Complexity of Abduction in Łukasiewicz Logic

2025-07-18

AI Summary

This paper investigates abductive explanation under multi-valued fuzzy settings, focusing on explaining truth-degree assertions (e.g., “the elevator is fully loaded”, “symptoms are severe”) in Łukasiewicz infinite-valued logic. To capture variable value ranges, the authors extend the logical language with interval literals—such as $p geq mathbf{c}$, $p leq mathbf{c}$, and their negations. They formally define and systematically analyze abduction tasks—including explanation enumeration, existence, and relevance—in this setting for the first time within Łukasiewicz logic. Their complexity analysis reveals a fine-grained hierarchy: abduction over clause fragments is strictly less complex than over the full language—a counterintuitive contrast to classical propositional logic. The work establishes foundational theoretical results and precise complexity characterizations for uncertainty-aware abductive reasoning in fuzzy knowledge representation and explainable AI.

Technology Category

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

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

We explore the problem of explaining observations in contexts involving statements with truth degrees such as `the lift is loaded', `the symptoms are severe', etc. To formalise these contexts, we consider infinitely-valued Łukasiewicz fuzzy logic. We define and motivate the notions of abduction problems and explanations in the language of Łukasiewicz logic expanded with `interval literals' of the form $pgeqmathbf{c}$, $pleqmathbf{c}$, and their negations that express the set of values a variable can have. We analyse the complexity of standard abductive reasoning tasks (solution recognition, solution existence, and relevance / necessity of hypotheses) in Łukasiewicz logic for the case of the full language and for the case of theories containing only disjunctive clauses and show that in contrast to classical propositional logic, the abduction in the clausal fragment has lower complexity than in the general case.

Problem

• Explaining observations with truth degrees in Łukasiewicz logic
• Analyzing abduction complexity in full vs clausal theories
• Defining abduction problems using interval literals in fuzzy logic

Innovation

• Uses Łukasiewicz fuzzy logic for abduction
• Introduces interval literals for variable constraints
• Analyzes complexity in clausal vs general cases