We introduce the 2-sorted counting logic $GC^k$ that expresses properties of hypergraphs. This logic has available k variables to address hyperedges, an unbounded number of variables to address vertices, and atomic formulas E(e,v) to express that a vertex v is contained in a hyperedge e. We show that two hypergraphs H, H' satisfy the same sentences of the logic $GC^k$ if, and only if, they are homomorphism indistinguishable over the class of hypergraphs of generalised hypertree width at most k. Here, H, H' are called homomorphism indistinguishable over a class C if for every hypergraph G in C the number of homomorphisms from G to H equals the number of homomorphisms from G to H'. This result can be viewed as a generalisation (from graphs to hypergraphs) of a result by Dvorak (2010) stating that any two (undirected, simple, finite) graphs H, H' are indistinguishable by the (k+1)-variable counting logic $C^{k+1}$ if, and only if, they are homomorphism indistinguishable on the class of graphs of tree width at most k.

翻译：我们引入了双排序计数逻辑 $GC^k$，用于表达超图的性质。该逻辑拥有 k 个变量来指代超边，以及无界数量的变量来指代顶点，并通过原子公式 E(e,v) 表示顶点 v 包含在超边 e 中。我们证明，两个超图 H 和 H' 满足相同的 $GC^k$ 逻辑语句当且仅当它们在广义超树宽不超过 k 的超图类上是同态不可区分的。这里，H 和 H' 被称为在类 C 上是同态不可区分的，如果对于 C 中的每个超图 G，从 G 到 H 的同态数量等于从 G 到 H' 的同态数量。该结果可视为 Dvorak (2010) 的一个结论（从图到超图）的推广，该结论指出：任意两个（无向、简单、有限）图 H, H' 在树宽不超过 k 的图类上是同态不可区分的，当且仅当它们无法被 (k+1) 变量计数逻辑 $C^{k+1}$ 区分。