Algebraic Semantics of Datalog with Equality
For database and logic programming researchers, it offers a novel algebraic semantics and free model construction for Datalog with equality, enabling new implementations.
The paper extends Datalog with equality and partial functions (RHL and PHL), and provides a new construction of free models using the small object argument, which generalizes Datalog evaluation and underpins the Eqlog engine.
We discuss the syntax and semantics of relational Horn logic (RHL) and partial Horn logic (PHL). RHL is an extension of the Datalog programming language that allows introducing and equating variables in conclusions. PHL is a syntactic extension of RHL by partial functions and one of the many equivalent notions of essentially algebraic theory. Our main contribution is a new construction of free models. We associate to RHL and PHL sequents classifying morphisms, which enable us to characterize logical satisfaction using lifting properties. We then obtain free and weakly free models using the small object argument. The small object argument can be understood as an abstract generalization of Datalog evaluation. It underpins the implementation of the Eqlog Datalog engine, which computes free models of PHL theories.