  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.
