We introduce the 2-sorted counting logic $GC^k$ that expresses properties of hypergraphs. This logic has available k "blue" variables to address edges, an unbounded number of "red" variables to address vertices of a hypergraph, and atomic formulas E(e,v) to express that vertex v is contained in edge 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：本文引入了用于表达超图性质的2-排序计数逻辑$GC^{k}$。该逻辑包含k个"蓝色"变量用于指代边，以及无界数量的"红色"变量用于指代超图的顶点，并通过原子公式E(e,v)表达顶点v包含于边e中。我们证明：两个超图H、H'对于逻辑$GC^{k}$的相同语句成立，当且仅当它们在广义超树宽不超过k的超图类上满足同态不可区分性。这里，H、H'被称为在类C上同态不可区分，若对C中任意超图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Dvořák（2010）定理（任意两个（无向、简单、有限）图H、H'可由（k+1）变量计数逻辑$C^{k+1}$区分当且仅当它们在树宽不超过k的图类上满足同态不可区分性）从图到超图的推广。