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：我们为近期的责任度量研究工作做出了贡献，这些度量量化了数据库事实对获取查询结果的贡献。与以往几乎完全专注于单调查询的工作不同，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“什么构成带否定原子查询的合理解释或相关性概念”这一问题出发，我们提出了两种方法：一种只为（正）数据库事实分配分数，另一种则同时考虑否定事实。我们的方法与先前Reshef等人研究的分数正交，可用于将先前为单调查询研究的分数（即剧烈Shapley和最小支持加权和，WSMS）推广到UCQ$^\lnot$。我们研究了所得度量的数据复杂性和组合复杂性，特别证明了WSMS度量对所有UCQ${}^\lnot$查询在数据复杂性上是可处理的，并进一步建立了适用于带否定合取查询适宜类别的组合复杂性可处理性。