Equational Logic and Set-Theoretic Models for Multi-Languages

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