拟阵自由性的分离
Separations of Matroid Freeness Properties
摘要：
布尔函数在超立方体上关于定义域的线性变换不变量的性质是性质测试中研究得最多的性质之一。在本文中，我们研究的基本类的线性不变性质称为拟阵自由性质。这些性质已被证明基本上符合所有可测试的线性不变性质，最近的一系列工作已经建立了越来越大的子类的可测试性。然而，一个悬而未决的问题是，最近被证明可测试的无限多的语法上不同的属性实际上是否对应于新的、语义上不同的属性。这是一个至关重要的问题，因为它也已被证明，存在这些属性的子类，对于这些属性，一个语法上不同的表示的无限集合坍缩成一个小的，有限的属性集合之一，所有以前已知的是可测试的。因此，一个重要的问题是理解拟阵自由度属性的语义，特别是当两个语法不同的属性是真正不同的时候。我们揭示了这个问题，通过开发一种方法来确定两个拟阵自由属性P和Q之间的关系。此外，我们还证明了拟阵自由性性质存在一个自然子类，使得对于该子类中的任何两个性质P和Q，必须保持强二分法：要么P包含在Q中，要么这两个性质“良好分离”。“作为这种方法的应用，我们展示了新的，无限层次的可测试拟阵自由度属性，使得在层次结构的每一级，都有函数远离所有位于层次结构较低层次的函数。我们的关键技术工具是一个明显的新概念的映射之间的线性拟阵，称为拟阵同态，这可能是独立的利益。
英文摘要：
Properties of Boolean functions on the hypercube invariant with respect to linear transformations of the domain are among the most well-studied properties in the context of property testing. In this paper, we study the fundamental class of linear-invariant properties called matroid freeness properties. These properties have been conjectured to essentially coincide with all testable linear-invariant properties, and a recent sequence of works has established testability for increasingly larger subclasses. One question left open, however, is whether the infinitely many syntactically different properties recently shown testable in fact correspond to new, semantically distinct ones. This is a crucial issue since it has also been shown that there exist subclasses of these properties for which an infinite set of syntactically different representations collapse into one of a small, finite set of properties, all previously known to be testable. An important question is therefore to understand the semantics of matroid freeness properties, and in particular when two syntactically different properties are truly distinct. We shed light on this problem by developing a method for determining the relation between two matroid freeness properties P and Q. Furthermore, we show that there is a natural subclass of matroid freeness properties such that for any two properties P and Q from this subclass, a strong dichotomy must hold: either P is contained in Q or the two properties are "well separated." As an application of this method, we exhibit new, infinite hierarchies of testable matroid freeness properties such that at each level of the hierarchy, there are functions that are far from all functions lying in lower levels of the hierarchy. Our key technical tool is an apparently new notion of maps between linear matroids, called matroid homomorphisms, that might be of independent interest.
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