Dichotomies for Queries with Negation in Probabilistic Databases

This article charts the tractability frontier of two classes of relational algebra queries in tuple-independent probabilistic databases. The first class consists of queries with join, projection, selection, and negation but without repeating relation symbols and union. The second class consists of quantified queries that express the following binary relationships among sets of entities: set division, set inclusion, set equivalence, and set incomparability. Quantified queries are expressible in relational algebra using join, projection, nested negation, and repeating relation symbols. Each query in the two classes has either polynomial-time or #P-hard data complexity and the tractable queries can be recognised efficiently. Our result for the first query class extends a known dichotomy for conjunctive queries without self-joins to such queries with negation. For quantified queries, their tractability is sensitive to their outermost projection operator: They are tractable if no attribute representing set identifiers is projected away and #P-hard otherwise.