Questions tagged [boolean-functions]
Questions about Boolean functions and their analysis
217
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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{ ...
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38
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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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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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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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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 ...
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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 ...
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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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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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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 ...
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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 ...
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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 ...
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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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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 ...
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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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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, ...
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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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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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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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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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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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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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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 ...
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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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$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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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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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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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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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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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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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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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 ...
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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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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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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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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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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 ...
- 235
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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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How tight is the XOR lemma?
The XOR lemma states that if you have a distribution $D$ on $\{0,1\}^n$, and all the Fourier coefficients of $2^n D$ are small, then it is close in $L_1$ to the uniform distribution. Specifically, ...
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Cover set of Boolean formulas with conjunctions
I want to cover a set of Boolean formulas (over the same variables) with disjunctive conjunctions. Here's an example with two formulas $p_1$ and $p_2$ over the set of variables $\{A, B, X, Y\}$:
I ...
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Characterizing the ANF of Single-Cycle Boolean Permutations
Given a function $F: \{0, 1\}^n \to \{0, 1\}^n$, we say that $F$ is a boolean permutation (also sometimes called a vectorial boolean function or an s-box in the literature) if $F$ is a bijection. We ...
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On the sensitivity conjecture?
The recent establishment of the relation $bs(f)=O(s(f)^4)$ goes through Gotsman,Linial .
Can the same approach get to $O(s(f)^2)$ or is there an essential limitation to the approach?
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Sensitivity and Low-Degree Approximation under Non-Uniform Distribution
I am searching for generalizations of analysis of Boolean functions when the input strings are distributed according to a general non-uniform distribution, possibly with arbitrary dependencies between ...
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Which monotone DNFs are evasive?
A Boolean function $\phi$ on variables $X$ is evasive if every decision tree for $\phi$ has height $|X|$. In other words, for any strategy that picks variables of $X$ and asks for their value, an ...
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Given a subset of of the hypercube and an affine transform of it, find the affine map
This is a follow up to this resolved question.
Suppose we are given a set of bitvectors $A\subseteq\mathbb{F}_2^d$ and an invertible affine transformed copy of it
$$B=\{Mx + s\mid x\in A\}$$
for some ...
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Given a subset of the hypercube and a copy translated by s, find s
Problem: Suppose we are given an $n$ element subset $A\subseteq\{0,1\}^d$ of the $d$ dimensional hypercube and a translated copy $B= A+s$ by some secret $s\in\{0,1\}^d$. Find $s$ as fast as possible ...
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Reverse Skolemization?
I'm wondering if there are any references on "reverse skolemization", that is, converting a formula with functions into one purely consisting of quantifiers by eliminating function applications.
I'm ...
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2
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Question about two matrices: Hadamard v. "the magical one" in the proof of the sensitivity conjecture
The recent and incredibly slick proof of the sensitivity conjecture relies on the explicit* construction of a matrix $A_n\in\{-1,0,1\}^{2^n\times 2^n}$, defined recursively as follows:
$$A_1 = \begin{...
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relations between the degrees of a boolean function and its absolute function
Given a boolean function $f:\{0,1\}^n\rightarrow\mathbb{R}$ of degree $d$, is there any upper bound in terms of $d$ on the degree of the function $|f|$, where $|f|(x)=|f(x)|$. Here the degree of $f$ ...
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Sampling monotone Boolean functions
I'm interested in sampling monotone increasing Boolean functions on $n$ input bits uniformly at random. I understand that this is equivalent to approximating the Dedekind numbers ($D_n = $ the number ...
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