We provide a combinatorial characterization of all testable properties of $k$-uniform hypergraphs ($k$-graphs for short). Here, a $k$-graph property $P$ is testable if there is a randomized algorithm which makes a bounded number of edge queries and distinguishes with probability $2/3$ between $k$-graphs that satisfy $P$ and those that are far from satisfying $P$. For the $2$-graph case, such a combinatorial characterization was obtained by Alon, Fischer, Newman and Shapira. Our results for the $k$-graph setting are in contrast to those of Austin and Tao, who showed that for the somewhat stronger concept of local repairability, the testability results for graphs do not extend to the $3$-graph setting. Our proof relies on a random subhypergraph sampling result proved in a companion paper.
翻译:我们提供了所有可测试特性的组合性特征的组合性特征,即美元单体高光谱(k$-countial graphes) 。 这里,如果存在一种随机化算法,使边缘查询数量固定起来,并有可能将2/3美元区分在满足美元和远远不能满足美元。对于这个2美元谱案,Aron、Fischer、Newman和Shapira都获得了这种组合性特征。 我们的美元组合性特征与Oustin和Tao的对比,Oustin和Tao的结果表明,对于比较强的当地可修复性概念来说,图表的可测试性结果不及于3美元的绘图设置。我们的证据依赖于随机的子精谱抽样结果在一份配套文件中得到了证明。