# Questions tagged [circuit-complexity]

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

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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 ...
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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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### Does small circuits for a NP-complete problem contradict ETH?

The remarks of the Theorem 4 in the paper "On the complexity of circuit satisfiability" claims that: if circuit satisfiability (CktSat) problem can be decided by deterministic circuits of $2^{o(n)}$ ...
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### Boolean Functions Where Sensitivity Equals Block Sensitivity

Some of the work on sensitivity vs. block sensitivity has been aimed at examining functions with as large a gap as possible between $s(f)$ and $bs(f)$ in order to resolve the conjecture that $bs(f)$ ...
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### Does Depth-First-Search admit a quasilinear time algorithm in mutitape Turing Machine model?

Depth-First-Search (DFS) has a quasilinear (i.e.,$\widetilde{O}(m+n)$) time algorithm in random access model (RAM). I am curious about whether DFS still admits a $\widetilde{O}(m+n)$ time algorithm in ...
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### Switching lemma for polynomials over $\mathbb{F}_2$

Suppose $f$ is in $\mathbb{F}_2[x_1,...,x_n]$ with total degree $d$. Q. Is there any kind of switching lemma or restriction lemma in which by applying the lemma on $f$ we can reduce the total ...
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### What is the communication complexity of approximating addition?

In my circuit complexity research, I came across the need to find the communication complexity of approximating addition. Specifically, the class of problems I am interested in is parametrized by four ...
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### VC dimension of polynomials over tropical semirings?

As in this question, I am interested the $\mathbf{BPP}$ vs. $\mathbf{P}$/$\mathrm{poly}$ problem for tropical $(\max,+)$ and $(\min,+)$ circuits. This question reduces to showing upper bounds for the ...
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### Upper bounds on the circut depth

Suppose $f:\{0,1\}^n \to \{0,1\}$ is a function such that it can be computed by a circuit of size $n^c$ for some constant $c>0$. Q. Is there any nontrivial upper bound on the depth of a circuit ...
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### Razborov-Smolensky polynomial argument on $\textrm{ACC}[q]$ where $q$ is a prime power

It seems to be a folklore that we can "handle" $\textrm{ACC}[q]$ circuits not only for prime $q$ but prime power $q$. For example, authors of this paper say that ... any constant depth circuit ...
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### Are Boolean circuits 'universal'

I have a question, but I don't seem to know enough computer science terminology in order to look up an answer. So I wonder if you guys could help a poor physicist like me. I would like to know if ...
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### Example demonstrating the power of non-deterministic circuits

A non-deterministic Boolean circuit has, in addition to the ordinary inputs $x = (x_1,\dots,x_n)$, a set of "non-deterministic" inputs $y=(y_1,\dots,y_m)$. A non-deterministic circuit $C$ accepts ...
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### Ruzzo-Simon-Tompa oracle access mechanism

In a paper on relativizing logspace computations, Ladner and Lynch construct an oracle relative to which $\mathsf{NL} \nsubseteq \mathsf{P}$. There are some more pathological examples in this vein in ...
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### Impagliazzo lemma, unclear detail in its proof

In Arora-Barak's book on page 378 in the proof of Impagliazzo's Hard Core lemma why did they choose the number 50 in this line: Set $t = \frac{50n}{\epsilon^2}$ ? How this choice then yields the size ...
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### Which $SIZE$-$DEPTH(s, d)$ classes with $log(s(n))^{d(n) - 1} = o(n)$ can we not separate by known methods?

Define $SIZE$-$DEPTH(s, d)$ to be the functions which are computed by circuit families of size $O(s(n))$ and less than depth $d(n)$. We know from Boppana's 1997 paper on the average sensitivity of ...