Logic of Hypotheses: from Zero to Full Knowledge in Neurosymbolic Integration

2025-09-25

AI Summary

This work addresses the fragmentation in neural-symbolic integration (NeSy) by unifying two dominant paradigms: expert-knowledge-driven rule injection and data-driven symbolic rule induction. We propose the Logic of Hypotheses (LoH), a novel formal language that extends propositional logic with learnable selection operators, and integrates Gödel fuzzy logic with Gödel’s encoding technique to enable end-to-end differentiable coupling of neural networks and symbolic reasoning. LoH supports progressive modeling—from zero prior knowledge to full symbolic grounding—while preserving interpretability, enabling arbitrary-degree knowledge injection, and admitting lossless discretization into Boolean functions. Via differentiable compilation, LoH formulas are translated into computational graphs amenable to backpropagation. Experiments on tabular datasets and visual tic-tac-toe demonstrate significant performance gains and yield high-fidelity, human-readable rules, validating LoH’s effectiveness, generalizability, and interpretability.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Knowledge Representation and Reasoning: Logic Programming
• Computer Vision: Visual Reasoning & Symbolic Representations

Application Category

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

Abstract

Neurosymbolic integration (NeSy) blends neural-network learning with symbolic reasoning. The field can be split between methods injecting hand-crafted rules into neural models, and methods inducing symbolic rules from data. We introduce Logic of Hypotheses (LoH), a novel language that unifies these strands, enabling the flexible integration of data-driven rule learning with symbolic priors and expert knowledge. LoH extends propositional logic syntax with a choice operator, which has learnable parameters and selects a subformula from a pool of options. Using fuzzy logic, formulas in LoH can be directly compiled into a differentiable computational graph, so the optimal choices can be learned via backpropagation. This framework subsumes some existing NeSy models, while adding the possibility of arbitrary degrees of knowledge specification. Moreover, the use of Goedel fuzzy logic and the recently developed Goedel trick yields models that can be discretized to hard Boolean-valued functions without any loss in performance. We provide experimental analysis on such models, showing strong results on tabular data and on the Visual Tic-Tac-Toe NeSy task, while producing interpretable decision rules.

Problem

• Unifying rule injection and induction in neurosymbolic integration
• Enabling flexible integration of learned rules with symbolic priors
• Allowing arbitrary knowledge specification degrees in neural models

Innovation

• LoH unifies rule injection and induction via choice operator
• LoH formulas compile into differentiable graphs via fuzzy logic
• Models discretize without loss using Goedel fuzzy logic