[Paper Review] Counting Homomorphisms from Hypergraphs of Bounded Generalised Hypertree Width: A Logical Characterisation

Benjamin Scheidt, Nicole Schweikardt|arXiv (Cornell University)|Jan 1, 2023

Advanced Graph Theory Research19 references2 citations

TL;DR

This paper introduces a 2-sorted counting logic GCk for hypergraphs, which characterizes homomorphism indistinguishability over hypergraphs of bounded generalized hypertree width. It proves that two hypergraphs are indistinguishable by homomorphism counts from all such hypergraphs if and only if they satisfy the same sentences in GCk, extending Dvoürák's result from graphs to hypergraphs.

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, 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 $GC^k$ 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 in 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 Dvorak (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 on the class of graphs of tree width at most k.

Motivation & Objective

• To extend Dvoürák's result on tree-width and counting logic from graphs to hypergraphs.
• To provide a logical characterization of homomorphism indistinguishability over hypergraphs of bounded generalized hypertree width.
• To define and analyze a new 2-sorted counting logic GCk with k variables for hyperedges and unbounded variables for vertices.
• To establish a logical equivalence between homomorphism indistinguishability and elementary equivalence in GCk for hypergraphs.
• To explore the expressive power and model-theoretic properties of GCk in the context of hypergraph decompositions.

Proposed method

• Introduce GCk, a 2-sorted counting logic with k 'blue' variables for hyperedges and unbounded 'red' variables for vertices, including atomic formulas E(e,v) for incidence and counting quantifiers ∃⩾n z over tuples of variables.
• Define incidence graphs I(H) for hypergraphs H and use them as models for GCk sentences.
• Introduce k-labeled incidence graphs and the class GLIk to represent structures with bounded generalized hypertree width.
• Establish a normal form RGCk for GCk sentences to simplify logical analysis and equivalence checking.
• Use entangled hypertree decompositions (ehds) as a key technical tool to relate logical indistinguishability to structural properties.
• Prove equivalence between GCk-definable properties and homomorphism counts over the class GHWk via a sequence of logical and structural lemmas, including a novel construction of witness hypergraphs via partial functions and guards.

Experimental results

Research questions

• RQ1 Can Dvoürák's result on tree-width and Ck+1 logic be lifted from graphs to hypergraphs?
• RQ2 What logical system characterizes homomorphism indistinguishability over hypergraphs of bounded generalized hypertree width?
• RQ3 How does the expressive power of GCk compare to other logical fragments in the context of hypergraph properties?
• RQ4 Is there a hypergraph analog of the k-dimensional Weisfeiler-Leman algorithm that matches the expressive power of GCk?
• RQ5 What is the computational complexity of determining generalized hypertree width or computing homomorphism counts over GHWk?

Key findings

• Two hypergraphs H and H′ are homomorphism indistinguishable over the class of hypergraphs of generalized hypertree width at most k if and only if they satisfy the same sentences in the logic GCk.
• The logic GCk is a 2-sorted counting logic with k variables for hyperedges and unbounded variables for vertices, supporting counting quantifiers over tuples of variables.
• The main result holds for both general and simple hypergraphs, with the logic GCk characterizing homomorphism counts over the class GHWk.
• The proof relies on a novel construction using entangled hypertree decompositions (ehds), which are shown to be equivalent to generalized hypertree decompositions for the purpose of homomorphism indistinguishability.
• A normal form RGCk is established for GCk, enabling the transformation of any GCk sentence into an equivalent form that facilitates model-checking and equivalence reasoning.
• The paper establishes that homomorphism indistinguishability over IEHWk (a subclass of GHWk based on entangled decompositions) coincides with that over GHWk, despite IEHWk being a strict subclass of GHWk for sufficiently large k.