The number of quantifiers needed to express first-order properties is captured by two-player combinatorial games called multi-structural (MS) games. We play these games on linear orders and strings, and introduce a technique we call "parallel play", that dramatically reduces the number of quantifiers needed in many cases. Linear orders and strings are the most basic representatives of ordered structures -- a class of structures that has historically been notoriously difficult to analyze. Yet, in this paper, we provide upper bounds on the number of quantifiers needed to characterize different-sized subsets of these structures, and prove that they are tight up to constant factors, including, in some cases, up to a factor of $1+\varepsilon$, for arbitrarily small $\varepsilon$.

翻译：描述一阶性质所需的量词数量可以通过称为多结构（MS）游戏的双人组合博弈来刻画。我们在线性序和字符串上开展这些游戏，并引入一种称为"平行游戏"的技术，该技术能在许多情况下显著减少所需的量词数量。线性序和字符串是有序结构中最基本的代表——这类结构历来被公认为极具分析难度。然而，本文给出了刻画这些结构不同规模子集所需量词数量的上界，并证明这些上界在常数因子范围内是紧的，在某些情况下甚至对任意小的ε可达1+ε因子紧界。