Responsibility Measures for Conjunctive Queries with Negation

2026-01-08

arXiv.org

AI Summary

Existing responsibility measures struggle to handle unions of conjunctive queries with negated atoms (UCQ^¬) due to their non-monotonicity, which renders traditional positive-fact-based metrics ineffective. This work proposes the first two responsibility measures tailored for UCQ^¬: one that evaluates only the contribution of positive facts, and another that additionally accounts for the influence of negative facts. We orthogonally extend established monotonic measures—such as drastic Shapley and Weighted Minimal Support Sets (WSMS)—to this non-monotonic setting. Leveraging logical semantics, variants of Shapley values, and WSMS, we establish that the proposed WSMS-based measure is data-complexity tractable for arbitrary UCQ^¬ and achieves combined-complexity tractability for specific classes of conjunctive queries, thereby laying a theoretical foundation for responsibility analysis in non-monotonic query settings.

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

• Responsible Web: Algorithmic accountability and transparency on the web
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs

Abstract

We contribute to the recent line of work on responsibility measures that quantify the contributions of database facts to obtaining a query result. In contrast to existing work which has almost exclusively focused on monotone queries, here we explore how to define responsibility measures for unions of conjunctive queries with negated atoms (UCQ${}^\lnot$). Starting from the question of what constitutes a reasonable notion of explanation or relevance for queries with negated atoms, we propose two approaches, one assigning scores to (positive) database facts and the other also considering negated facts. Our approaches, which are orthogonal to the previously studied score of Reshef et al., can be used to lift previously studied scores for monotone queries, known as drastic Shapley and weighted sums of minimal supports (WSMS), to UCQ$^\lnot$. We investigate the data and combined complexity of the resulting measures, notably showing that the WSMS measures are tractable in data complexity for all UCQ${}^\lnot$ queries and further establishing tractability in combined complexity for suitable classes of conjunctive queries with negation.

Problem

• responsibility measures
• conjunctive queries with negation
• query explanation
• database facts
• UCQ^¬

Innovation

• responsibility measures
• conjunctive queries with negation
• Shapley values
• weighted sums of minimal supports
• data complexity