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oai:drops-oai.dagstuhl.de:18613 2024-03-06T11:02:16Z ddc:004 open_access

Counting Homomorphisms from Hypergraphs of Bounded Generalised Hypertree Width: A Logical Characterisation Scheidt, Benjamin Schweikardt, Nicole counting logics guarded logics homomorphism counting hypertree decompositions hypergraphs incidence graphs quantum graphs 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Benjamin Scheidt and Nicole Schweikardt 2023 Is Part Of LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.MFCS.2023.79 urn:nbn:de:0030-drops-186131 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.79 eng https://creativecommons.org/licenses/by/4.0/legalcode