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

Abstract

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 of a hypergraph, and atomic formulas E(e, v) to express that vertex v is contained in 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 ∈ 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 Dvořák (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 over the class of graphs of tree-width at most k.