Counting Homomorphisms from Hypergraphs of Bounded Generalised Hypertree Width: A Logical Characterisation

We introduce the 2-sorted counting logic GCk 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 GCk 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 Ck+1 if, and only if, they are homomorphism indistinguishable on the class of graphs of tree width at most k.