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

Authors Benjamin Scheidt , Nicole Schweikardt

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ACM Subject Classification
  • Theory of computation → Finite Model Theory
  • Mathematics of computing → Hypergraphs
Keywords
  • counting logics
  • guarded logics
  • homomorphism counting
  • hypertree decompositions
  • hypergraphs
  • incidence graphs
  • quantum graphs

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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.

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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) https://doi.org/10.4230/LIPIcs.MFCS.2023.79

Author Details

Benjamin Scheidt
  • Humboldt-Universität zu Berlin, Germany
Nicole Schweikardt
  • Humboldt-Universität zu Berlin, Germany

Funding

  • Schweikardt, Nicole: Partially supported by the ANR project EQUUS ANR-19-CE48- 0019; funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – project number 431183758 (gefördert durch die Deutsche Forschungsgemeinschaft (DFG) – Projektnummer 431183758).

Acknowledgements

We thank Isolde Adler for pointing us to the results in [Isolde Adler, 2006; Isolde Adler, 2004; Adler et al., 2007], which led to Theorem 2.3.

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