eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-08-21
79:1
79:15
10.4230/LIPIcs.MFCS.2023.79
article
Counting Homomorphisms from Hypergraphs of Bounded Generalised Hypertree Width: A Logical Characterisation
Scheidt, Benjamin
1
https://orcid.org/0000-0003-2379-3675
Schweikardt, Nicole
1
https://orcid.org/0000-0001-5705-1675
Humboldt-Universität zu Berlin, Germany
We introduce the 2-sorted counting logic GC^k and its restriction RGC^k that express properties of hypergraphs. These logics have 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 a vertex v is contained in a hyperedge e. We show that two hypergraphs H,H' satisfy the same sentences of the logic RGC^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 lifting (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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol272-mfcs2023/LIPIcs.MFCS.2023.79/LIPIcs.MFCS.2023.79.pdf
counting logics
guarded logics
homomorphism counting
hypertree decompositions
hypergraphs
incidence graphs
quantum graphs