Equational Logic and Set-Theoretic Models for Multi-Languages

Interoperability is the capability of two languages to interact within a single system: HTML, CSS, and JavaScript can work together to render webpages. Some object oriented languages have interoperability via a virtual machine host (.NET CLI compliant languages in the Common Language Runtime). A high-level language can be interoperable with a low-level one (Apple’s Swift and Objective-C). While there has been some research in the foundations of interoperability there is little supporting theory. This paper is based upon our existing work on combining languages to produce so-called multi-languages. Here, we define an equational logic for deducing valid equations, from axioms that postulate properties of the multi-language. We define set-theoretic multi-language algebras as models, and provide algebraic constructions such as congruences and quotient algebras. Such models, and the constructions, provide the ingredients for the main deliverable, soundness and completeness for the equational logic. We illustrate the basic ideas with a running example.