SQL, Homomorphisms and Constraint Satisfaction Problems

The post mentions the idea that querying a database D can be understood algebraically as enumerating all morphisms Q -> D, where Q is the "classifying" database of the query, i.e. a minimal database instance that admits a single "generic" result of the query. You can use this to give a neat formulation of Datalog evaluation. A Datalog rule then corresponds a morphism P -> H, where P is the classifying database instance of the rule body and H is the classifying database instance for matches of both body and head. For example, for the the transitivity rule

  edge(x, z) :- edge(x, y), edge(y, z).

you'd take for P the database instance containing two rows (a_1, a_2) and (a_2, a_3), and the database instance H contains additionally (a_1, a_3). Now saying that a Database D satisfies this rule means that every morphism P -> D (i.e., every match of the premise of the rule) can be completed to a commuting diagram

  P --> D
  |    ^
  |   /
  ⌄  /
  Q 

where the additional map is the arrow Q -> D, which corresponds to a match of both body and head.

This kind of phenomenon is known in category theory as a "lifting property", and there's rich theory around it. For example, you can show in great generality that there's always a "free" way to add data to a database D so that it satisfies the lifting property (the orthogonal reflection construction/the small object argument). Those are the theoretical underpinnings of the Datalog engine I'm sometimes working on [1], and there they allow you to prove that Datalog evaluation is also well-defined if you allow adjoining new elements during evaluation in a controlled way. I believe the author of this post is involved in the egglog project [2], which might have similar features as well.

[1] https://github.com/eqlog/eqlog [2] https://github.com/egraphs-good/egglog