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DOI: 10.4230/LIPIcs.MFCS.2023.79
Counting Homomorphisms from Hypergraphs of Bounded Generalised Hypertree Width: A Logical Characterisation

Authors: Benjamin Scheidt and Nicole Schweikardt

Published in: LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Abstract
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.
Cite as

Benjamin Scheidt and Nicole Schweikardt. Counting Homomorphisms from Hypergraphs of Bounded Generalised Hypertree Width: A Logical Characterisation. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 79:1-79:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{scheidt_et_al:LIPIcs.MFCS.2023.79,
  author =	{Scheidt, Benjamin and Schweikardt, Nicole},
  title =	{{Counting Homomorphisms from Hypergraphs of Bounded Generalised Hypertree Width: A Logical Characterisation}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{79:1--79:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-292-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{272},
  editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.79},
  URN =		{urn:nbn:de:0030-drops-186131},
  doi =		{10.4230/LIPIcs.MFCS.2023.79},
  annote =	{Keywords: counting logics, guarded logics, homomorphism counting, hypertree decompositions, hypergraphs, incidence graphs, quantum graphs}
}
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