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.

翻译：本文对近期关于责任度量的研究做出贡献，该度量旨在量化数据库事实对获得查询结果的贡献。与现有工作几乎完全集中于单调查询不同，本文探讨了如何为带否定原子的合取查询并集（UCQ${}^\lnot$）定义责任度量。从"何为带否定原子查询的合理解释或相关性概念"这一基本问题出发，我们提出了两种方法：一种仅对（正）数据库事实分配分数，另一种则同时考虑否定事实。这些方法与Reshef等人先前研究的分数正交，可用于将单调查询中已研究的度量——即剧烈Shapley与最小支持加权和（WSMS）——推广至UCQ$^\lnot$。我们研究了所得度量的数据复杂度与联合复杂度，特别证明了WSMS度量对所有UCQ${}^\lnot$查询在数据复杂度上均是易处理的，并进一步为带否定的合取查询的适当类别建立了联合复杂度的可处理性。