一阶逻辑中的随机图有多复杂？
How Complex are Random Graphs in First Order Logic?
摘要：
不难写出一个一阶公式，它对给定的图G为真，但对任何不同构于G的图为假。在这样的公式中嵌套量词的最小数目$（G）可以作为G的“一阶复杂性”的度量。在这里，这个参数的随机图进行了研究。当边缘概率p为常数时，我们渐近地确定它;事实上，D（G）是log n阶的。对于非常稀疏的图，它的大小是Θ（n）。另一方面，对于某些（精心选择的）p值，参数D（G）可以下降到非常缓慢增长的函数log^* n，即塔函数的逆。然而，总体情况仍然是一个谜。
英文摘要：
It is not hard to write a first order formula which is true for a given graph G but is false for any graph not isomorphic to G. The smallest number $(G) of nested quantifiers in a such formula can serve as a measure for the ``first order complexity'' of G. Here, this parameter is studied for random graphs. We determine it asymptotically when the edge probability p is constant; in fact, D(G) is of order log n then. For very sparse graphs its magnitude is Θ(n). On the other hand, for certain (carefully chosen) values of p the parameter D(G) can drop down to the very slow growing function log^* n, the inverse of the tower-function. The general picture, however, is still a mystery.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
