Questions about Boolean functions and their analysis

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7
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95 views

How to find the “hard” probability distribution on the input for recursive boolean functions?

Background: Decision tree complexity or query complexity is a simple model of computation defined as follows. Let $f:\{0,1\}^n\to \{0,1\}$ be a Boolean function. The deterministic query complexity of ...
3
votes
1answer
114 views

Lower bounds for Polynomials computing the boolean functions

Expressing a boolean function $f$ $:\{ 0,1 \}^{n} \rightarrow \{0,1 \}$ using a polynomial $P(x_{1},...,x_{n})$, where $x_{1},...,x_{n}$ may be integer, finite fields, or other fields. One of the ...
2
votes
1answer
63 views

Satisfiability for various branching programs

Clearly, SAT for CNF, formula, or many classes of boolean circuits have strong significance on the entire theory community. Branching program, or BDD is one of the most basic and popular ...
8
votes
1answer
146 views

Random restrictions and the connection to total influence of Boolean functions

Say we have a Boolean function $f:\{-1,1\}^n\rightarrow \{-1,1\}$ and we apply $\delta$-random restriction on $f$. In addition, say that the decision tree $T$ that computes $f$ shrinks to size $O(1)$ ...
11
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1answer
174 views

Given $f:\{0,1\}^n \rightarrow \{-1,1\}$, find a subcube with large volume and large average value

Here is a problem with a similar flavor to learning juntas: Input: A function $f: \{0,1\}^n \rightarrow \{-1,1\}$, represented by a membership oracle, i.e. an oracle that given $x$, returns $f(x)$. ...
3
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0answers
53 views

Social choice theory, preference aggregation data sets

I do computational research on preference aggregation. I am quite interested in Kemeny Optimal Aggregation. However I do not find much useful data for preference aggregation in context of social ...
5
votes
2answers
61 views

L1 - embeddability of metrics supported on the Hypercube

I am quite new to the area of metric embeddings so this question might turn out to be extremely easy. Consider a metric supported on the edges of a boolean hypercube. By supported I mean every edge ...
4
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0answers
105 views

A possible application of TCS to EE: disruption-resistant circuits

Consider the following problem. We're given a circuit $C$ with $n$ binary inputs and $n$ binary outputs, computing some boolean function $f_C : \mathbb{Z}_2^n \rightarrow \mathbb{Z}_2^n$. We assume ...
14
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2answers
187 views

On the status of learnability inside $\mathsf{TC}^0$

I'm trying to understand the complexity of functions expressible via threshold gates and this led me to $\mathsf{TC}^0$. In particular, I'm interested what's currently known about learning inside ...
5
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0answers
125 views

Fourier Analysis on checking whether there exists a vector in hypercube orthogonal to a set of vectors?

I know virtually noting about Fourier Analysis and I'd like to know whether it's worth to learn this topic for my problem. My problem is: Given vectors $h_1,\cdots,h_k\in\{+1,-1\}^n$ where $k < ...
6
votes
2answers
258 views

Circuit complexity of Majority function

Let $f: \{0,1\}^n \to \{0,1\}$ be the majority function, i.e. $f(x) = 1$ if and only if $\sum_{i = 1}^n x_i > n/2$. I was wondering if there was a simple proof of the following fact (by "simple" I ...
3
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0answers
79 views

Deciding whether a binary multiplicity automaton has empty language

Multiplicity automatons (see here) is an interesting model. They have the (almost) same syntax as a non-deterministic finite automatons, but instead of deciding whether a word belongs to a language, ...
0
votes
1answer
92 views

How to go about finding the most “complex” function?

Intro Hi, I'm a hobbyist, with no formal education, tinkering with SAT solving and boolean algebra minimization. So expect bad terminology. I hope you will forgive me I'm asking a wrong question in a ...
3
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0answers
108 views

An identity about the Majority function?

Let $f:\{-1,1\}^n \rightarrow \{-1,1\}$ be any boolean function. Let $Maj_n$ represent the majority function. Let $\langle f,g \rangle = E[f(x)g(x)]$ and $\mathcal{I}(f) = E_x[\# i, s.t. f(x)\neq f(x ...
0
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0answers
94 views

Determining if a function is constant or not using period finding

Consider an arbitrary boolean function $$f: {\lbrace 0,1 \rbrace}^n \rightarrow \lbrace 0,1 \rbrace$$ which we write as: $$f(x_1, x_2 ... x_n) $$ where each $x_i$ is a boolean variable We note ...
2
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0answers
36 views

Is there a relation between $l_p$-norms of functions with same Fourier spectra but w.r.t different measures on the Hamming cube?

Informally, I want to ask if two functions $f$ and $g$ on the Hamming cube have the same Fourier spectra but w.r.t different measure and basis, then is $||f||_p$ related to $||g||_p$? (Where each ...
4
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0answers
100 views

Boolean Functions Where Sensitivity Equals Block Sensitivity

Some of the work on sensitivity vs. block sensitivity has been aimed at examining functions with as large a gap as possible between $s(f)$ and $bs(f)$ in order to resolve the conjecture that $bs(f)$ ...
4
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1answer
163 views

any connection between binary/integer multiplication and matrix multiplication?

is there a connection between the inherent complexity of binary/integer multiplication algorithms and matrix multiplication algorithms? if so what is a ref that outlines/discusses it? some ...
5
votes
1answer
180 views

Lower bounds for formulae sizes for addition

I am interested in the conversion of $\sum_{i=1}^n x_i = y$ to 3-CNF. Here $x_i$ is a binary 0/1 variable and $y$ is some positive integer. There are a number of practical methods for doing this, ...
2
votes
2answers
263 views

Are Boolean circuits 'universal'

I have a question, but I don't seem to know enough computer science terminology in order to look up an answer. So I wonder if you guys could help a poor physicist like me. I would like to know if ...
5
votes
0answers
116 views

An alternative proof of hypercontractivity of the Becker-Bonami operator

The Becker-Bonami operator $T_p$ on a function $f(x) : \{0,1\}^d \to \mathbb{R}$ is defined as follows. Let $\nu(x)$ be a perturbation of $x$ so that every bit remains the same with probability $p$ or ...
6
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0answers
82 views

Problems that reduce to or are abstracted by the learning juntas problem

What problems are either abstracted by or reduce to the learning juntas problem? (An example of a real-world problem abstracted by the learning juntas problem is the Identification of genetic loci ...
12
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1answer
369 views

Hardness of noisy Boolean functions

Let $f$ be a Boolean function of $n$ Boolean variables. Let $g(x)=T_\epsilon (f) (x)$ be the expected value of $f(y)$ when $y$ is obtained from $x$ by flipping each coordinate with probability ...
14
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0answers
189 views

Random functions of low degree as a real polynomial

Is there a (reasonable) way to sample a uniformly random boolean function $f:\{0,1\}^n \to \{0,1\}$ whose degree as a real polynomial is at most $d$? EDIT: Nisan and Szegedy have shown that a ...
12
votes
1answer
255 views

Beigel-Tarui transformation of ACC cricuits

I am reading the appendix about ACC lower bounds for NEXP in Arora and Barak's Computational Complexity book. http://www.cs.princeton.edu/theory/uploads/Compbook/accnexp.pdf One of the key lemmas is a ...
6
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2answers
157 views

Attribute efficiently learning the relevant attributes of juntas with membership queries

Can the relevant attributes of k-juntas be learned attribute efficiently given a membership query oracle? What's the best known lower bound for this problem?
3
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0answers
56 views

Estimating the average contiguous sequence length of a balanced boolean function

Given a balanced Boolean function $f:\{0,1\}^n\mapsto\{0,1\}$, is there a sampling strategy that needs less than $O(2^n)$ samples to estimate the average length of contiguous sequences of 1's? ...
1
vote
1answer
137 views

Succinct Representation and Communication complexity

Succinct representation is often used to define NEXP or EXP complete problems. For example, when a graph is given as a circuit to compute the existence of edge between vertex $i,j$ for indices of ...
2
votes
0answers
124 views

Natural Proofs and methods for polylog depth circuit lower bounds

I have a question about the following question and its answer. Status on circuit lower bounds for polylog-bounded depth circuits In the above question, it is asked about methods to prove lower ...
3
votes
1answer
112 views

Lower Bound for the Parity Learning Problem

What are known lower bounds for the time and query complexity of the problem of learning parities with an adaptive membership query oracle? To be clear the concept space $C$ is $\{x\in \{0,1\}^n \, \, ...
3
votes
1answer
115 views

Inferring simplest method to convert bit array 1 to bit array 2

Consider the set of all bit arrays of length n. Now consider the set of all 1-to-1 functions that map from this set to this set. Now select a single function out of the latter set. Is there any ...
2
votes
1answer
133 views

Locally monotone Boolean function

I am unable to understand the definition of locally monotone Boolean function which is defined in Gotsman and Linial, "Spectral Properties of Threshold Functions", 1994, p. 40: A function $f$ is ...
-3
votes
1answer
149 views

computing with gates on polar coordinates, functionally complete wrt boolean functions?

this question is inspired by a particular somewhat "natural" physical system specifically constructed to mimic another complex highly-studied physical computing system. (some may astutely guess at ...
1
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0answers
93 views

Extended Formulaiton and Integer Programming

An extended formulation (EF) of a polytope $P\subseteq \mathbb{R}^d$ is a system of linear constraints $Ex + Fy = g, y\geq 0$ in variables $(x,y)\in \mathbb{R}^{d+r}$ where $E,F$ are real matrices ...
1
vote
1answer
94 views

Expressing a set of 0-1 strings by Extended Formulation

Extended Formulation of a polytope $P \subseteq \mathbb{R}$ is a system of linear constraints which satisfies the following condition: $x \in P \iff \exists y\in \mathbb{R}^{d} \mbox{ such that } Ax ...
3
votes
1answer
128 views

About Closure under Resolution

The question looks very simple, that is why I posted it first on MathSE, unsuccesfully - no answer for 12 days. I tried to find a short and elegant answer to the question, but I haven't succeed yet. ...
0
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0answers
76 views

Bound for the spectral norm of a boolean function [duplicate]

As we know that the upper bound for the spectral norm of a Boolean function on $n$ variables is $2^{n/2}$. Is it possible to improve this bound..? Can somebody provide me an example of a Boolean ...
5
votes
2answers
432 views

Is the the spectral norm of a Boolean function bounded by the degree of its Fourier expansion?

Let $f: \{-1,1\}^n \rightarrow \{-1,1\}$ be a Boolean function. The Fourier expansion of $f$ is $$f(T) = \sum_{S \subseteq [n]} \widehat{f}(S)\ \chi_S(T)$$ where $\widehat{f}(S)$ are real numbers ...
2
votes
2answers
201 views

Size of decision tree for f is polynomial in the DNF size of f and CNF size of f

I've been having hard time with proving the following claim: Let $f:\{T,F\}^n\rightarrow \{T,F\}$ be a boolean function. Let $size_{DT}(f)$ denote the number of leaves in the smallest (w.r.t the ...
16
votes
2answers
297 views

An extension of the noise operator

In a problem I am currently working on, an extension of the noise operator arises naturally, and I was curious whether there has been prior work. First let me revise the basic noise operator ...
2
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0answers
72 views

Exhibiting an adversary to prove a boolean function is evasive

I'm trying to collect examples of exhibiting an adversary to prove that a Boolean function is evasive. I know of several examples of graph properties for which adversary methods have been used, i.e. ...
16
votes
2answers
473 views

Robustness of splitting a junta

We say that a Boolean function $f: \{0,1\}^n \to \{0,1\}$ is a $k$-junta if $f$ has at most $k$ influencing variables. Let $f: \{0,1\}^n \to \{0,1\}$ be a $2k$-junta. Denote the variables of $f$ by ...
3
votes
2answers
315 views

Factoring Cartesian bitwise join of bit vectors

(This question has been substantially revised in an attempt to word it clearly.) I am wondering if anyone has seen this problem. Let $[n] = \{1,\ldots,n\}$ for an integer $n$. Consider two finite ...
4
votes
0answers
170 views

Sparse Boolean Function and Other Boolean Functions

Let $s$ be a Sparse boolean function $s:\{0,1\}^{n}\rightarrow \{0,1\}$ such that $|s^{-1}(1)| \leq 2^{n\delta}, 0 < \delta <1$ The majority function $MAJ_{n}$ takes value 1 if and only if the ...
17
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2answers
340 views

Are all the functions whose fourier weight is concentrated on the small sized sets computed by AC0 circuits?

Are all the functions whose fourier weight is concentrated on the small sized sets(or terms with low degree) computed by $\mathsf{AC}^0$ circuits ?
3
votes
1answer
277 views

Length of a minimal DNF Boolean expression

Consider an disjunctive normal form boolean expression on $n$ variables. What is the upper bound on the number of terms in a minimal equivalent DNF expression? That is, given an arbitrary DNF ...
41
votes
2answers
3k views

Why does Fourier analysis of Boolean functions “work”?

Over the years I have gotten used to seeing many TCS theorems proved using discrete Fourier analysis. The Walsh-Fourier (Hadamard) transform is useful in virtually every subfield of TCS, including ...
7
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0answers
226 views

Can one prove $\mathsf{PARITY} \notin \mathsf{AC}^0$ using Linial-Mansour-Nisan theorem and the knowledge of fourier spectrum of $\mathsf{PARITY}$?

Result 1: Linial-Mansour-Nisan theorem says that the fourier weight of the functions computed by the $\mathsf{AC}^0$ circuits is concentrated on the subsets of small size with high probability. ...
0
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208 views

Boolean circuits and digraphs

It is well known that connecting NAND gates allows the construction of arbitrary circuits. Furthermore, a NAND gate can be represented as a digraph with four vertices (in order, the two inputs, the ...
10
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1answer
240 views

The entropy of a noisy distribution

Say we have a function $f:\mathbb{Z}_2^n \to \mathbb{R}$ such that $$\forall x\in \mathbb{Z}_2^n \quad f(x) \in \left\{\frac{1}{2^n}, \frac{2}{2^n}, \ldots, \frac{2^n}{2^n} \right\},$$ and $f$ is a ...