The Logical Expressiveness of Topological Neural Networks

2026-04-21

AI Summary

The logical expressiveness of topological neural networks (TNNs) has thus far lacked a precise characterization. This work addresses this gap by introducing a novel existential counting logic, denoted TCₖ, featuring pairwise counting quantifiers over simplicial complexes, and establishing a three-way equivalence among the k-CCWL isomorphism test on combinatorial complexes, the logic TCₖ₊₂, and the topological (k+2)-pebble game. This equivalence provides the first rigorous theoretical framework for the logical expressivity of TNNs, precisely delineating the class of binary classifiers they can represent. Moreover, it reveals deep connections between TNNs, higher-order Weisfeiler–Leman tests, and counting logics, thereby advancing our understanding of the expressive power of topological deep learning architectures.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Statistical Relational/Logic Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Graph neural networks (GNNs) are the standard for learning on graphs, yet they have limited expressive power, often expressed in terms of the Weisfeiler-Leman (WL) hierarchy or within the framework of first-order logic. In this context, topological neural networks (TNNs) have recently emerged as a promising alternative for graph representation learning. By incorporating higher-order relational structures into message-passing schemes, TNNs offer higher representational power than traditional GNNs. However, a fundamental question remains open: what is the logical expressiveness of TNNs? Answering this allows us to characterize precisely which binary classifiers TNNs can represent. In this paper, we address this question by analyzing isomorphism tests derived from the underlying mechanisms of general TNNs. We introduce and investigate the power of higher-order variants of WL-based tests for combinatorial complexes, called $k$-CCWL test. In addition, we introduce the topological counting logic (TC$_k$), an extension of standard counting logic featuring a novel pairwise counting quantifier $ \exists^{N}(x_i,x_j)\, \varphi(x_i,x_j), $ which explicitly quantifies pairs $(x_i, x_j)$ satisfying property $\varphi$. We rigorously prove the exact equivalence: $ \text{k-CCWL} \equiv \text{TC}_{k{+}2} \equiv \text{Topological }(k{+}2)\text{-pebble game}.$ These results establish a logical expressiveness theory for TNNs.

Problem

• topological neural networks
• logical expressiveness
• Weisfeiler-Leman test
• counting logic
• graph representation learning

Innovation

• Topological Neural Networks
• Logical Expressiveness
• Weisfeiler-Leman Test
• Counting Logic
• Combinatorial Complexes