Questions tagged [boolean-functions]

Questions about Boolean functions and their analysis

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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 ...
8 votes
1 answer
189 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 ...
0 votes
0 answers
53 views

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 $...
2 votes
0 answers
89 views

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 $...
5 votes
0 answers
82 views

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 ...
0 votes
0 answers
61 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 ...
2 votes
2 answers
239 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 ...
3 votes
1 answer
133 views

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 ...
23 votes
4 answers
1k views

Monotone arithmetic circuits

The state of our knowledge about general arithmetic circuits seems to be similar to the state of our knowledge about Boolean circuits, i.e. we don't have good lower-bounds. On the other hand we have ...
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 ...
1 vote
1 answer
182 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 ...
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 ...
4 votes
0 answers
80 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, ...
17 votes
1 answer
3k 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 $bs(f)$ ...
5 votes
1 answer
274 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" ...
10 votes
1 answer
327 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 $S = \emptyset$. ...
3 votes
0 answers
104 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 votes
1 answer
356 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
1 answer
351 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 votes
0 answers
112 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 ...
14 votes
3 answers
1k views

Checking formulas with two quantifiers ($\forall \exists$) - 2QBF

SAT solvers give a powerful way to check the validity of a boolean formula with one quantifier. For instance, to check the validity of $\exists x . \varphi(x)$, we can use a SAT solver to determine ...
11 votes
1 answer
509 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 votes
1 answer
67 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
1 answer
230 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 ...
4 votes
1 answer
414 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 $\...
5 votes
0 answers
116 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
1 answer
262 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, ...
3 votes
1 answer
210 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 ...
2 votes
0 answers
118 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 ...
2 votes
0 answers
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 ...
8 votes
1 answer
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?
4 votes
0 answers
133 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 ...
13 votes
0 answers
263 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
1 answer
156 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
2 answers
341 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 ...
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 $\...
23 votes
2 answers
720 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{...
6 votes
0 answers
316 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 ...
7 votes
3 answers
2k 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 ...
1 vote
0 answers
58 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 $f$ ...
4 votes
1 answer
232 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
0 answers
474 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|+...
7 votes
1 answer
221 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 ...
30 votes
1 answer
2k views

Fourier coefficients Boolean Functions described by Bounded Depth Circuits with AND OR and XOR gates

Let $f$ be a Boolean function and let's think about f as a function from $\{-1,1\}^n$ to $\{ -1,1 \}$. In this language the Fourier expansion of f is simply the expansion of f in terms of square free ...
4 votes
1 answer
221 views

Möbius values of CNF and DNF lattices of a monotone Boolean function

Let $\phi$ be a monotone Boolean function on a set of variables $\langle k \rangle := \{0,\ldots,k\}$ such that $\phi$ depends on all the variables in $\langle k \rangle$ (that is, for every variable $...
3 votes
0 answers
122 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 $C$...
9 votes
1 answer
223 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 ...
6 votes
1 answer
190 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 ...
21 votes
1 answer
437 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 ...
16 votes
1 answer
294 views

Which monotone Boolean functions are representable as thresholds on sums?

I will introduce my problem with an example. Say you are designing an exam, which consists of a certain set of $n$ independent questions (that the candidates can get either right or wrong). You want ...

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