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个变量处理超边，无界变量处理顶点，以及原子公式E(e,v)表示顶点v包含于超边e中。我们证明，两个超图H, H'满足相同的$GC^k$逻辑句子当且仅当它们在广义超树宽至多为k的超图类上是同态不可区分的。这里，H, H'在类C上被称为同态不可区分，如果对C中每个超图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）结果的推广（从图到超图），Dvorak结果表明任意两个（无向、简单、有限）图H, H'被(k+1)变量计数逻辑$C^{k+1}$不可区分当且仅当它们在树宽至多为k的图类上是同态不可区分的。