Implementing a Category-Theoretic Framework for Typed Abstract Syntax

TL;DR

This paper extends a category-theoretic framework for abstract syntax to simply-typed languages, incorporating bicategorical semantics and mechanized in UniMath, enabling automated, certified language implementations.

Contribution

It generalizes previous untyped syntax constructions to typed languages using bicategorical semantics and extends mechanized libraries in UniMath for multi-sorted syntax.

Findings

Successfully generalized syntax construction to simply-typed languages.

Extended UniMath library to handle multi-sorted syntax.

Provided a modular, mechanized framework for language implementation.

Abstract

In previous work ("From signatures to monads in UniMath"), we described a category-theoretic construction of abstract syntax from a signature, mechanized in the UniMath library based on the Coq proof assistant. In the present work, we describe what was necessary to generalize that work to account for simply-typed languages. First, some definitions had to be generalized to account for the natural appearance of non-endofunctors in the simply-typed case. As it turns out, in many cases our mechanized results carried over to the generalized definitions without any code change. Second, an existing mechanized library on $\omega$-cocontinuous functors had to be extended by constructions and theorems necessary for constructing multi-sorted syntax. Third, the theoretical framework for the semantical signatures had to be generalized from a monoidal to a bicategorical setting, again to account for non-endofunctors arising in the typed case. This uses actions of endofunctors on functors with given source, and the corresponding notion of strong functors between actions, all formalized in UniMath using a recently developed library of bicategory theory. We explain what needed to be done to plug all of these ingredients together, modularly. The main result of our work is a general construction that, when fed with a signature for a simply-typed language, returns an implementation of that language together with suitable boilerplate code, in particular, a certified monadic substitution operation.