A celebrated theorem of Chandra and Merlin states that Conjunctive Queries inclusion is decidable. Its proof transforms logical formulas to hypergraphs: each query has a natural model – a kind of graph – and query inclusion reduces to the existence of a graph homomorphism between natural models.
In this talk, we introduce the diagrammatic language Graphical Conjunctive Queries (GCQ) and show that it has the same expressivity as Conjunctive Queries. GCQ terms are string diagrams, and their algebraic structure allows us to derive a sound and complete axiomatisation of query inclusion, which turns out to be exactly Carboni and Walters’ notion of cartesian bicategory of relations. Our completeness proof exploits the combinatorial nature of string diagrams as (certain cospans of) hypergraphs: Chandra and Merlin’s insights inspire a theorem that relates such cospans with spans. Completeness and decidability of the (in)equational theory of GCQ follow as a corollary. Categorically speaking, our contribution is a model-theoretic completeness theorem of free cartesian bicategories (on a relational signature) for the category of sets and relations.