Neural-Symbolic Logic Query Answering in Non-Euclidean Space

Large Language Models (LLMs) have demonstrated remarkable proficiency in natural language understanding, yet they struggle with strict multi-step reasoning, frequently suffering from hallucinations and inconsistency. Existing solutions like Chain-of-Thought (CoT) lack rigorous verification mechanisms, while standard Retrieval-Augmented Generation (RAG) often misses the complex, structural dependencies inherent in logical tasks. To bridge this gap, we propose a Neuro-Symbolic architecture that integrates a Logical Knowledge Graph (LKG) with dynamic solver routing. Specifically, we introduce an ontology-based LKG that treats logical rules and constraints as first-class topological nodes, enabling explicit modeling of dependencies extracted from text. We further design a Logic Router to dynamically dispatch tasks to the optimal symbolic engine, which is supported by a topology-aware hybrid retrieval mechanism. Experimental results on logical reasoning benchmarks demonstrate that our framework significantly outperforms state-of-the-art prompting and RAG baselines, delivering higher accuracy and verifiable reasoning paths.

Standard dense retrievers lack a native calculus for multi-atom logical constraints. We introduce Neuro-Symbolic Fuzzy Logic (NSFL), a framework that adapts formal t-norms and t-conorms to neural embedding spaces without requiring retraining. NSFL operates as a first-order hybrid calculus: it anchors logical operations on isolated zero-order similarity scores while actively steering representations using Neuro-Symbolic Deltas (NS-Delta) -- the first-order marginal differences derived from contextual fusion. This preserves pure atomic meaning while capturing domain reliance, preventing the representation collapse and manifold escape endemic to traditional geometric baselines. For scalable real-time retrieval, Spherical Query Optimization (SQO) leverages Riemannian optimization to project these fuzzy formulas into manifold-stable query vectors. Validated across six distinct encoder configurations and two modalities (including zero-shot and SOTA fine-tuned models), NSFL yields mAP improvements up to +81%. Notably, NSFL provides an additive 20% average and up to 47% boost even when applied to encoders explicitly fine-tuned for logical reasoning. By establishing a training-free, order-aware calculus for high-dimensional spaces, this framework lays the foundation for future dynamic scaling and learned manifold logic.

Large pretrained language models and neural reasoning systems have advanced many natural language tasks, yet they remain challenged by knowledge-intensive queries that require precise, structured multi-hop inference. Knowledge graphs provide a compact symbolic substrate for factual grounding, but integrating graph structure with neural models is nontrivial: naively embedding graph facts into prompts leads to inefficiency and fragility, while purely symbolic or search-heavy approaches can be costly in retrievals and lack gradient-based refinement. We introduce NeuroSymActive, a modular framework that combines a differentiable neural-symbolic reasoning layer with an active, value-guided exploration controller for Knowledge Graph Question Answering. The method couples soft-unification style symbolic modules with a neural path evaluator and a Monte-Carlo style exploration policy that prioritizes high-value path expansions. Empirical results on standard KGQA benchmarks show that NeuroSymActive attains strong answer accuracy while reducing the number of expensive graph lookups and model calls compared to common retrieval-augmented baselines.