# Questions tagged [circuits]

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### Monotone circuit representations of paths in a graph?

Consider a directed graph $G = (V, E)$ with a source $s \in V$ and sink $t \in V$. From $G$, I can define a monotone Boolean function $\phi_G$ on the set of variables $E$, in the following way: every ...
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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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### How to find for each 3-input boolean function the minimum number of NAND operators needed to compute it [closed]

I need to know for each of the $2^{2^3}$ boolean functions with $3$ inputs the smallest boolean circuit made only of NAND gates computing it (smallest in terms of the number gates). I would be glad ...
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### Correlation between noise resilience and output distribution of Boolean circuits

Given a randomly generated AND/OR tree (and negations), we can calculate the probability that the circuit will represent a specific Boolean function up to 3 input literals. Starting from 4 (or at ...
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### How to build comparison operator (comparator) in an arithmetic circuit

I am trying to convert a basic program into an arithmetic circuit. I am stuck on the step of converting the greater than operator into an arithmetic circuit. To be specific, I do not know how to ...
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### Is the unbounded fan-in model realistic?

Does the unbounded fan-in circuit model apply in "practical" settings? In other words, are there real-world realisable computers with unbounded fan-in gates? As I understand, standard silicon ASICs ...
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### Is Circuit Minimization $P$-hard under logspace reductions?

By Circuit Minimization, I am referring to the following decision problem. Circuit Minimization Input: A bit string $x$ and a number $k$. Question: Does there exist a Boolean Circuit $C$...
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### Is topological conventional computation possible?

A function $f:X^{2}\rightarrow X^{2}$ is said to satisfy the Yang-Baxter equation if (f\times\mathrm{Id}_{X})\circ(\mathrm{Id}_{X}\times f)\circ(f\times \mathrm{Id}_{X})=(\mathrm{Id}_{X}\times f)\...
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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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### Is there any notion of sensitivity for probabilistic Boolean functions?

Sensitivity is defined here. Denoting the neighbors of $x$ in the Boolean cube as $N(x)$, we define the sensitivity to be $s(f, x) = \sum_{y \in N(x)} I(f(x) \neq f(y))$, where $I$ is $1$ if the ...
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### Some questions about the depth hierarchy for threshold circuits [closed]

(I am hugely editing the question. My initial question was if lowerbounds on threshold circuits say anything about P/NP and it seems that they dont. Irrespective of P/NP its an independently true fact ...
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### Some consequences of the Roychowdhury-Orlitsky-Siu result from 1994

This pertains to the proof of theorem 1.1 in this paper, http://dl.acm.org/citation.cfm?id=2897636 So Roychowdhury-Orlitsky-Siu had shown that the number of depth $2$ linear threshold gate circuits ...
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### Has there been a study of circuits operating on arrays?

Much ink has been spilled studying the theory surrounding computation by combinatorial circuits operating on bits or boolean values - with AND, OR and NOT gates (as those are enough to implement any ...
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### Piecewise Linear Circuit Identity Testing

A circuit is used to express a piecewise linear function of one variable $\ f:\mathbb{Q}\to\mathbb{Q}$ The component gates are: add the outputs of two other gates together scale the output of one ...