Demetres Christofides
Recently, Keller and Pilpel conjectured that the influence of a monotone Boolean function does not decrease if we apply to it an invertible linear transformation. Our aim in this short note is to prove this conjecture.
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We present a few algorithms and methods to count fixes of permutations acting on monotone Boolean functions. Some of these methods was used by Pawelski \cite{P} to compute the number of inequivalent monotone Boolean functions with 8 variables.
We derive asymptotic formulas for the Lyapunov exponent of almost all monotone Boolean networks. The formulas are different depending on whether the number of variables of the constituent Boolean functions, or equivalently, the connectivity of the Boolean network, is even or odd.
In this paper, the author presents algorithms that allow determining the number of fixed points in permutations of a set of monotone Boolean functions. Then, using Burnside's lemma, the author determines the number of inequivalent monotone Boolean functions of 8 variables. The number obtained is 1,392,195,548,889,993,358.
On the number of inequivalent monotone Boolean functions of 9 variables
本文首次计算出9变量不等价单调布尔函数的精确数量，结果为789,204,635,842,035,040,527,740,846,300,252,680。
Counting interval sizes in the poset of monotone Boolean functions
研究单调布尔函数偏序集 $D_n$ 中区间大小计数的计算问题，提出一种资源感知算法，成功实现了 $D_7$ 中区间大小的计算。
The study of partial clones on $\mathbf{2}:=\{0,1\}$ was initiated by R. V. Freivald. In his fundamental paper published in 1966, Freivald showed, among other things, that the set of all monotone partial functions and the set of all self-dual partial functions are both maximal partial clones on $\mathbf{2}$. Several papers dealing with intersections of maximal partial clones on $\mathbf{2}$ have appeared after Freivald work. It is known that there are infinitely many partial clones that contain the set of all monotone self-dual partial functions on $\mathbf{2}$, and the problem of describing t…
We propose local versions of monotonicity for Boolean and pseudo-Boolean functions: say that a pseudo-Boolean (Boolean) function is p-locally monotone if none of its partial derivatives changes in sign on tuples which differ in less than p positions. As it turns out, this parameterized notion provides a hierarchy of monotonicities for pseudo-Boolean (Boolean) functions. Local monotonicities are shown to be tightly related to lattice counterparts of classical partial derivatives via the notion of permutable derivatives. More precisely, p-locally monotone functions are shown to have p-permutable…
We prove that the fully asynchronous dynamics of a Boolean network $f:\{0,1\}^n\to\{0,1\}^n$ without negative loop can be simulated, in a very specific way, by a monotone Boolean network with $2n$ components. We then use this result to prove that, for every even $n$, there exists a monotone Boolean network $f:\{0,1\}^n\to\{0,1\}^n$, an initial configuration $x$ and a fixed point $y$ of $f$ such that: (i) $y$ can be reached from $x$ with a fully asynchronous updating strategy, and (ii) all such strategies contains at least $2^{\frac{n}{2}}$ updates. This contrasts with the following known prope…
Counting self-dual monotone Boolean functions
本文研究自对偶单调布尔函数的计数问题，通过推导多种计数算法，成功确认了第9个Hosten-Morris数λ_9的具体数值。
Monotone Boolean functions (MBFs) are Boolean functions $f: {0,1}^n \rightarrow {0,1}$ satisfying the monotonicity condition $x \leq y \Rightarrow f(x) \leq f(y)$ for any $x,y \in {0,1}^n$. The number of MBFs in n variables is known as the $n$th Dedekind number. It is a longstanding computational challenge to determine these numbers exactly - these values are only known for $n$ at most 8. Two monotone Boolean functions are inequivalent if one can be obtained from the other by renaming the variables. The number of inequivalent MBFs in $n$ variables was known only for up to $n = 6$. In this pape…
We prove two conjectures on correlation inequalities for functions that are linear combinations of unimodal Boolean monotone nondecreasing functions
Clonoids of Boolean functions with a monotone or discriminator source clone
该研究拓展Sparks定理，确定了目标克隆为投影克隆时布尔函数克隆体格的基数，并刻画了源克隆为单调克隆或含判别器函数时的布尔函数克隆体结构。
The theory of cumulants is revisited in the "Rota way", that is, by following a combinatorial Hopf algebra approach. Monotone, free, and boolean cumulants are considered as infinitesimal characters over a particular combinatorial Hopf algebra. The latter is neither commutative nor cocommutative, and has an underlying unshuffle bialgebra structure which gives rise to a shuffle product on its graded dual. The moment-cumulant relations are encoded in terms of shuffle and half-shuffle exponentials. It is then shown how to express concisely monotone, free, and boolean cumulants in terms of each oth…
The 93 minions of Boolean functions stable under left composition with the clone of self-dual monotone functions are described. As an easy consequence, all $(C_1,C_2)$-stable classes of Boolean functions are determined for an arbitrary clone $C_1$ and for any clone $C_2$ containing the clone of self-dual monotone functions.
Identification of a monotone Boolean function with $k$ "reasons" as a combinatorial search problem
研究识别具有 $k$ 个“原因”的单调布尔函数所需查询次数，给出固定 $k$ 的渐近结果及非自适应情形下大多数 $k$ 和 $n$ 的精确值。
Counting unate and balanced monotone Boolean functions
本研究将n变量单边函数、平衡单边函数的计数问题分别归约为单调函数、平衡单调函数的计数问题，结合枚举法得到了不同变量规模下的函数数量、等价类数量及非退化函数数量。
Consider a monotone Boolean function $f:\{0,1\}^n\to\{0,1\}$ and the canonical monotone coupling $\{η_p:p\in[0,1]\}$ of an element in $\{0,1\}^n$ chosen according to product measure with intensity $p\in[0,1]$. The random point $p\in[0,1]$ where $f(η_p)$ flips from $0$ to $1$ is often concentrated near a particular point, thus exhibiting a threshold phenomenon. For a sequence of such Boolean functions, we peer closely into this threshold window and consider, for large $n$, the limiting distribution (properly normalized to be nondegenerate) of this random point where the Boolean function switche…
A key fact in the theory of Boolean functions $f : \{0,1\}^n \to \{0,1\}$ is that they often undergo sharp thresholds. For example: if the function $f : \{0,1\}^n \to \{0,1\}$ is monotone and symmetric under a transitive action with $\E_p[f] = \eps$ and $\E_q[f] = 1-\eps$ then $q-p \to 0$ as $n \to \infty$. Here $\E_p$ denotes the product probability measure on $\{0,1\}^n$ where each coordinate takes the value $1$ independently with probability $p$. The fact that symmetric functions undergo sharp thresholds is important in the study of random graphs and constraint satisfaction problems as well…
Simple games with minimum
本文研究具有唯一最小获胜向量的简单博弈（单调布尔函数），推导出此类博弈数量的闭式公式及其以玩家数和等价类表示的维度。
Compounding submodular monotone (i.e. 2-alternating) set functions on a finite set preserves this property, as shown in 2010. A natural generalization to k-alternating functions was presented in 2018, however hardly readable because of page long formulas. We give an easier proof of a more general result, exploiting known properties of higher order monotonic functions.
We describe monotone maps between subsequences of the Farey sequences.
The asynchronous automaton associated with a Boolean network $f:\{0,1\}^n\to\{0,1\}^n$ is considered in many applications. It is the finite deterministic automaton with set of states $\{0,1\}^n$, alphabet $\{1,\dots,n\}$, where the action of letter $i$ on a state $x$ consists in either switching the $i$th component if $f_i(x)\neq x_i$ or doing nothing otherwise. This action is extended to words in the natural way. We then say that a word $w$ fixes $f$ if, for all states $x$, the result of the action of $w$ on $x$ is a fixed point of $f$. In this paper, we ask for the existence of fixing words,…
A Boolean function $f:\{0,1\}^n \mapsto \{0,1\}$ is said to be $\eps$-far from monotone if $f$ needs to be modified in at least $\eps$-fraction of the points to make it monotone. We design a randomized tester that is given oracle access to $f$ and an input parameter $\eps>0$, and has the following guarantee: It outputs {\sf Yes} if the function is monotonically non-decreasing, and outputs {\sf No} with probability $>2/3$, if the function is $\eps$-far from monotone. This non-adaptive, one-sided tester makes $O(n^{7/8}\eps^{-3/2}\ln(1/\eps))$ queries to the oracle.
We establish monotone bijections between subsequences of the Farey sequences and the halfsequences of Farey subsequences associated with elements of the Boolean lattices.
We establish monotone bijections between the Farey sequences of order m and the halfsequences of Farey subsequences associated with the rank m elements of the Boolean lattice of subsets of a 2m-set. We also present a few related combinatorial identities.
General limit theorems for mixtures of free, monotone, and boolean independence
本文在$\u27b1$-无卷积框架下研究三类独立性混合的有向图卷积运算$\boxplus_G$，给出有向图序列下极限存在的充分条件，涵盖多种经典极限定理，并验证多类示例，得到多正则有向图新极限定理。
The asynchronous dynamics associated with a Boolean network $f : \{0,1\}^n \to \{0,1\}^n$ is a finite deterministic automaton considered in many applications. The set of states is $\{0,1\}^n$, the alphabet is $[n]$, and the action of letter $i$ on a state $x$ consists in either switching the $i$th component if $f_i(x)\neq x_i$ or doing nothing otherwise. This action is extended to words in the natural way. We then say that a word $w$ {\em fixes} $f$ if, for all states $x$, the result of the action of $w$ on $x$ is a fixed point of $f$. A whole family of networks is fixable if its members are a…
We introduce a new notion of influence for symmetric convex sets over Gaussian space, which we term "convex influence". We show that this new notion of influence shares many of the familiar properties of influences of variables for monotone Boolean functions $f: \{\pm1\}^n \to \{\pm1\}.$ Our main results for convex influences give Gaussian space analogues of many important results on influences for monotone Boolean functions. These include (robust) characterizations of extremal functions, the Poincaré inequality, the Kahn-Kalai-Linial theorem, a sharp threshold theorem of Kalai, a stability ve…
Given a digraph $G$, a lot of attention has been deserved on the maximum number $ϕ(G)$ of fixed points in a Boolean network $f:\{0,1\}^n\to\{0,1\}^n$ with $G$ as interaction graph. In particular, a central problem in network coding consists in studying the optimality of the classical upper bound $ϕ(G)\leq 2^τ$, where $τ$ is the minimum size of a feedback vertex set of $G$. In this paper, we study the maximum number $ϕ_m(G)$ of fixed points in a {\em monotone} Boolean network with interaction graph $G$. We establish new upper and lower bounds on $ϕ_m(G)$ that depends on the cycle structure of $…
Unlike classical and free independence, the boolean and monotone notions of independence lack of the property of independent constants. In the scalar case, this leads to restrictions for the central limit theorems, as observed by F. Oravecz. We characterize the property of independent constants from a combinatorial point of view, based on cumulants and set partitions. This characterization also holds for the operator-valued extension. Our considerations lead rather directly to very mild variations of boolean and monotone cumulants, where constants are now independent. These alternative probabi…
We examine a hierarchy of equivalence classes of quasi-random properties of Boolean Functions. In particular, we prove an equivalence between a number of properties including balanced influences, spectral discrepancy, local strong regularity, homomorphism enumerations of colored or weighted graphs and hypergraphs associated with Boolean functions as well as the $k$th-order strict avalanche criterion amongst others. We further construct families of quasi-random boolean functions which exhibit the properties of our equivalence theorem and separate the levels of our hierarchy.
We show improved monotonicity testers for the Boolean hypercube under the $p$-biased measure, as well as over the hypergrid $[m]^n$. Our results are: 1. For any $p\in (0,1)$, for the $p$-biased hypercube we show a non-adaptive tester that makes $\tilde{O}(\sqrt{n}/\varepsilon^2)$ queries, accepts monotone functions with probability $1$ and rejects functions that are $\varepsilon$-far from monotone with probability at least $2/3$. 2. For all $m\in\mathbb{N}$, we show an $\tilde{O}(\sqrt{n}m^3/\varepsilon^2)$ query monotonicity tester over $[m]^n$. We also establish corresponding directed isoper…
We give a combinatorial proof of the result of Kahn, Kalai, and Linial, which states that every balanced boolean function on the $n$-dimensional boolean cube has a variable with influence of at least Omega(\frac{log n}{n}). The methods of the proof are then used to recover additional isoperimetric results for the cube, with improved constants. We also state some conjectures about optimal constants and discuss their possible implications
In this paper we introduce a method which allows us to study properties of the random uniform simplicial complex. That is, we assign equal probability to all simplicial complexes with a given number of vertices and then consider properties of a complex under this measure. We are able to determine or present bounds for a number of topological and combinatorial properties. We also study the random pure simplicial complex of dimension $d$, generated by letting any subset of size $d+1$ of a set of $n$ vertices be a facet with probability $p$ and considering the simplicial complex generated by thes…
In this paper we consider the influences of variables on Boolean functions in general product spaces. Unlike the case of functions on the discrete cube where there is a clear definition of influence, in the general case at least three definitions were presented in different papers. We propose a family of definitions for the influence, that contains all the known definitions, as well as other natural definitions, as special cases. We prove a generalization of the BKKKL theorem, which is tight in terms of the definition of influence used in the assertion, and use it to generalize several known r…
We present a regularity lemma for Boolean functions $f:\{-1,1\}^n \to \{-1,1\}$ based on noisy influence, a measure of how locally correlated $f$ is with each input bit. We provide an application of the regularity lemma to weaken the conditions on the Majority is Stablest Theorem. We also prove a "homogenized" version stating that there is a set of input bits so that most restrictions of $f$ on those bits have small noisy influences. These results were sketched out by [OSTW10], but never published. With their permission, we present the full details here.
We show that on every product probability space, Boolean functions with small total influences are essentially the ones that are almost measurable with respect to certain natural sub-sigma algebras. This theorem in particular describes the structure of monotone set properties that do not exhibit sharp thresholds. Our result generalizes the core of Friedgut's seminal work [Ehud Friedgut. Sharp thresholds of graph properties, and the k-sat problem. J. Amer. Math. Soc., 12(4):1017-1054, 1999.] on properties of random graphs to the setting of arbitrary Boolean functions on general product probabil…
On the Analysis of Boolean Functions and Fourier-Entropy-Influence Conjecture
本文研究布尔函数分析与Fourier-Entropy-Influence猜想，除经典结果外，提出并证明布尔函数$f$的熵可被$O(I(f))+O(\sum\limits_{k}I_k(f)\log (1/I_k(f)))$上界界定的新结果。
Log-Sobolev under random monotone censoring
本文证明布尔立方体的对数Sobolev不等式在随机单调删失下保持稳定，并推导出典型单调子集上的超收缩性、高斯集中和O(n log n)混合时间，解决了Ding和Mossel的猜想。
Let $U_{k,N}$ denote the Boolean function which takes as input $k$ strings of $N$ bits each, representing $k$ numbers $a^{(1)},\dots,a^{(k)}$ in $\{0,1,\dots,2^{N}-1\}$, and outputs 1 if and only if $a^{(1)} + \cdots + a^{(k)} \geq 2^N.$ Let THR$_{t,n}$ denote a monotone unweighted threshold gate, i.e., the Boolean function which takes as input a single string $x \in \{0,1\}^n$ and outputs $1$ if and only if $x_1 + \cdots + x_n \geq t$. We refer to circuits that are composed of THR gates as monotone majority circuits. The main result of this paper is an exponential lower bound on the size of b…
On the Fourier Entropy-Influence Conjecture for Boolean Plateaued Functions
本文通过证明一个关键不等式，解决了布尔平缓函数和部分弯曲函数的傅里叶熵-影响猜想，并确定了相应的最优常数分别为4和2。
In this paper we consider a variant of the well-known Achlioptas process for graphs adapted to monotone Boolean functions. Fix a number of choices $r\in \mathbb N$ and a sequence of increasing functions $(f_n)_{n\ge 1}$ such that, for every $n\ge 1$, $f_n:\{0,1\}^n\mapsto \{0,1\}$. Given $n$ bits which are all initially equal to 0, at each step $r$ 0-bits are sampled uniformly at random and are proposed to an agent. Then, the agent selects one of the proposed bits and turns it from 0 to 1 with the goal to reach the preimage of 1 as quickly as possible. We nearly characterize the conditions und…
The influence of the $k$'th coordinate on a Boolean function $f:\{0,1\}^n \rightarrow \{0,1\}$ is the probability that flipping $x_k$ changes the value $f(x)$. The total influence $I(f)$ is the sum of influences of the coordinates. The well-known `Junta Theorem' of Friedgut (1998) asserts that if $I(f) \leq M$, then $f$ can be $ε$-approximated by a function that depends on $O(2^{M/ε})$ coordinates. Friedgut's theorem has a wide variety of applications in mathematics and theoretical computer science. For a biased function with $E[f]=μ$, the edge isoperimetric inequality on the cube implies that…
We pursue a systematic study of the following problem. Let f:{0,1}^n -> {0,1} be a (usually monotone) Boolean function whose behaviour is well understood when the input bits are identically independently distributed. What can be said about the behaviour of the function when the input bits are not completely independent, but only k-wise independent, i.e. every subset of k bits is independent? more precisely, how high should k be so that any k-wise independent distribution "fools" the function, i.e. causes it to behave nearly the same as when the bits are completely independent? We analyze sever…
Sparse juntas on the biased hypercube
本文证明了有偏超立方体上在L2范数下接近低次多项式的布尔函数必接近稀疏junta函数或稀疏DNF公式，并确定了最优常数及最优函数类。
Cumulant-cumulant relations in free probability theory from Magnus' expansion
本文利用 pre-Lie 代数上的 Magnus 展开，结合连续 Baker-Campbell-Hausdorff 公式，给出了用自由和布尔累积量闭式表达多元单调累积量的方法，并引入了非交叉划分上的新构造与统计量。
Further evidence towards the Fourier Entropy-Influence conjecture
本文通过引入关键不等式和停止二叉树概念，证明了若干新布尔函数类满足傅里叶熵-影响猜想。
We develop a new technique for proving concentration inequalities which relate between the variance and influences of Boolean functions. Using this technique, we 1. Settle a conjecture of Talagrand [Tal97] proving that $$\int_{\left\{ -1,1\right\} ^{n}}\sqrt{h_{f}\left(x\right)}dμ\geq C\cdot\mathrm{var}\left(f\right)\cdot\left(\log\left(\frac{1}{\sum\mathrm{Inf}_{i}^{2}\left(f\right)}\right)\right)^{1/2},$$ where $h_{f}\left(x\right)$ is the number of edges at $x$ along which $f$ changes its value, and $\mathrm{Inf}_{i}\left(f\right)$ is the influence of the $i$-th coordinate. 2. Strengthen se…
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