabilistic Polynomials

Has there been any research done on proving sparsity lower bounds for probabilistic polynomials (over the Reals) for Majority? A probabilistic polynomial is a distribution of polynomials $D$ such that ...
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1 vote
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### Can NP-complete language be in $mP/poly$?

Can NP-complete language be in monotone $P/poly$?
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### Pebble games and conversions to bounded width circuits

Questions: Are there references which mention the relation between pebble games and conversions to bounded width circuits? Here, "conversions to bounded width circuits" means that circuits ...
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### Complexity of approximating boolean functions with circuits

Let $f$ be a boolean function on $n$ variables - say we want to find the smallest circuit $C$ where $C(x)=f(x)$ for all but an $\epsilon$ fraction of inputs $x \in \{0,1\}^n$. What is known about the ...
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### Does advice reduce depth?

Specifically I'm thinking about NC$^1$/poly and NC$^1$/rpoly (randomized advice). Are there any statements like "If $\{C_n\}$ is a family of NC$^1$/(r)poly circuits with depth $C\log n$, then ...
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### $\mathrm{AC}^0$ upper bound for Hamming weight

Consider Theorem 11 of this paper (S. Aaronson, BQP and the Polynomial Hierarchy), which says: Any depth $d$ circuit that accepts all $n$ bit strings of Hamming weight $\frac{n}{2} + 1$ and rejects ...
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### Does every Boolean function of degree $n$ decompose as the (XOR) product of two functions of complementary degrees?

Say $f: \{-1,+1\}^n \rightarrow \{-1, +1\}$ is a Boolean function of (Fourier) degree $n$. Is it true that there exist non-constant Boolean functions $g$, $h$ of degrees $a$ and $b$ respectively such ...
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1 vote
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### Composition theorem for randomized communication complexity

I am currently organizing the literature of composition theorem, and I found the paper by https://www.research.cs.rutgers.edu/~troyjlee/Composition.pdf, in their theorem 5, I find  R_{1/4} (f \circ ...
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### Proof of $DLOGTIME-CC^0 = MOD[<,bit]$

Let $CC^0[m]$ be the class of constant-depth, polynomial-sized circuits consisting entirely of $MOD_m$ gates, which put out $1$ iff the sum of their inputs $\equiv 0~(\textrm{mod}~m)$. In the same way ...
• 802
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### On solving Planar Circuit SAT

This enquiry is three-sided. Side 1 - State of the art Which is the best known algorithm for $\text{PLANAR-CIRCUIT-SAT}$? Which is the best known algorithm for $\text{PLANAR-CIRCUIT-SAT}$ assuming ...
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1 vote
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### Multiplicative-depth 1 composition of arithmetic circuits

I am trying to find information about the following problem. Let $C_1$ and $C_2$ be any two poly-size arithmetic circuits on input vectors x₁ and x₂ correspondingly. Assume that a third arithmetic ...
• 401
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### What is FO(REGULAR)? (The descriptive complexity equivalent of NC1)

According to Immerman's Descriptive Complexity diagram, there is a logic called $\mathsf{FO(REGULAR)}$ which captures $\mathsf{NC}^1$. However, I can't find the reference where this logic is defined. ...
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### Improving boolean circuits w.r.t. a probability distribution

This is a reference request. Consider the following problem on boolean circuits [ 1 ]: Given: Boolean circuit $B$ and probability distribution $\mathbb{P}$ on inputs to $B$. Task: Find one or more ...
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### Is there a universal gate set for classical probabilistic computing?

We know that NAND gates are universal for deterministic classical circuits, Toffoli gates are universal for reversible deterministic classical circuits, and Clifford+T is universal for quantum ...
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1 vote
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### 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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This result by Tavenas, Koiran and others show that any polynomial computed by a circuit of size $s$ is computed by a depth-4 homogenous circuit of size $s^{\sqrt{d}}$. Are there any similar results ...