On Homomorphism Indistinguishability and Hypertree Depth
Benjamin Scheidt
TL;DR
We address the problem of extending Grohe's characterization from graphs to hypergraphs. By developing $k$-labeled incidence graphs and an inductive framework for strict hypertree depth, the paper proves that GC$^k$ sentences with guard depth at most $k$ distinguish hypergraphs exactly when they are homomorphism indistinguishable over $\mathsf{SHD}_k$, and that $\operatorname{shd}(\\mathcal{H}) \le \operatorname{hd}(\mathcal{H}) + 1$ (in particular $\operatorname{hd}(\mathcal{H}) \le \operatorname{shd}(\mathcal{H}) \le \operatorname{hd}(\mathcal{H}) + 1$). The work clarifies the relationship between hypertree depth variants and incidence-graph formalisms, and provides a principled generalisation of tree depth to hypergraphs. This advances a unified theory connecting hypergraph logics, homomorphism counting, and depth-based structural decompositions, with potential implications for hypergraph algorithms and logic-based abstraction in databases and AI.
Abstract
$GC^k$ 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, Gavenčiak and Klimošová (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.