An AI Monkey Gets Grapes for Sure -- Sphere Neural Networks for Reliable Decision-Making

TL;DR

The paper addresses the reliability gap in neural reasoning for syllogistic decision-making by comparing LLM-based, supervised, and explicit model construction approaches. It presents Sphere Neural Networks that encode concepts as circles on an $n$-dimensional sphere and use Euler diagrams to perform satisfiability-based reasoning, including complement circles. The main contributions include mastering 16 syllogistic-styled tasks with SphNN across diverse sphere dimensions, showing robustness and symbolic-level rigor, and contrasting this with Euler Net which achieves high accuracy on specific tasks but suffers catastrophic forgetting and pattern sensitivity. The work argues that explicit, interpretable model construction provides more reliable neural reasoning for high-stakes domains and outlines paths for future integration with LLMs and supervised models to achieve safe, interpretable AI.

Abstract

This paper compares three methodological categories of neural reasoning: LLM reasoning, supervised learning-based reasoning, and explicit model-based reasoning. LLMs remain unreliable and struggle with simple decision-making that animals can master without extensive corpora training. Through disjunctive syllogistic reasoning testing, we show that reasoning via supervised learning is less appealing than reasoning via explicit model construction. Concretely, we show that an Euler Net trained to achieve 100.00% in classic syllogistic reasoning can be trained to reach 100.00% accuracy in disjunctive syllogistic reasoning. However, the retrained Euler Net suffers severely from catastrophic forgetting (its performance drops to 6.25% on already-learned classic syllogistic reasoning), and its reasoning competence is limited to the pattern level. We propose a new version of Sphere Neural Networks that embeds concepts as circles on the surface of an n-dimensional sphere. These Sphere Neural Networks enable the representation of the negation operator via complement circles and achieve reliable decision-making by filtering out illogical statements that form unsatisfiable circular configurations. We demonstrate that the Sphere Neural Network can master 16 syllogistic reasoning tasks, including rigorous disjunctive syllogistic reasoning, while preserving the rigour of classical syllogistic reasoning. We conclude that neural reasoning with explicit model construction is the most reliable among the three methodological categories of neural reasoning.

Figures (8)

• Figure 1: (a) The architecture of Euler Net. The inputs are two images, representing two premises of Aristotelian syllogistic reasoning. The green circle inside the blue circle represents "all green are blue"; the red circle inside the green circle represents "all red are green". Its output represents "all red are blue". (b) The transformer component of LLMs learns to predict the missing word. (c) The architecture of Sphere Neural Networks (SphNN) for syllogistic reasoning. SphNN transforms syllogistic statements into spatial statements between spheres, constructs a unified configuration, and draws conclusions by inspecting the constructed configuration.
• Figure 2: There are four jars, A, B, C, and D, in front of monkeys. (a) A blackboard covers jars A and B. A grape is dropped into one of jars A or B behind the blackboard; (b) Another blackboard covers jars C and D. Another grape is dropped into one of jars C or D behind the blackboard; (c) After removing the blackboards, monkeys will see four jars, each having $50\%$ probability to contain a grape. If jar A is empty, clever monkeys will lift jar B. (d) The decision process of the clever monkeys can be abstracted as disjunctive syllogistic reasoning; (e) a diagrammatic representation for disjunctive syllogistic reasoning; (f) sphere neural networks and supervised neural networks may simulate disjunctive syllogistic reasoning.
• Figure 3: Basic diagrammatic representation for syllogistic statements. (1) that all X are Y is represented by (a) $X\subset Y$; (2) that some X are Y is represented by (a) $X\subset Y$ or (b) $Y\subset X$ or (c) $X\cap Y\neq\emptyset$; (3) that no X are Y is represented by (d) $X\cap Y=\emptyset$; (4) that some X are not Y is represented by (b) $Y\subset X$ or (c) $X\cap Y\neq\emptyset$ or (d) $X\cap Y=\emptyset$.
• Figure 4: (a) An Euler diagram on the surface of a 3-d sphere. (b) The surface of a 2-d sphere is a circle. A sphere on this surface is a curve, e.g., $F(x)$, $\neg F(x)$; (c) $\forall x\cdot F(x)\rightarrow G(x)\lor H(x)$: the $F(x)$ arc is part of $G(x)$ arc or $H(x)$ arc; (d) $\forall x\cdot F(x)\rightarrow \neg G(x)$: $F(x)$ arc is part of $\neg G(x)$ arc; (e) $\forall x\cdot F(x)\rightarrow H(x)$: $F(x)$ arc is part of $H(x)$ arc.
• Figure 5: (a) The neighbourhood transition map between circles on the surface of a sphere. $\mathbf{PO}$ stands for "partial overlap", $\mathbf{EQ}$ stands for "equal with", $\overline{\mathbf{P}}$ stands for "inverse part of" or "contain"; (b) the circle configuration of disjunctive syllogistic reasoning: (Premise 1) a grape $\bigcirc_g$ is either in jar A or in jar B, $\mathbf{P}(\bigcirc_g,\bigcirc_A)\lor\mathbf{P}(\bigcirc_g,\bigcirc_B)$; (Premise 2) the grape is not in jar A, $\mathbf{P}(\bigcirc_g, \overline{\bigcirc_A})$; (Conclusion) the grape is in jar B, $\mathbf{P}(\bigcirc_g, \bigcirc_B)$.