Answering Conjunctive Queries with Safe Negation and Inequalities over RDFS Knowledge Bases

Expressing negative conditions is a crucial feature of query languages for knowledge bases (KBs). Answering such queries over ontological KBs, however, is a very challenging task that becomes undecidable even for lightweight Description Logic (DL) ontologies. Such negative results hold even for Conjunctive Queries (CQs) equipped with basic forms of negative conditions such as the so-called safe negation or inequality atoms. One ontology language that is seemingly unaffected by these results is (the DL counterpart of) RDFS even if equipped with disjointness axioms. Answering CQs with inequalities over such ontologies is known to be Pi^p_2-complete, if the number of inequality atoms is unbounded, and NP-complete if we limit this number to one. Notably, these results leave open the cases of CQs with a fixed number greater than two of inequality atoms. Additionally, such a thorough analysis is missing for CQs with safe negation. In this paper, we embark in a refined analysis of the combined complexity of answering CQs with inequality atoms and safe negation over RDFS ontologies augmented with disjointness axioms. Firstly, we provide a unified Pi^p_2 query answering algorithm for the general problem. Secondly, we confirm the generally held conjecture according to which answering CQs with two inequality atoms over such ontologies is already Pi^p_2-hard. This result closes an important gap in the current literature and has an impact on the widely influential problem of query containment. Lastly, for CQs with safe negation, we prove a behavior similar to that of CQs with inequality atoms. Specifically, we show that answering CQs with at most one negated atom can be done in NP, while allowing at most two negated atoms is sufficient to obtain Pi^p_2-hardness.