Tight Bounds of Circuits for Sum-Product Queries

In this paper, we ask the following question: given a Boolean Conjunctive Query (CQ), what is the smallest circuit that computes the provenance polynomial of the query over a given semiring? We answer this question by giving upper and lower bounds. Notably, it is shown that any circuit F that computes a CQ over the tropical semiring must have size log |F| ≥ (1-ε) · da-entw for any ε >0, where da-entw is the degree-aware entropic width of the query. We show a circuit construction that matches this bound when the semiring is idempotent. The techniques we use combine several central notions in database theory: provenance polynomials, tree decompositions, and disjunctive Datalog programs. We extend our results to lower and upper bounds for formulas (i.e., circuits where each gate has outdegree one), and to bounds for non-Boolean CQs.