Equational Logic and Categorical Semantics for Multi-Languages
We give ways in which two languages can be blended, and interoperability reasoned about using equations over the blended language. Formally, multi-language equational logic is defined within which one may deduce valid equations starting from a collection of axioms that postulate properties of the combined language. Thus we have the notion of a multi-language theory and much of the paper is devoted to exploring the properties of these theories. This is accomplished by way of category theory, giving us a very general and flexible semantics, and hence a nice collection of models. Classifying categories are constructed, and hence equational theories furnish each categorical model with an internal language; from this we can also establish soundness and completeness. A set-theoretic semantics follows as an instance, itself sound and complete. The categorical semantics is based on some pre-existing research, but we give a presentation that we feel is easier and simpler to work with, improves and mildly extends current research, and in particular is well suited to computer scientists. Throughout the paper we prove some interesting properties of the new semantic machinery. We provide a small running example throughout the paper to illustrate our ideas, and a more complex example in conclusion.