Questions tagged [boolean-functions]

Questions about Boolean functions and their analysis

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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 votes
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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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52 views

Does every Boolean function of degree $n$ decompose as the (XOR) product of two functions of complementary degrees?

Say $f: \{-1,+1\}^n \rightarrow \{-1, +1\}$ is a Boolean function of (Fourier) degree $n$. Is it true that there exist non-constant Boolean functions $g$, $h$ of degrees $a$ and $b$ respectively such ...
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3 votes
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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 votes
0 answers
98 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 ...
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2 votes
2 answers
235 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 ...
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1 vote
1 answer
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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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7 votes
1 answer
148 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 ...
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3 votes
1 answer
151 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 ...
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4 votes
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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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6 votes
1 answer
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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 votes
1 answer
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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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  • 187
11 votes
1 answer
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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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-1 votes
1 answer
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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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4 votes
1 answer
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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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5 votes
0 answers
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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
410 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 answer
206 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 ...
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10 votes
1 answer
318 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 $S = \emptyset$. ...
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4 votes
1 answer
258 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, ...
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  • 235
2 votes
0 answers
112 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 ...
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4 votes
0 answers
132 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 ...
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8 votes
1 answer
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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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2 votes
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146 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 ...
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13 votes
0 answers
256 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 ...
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4 votes
1 answer
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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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7 votes
2 answers
339 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 ...
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6 votes
0 answers
307 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 ...
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23 votes
2 answers
712 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{...
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1 vote
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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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7 votes
1 answer
207 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 ...
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4 votes
1 answer
227 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 ...
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18 votes
0 answers
466 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|+...
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3 votes
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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 $C$...
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5 votes
1 answer
187 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 ...
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  • 51
6 votes
1 answer
130 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} [ (-...
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  • 383
3 votes
0 answers
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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} \...
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9 votes
1 answer
219 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 ...
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1 vote
0 answers
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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)\...
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2 votes
0 answers
67 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 ...
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2 votes
1 answer
221 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}...
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6 votes
2 answers
956 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) ...
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  • 360
3 votes
1 answer
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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-...
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  • 133
6 votes
0 answers
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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'...
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1 vote
1 answer
151 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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1 vote
0 answers
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 ...
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  • 1,443
11 votes
3 answers
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 $\...
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