Sphere Neural-Networks for Rational Reasoning
Tiansi Dong, Mateja Jamnik, Pietro Liò
TL;DR
Sphere Neural Networks (SphNN) introduce a deterministic, neuro-symbolic framework that represents concepts as spheres and set-theoretic relations via boundary interactions, enabling one-epoch satisfiability checks for long-chained syllogisms with $O(N)$ complexity. By embedding syllogistic statements as spatial relations and using a Kolmogorov-Arnold-style neural transition map within a three-layer GPS-like architecture, SphNN performs model-construction and inspection to verify validity, bridging vector semantics with explicit set-theoretic boundaries. The approach demonstrates deterministic neural syllogistic reasoning, supports neuro-symbolic unification with latent vector embeddings, and extends toward spatio-temporal, probabilistic, and even humour reasoning, offering a potential neural implementation of Herbert A. Simon’s neural scissors. Empirical results show SphNN can construct counter-models for invalid deductions and, via interaction with LLMs like ChatGPT, can improve reasoning reliability, albeit with higher computational cost. Overall, SphNN provides a principled path to deterministic, explainable neural reasoning that complements and constrains large language models, grounding high-level cognition in geometric, boundary-based representations with scalable theoretical guarantees.
Abstract
The success of Large Language Models (LLMs), e.g., ChatGPT, is witnessed by their planetary popularity, their capability of human-like communication, and also by their steadily improved reasoning performance. However, it remains unclear whether LLMs reason. It is an open problem how traditional neural networks can be qualitatively extended to go beyond the statistic paradigm and achieve high-level cognition. Here, we present a novel qualitative extension by generalising computational building blocks from vectors to spheres. We propose Sphere Neural Networks (SphNNs) for human-like reasoning through model construction and inspection, and develop SphNN for syllogistic reasoning, a microcosm of human rationality. SphNN is a hierarchical neuro-symbolic Kolmogorov-Arnold geometric GNN, and uses a neuro-symbolic transition map of neighbourhood spatial relations to transform the current sphere configuration towards the target. SphNN is the first neural model that can determine the validity of long-chained syllogistic reasoning in one epoch without training data, with the worst computational complexity of O(N). SphNN can evolve into various types of reasoning, such as spatio-temporal reasoning, logical reasoning with negation and disjunction, event reasoning, neuro-symbolic unification, and humour understanding (the highest level of cognition). All these suggest a new kind of Herbert A. Simon's scissors with two neural blades. SphNNs will tremendously enhance interdisciplinary collaborations to develop the two neural blades and realise deterministic neural reasoning and human-bounded rationality and elevate LLMs to reliable psychological AI. This work suggests that the non-zero radii of spheres are the missing components that prevent traditional deep-learning systems from reaching the realm of rational reasoning and cause LLMs to be trapped in the swamp of hallucination.