$GC^k$ is a logic introduced by Scheidt and Schweikardt (2023) to express properties of hypergraphs. It is similar to first-order logic with counting quantifiers ($C$) adapted to the hypergraph setting. It has distinct sets of variables for vertices and for hyperedges and requires vertex variables to be guarded by hyperedge variables on every quantification. We prove that two hypergraphs $G$, $H$ satisfy the same sentences in the logic $GC^k$ with guard depth at most $k$ if, and only if, they are homomorphism indistinguishable over the class of hypergraphs of strict hypertree depth at most $k$. This lifts the analogous result for tree depth $\leq k$ and sentences of first-order logic with counting quantifiers of quantifier rank at most $k$ due to Grohe (2020) from graphs to hypergraphs. The guard depth of a formula is the quantifier rank with respect to hyperedge variables, and strict hypertree depth is a restriction of hypertree depth as defined by Adler, Gaven\v{c}iak and Klimo\v{s}ov\'a (2012). To justify this restriction, we show that for every $H$, the strict hypertree depth of $H$ is at most 1 larger than its hypertree depth, and we give additional evidence that strict hypertree depth can be viewed as a reasonable generalisation of tree depth for hypergraphs.

翻译：$GC^k$ 是由 Scheidt 和 Schweikardt（2023）引入的一种用于描述超图性质的逻辑。它类似于一阶逻辑中针对超图设置调整的计数量词（$C$）逻辑。该逻辑为顶点和超边分别设置不同的变量集合，并要求在每次量化时顶点变量必须由超边变量进行守护。我们证明：两个超图 $G$、$H$ 在守护深度不超过 $k$ 的 $GC^k$ 逻辑中满足相同句子当且仅当它们在严格超树深度不超过 $k$ 的超图类上满足同态不可区分性。这将对偶于 Grohe（2020）从图到超图的推广结果（图树深度 $\leq k$ 与量词秩不超过 $k$ 的计数量词一阶逻辑句子）。公式的守护深度是超边变量上的量词秩，而严格超树深度是 Adler、Gavenčiak 和 Klimošová（2012）定义的超树深度的一种限制。为了证明这种限制的合理性，我们表明每个超图 $H$ 的严格超树深度最多比其超树深度大 1，并进一步证明严格超树深度可被视为超图树深度的合理推广。