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^¬s). After first investigating the question of what constitutes a reasonable notion of qualitative 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. [Alon Reshef et al., 2020], can be used to lift previously studied scores for monotone queries, known as drastic Shapley and weighted sums of minimal supports (WSMS), to UCQ^¬s. 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^¬s and further establishing tractability in combined complexity for suitable classes of conjunctive queries with negation.

We consider the following fundamental problem: given a database D, Boolean conjunctive query (CQ) q, and fact f from D, decide whether f is relevant to q w.r.t. D, i.e. does f belong to a minimal subset S of D that makes q hold. Despite being of central importance to query answer explanation, the combined complexity of deciding query relevance has not been studied in detail, leaving open what makes this problem hard, and which restrictions can yield lower complexity. Relevance has already been shown to be harder than query evaluation: namely, it is Sigma^p_2 -complete for CQs, even over a binary signature. We further observe that NP-hardness applies already to (acyclic) chain CQs. Our work identifies self-joins (multiple atoms with the same relation) as the culprit. Indeed, we prove that if we forbid or bound the occurrence of self-joins, then relevance has the same complexity as query evaluation, namely, NP (without structural restrictions) and LogCFL (for bounded hypertreewidth classes). In the ontology setting, we establish an analogous result for ontology-mediated queries consisting of a CQ and DL-Lite_R ontology, namely that relevance is no harder than query answering provided that we bound the interaction width (which generalizes both self-join width and a recently introduced ‘interaction-free’ condition). Our results thus pinpoint what makes relevance harder than query evaluation and identify natural classes of queries which admit efficient relevance computation.

In this work, we explore the use of the Shapley value in ontology-mediated query answering (OMQA) and provide a detailed complexity analysis of Shapley value computation (SVC) in the OMQA setting. In particular, we establish a FP/#P-hard dichotomy for SVC for ontology-mediated queries (T,q) composed of an ontology T formulated in the description logic ELHI-bot and a connected constant-free homomorphism-closed query q. We further strengthen the #P-hardness side of the dichotomy to cover possibly disconnected queries with constants. Our results exploit recently discovered connections between SVC and probabilistic query evaluation and allow us to generalize existing results on probabilistic OMQA.

Recent work on quantitative approaches to explaining query answers employs responsibility measures to assign scores to facts in order to quantify their respective contributions to obtaining a given answer. In this paper, we study the complexity of computing such responsibility scores in the setting of ontology-mediated query answering, focusing on a very recently introduced family of Shapley-value-based responsibility measures defined in terms of weighted sums of minimal supports (WSMS). By exploiting results from the database setting, we can show that such measures enjoy polynomial data complexity for classes of ontology-mediated queries that are first-order-rewritable, whereas the problem becomes #P-hard when the ontology language can encode reachability queries (via axioms like ∃R.A ⊑ A). To better understand the tractability frontier, we next explore the combined complexity of WSMS computation. We prove that intractability applies already to atomic queries if the ontology language supports conjunction, as well as to unions of ‘well-behaved’ conjunctive queries, even in the absence of an ontology. By contrast, our study yields positive results for common DL-Lite dialects: by means of careful analysis, we identify classes of structurally restricted conjunctive queries (which intuitively disallow undesirable interactions between query atoms) that admit tractable WSMS computation.

The Shapley value, originating from cooperative game theory, has been employed to define responsibility measures that quantify the contributions of database facts to obtaining a given query answer. For non-numeric queries, this is done by considering a cooperative game whose players are the facts and whose wealth function assigns 1 or 0 to each subset of the database, depending on whether the query answer holds in the given subset. While conceptually simple, this approach suffers from a notable drawback: the problem of computing such Shapley values is #P-hard in data complexity, even for simple conjunctive queries. This motivates us to revisit the question of what constitutes a reasonable responsibility measure and to introduce a new family of responsibility measures -- weighted sums of minimal supports (WSMS) -- which satisfy intuitive properties. Interestingly, while the definition of WSMSs is simple and bears no obvious resemblance to the Shapley value formula, we prove that every WSMS measure can be equivalently seen as the Shapley value of a suitably defined cooperative game. Moreover, WSMS measures enjoy tractable data complexity for a large class of queries, including all unions of conjunctive queries. We further explore the combined complexity of WSMS computation and establish (in)tractability results for various subclasses of conjunctive queries.

The Shapley value, originally introduced in cooperative game theory for wealth distribution, has found use in KR and databases for the purpose of assigning scores to formulas and database tuples based upon their contribution to obtaining a query result or inconsistency. In the present paper, we explore the use of Shapley values in ontology-mediated query answering (OMQA) and present a detailed complexity analysis of Shapley value computation (SVC) in the OMQA setting. In particular, we establish a FP / #P-hard dichotomy for SVC for ontology-mediated queries (T, q) composed of an ontology T is formulated in the description logic ELHIbot and a connected constant-free homomorphism-closed query q. We further show that the #P-hardness side of the dichotomy can be strengthened to cover possibly disconnected queries with constants. Our results exploit recently discovered connections between SVC and probabilistic query evaluation and allow us to generalize existing results on probabilistic OMQA.

The Shapley value provides a natural means of quantifying the contributions of facts to database query answers. In this work, we seek to broaden our understanding of Shapley value computation (SVC) in the database setting by revealing how it relates to Fixed-size Generalized Model Counting (FGMC), which is the problem of computing the number of sub-databases of a given size and containing a given set of assumed facts that satisfy a fixed query. Our focus will be on explaining the difficulty of SVC via FGMC, and to this end, we identify general conditions on queries which enable reductions from FGMC to SVC. As a byproduct, we not only obtain alternative explanations for existing hardness results for SVC, but also new complexity results. In particular, we establish FP-#P complexity dichotomies for constant-free unions of connected CQs and connected homomorphism-closed graph queries. We also consider some variants of the SVC problem, by disallowing assumed facts or quantifying the contributions of constants rather than facts.