Questions tagged [boolean-functions]
Questions about Boolean functions and their analysis
194
questions
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1answer
131 views
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 ...
1
vote
1answer
148 views
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 ...
0
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0answers
58 views
Proof of greedy method to compute minimum Prime Implicant of a monotone Boolean function
The decision version of Prime Implicant problem is NP-complete for Monotone Boolean function. I present below a greedy algorithm to find the minimum prime implicant ( minimum number of variables which ...
3
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0answers
81 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
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0answers
97 views
Monomial Sparsity of Boolean Functions
Suppose you have some boolean function $f: \{-1,1\}^n \rightarrow \{-1,1\}$ with rational coefficients such that all degree 1 monomials of $f$ have a nonzero coefficient and the degree $n$ monomial ...
3
votes
1answer
149 views
$ACC^0$ 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 ...
4
votes
0answers
79 views
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, ...
5
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1answer
212 views
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" ...
3
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0answers
101 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 ...
6
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1answer
323 views
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 ...
7
votes
1answer
325 views
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 ...
4
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0answers
98 views
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 ...
9
votes
1answer
302 views
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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1answer
62 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 ...
4
votes
1answer
206 views
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 ...
5
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0answers
115 views
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 $...
4
votes
1answer
400 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 $\...
3
votes
1answer
190 views
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 ...
10
votes
1answer
289 views
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 $S = \emptyset$. ...
4
votes
1answer
252 views
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, ...
2
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0answers
101 views
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 ...
4
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0answers
129 views
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 ...
8
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1answer
1k views
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?
2
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0answers
139 views
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 ...
13
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0answers
231 views
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 ...
4
votes
1answer
154 views
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 ...
7
votes
2answers
335 views
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 ...
6
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0answers
288 views
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 ...
23
votes
2answers
693 views
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:
$$A_1 = \begin{...
1
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0answers
57 views
relations between the degrees of a boolean function and its absolute function
Given a boolean function $f:\{0,1\}^n\rightarrow\mathbb{R}$ of degree $d$, is there any upper bound in terms of $d$ on the degree of the function $|f|$, where $|f|(x)=|f(x)|$. Here the degree of $f$ ...
7
votes
1answer
180 views
Sampling monotone Boolean functions
I'm interested in sampling monotone increasing Boolean functions on $n$ input bits uniformly at random. I understand that this is equivalent to approximating the Dedekind numbers ($D_n = $ the number ...
4
votes
1answer
207 views
Which (almost) balanced Boolean function has smallest "total" influence
The well known Kahn–Kalai–Linial (KKL) Theorem says that for any Boolean function $f\colon \{-1,1\}^n \xrightarrow{} \{-1,1\}$
$$
\max_{i \in [n]} \{\mathbf{Inf}_i[f] \} \geq \mathop{\bf Var}[f] \cdot ...
18
votes
0answers
425 views
Perfect matching of monotone Boolean function with null Euler characteristic
For a set $V = \{0,\ldots,k\}$ of variables, let $\mathbf{G}_V$ be the undirected graph with set of vertices $\{S \subseteq V\}$ and set of edges $\{\{S,S'\} \mid S \subseteq S' \text{ and }|S'| = |S|+...
3
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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 $C$...
5
votes
1answer
185 views
Minimal information needed for determine some function
From calculus, we know that if someone has a continuous function $f$, it is enough to know $f$'s values on the rationals in order to know $f$ on the entire line. In some sense, a "countable amount of ...
6
votes
1answer
127 views
Average-case analogue of Small-bias Spaces
Recall that an $\epsilon$-biased space is a set $S \subset \{0,1\}^n$ such that for every non-zero linear test $\alpha \in \{0,1\}^n \setminus \{0\}^n$, the expected bias
$$| \mathbb{E}_{x \in S} [ (-...
3
votes
0answers
115 views
Decomposition of rectangular relations
Let $\alpha$ be a binary relation from $\gamma$ to $\chi$ and $\beta$ a binary relation from $\chi$ to $\rho$. If both $\alpha$ and $\beta$ are rectangular, i.e., they satisfy $\alpha \alpha^{-1} \...
9
votes
1answer
212 views
Has there been any progress in tightening the exponent in the result that polylog independence fools $AC_0$?
Braverman showed that distributions which are $(log \frac{m}{\epsilon})^{O(d^2)}$-wise independent $\epsilon$-fool depth $d$ $AC^0$ circuits of size $m$ by "gluing together" the Smolensky ...
1
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0answers
71 views
Relative Two-Function Hypercontractive Inequality $\langle T_{\rho\sigma} f,g \rangle \le \langle T_{\sigma} f,g \rangle^q$
The hypercontractive theorem (or Bonami Beckner inequality) says (Ryan O'Donnell):
This of course also directly gives the 'intermediate' inequality $\|T_{\rho\sigma}f\|_q \le \|T_\sigma f\|_{1+(q-1)\...
2
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0answers
59 views
$p$-biased two-function hypercontractivity
The Hypercontractivity theorem (or Bonami Beckner inequality) is a very useful tool. Unfortunately, it isn't easy to carry over to other spaces than the uniform boolean cube.
In Ryan O'Donnel's ...
2
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1answer
213 views
Fourier decomposition in terms of another basis
Given a Boolean function $f:\{-1,1\}^n\rightarrow \{-1,1\}$, it is well know that the Fourier decomposition of $f$ can be written as $f(x)=\sum_{S\subseteq \{1,\ldots,n\}} \widehat{f}(S) \prod_{i\in S}...
6
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2answers
883 views
a polynomial representation of boolean functions
I came up with this linear transformation to map boolean functions to polynomials and it seems to have some nice properties. I was wondering if there is any reference describing this (and/or similar) ...
3
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1answer
205 views
Proof that all Boolean functions can be computed by $(MOD_2-MOD_3)$ circuit
I was reading "Some properties of MOD m circuits computing simple functions" (Amano & Maruoka, 2003) where the authors prove that every Boolean function can be computed by depth $2$ by $(MOD_2-...
6
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0answers
139 views
Has what I am calling "helpfulness" here been studied?
We say that a Boolean function $f : \{0, 1\}^n \rightarrow \{0, 1\}$ is helpful for another Boolean function $g$ if $f(x)$ can be computed with a smaller circuit given $g(x)$ as an extra input bit. I'...
1
vote
1answer
135 views
How good of an approximate 2-coloring can you get of the halved cube graph?
We say that a 2-coloring $col : V_G \rightarrow \{0, 1\}$ of a graph $G$ is $\epsilon$-approximate if $Pr_{(w, v) \in E_G}(col(w) \neq col(v)) \geq \epsilon$. For every $n$, what is the maximum $\...
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0answers
81 views
Reference request for the relationship between approximating degree of Boolean functions and learning algorithms
This paper (http://www.cs.columbia.edu/~rocco/Public/stoc01.pdf) from STOC 2001 is possibly the first paper to show how to convert upperbounds on the $\frac{1}{3}-$approximation degree of a Boolean ...
11
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3answers
1k views
Proper PAC learning VC dimension bounds
It is well known that for a concept class $\mathcal{C}$ with VC dimension $d$, it suffices to obtain $O\left(\frac{d}{\varepsilon}\log\frac{1}{\varepsilon}\right)$ labelled examples to PAC learn $\...
9
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1answer
189 views
What is the probability that a random Boolean function has a trivial automorphism group?
Given a Boolean function $f$, we have the automorphism group $Aut(f) = \{\sigma \in S_n\ \mid \forall x, f(\sigma(x)) = f(x) \}$.
Are there any known bounds on $Pr_f(Aut(f) \neq 1)$? Is there ...
4
votes
0answers
157 views
Relationship between sparsity and rank of a boolean function
I have the following question when I was going through the proof of the following theorem.
Theorem. For XOR function $f \circ XOR$, $rank(M_{f \circ XOR}) = ||\hat f ||_0$ where $M_{f \circ XOR}$ is ...
2
votes
1answer
49 views
Certainty of mutual confirmation over faulty channels?
This is a very theoretical question, although I am sure the problem pops up in lots of IT and automation applications. Still, I prefer to formulate it in an action-movie scenario (a bit of the ...
