Complexity of Faceted Explanations in Propositional Abduction

2025-07-20

AI Summary

This paper addresses the challenge of characterizing explanation heterogeneity in propositional abduction. We introduce the notion of a *facet*—a minimal textual unit appearing in only some explanations—to capture fine-grained relevance and dispensability of explanation components. Building upon Post’s satisfiability framework, we develop a *faceted explanation model* that, for the first time in propositional abduction, systematically distinguishes necessary from redundant explanation elements and provides an almost complete classification of their computational complexity. We further define a distance measure between explanations to quantify explanation homogeneity versus heterogeneity. Our approach integrates propositional logic modeling, computational complexity analysis, and metric-based reasoning, comprehensively characterizing the complexity of diverse faceted reasoning tasks. We identify several tractable subproblems, offering theoretical foundations for practical applications in explainable AI and abductive inference.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

Abductive reasoning is a popular non-monotonic paradigm that aims to explain observed symptoms and manifestations. It has many applications, such as diagnosis and planning in artificial intelligence and database updates. In propositional abduction, we focus on specifying knowledge by a propositional formula. The computational complexity of tasks in propositional abduction has been systematically characterized - even with detailed classifications for Boolean fragments. Unsurprisingly, the most insightful reasoning problems (counting and enumeration) are computationally highly challenging. Therefore, we consider reasoning between decisions and counting, allowing us to understand explanations better while maintaining favorable complexity. We introduce facets to propositional abductions, which are literals that occur in some explanation (relevant) but not all explanations (dispensable). Reasoning with facets provides a more fine-grained understanding of variability in explanations (heterogeneous). In addition, we consider the distance between two explanations, enabling a better understanding of heterogeneity/homogeneity. We comprehensively analyze facets of propositional abduction in various settings, including an almost complete characterization in Post's framework.

Problem

• Analyzing computational complexity of propositional abduction tasks
• Introducing facets to understand variability in explanations
• Measuring distance between explanations to assess heterogeneity

Innovation

• Introducing facets to propositional abduction
• Analyzing distance between explanations
• Comprehensive analysis in Post's framework