Questions tagged [circuits]
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29
questions
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26 views
Can we pick a basis for synchronous circuits which coarsens towards the target partition at every layer?
Given a Boolean function $f : \{0, 1\}^n \rightarrow \{0, 1\}$ and a Boolean basis for circuit gates $B$ (for instance $B = \{AND, OR\}$), we can construct the set of size optimal synchronous (Harper ...
3
votes
0answers
95 views
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 ...
7
votes
1answer
134 views
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 ...
3
votes
0answers
104 views
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 ...
-1
votes
1answer
64 views
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 ...
2
votes
1answer
93 views
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 ...
0
votes
1answer
487 views
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 ...
3
votes
2answers
318 views
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 ...
3
votes
0answers
117 views
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 $C$...
2
votes
0answers
52 views
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)\...
11
votes
1answer
345 views
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 $\...
4
votes
0answers
50 views
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 ...
-2
votes
1answer
108 views
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 $...
1
vote
0answers
227 views
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 ...
9
votes
0answers
114 views
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_{...
2
votes
0answers
50 views
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 ...
4
votes
0answers
79 views
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 ...
2
votes
0answers
69 views
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\}^...
4
votes
2answers
145 views
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 ...
-3
votes
1answer
234 views
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 ...
2
votes
1answer
89 views
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 ...
6
votes
1answer
233 views
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 ...
10
votes
1answer
343 views
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,...
5
votes
0answers
99 views
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 ...
0
votes
0answers
190 views
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 ...
12
votes
1answer
512 views
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 $...
2
votes
1answer
184 views
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 ...
5
votes
0answers
284 views
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 ...
4
votes
2answers
506 views
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/...