We study the expressive power of first-order logic with counting quantifiers, especially the $k$-variable and quantifier-rank-$q$ fragment $\mathsf{C}^k_q$, using homomorphism indistinguishability. Recently, Dawar, Jakl, and Reggio (2021) proved that two graphs satisfy the same $\mathsf{C}^k_q$-sentences if and only if they are homomorphism indistinguishable over the class $\mathcal{T}^k_q$ of graphs admitting a $k$-pebble forest cover of depth $q$. Their proof builds on the categorical framework of game comonads developed by Abramsky, Dawar, and Wang (2017). We reprove their result using elementary techniques inspired by Dvo\v{r}\'ak (2010). Using these techniques we also give a characterisation of guarded counting logic. Our main focus, however, is to provide a graph theoretic analysis of the graph class $\mathcal{T}^k_q$. This allows us to separate $\mathcal{T}^k_q$ from the intersection of the graph class $\mathcal{TW}_{k-1}$, that is graphs of treewidth less or equal $k-1$, and $\mathcal{TD}_q$, that is graphs of treedepth at most $q$ if $q$ is sufficiently larger than $k$. We are able to lift this separation to the semantic separation of the respective homomorphism indistinguishability relations. A part of this separation is to prove that the class $\mathcal{TD}_q$ is homomorphism distinguishing closed, which was already conjectured by Roberson (2022).

翻译：我们研究带有计数量词的一阶逻辑的表达能力，特别是通过同态不可区分性来探讨 $k$-变量且量词秩为 $q$ 的片段 $\mathsf{C}^k_q$。近期，Dawar、Jakl 和 Reggio（2021年）证明，两个图满足相同的 $\mathsf{C}^k_q$-句子当且仅当它们在具有 $q$ 深度 $k$-卵石森林覆盖的图类 $\mathcal{T}^k_q$ 上同态不可区分。他们的证明基于 Abramsky、Dawar 和 Wang（2017年）发展的博弈余单子范畴框架。我们受 Dvo\v{r}\'ak（2010年）启发，使用初等方法重新证明该结果。利用这些方法，我们还给出了守卫计数逻辑的特征刻画。然而，我们的主要关注点是对图类 $\mathcal{T}^k_q$ 进行图论分析。这使我们能够将 $\mathcal{T}^k_q$ 与图类 $\mathcal{TW}_{k-1}$（即树宽小于等于 $k-1$ 的图）和 $\mathcal{TD}_q$（即树深至多为 $q$ 的图）的交集分离开，前提是 $q$ 充分大于 $k$。我们能够将此分离提升至相应同态不可区分性关系的语义分离。该分离的一部分是证明图类 $\mathcal{TD}_q$ 是同态区分封闭的，这已由 Roberson（2022年）猜测。