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.