It was proved few years ago that classes of Boolean functions definable by means of functional equations \cite{EFHH}, or equivalently, by means of relational constraints \cite{Pi2}, coincide with initial segments of the quasi-ordered set $(Ω, \leq)$ made of the set $Ω$ of Boolean functions, suitably quasi-ordered. The resulting ordered set $(Ω/\equiv, \sqsubseteq)$ embeds into $([ω]^{<ω}, \subseteq)$, the set -ordered by inclusion- of finite subsets of the set $ω$ of integers. We prove that $(Ω/\equiv, \sqsubseteq)$ also embeds $([ω]^{<ω}, \subseteq)$. We prove that initial segments of $(Ω, \leq)$ which are definable by finitely many obstructions coincide with classes defined by finitely many equations. This gives, in particular, that the classes of Boolean functions with a bounded number of essential variables are finitely definable. As an example, we provide a concrete characterization of the subclasses made of linear functions.
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.
Algorithmic decidability is established for two order-theoretic properties of downward closed subsets defined by finitely many obstructions in two infinite posets. The properties under consideration are: (a) being atomic, i.e. not being decomposable as a union of two downward closed proper subsets, or, equivalently, satisfying the joint embedding property; and (b) being well quasi-ordered. The two posets are: (1) words over a finite alphabet under the consecutive subword ordering; and (2) finite permutations under the consecutive subpermutation ordering. Underpinning the four results are chara…
An integer packing set is a set of non-negative integer vectors with the property that, if a vector $x$ is in the set, then every non-negative integer vector $y$ with $y \leq x$ is in the set as well. Integer packing sets appear naturally in Integer Optimization. In fact, the set of integer points in any packing polyhedron is an integer packing set. The main result of this paper is that integer packing sets, ordered by inclusion, form a well-quasi-ordering. This result allows us to answer a question recently posed by Bodur et al. In fact, we prove that the k-aggregation closure of any packing …
Well-Quasi-Ordering of Matrices under Schur Complement and Applications to Directed Graphs
在[Rank-Width and Well-Quasi-Ordering of Skew-Symmetric or Symmetric Matrices，arXiv：1007.3807v1]中，Oum证明了，对于一个固定的有限域$\mathbf{F}$，在有界$\mathbf{F}$ -rank-width的$\mathbf{F}$上的（反）对称矩阵的任何无限序列$M_1,M_2,...$都有一对$i< j$，使得$M_i$同构于$M_j$的主枢轴变换的主子矩阵。我们将这个结果推广到Rao和我在[边着色图的秩宽度，arXiv：0709.1433v4]中介绍的$σ$对称矩阵。（斜）对称矩阵是PDMATHAMR-对称矩阵的特殊情况。作为一个副产品，我们得到，对于每一个无限序列$G_1,G_2,...$的有界秩宽度的有向图，存在一对$i<j$，使得$G_i$是一个非子式$G_j$。另一个推论是，PDMATHAMR对称矩阵的非奇异主子矩阵形成δ拟阵。我们以这种方式扩展了Bouchet的Delta拟阵可表示性的概念。
Computing the partition function of a polynomial on the Boolean cube
对于多项式f：{-1，1}^n --> C，我们将配分函数定义为e^{lambda f（x）}在{-1，1}^n中所有点x上的平均值，其中C中的lambda是一个参数。我们提出了一个拟多项式算法，在给定f，lambda和λ>0的情况下，只要|lambda| < 1/（2 L sqrt{deg f}），该算法在N^{O（ln n -ln n）}时间内以相对误差λ逼近配分函数，其中L=L（f）是从上面限定f的Lipschitz常数的参数，N是f中单项式的个数.作为推论，我们获得了一个准多项式算法，给定系数为+1和-1的f，并且每个变量输入不超过4个单式，该算法在O（sqrt{deg f}/delta）的因子内逼近{-1，1}^n上f的最大值，如果对于某些0< delta 4，最大值为N delta<1. If every variable enters not more than k monomials for some fixed k >，则当delta >（k-1）/k时，我们能够建立类似的结果。
