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Responsibility Measures for Conjunctive Queries with Negation

Meghyn Bienvenu, Diego Figueira, Pierre Lafourcade

TL;DR

This work extends responsibility measures from monotone queries to UCQ$^{\lnot}$ by introducing two explanatory paradigms: signed facts and positive supports. It shows how to adapt Shapley-based scores (MS-Shapley and drastic-Shapley) to UCQ$^{\lnot}$ via reductions to monotone counterparts on transformed queries, establishing data- and some combined-complexity results under structural restrictions. The paper proves tractability for many cases (e.g., AC0 data-complexity for signed/positive relevance, FP results for safe or guarded negation classes) and clarifies the relationships and trade-offs between signed-relevance, positive-relevance, and impact-based notions. It also situates WSMS as a broader framework that can accommodate other tractable weight functions, enabling flexible, scalable explanations for query results in the presence of negation with potential extension to exogenous facts.

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.

Responsibility Measures for Conjunctive Queries with Negation

TL;DR

This work extends responsibility measures from monotone queries to UCQ by introducing two explanatory paradigms: signed facts and positive supports. It shows how to adapt Shapley-based scores (MS-Shapley and drastic-Shapley) to UCQ via reductions to monotone counterparts on transformed queries, establishing data- and some combined-complexity results under structural restrictions. The paper proves tractability for many cases (e.g., AC0 data-complexity for signed/positive relevance, FP results for safe or guarded negation classes) and clarifies the relationships and trade-offs between signed-relevance, positive-relevance, and impact-based notions. It also situates WSMS as a broader framework that can accommodate other tractable weight functions, enabling flexible, scalable explanations for query results in the presence of negation with potential extension to exogenous facts.

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). 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. 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 queries and further establishing tractability in combined complexity for suitable classes of conjunctive queries with negation.
Paper Structure (13 sections, 5 theorems, 5 equations, 3 figures)

This paper contains 13 sections, 5 theorems, 5 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Responsibility Measures via Signed Supports
  4. Relevance and Responsibility of Signed Facts
  5. Computing MS-Shapley Values for Signed Facts
  6. Computing Drastic-Shapley Values for Signed Facts
  7. Responsibility Measures via Positive Supports
  8. Relevance and Responsibility of Positive Facts
  9. Computing MPS-Shapley Values for Positive Facts
  10. Computing Drastic-Shapley Values for Positive Facts
  11. Related Work on Shapley Values for Queries with Negation
  12. Concluding Remarks
  13. Weighted sums of minimal supports

Key Result

Lemma 5

For any class $\mathcal{C} \subseteq \text{"UCQneg"}$ of "bounded negative arity", there are polynomial-time reductions from $\ShapleypmMinSup{\mathcal{C}}$ to $\msShapley{\mathcal{C}^\pm}$ and from $\ShapleypmDrastic{\mathcal{C}}$ to $\drShapley{\mathcal{C}^\pm}$, where $\mathcal{C}^\pm = \{\queryp

Figures (3)

  • Figure 1: Example database and queries, based on recipes from escoffierGuideCulinaireAidememoire1903. All edges are $I$-tuples, and recipe names Matelote Meunière and Matelote Pochouse are abbreviated to $mm$ and $mp$ respectively.
  • Figure 2: Query and database from \ref{['ex:signed-example']}, together with the "minimal" "signed supports" $S_1$ and $S_2$. Solid black arrows represent "positive facts" and dotted red arrows "negative facts".
  • Figure 3: Construction of minimal support of unbounded size for "UCQneg" under alternative semantics for \ref{['ex:unbounded-minimal-supports']}.

Theorems & Definitions (12)

  • Remark 1: Exogenous facts
  • Example 2
  • Example 3
  • Example 4
  • Lemma 5
  • Proposition 6
  • Proposition 7
  • Example 8
  • Example 9
  • Example 10: Unbounded supports under alternative semantics
  • ...and 2 more