Approximation Fixpoint Theory with Refined Approximation Spaces

2025-06-19

AI Summary

Existing Approximation Fixpoint Theory (AFT) relies on interval approximations over lattices, limiting its ability to capture fine-grained semantic fixed points in nonmonotonic reasoning and rendering it inadequate for several simple yet critical cases. To address this, we introduce the notion of *generalized approximation spaces*, transcending the traditional lattice-based interval structure by supporting hierarchical organization and cross-space mappings. Grounded in lattice theory, fixed-point theory, and abstract interpretation, we develop a more expressive and broadly applicable AFT semantic framework. Our model rigorously characterizes entailment relationships among major nonmonotonic semantics—including Kripke–Kleene, well-founded, and stable semantics—and significantly enhances modeling capability for cases beyond the scope of classical AFT. This advances the meta-theoretical foundations of logic programming and answer-set programming, offering greater expressiveness and conceptual robustness.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• 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
• Search and Retrieval-Augmented AI: Agentic search

Abstract

Approximation Fixpoint Theory (AFT) is a powerful theory covering various semantics of non-monotonic reasoning formalisms in knowledge representation such as Logic Programming and Answer Set Programming. Many semantics of such non-monotonic formalisms can be characterized as suitable fixpoints of a non-monotonic operator on a suitable lattice. Instead of working on the original lattice, AFT operates on intervals in such lattice to approximate or construct the fixpoints of interest. While AFT has been applied successfully across a broad range of non-monotonic reasoning formalisms, it is confronted by its limitations in other, relatively simple, examples. In this paper, we overcome those limitations by extending consistent AFT to deal with approximations that are more refined than intervals. Therefore, we introduce a more general notion of approximation spaces, showcase the improved expressiveness and investigate relations between different approximation spaces.

Problem

• Extend Approximation Fixpoint Theory for refined approximation spaces
• Overcome limitations in non-monotonic reasoning semantics
• Generalize approximation spaces to improve expressiveness

Innovation

• Extends AFT with refined approximation spaces
• Introduces general notion of approximation spaces
• Investigates relations between approximation spaces