Equational Logic and Set-Theoretic Models for Multi-Languages

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