String Diagrams for $\lambda$-calculi and Functional Computation

TL;DR

This tutorial introduces string diagrams and hierarchical hypergraph syntax as a novel, category-theoretic approach to modeling lambda calculus and functional programming, emphasizing applications in semantics and program transformations.

Contribution

It develops a hierarchical hypergraph syntax from categorical models, providing an alternative to traditional syntax for lambda calculus and functional programming.

Findings

Hierarchical hypergraph syntax improves over linear term syntax.

Categorical models facilitate advanced program transformations.

Application to semantics and type inference enhances language analysis.

Abstract

This tutorial gives an advanced introduction to string diagrams and graph languages for higher-order computation. The subject matter develops in a principled way, starting from the two dimensional syntax of key categorical concepts such as functors, adjunctions, and strictification, and leading up to Cartesian Closed Categories, the core mathematical model of the lambda calculus and of functional programming languages. This methodology inverts the usual approach of proceeding from syntax to a categorical interpretation, by rationally reconstructing a syntax from the categorical model. The result is a graph syntax -- more precisely, a hierarchical hypergraph syntax -- which in many ways is shown to be an improvement over the conventional linear term syntax. The rest of the tutorial focuses on applications of interest to programming languages: operational semantics, general frameworks for type inference, and complex whole-program transformations such as closure conversion and automatic differentiation.