The number of quantifiers needed to express first-order (FO) properties is captured by two-player combinatorial games called multi-structural games. We analyze these games on binary strings with an ordering relation, using a technique we call parallel play, which significantly reduces the number of quantifiers needed in many cases. Ordered structures such as strings have historically been notoriously difficult to analyze in the context of these and similar games. Nevertheless, in this paper, we provide essentially tight upper bounds on the number of quantifiers needed to characterize different-sized subsets of strings. The results immediately give bounds on the number of quantifiers necessary to define several different classes of Boolean functions. One of our results is analogous to Lupanov's upper bounds on circuit size and formula size in propositional logic: we show that every Boolean function on $n$-bit inputs can be defined by a FO sentence having $(1 + \varepsilon)n\log(n) + O(1)$ quantifiers, and that this is essentially tight. We reduce this number to $(1 + \varepsilon)\log(n) + O(1)$ when the Boolean function in question is sparse.

翻译：表达一阶逻辑性质所需的量词数量可通过一种称为多结构博弈的双人组合博弈来刻画。我们在具有序关系的二进制串上分析这些博弈，采用一种称为并行博弈的技术，该技术在许多情况下能显著减少所需量词的数量。在历史上，字符串这类有序结构在此类及类似博弈的背景下一直以难以分析而著称。尽管如此，本文中我们为刻画不同规模字符串子集所需量词的数量提供了本质上紧的上界。这些结果直接给出了定义多个不同布尔函数类别所需量词数量的界。我们的一个结果类似于命题逻辑中电路规模和公式规模的卢帕诺夫上界：我们证明每个定义在$n$比特输入上的布尔函数都可由一个具有$(1 + \varepsilon)n\log(n) + O(1)$个量词的一阶逻辑句子定义，且该界在本质上是紧的。当所讨论的布尔函数是稀疏函数时，我们将此数量减少至$(1 + \varepsilon)\log(n) + O(1)$。