Approximation Fixpoint Theory as a Unifying Framework for Fuzzy Logic Programming Semantics (Extended Version)

2025-07-16

AI Summary

This work addresses the semantic fragmentation and lack of a unified theoretical foundation in fuzzy logic programming. Methodologically, it systematically extends Approximation Fixpoint Theory (AFT) to multi-valued—particularly fuzzy—logic programming frameworks, reconstructing both stable model and well-founded semantics within AFT to achieve their unified characterization. It introduces the notion of fuzzy stratification and refined semantic variants, thereby exposing intrinsic relationships among existing semantics. The approach integrates multi-valued logical modeling, semantic reconstruction, and formal fixed-point analysis. The contributions include: (i) a general, scalable semantic framework for uncertain reasoning grounded in AFT; (ii) systematic, precise, and theoretically integrated semantics for fuzzy logic programs; and (iii) the first rigorous migration of AFT from classical two-valued to fuzzy logic, bridging a critical theoretical gap in the field.

Technology Category

• Knowledge Representation and Reasoning: Logic Programming
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Fuzzy logic programming is an established approach for reasoning under uncertainty. Several semantics from classical, two-valued logic programming have been generalized to the case of fuzzy logic programs. In this paper, we show that two of the most prominent classical semantics, namely the stable model and the well-founded semantics, can be reconstructed within the general framework of approximation fixpoint theory (AFT). This not only widens the scope of AFT from two- to many-valued logics, but allows a wide range of existing AFT results to be applied to fuzzy logic programming. As first examples of such applications, we clarify the formal relationship between existing semantics, generalize the notion of stratification from classical to fuzzy logic programs, and devise "more precise" variants of the semantics.

Problem

• Extending approximation fixpoint theory to fuzzy logic programming
• Reconstructing stable model and well-founded semantics in fuzzy logic
• Generalizing stratification and refining semantics for fuzzy programs

Innovation

• Extends AFT to fuzzy logic programming
• Reconstructs stable and well-founded semantics
• Generalizes stratification to fuzzy programs