Questions tagged [boolean-functions]
Questions about Boolean functions and their analysis
229 questions
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Pairwise (partial) equivalence of boolean functions
I have a bunch of boolean functions, say $b_1,b_2,\dots,b_k \colon \{0,1\}^m \to \{0,1\}^n$, all given in terms of circuits.
I want to determine for which inputs they pairwise agree, that is, I want ...
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7
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1
answer
491
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Complexity of computing minimum unsatisfiable core
Given a Boolean formula $\phi$ in conjunctive normal form (CNF), an unsatisfiable core of $\phi$ is a subset $\phi'$ of the clauses of $\phi$ which is unsatisfiable. We say that $\phi'$ is a minimum ...
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0
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75
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Learning a boolean function using decision tree with small number of queries
I am working on a problem and I am looking to solve the following subproblem : Given a "restrictive" blackbox access to boolean function $\phi$, output a "small-sized" CNF that ...
-1
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1
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209
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How close is a Boolean function from being a linear boolean function?
I want to prove that for every boolean function $f:\mathbb{F}_2^n\rightarrow \mathbb{F}_2$ the (normalized) Hamming distance from a certain linear boolean function (i.e. a boolean function $f:\mathbb{...
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Constructing vector valued boolean circuits from boolean circuits
This is a reference request. I'm
interested in the compositional construction of small boolean circuits
for vector-valued boolean functions $\phi : \mathbb{B}^m \rightarrow
\mathbb{B}^n$ for $n >...
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5
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0
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76
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Hardness of Computing Tribes-DNF by Decision Trees
In this paper on "The Polynomial Hierarchy, Random Oracles, and Boolean Circuits", Fact (3.2) states that it is impossible for a polylogarithmic depth decision tree to compute the Tribes-DNF ...
- 51
2
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0
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77
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Distance between Fourier distributions of independent random Boolean functions
For a boolean function $f: \{-1,+1\}^n \to \{-1,+1\}$, the squared Fourier coefficients $\{\hat{f}(S)^2\}_{S \subseteq \{0,1\}^n}$ form a probability distribution. I want to know what the total ...
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56
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"Inverting" the fourier spectrum representation of a boolean function to recover a circuit representaiton
Given a boolean circuit, or an equivalent boolean expression, we can compute its fourier spectrum to yield a real-valued (multilinear) polynomial representation. What about the other way around? ...
3
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0
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58
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Name for the dimension of subspace of support of boolean function?
The support of a boolean function is defined as the set of $x$ such that $f(x) = 1$. I want to quantify the dimension of the support of the function (i.e. how large the subspace is that is made up of ...
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3
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362
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Relationship between size of Boolean functions and DFAs
Are there any works that study the relationship between Boolean functions and the size of the minimal DFAs required to represent those Boolean functions? Boolean functions refer to the usual ...
- 181
1
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0
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60
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Number of ${\tt NAND}$-gates needed to build any $n$-ary Boolean function
Are there positive integers $k,c$ with the following property?
Whenever $n$ is a positive integer, then any function $$f:\{0,1\}^n\to\{0,1\}$$ can be built using at most $n^k+c$ ${\tt NAND}$-gates ...
0
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0
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57
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Is it possible to estimate the positive outcomes of a boolean function using an optimized version of Goldreich-Levin?
Let $\mathcal{X} = \{-1,1\}^n$ and $h: \mathcal{X} \to \{-1,1\}$, h can be expanded in the basis of monomials for the uniform distribution, or also can have a distribution free expansion (Gram-Schmidt ...
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3
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1
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482
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Intuition on Lupanov's Upper Bound on Circuit Size
The following result, by Lupanov, is a classic in the theory of Boolean function complexity:
Theorem: For every boolean function $f$ of $n$ variables:
$$C(f) \leq (1 + \alpha_n)\frac{2^n}{n}, \text{ ...
- 133
2
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0
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57
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Does Goldreich-Levin algorithm for finding large Fourier coefficients have time complexity upper bound = sample complexity upper bound?
I'm currently working on finding better bounds for Goldreich-Levin algorithm for estimating large Fourier coefficients of a boolean function.
I was surprised seeing that the upper bounds for time ...
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1
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0
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70
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Is there an efficient Goldreich-Levin algorithm that generalizes to agnostic PAC setting?
Goldreich Levin algorithm is an algorithm that based on some assumption (boundness on Fourier coefficients) outputs the indices for most significant Fourier coefficients of a boolean function, however ...
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1
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0
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79
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Testing positivity of a function by an IP system?
We are given a polynomial function $f:\{0,1\}^n\to\mathbb{R}$ with $\text{deg}(f)\leq d$ ($d$ is constant), and $\epsilon>0$; $f$ here is presented by its coefficients (the degree is constant, so ...
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3
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0
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105
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Influence for boolean functions on larger domains
Most of the literature on boolean function complexity considers boolean functions on $\{0,1\}^n$, but I am not finding very much about functions over larger (finite) domains. Specifically, fix a ...
- 1,242
4
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1
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198
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Concrete version of KKL Theorem
The Kahn–Kalai–Linial (KKL) Theorem says that for any balanced Boolean function $f:\{−1,1\}^n→\{−1,1\}$ we have $\max_i {\bf Inf}_i(f) = \Omega\left(\frac{\log n}{n}\right)$. I am looking for a ...
- 1,242
1
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0
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56
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Maximum degree of the Sum of Squares certificate of a non-negative degree d polynomial on the boolean hypercube
Let $f: \{0, 1\}^n \rightarrow \mathbb R$ be a polynomial on the boolean hypercube. If $f$ is non-negative $(f \geq 0)$ i.e. $f(x) \geq 0, \forall x \in \{0, 1\}^n$ then $f$ always has a degree $2n$ ...
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0
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2
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191
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Finding an $\epsilon$-concentrated collection with size in terms of spectral $1$-norm
$\newcommand{\R}{\mathbb{R}}$
This question is about Problem 3.16 in Ryan O'Donnell's Analysis of Boolean Functions book. The problem is stated as follows:
Let $f : \{-1,1\}^n\to\R$ and let $\epsilon&...
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1
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If boolean function $f$ is computable by a k-CNF and an l-DNF then it can be computed by a decision tree of depth at most kl
I have seen it stated that if boolean function $f$ is computable by a $k$-CNF and an $l$-DNF then it can be computed by a decision tree of depth at most $kl$. However, I am not able to see why this is ...
0
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1
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160
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What is the simplest one-way function (in terms of boolean circuit complexity)?
What is the simplest known one-way function?
By simplest, I mean, when implemented as boolean logic, the number of AND/OR/NOT gates needed is minimal (smallest circuit complexity).
(I'm trying to find ...
2
votes
1
answer
239
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Value of studying boolean function complexity through circuits complexity nowadays
Apparently boolean function complexity analysis through circuit complexity has a limit (as they are natural proofs), and this means it is not possible to proof $P \not= NP$ unless there are no one-way ...
2
votes
1
answer
170
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Counting argument for LTF circuits
In Boolean circuit complexity, Shanon's counting argument shows that a random Boolean function on $n$-input bits requires a circuit of size $\Omega(2^n/n)$ to be computed by a circuit made of AND, OR ...
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8
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1
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172
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Reductions and projections in circuit complexity
I'm struggling to find a good reference that defines the difference between projection and monotone projection in the context of Boolean functions and circuit complexity.
My understanding is that a ...
- 1,049
7
votes
1
answer
201
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Complexity of Maximizing Hamming Distances Below a Threshold
Problem Statement
Is the following problem NP-Complete?
Input: A collection $S$ of binary strings, with each string of length $m$.
Goal: Compute a binary string $s^*$ of length $m$ that mazimizes the ...
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2
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0
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67
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Learning boolean functions with input-ouput examples and side-information
The Kushilevitz-Mansour, "low-degree", and Goldreich-Levin algorithms aim to learn a function $f: \{0,1\}^n \rightarrow \{0,1\}$ from a sufficiently large set of input-output examples $(x_i, ...
0
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0
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100
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Boolean vs algebraic circuits difference
Valiant, Skyum, Berkowitz and Rackoff in https://epubs.siam.org/doi/10.1137/0212043 showed that $VP=VNC^2$, namely, that arithmetic circuits can be parallelized.
What is the central reason such a ...
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0
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61
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Is there a name/terminology for binary codes with evenly spaced number of ones?
I am generating a random binary matrix $A \in \{0, 1\}^{m \times n}$ with the number of ones in each row set to evenly spaced numbers from an interval. For example, if $n=50$, the number of ones for $...
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Natural proofs and size of propositional formulas
Given a formula $\phi$ of propositional logic, we define its size $|\phi|$ as the number of proposition symbols that $\phi$ contains (counted with multiplicity). For example, $|(p \land p)| = 2$.
Let $...
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5
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0
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145
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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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3
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1
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217
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XORSAT to HornSAT reduction
I am trying to write a practical piece of code that solves a XORSAT by first reducing it to HornSAT and then solving the HornSAT (instead of doing Gaussian Elimination over F2). The reason for this ...
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3
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0
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108
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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 ...
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2
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2
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369
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What is the polynomial representation of the Hamming weight function?
For any function $f: \{1,-1\}^n \rightarrow \{1,-1\}$, there is a unique multilinear polynomial $p \in \mathbb{R}[x_1,\dots, x_n]$ for which $p(x)=f(x)$ for all $x \in \{1,-1\}^n$ (see e.g. Lemma 4.1 ...
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1
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1
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KRW Conjecture: separation of NC^1 and P
More than a real question this is a recap of something I have been studying. I hope someone will help me getting things straight, so any correction or thought about the following reasoning is more ...
8
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1
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244
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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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3
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1
answer
164
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$ACC^0$ implementation of a boolean function
Consider the symmetric boolean function
$$F(x_1,\dots,x_n)=1\iff\sum_{i=1}^nx_i\mbox{ is a square}.$$
It is implementable in $TC^0$.
Is there an $ACC^0$ implementation?
The reason I ask is there seems ...
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0
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88
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Harmonic analysis of sequences of Boolean functions (i.e. of words in $(\{0,1\}^n)^*$)
Is there any research on harmonic analysis of sequences of Boolean functions, which represent the application of a Boolean function on a word in $(\{0,1\}^n)^*$?
I'm looking for any reference on this, ...
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1
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488
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A boolean function $f: \{0,1\}^n \rightarrow \{0,1\}$ is chosen at random from all $2^{2^n}$ such $f$. What do the Fourier coefficients look like?
As in the title.
I'm not sure where to start here. My guess is that in expectation at least a constant fraction are non zero, and as a result there would exist some "large" coefs. and some "small" ...
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0
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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 ...
- 11.6k
6
votes
1
answer
460
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Is the basis of parity functions the only orthonormal basis for Boolean functions?
Is there another orthonormal basis of functions for Boolean functions? Or, more specifically, besides the parity functions, is there another explicit function (which is common and has a name) that can ...
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7
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1
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391
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Complexity of constructing minimum depth decision trees
I am interested in the computational complexity of
Problem 1: Given a finite, non-empty set $J$, given $A, B \subseteq \{0,1\}^J$ such that $A \cap B = \emptyset$, and given $n \in \mathbb{N}$, does ...
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0
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149
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Status of the Junta Problem (soft question)
Does the learning theory community in general believe that juntas can be learned in polynomial time?
The naive algorithm works in quasi-polynomial time. MOS's paper shows how to solve the junta ...
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13
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1
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922
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What is the complexity of checking equivalence of two boolean formulae without NOT symbol?
Suppose I have two boolean formulae (propositions) $P_1$, and $P_2$ (can be assumed to be in CNF) over the same variables and such that there are no "NOT" symbols used. I.e. only conjunction and ...
-1
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1
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87
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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
5
votes
2
answers
403
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Level $k$ bounds in Analysis of Boolean functions
In Ryan O'Donnell's book Analysis of Boolean functions, following Corollary 9.25 the following appears:
If $f\colon \{-1,1\}^n \to \{0,1\}$, and we have $\mathbb{E}[f] = \alpha$, then for any integer ...
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0
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117
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Majority function stability under deletion and addition of entries
It is well known that the majority function is stable under random flipping of bits. That is, if $v$ is a random binary vector, and then we re-sample each bit of $v$ with probability $\delta$ and get $...
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4
votes
1
answer
439
views
Maximization of Mutual Information
Let $X\in\{0,1\}^d$ be a Boolean vector and $Y, Z\in\{0,1\}$ are Boolean variables. Assume that there is a joint distribution $\mathcal{D}$ over $Y, Z$ and we'd like to find a joint distribution $\...
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3
votes
1
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231
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Dual to hypercontractive inequality
Recall the hypercontractive inequality:
Let $\rho = \sqrt{\frac{p-1}{q-1}}$, then $||T_\rho(f)||_q \leq ||f||_p$
In https://www.cs.cmu.edu/~odonnell/papers/analysis-survey.pdf it is stated that the ...
- 245
10
votes
1
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367
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Lighting up all elements of a poset by toggling upsets
I consider the following game on a finite poset $(P, <)$. At each point of the game, I have a set of elements $S$ of the poset which are "on", and all others are "off". Initially $S = \emptyset$. ...
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