Variable binding and substitution for (nameless) dummies

Abstract

By abstracting over well-known properties of De Bruijn’s representation with nameless dummies, we design a new theory of syntax with variable binding and capture-avoiding substitution. We propose it as a simpler alternative to Fiore, Plotkin, and Turi’s approach, with which we establish a strong formal link. We also show that our theory easily incorporates simple types and equations between terms.

A unified treatment of structural definitions on syntax for capture-avoiding substitution, context application, named substitution, partial differentiation, and so on

“…Our framework, however, applies just as well in other settings. To illustrate this, in this subsection, we transpose the example of capture-avoiding substitution to the De Bruijn-style setting of [Hirschowitz et al 2022b].…”

“…To summarise the idea: just as the presheaf-based approach equipes the nested datatypes representation [Bird and Paterson 1999] with initial algebra semantics, the setting of [Hirschowitz et al 2022b] does the same for De Bruijn representation [De Bruijn 1972]. In the presheaf-based approach, terms are indexed by sets of potential free variables, and by convention the bound variable is always the greatest one, typically +1 for : ( + 1) → ( ).…”