SAT-Based Bounded Fitting for the Description Logic ALC

2025-07-29

AI Summary

This paper investigates the Bounded Fitting problem for the description logic ALC and its syntactic fragments: learning a minimal ALC concept expression consistent with a given finite set of positive and negative examples. We establish, for the first time, that this problem remains NP-complete even for a single positive and a single negative example—across ALC and its major fragments. To provide provable probabilistic guarantees, we formulate the problem within Valiant’s PAC learning framework and propose a symbolic approach based on propositional encoding and SAT solving. Key optimizations—including structured encoding, redundancy pruning, and instance preprocessing—significantly improve scalability. We implement the first prototype tool supporting bounded fitting for the full ALC language. Empirical evaluation demonstrates its effectiveness and competitiveness against state-of-the-art approaches.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Description Logics
• Machine Learning: Statistical Relational/Logic Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• 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
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Bounded fitting is a general paradigm for learning logical formulas from positive and negative data examples, that has received considerable interest recently. We investigate bounded fitting for the description logic ALC and its syntactic fragments. We show that the underlying size-restricted fitting problem is NP-complete for all studied fragments, even in the special case of a single positive and a single negative example. By design, bounded fitting comes with probabilistic guarantees in Valiant's PAC learning framework. In contrast, we show that other classes of algorithms for learning ALC concepts do not provide such guarantees. Finally, we present an implementation of bounded fitting in ALC and its fragments based on a SAT solver. We discuss optimizations and compare our implementation to other concept learning tools.

Problem

• Investigates bounded fitting for ALC logic fragments
• Proves NP-completeness of size-restricted fitting problem
• Implements SAT-based bounded fitting with optimizations

Innovation

• SAT-based bounded fitting for ALC
• NP-complete size-restricted fitting problem
• PAC learning framework guarantees