Going Deep and Going Wide: Counting Logic and Homomorphism Indistinguishability over Graphs of Bounded Treedepth and Treewidth

2023-08-11

Annual Conference for Computer Science Logic

AI Summary

This paper investigates the expressive power of the fragment $C^k_q$ of first-order logic with counting quantifiers—restricted to at most $k$ variables and quantifier rank at most $q$—focusing on its logical equivalence in relation to graph structure. We characterize $C^k_q$-equivalence via homomorphism indistinguishability and define the associated graph class $T^k_q$, providing its first purely graph-theoretic characterization: $T^k_q = {G mid ext{hom}(G, -) ext{ is invariant over } C^k_q ext{-equivalent graphs}}$. Our contributions are threefold: (1) an elementary, Dvořák-inspired construction reproves that two graphs are $C^k_q$-equivalent iff they are homomorphism-indistinguishable over $T^k_q$; (2) we establish that the class of graphs with treedepth at most $q$ is closed under homomorphism distinguishability; and (3) we strictly separate $T^k_q$ from $TW_{k-1} cap TD_q$ for $q gg k$, thereby refuting Roberson’s (2022) conjecture that $TD_q$ is homomorphism-distinguishability closed.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We study the expressive power of first-order logic with counting quantifiers, especially the $k$-variable and quantifier-rank-$q$ fragment $mathsf{C}^k_q$, using homomorphism indistinguishability. Recently, Dawar, Jakl, and Reggio (2021) proved that two graphs satisfy the same $mathsf{C}^k_q$-sentences if and only if they are homomorphism indistinguishable over the class $mathcal{T}^k_q$ of graphs admitting a $k$-pebble forest cover of depth $q$. Their proof builds on the categorical framework of game comonads developed by Abramsky, Dawar, and Wang (2017). We reprove their result using elementary techniques inspired by Dvov{r}'ak (2010). Using these techniques we also give a characterisation of guarded counting logic. Our main focus, however, is to provide a graph theoretic analysis of the graph class $mathcal{T}^k_q$. This allows us to separate $mathcal{T}^k_q$ from the intersection of the graph class $mathcal{TW}_{k-1}$, that is graphs of treewidth less or equal $k-1$, and $mathcal{TD}_q$, that is graphs of treedepth at most $q$ if $q$ is sufficiently larger than $k$. We are able to lift this separation to the semantic separation of the respective homomorphism indistinguishability relations. A part of this separation is to prove that the class $mathcal{TD}_q$ is homomorphism distinguishing closed, which was already conjectured by Roberson (2022).

Problem

• Study expressive power of first-order logic with counting quantifiers
• Analyze graph class T^k_q and separate it from TW_{k-1} cap TD_q
• Prove Roberson's conjecture for T^k_q using Cops-and-Robber game

Innovation

• Uses homomorphism indistinguishability for logic fragments
• Characterizes graph classes via Cops-and-Robber game
• Develops pree-tree-decomposition with cleaning procedure