From membership to relations: A neuro-fuzzy framework for learning relational predicates

Traditional neuro-fuzzy systems represent rule antecedents through membership-based conditions, restricting fuzzy rules primarily to equality-like expressions (e.g., temperature is low ). This limits their ability to express richer linguistic relations such as at least low , at most low , or not low . To address this limitation, we introduce the General Intuitive Fuzzy Term (GIFT), a unified adaptive fuzzy representation that models equality, inequality, and order-based relations within a single differentiable formulation. Building upon GIFT, we propose the Relational Linguistic Neuro-Fuzzy Inference System (ReLaFIS), which learns relational operators over linguistic terms and constructs fuzzy rules with relational antecedents. This extension enhances the expressive power of fuzzy rules while preserving their linguistic interpretability. Moreover, ReLaFIS introduces an indirect antecedent relaxation mechanism that allows competing relational operators to be assigned comparable importance. This mechanism enables the emergence of incomplete fuzzy rules and implicit feature suppression, resulting in more compact rule bases without requiring explicit feature selection. Unlike conventional Takagi–Sugeno–Kang (TSK) models, in which consequent functions are parameterized independently of antecedent geometry, the proposed framework constructs local consequents in a rule-centered coordinate system normalized by the widths of the corresponding antecedent linguistic terms. This enables geometry-aware function approximation within individual local fuzzy regions. In addition, a reconstruction-based self-supervised learning strategy is introduced to guide the learning of antecedent parameters under the proposed relational semantics, improving the stability of model training. To quantify the diversity of relational semantics expressed by fuzzy rules, we further introduce Antecedent Relation Entropy (ARE), an information-theoretic metric that measures the distribution of relational predicates across rule antecedents. Experimental results on a collection of benchmark classification problems demonstrate that ReLaFIS achieves competitive or superior predictive performance while substantially reducing the number of fuzzy rules, highlighting its ability to learn compact and linguistically rich relational neuro-fuzzy models.