Modules over relative monads for syntax and semantics

TL;DR

This paper introduces a formal algebraic framework using relative monads and modules to define syntax and semantics of languages with variable binding, including reduction rules, and implements it in Coq for certified language generation.

Contribution

It develops a novel algebraic approach to language syntax and semantics via 2-signatures and relative monads, with formal proof in Coq.

Findings

Defines 2-signatures for syntax and reduction rules

Constructs initial models with substitution and reduction

Provides a Coq formalization for language generation

Abstract

We give an algebraic characterization of the syntax and semantics of a class of languages with variable binding. We introduce a notion of 2-signature: such a signature specifies not only the terms of a language, but also reduction rules on those terms. To any 2-signature $S$ we associate a category of "models" of $S$. This category has an initial object, which integrates the terms freely generated by $S$, and which is equipped with reductions according to the inequations given in $S$. We call this initial object the language generated by $S$. Models of a 2--signature are built from relative monads and modules over such monads. Through the use of monads, the models---and in particular, the initial model---come equipped with a substitution operation that is compatible with reduction in a suitable sense. The initiality theorem is formalized in the proof assistant Coq, yielding a machinery which, when fed with a 2-signature, provides the associated programming language with reduction relation and certified substitution.