Questions tagged [circuits]
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33
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$NP\subseteq P/poly\implies PH\subseteq P/poly$ [closed]
We know if $NP\subseteq P/poly$ then $PH=\Sigma_2$ Then We want to show that $\Sigma_2-SAT\in P/poly$. Now $$\phi\in \Sigma_2-SAT\iff \exists\ x\in\{0,1\}^{p_1(n)}\ \forall\ y\in \{0,1\}^{p_2(n)},\ \...
- 288
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Size vs. Time Hierarchy Theorems
I'm struggling with the following idea, but I cannot see where my mistake is.
We know, from the Time Hierachy Theorems, that there are problems that require, say, double exponential time to decide. ...
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Can we do computing without electricity?
If electricity was never discovered, could we still make funcional computers to perform some level of useful tasks?
Or won't we be able to build anything comparable in terms of computer power?
- 101
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Solving sampling problems with circuits?
If I allow a circuit family (say, poly size, polylog depth) poly($n$) bits of randomized advice, then I can ask if its output samples from certain distributions or not. However I don't know what the ...
- 121
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Boolean logic: What is the name of this trick to replace explicit negations by implications?
Consider a Boolean circuit $C$ composed of some finite set of input variables $A_1,\ldots, A_n$ and the connectives $\lor\land\neg\rightarrow$ (with $X\rightarrow Y=\neg X\lor Y$) (update: assume that ...
- 125
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105
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Do random functions have synchronous, alternating circuits with non-injective first layers?
After discussing in the comments, I think a clearer definition of the question is as follows: for a random function $f : \{0, 1\}^n \rightarrow \{0, 1\}$, what is the probability that there exists a ...
- 1,572
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223
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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 ...
- 8,896
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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 ...
- 11.4k
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1
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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 ...
- 1
2
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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 ...
- 19
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443
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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 ...
- 319
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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$...
- 5,127
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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)\...
- 596
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How small can be a layered boolean circuit for a function with circuit complexity $s$?
Consider a function $f$ computed by a boolean circuit $C$ with $n$ inputs of size $s(n) = \mathsf{poly}(n)$ over the basis $\{\mathsf{XOR},\mathsf{AND},\mathsf{NOT}\}$ (with indegree 2 for the $\...
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Do there exists reversible gate sets of intermediate growth?
Suppose that $f_{1},...,f_{k}:\{0,1\}^{r}\rightarrow\{0,1\}^{r}$ are bijective functions.
For all $n\geq r$, let $G_{f_{1},...,f_{k};r}=\subseteq S(\{0,1\}^{n})$ be the subgroup generated by
i. the ...
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How to construct a branching program for a given function or formula?
Can someone throw some light on how to create a branching program from a given function? I have followed the definition of $BP$ on wiki and here. But I could not find any way to convert a function to $...
- 111
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238
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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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Expected value of the evaluation of Boolean circuits of depth $2n$
I am not an expert on circuits and I wonder whether the following problem was already studied (and possibly solved). Any reference or suitable method to solve this question would be welcome.
Let $C_{...
- 4,771
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53
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Arithmetic circuits with restrictions on occurrence of pairs of variables
I am curious if the following model was studied or has some obvious lower bounds:
We want to compute a polynomial $P(x_1,x_2, \dots , x_n)$. Suppose we have a graph G on $n$ nodes that we are going ...
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Circuits computing functions of inputs smaller than $n$
The usual circuit complexity concerns circuits where circuit $C_n$ computes function $f_n$. I am interested in circuits such that $C_n$ can compute $f_i$ for all $i \leq n$. I am assuming that the ...
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Can we compute encodings of binary strings under arbitrary permutation groups?
Given a permutation group $G \leq S_n$, can you construct non-uniformly a circuit computing a function $f : \{0, 1\}^n \rightarrow \{0, 1\}^{ceil(log|\{0, 1\}^n/G_n|)}$ with size $O_n(\frac{|\{0, 1\}^...
- 1,572
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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 ...
- 1,572
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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 ...
- 1,443
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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 ...
- 1,443
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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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Evaluate boolean circuit on batch of similar inputs
Suppose I have a boolean circuit $C$ that computes some function $f:\{0,1\}^n \to \{0,1\}$. Assume the circuit is composed of AND, OR, and NOT gates with fan-in and fan-out at most 2.
Let $x \in \{0,...
- 11.7k
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Switching between representations of boolean functions between circuits and Fourier expansions
I'm currently learning about the analysis of boolean functions (mainly based on their Fourier coefficients) by reading this excellent resource
There, boolean functions are represented as linear ...
- 51
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217
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Greater-Than operator using an Arithmetic Circuit
How can I transform the term $x>C$ (i.e. the term assumes the value $1$ if $x>C$ and assumes the value $0$ otherwise) to an arithmetic circuit that computes it?
Where $x$ is the input to the ...
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Does Kannan's theorem imply that NEXPTIME^NP ⊄ P/poly?
I was reading a paper of Buhrman and Homer “Superpolynomial Circuits, Almost Sparse Oracles and the Exponential Hierarchy”.
On the bottom of page 2 they remark that the results of Kannan imply that $...
Lorraine
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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 ...
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Sketch of Razborov's paper "On the method of approximations"
(The following question has bothered me for many years.) Razborov seems to have obtained some of the strongest/award winning lower bounds on circuits found in the field over many years, largely ...
- 11k
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P/poly vs NP separation based on circuit trees instead of DAGs
there are various theorems that relate major complexity class separations to circuit family DAGs sizes, in particular for P/poly vs NP. in contrast,
are there theorems/conjectures that relate P/...
- 11k