On the algebraic structure of declarative programming languages

We develop an algebraic framework, Logic Programming Doctrines, for the syntax, proof theory, operational semantics and model theory of Horn Clause logic programming based on indexed premonoidal categories. Our aim is to provide a uniform framework for logic programming and its extensions capable of incorporating constraints, abstract data types, features imported from other programming language paradigms and a mathematical description of the state space in a declarative manner. We define a new way to embed information about data into logic programming derivations by building a sketch-like description of data structures directly into an indexed category of proofs. We give an algebraic axiomatization of bottom-up semantics in this general setting, describing categorical models as fixed points of a continuous operator.