NSFL: A Post-Training Neuro-Symbolic Fuzzy Logic Framework for Boolean Operators in Neural Embeddings

Vladi Vexler; Ofer Idan; Gil Lederman; Dima Sivov

cs.IRApr 12, 2026

NSFL: A Post-Training Neuro-Symbolic Fuzzy Logic Framework for Boolean Operators in Neural Embeddings

Authors: Vladi Vexler, Ofer Idan, Gil Lederman, Dima Sivov

Organizations: Huawei Tel-Aviv Research Center

Abstract

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.

Figures & tables

Panel of Figure 1 — Figure 1: Comparison of embedding manifold operations: (a) Euclidean operations cause Manifold Escape and signal loss; (b) NSFL Inhibitory steering preserves context $A$ while excluding $B$ ; (c) NSFL Conjunctive steering resolves the Semantic Bottleneck by pulling the query toward the weaker atom $B$ .

Symbol	Description
$\mathcal{D}=\{D_{1},\dots,D_{m}\}$	A corpus of data items embedded as vectors $\mathbf{v}_{D_{i}}\in\mathbb{R}^{d}$ .
$L_{A},L_{B},L_{C}$	Semantic query atoms (e.g., text labels or raw concepts).
$\cup$	The Fusion Operator (FO): $L_{AB}=L_{A}\cup L_{B}$ . Merges atoms into a joint label.
$L_{M}$	Monolithic Query: The full boolean expression as a single natural language string.
$\mathbf{v}_{X}$	Embedding vector of query component $X\in\{A,B,AB,M\}$ .
$S_{X}^{i}$	Similarity score between data item $i$ and query $X$ , normalized to $[0,1]$ .

Table 1: Summary of Notation and Definitions.

Panel of Table 2 — Table 2: Empirical $\Delta$ -Extraction and Monolithic Negation Failure (BLIP-Base).

Figure 2: Comparison of zero-order similarity scores and first-order Neuro-Symbolic Deltas across retrieval candidates.

Operator	Fuzzy Logic ( $S^{\text{sm}}$ )	Stability Trigger	Boundary Value ( $S^{\text{stable}}$ )
AND ( $A\land B$ )	$2S_{AB}^{i}-\max(S_{A}^{i},S_{B}^{i})$	$S_{AB}^{i}>S_{A}^{i}+S_{B}^{i}$	$S_{AB}^{i}$ (Reinforcement)
NOT ( $A\land\neg B$ )	$S_{A}^{i}-\tilde{S}_{B}^{i}(S_{AB}^{i}-S_{A}^{i})$	$S_{AB}^{i}<S_{A}^{i},S_{B}^{i}$	$S_{AB}^{i}$ (Departure)
OR ( $A\lor B$ )	$\max(S_{A}^{i},S_{B}^{i},S_{AB}^{i},S_{M}^{i})$	$S_{AB}^{i}<S_{A}^{i},S_{B}^{i}$	$\min(S_{AB}^{i},S_{M}^{i})$ (Departure)

Table 3: Comprehensive Summary of NSFL Operators: Fuzzy Logic Formulations and Boundary Stability.

		Configuration		Avg. mAP@100
Method	Variant	Pre-Opt.	Post-Rerank	QUEST	COCO-L
Baseline	Monolithic	—	—	0.065	0.328
NSFL	Rerank only	—	✓	0.079	0.438
	Opt. only	✓	—	0.069	0.378
	Hybrid	✓	✓	0.079	0.438
Geometric	Rerank only	—	✓	0.068	0.418

Table 5: Systematic ablation of modular tiers: mAP@100 averaged over the six logical query types, comparing Pre-ANNS Optimization and Post-ANNS Reranking on BGE/QUEST (text) and BLIP-Large/COCO-Logic (cross-modal). Best non-baseline value per dataset in bold . See Table 8 in Appendix 13 for the per-logic-type breakdown.

Parameter	Implementation Detail	Value
$\alpha$	Learning Rate (Step Size)	$0.2$
$Steps$	Maximum Iterations	$100$
$Patience$	Early Stopping (No improvement)	$10$
$tol$	Convergence Tolerance	$10^{-6}$
init	Initialization Strategy	Gaussian ( $\mathcal{N}(0,1)$ )
$Manifold$	Geometry	$\mathcal{S}^{d-1}$ (Unit Sphere)

Table 6: Hyperparameter Configuration for SQO Implementation

Metric	Method	$A\land B$	$A\land B\land C$	$A\land\neg B$	$A\land B\land\neg C$	$A\lor B$	$A\lor B\lor C$
R@20	Baseline	0.178	0.198	0.172	0.089	0.298	0.225
R@20	NSFL	0.185	0.206	0.200	0.122	0.338	0.283
R@100	Baseline	0.338	0.406	0.397	0.237	0.504	0.423
R@100	NSFL	0.350	0.415	0.423	0.284	0.550	0.491

Table 7: Recall@K Comparison on QUEST using LogiCoL-e5-v2. Baseline denotes monolithic retrieval; NSFL denotes reranking with our proposed operators.

BGE / QUEST
Encoder	Method	$A\land B$	$A\land B\land C$	$A\land\neg B$	$A\land B\land\neg C$	$A\lor B$	$A\lor B\lor C$	Avg.
BGE	Baseline	0.038	0.042	0.050	0.015	0.135	0.107	0.065
	NSFL — Rerank only	0.053	0.044	0.075	0.025	0.144	0.135	0.079
	NSFL — Opt only	0.040	0.039	0.073	0.021	0.135	0.107	0.069
	NSFL — Hybrid	0.052	0.044	0.076	0.024	0.147	0.129	0.079
	GEO — Rerank only	0.021	0.023	0.067	0.018	0.144	0.135	0.068

Table 8: Per-logic-type ablation of NSFL and Geometric variants on BGE/QUEST and BLIP-Large/COCO-Logic. Bold marks the best non-baseline value per column within each encoder block.

Template	Baseline	NSFL	$\Delta$	95% CI	p-value
$A\land B$	0.038	0.053	+0.014	[+0.004, +0.025]	0.044
$A\land B\land C$	0.042	0.044	+0.002	[-0.005, +0.009]	0.018
$A\land\neg B$	0.050	0.075	+0.025	[+0.017, +0.035]	$<10^{-11}$
$A\land B\land\neg C$	0.015	0.025	+0.009	[+0.005, +0.014]	$<10^{-4}$
$A\lor B$	0.135	0.144	+0.009	[-0.001, +0.019]	0.128
$A\lor B\lor C$	0.107	0.135	+0.027	[+0.019, +0.036]	$<10^{-10}$

Table 9: Statistical significance analysis for BGE-Large-v1.5 on QUEST (per-query Wilcoxon signed-rank test).

Template	Baseline	NSFL	$\Delta$	95% CI	p-value
$A\land B$	0.140	0.177	+0.038	[+0.018, +0.061]	$<10^{-4}$
$A\land B\land C$	0.107	0.127	+0.020	[+0.010, +0.032]	$<10^{-5}$
$A\land\neg B$	0.244	0.386	+0.142	[+0.117, +0.168]	$<10^{-15}$
$A\land B\land\neg C$	0.149	0.256	+0.106	[+0.074, +0.143]	$<10^{-10}$
$A\lor B$	0.240	0.285	+0.045	[+0.035, +0.055]	$<10^{-16}$
$A\lor B\lor C$	0.155	0.206	+0.051	[+0.043, +0.058]	$<10^{-17}$

Table 10: Statistical significance analysis for BLIP-Large on COCO-Logic (per-query Wilcoxon signed-rank test).

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Abstract

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