Given two $n$-element structures, $\mathcal{A}$ and $\mathcal{B}$, which can be distinguished by a sentence of $k$-variable first-order logic ($\mathcal{L}^k$), what is the minimum $f(n)$ such that there is guaranteed to be a sentence $\phi \in \mathcal{L}^k$ with at most $f(n)$ quantifiers, such that $\mathcal{A} \models \phi$ but $\mathcal{B} \not \models \phi$? We will present various results related to this question obtained by using the recently introduced QVT games. In particular, we show that when we limit the number of variables, there can be an exponential gap between the quantifier depth and the quantifier number needed to separate two structures. Through the lens of this question, we will highlight some difficulties that arise in analysing the QVT game and some techniques which can help to overcome them. We also show, in the setting of the existential-positive fragment, how to lift quantifier depth lower bounds to quantifier number lower bounds. This leads to almost tight bounds.

翻译：给定两个$n$元结构$\mathcal{A}$和$\mathcal{B}$，它们可通过$k$变量一阶逻辑（$\mathcal{L}^k$）的句子区分，那么保证存在一个至多包含$f(n)$个量词的句子$\phi \in \mathcal{L}^k$使得$\mathcal{A} \models \phi$而$\mathcal{B} \not \models \phi$的最小$f(n)$是多少？我们将展示利用最近引入的QVT博弈获得的与此问题相关的若干结果。特别地，我们证明：当限制变量个数时，区分两个结构所需的量词深度与量词数量之间可能存在指数级差距。通过该问题的视角，我们将揭示分析QVT博弈时出现的一些困难以及克服这些困难的若干技巧。此外，在存在-正片段设定下，我们展示了如何将量词深度下界提升为量词数量下界。这导致了几乎紧致的界限。