Abductive diagnosis is an important method for identifying possible causes which explain a given set of observations. Unfortunately, abduction suffers from the fact that most of the algorithmic problems in this area are intractable. We have recently obtained very promising results for a strongly related problem in the database area. Specifically, the PRIMALITY problem becomes efficiently solvable and highly parallelizable if the underlying functional dependencies have bounded treewidth (Gottlob, Pichler, & Wei 2006b). In the current paper, we show that these favorable results can be carried over to logic-based abduction. In fact, we even show a further generalization of these results.
Ontology-mediated querying and querying in the presence of constraints are two key database problems where tuple-generating dependencies (TGDs) play a central role. In ontology-mediated querying, TGDs can formalize the ontology and thus derive additional facts from the given data, while in querying in the presence of constraints, they restrict the set of admissible databases. In this work, we study the limits of efficient query evaluation in the context of the above two problems, focussing on guarded and frontier-guarded TGDs and on UCQs as the actual queries. We show that a class of ontology-mediated queries (OMQs) based on guarded TGDs can be evaluated in FPT iff the OMQs in the class are equivalent to OMQs in which the actual query has bounded treewidth, up to some reasonable assumptions. For querying in the presence of constraints, we consider classes of constraint-query specifications (CQSs) that bundle a set of constraints with an actual query. We show a dichotomy result for CQSs based on guarded TGDs that parallels the one for OMQs except that, additionally, FPT coincides with PTime combined complexity. The proof is based on a novel connection between OMQ and CQS evaluation. Using a direct proof, we also show a similar dichotomy result, again up to some reasonable assumptions, for CQSs based on frontier-guarded TGDs with a bounded number of atoms in TGD heads. Our results on CQSs can be viewed as extensions of Grohe's well-known characterization of the tractable classes of CQs (without constraints). Like Grohe's characterization, all the above results assume that the arity of relation symbols is bounded by a constant. We also study the associated meta problems, i.e., whether a given OMQ or CQS is equivalent to one in which the actual query has bounded treewidth.
We establish complexities of the conjunctive query entailment problem for classes of existential rules (i.e. Tuple-Generating Dependencies or Datalog+/- rules). Our contribution is twofold. First, we introduce the class of greedy bounded treewidth sets (gbts), which covers guarded rules, and their known generalizations, namely (weakly) frontier-guarded rules. We provide a generic algorithm for query entailment with gbts, which is worst-case optimal for combined complexity with bounded predicate arity, as well as for data complexity. Second, we classify several gbts classes, whose complexity was unknown, namely frontier-one, frontier-guarded and weakly frontier-guarded rules, with respect to combined complexity (with bounded and unbounded predicate arity) and data complexity.
Answer Set Programming (ASP) is an active research area of artificial intelligence. We consider the problem projected answer set counting (#PDA) for disjunctive propositional ASP. #PDA asks to count the number of answer sets with respect to a given set of projected atoms, where multiple answer sets that are identical when restricted to the projected atoms count as only one projected answer set. Our approach exploits small treewidth of the primal graph of the input instance. Finally, we state a hypothesis (3ETH) that one cannot solve 3-QBF in polynomial time in the instance size while being significantly better than triple exponential in the treewidth. Taking 3ETH into account, we show that one can not expect to solve #PDA significantly better than triple exponential in the treewidth.