Abstract clones for abstract syntax

TL;DR

This paper introduces a formal, syntax-independent framework for simple type theories using abstract clones, enabling algebraic reasoning about syntax and properties like normalization without presheaf complexities.

Contribution

It develops a clone-theoretic approach to simple type theories, providing free algebra constructions and an induction principle for syntax-independent proofs.

Findings

Provides a clone-based algebraic framework for simple type theories

Constructs free algebras and induction principles within this framework

Facilitates syntax-independent proofs of properties like adequacy and normalization

Abstract

We give a formal treatment of simple type theories, such as the simply-typed $\lambda$-calculus, using the framework of abstract clones. Abstract clones traditionally describe first-order structures, but by equipping them with additional algebraic structure, one can further axiomatize second-order, variable-binding operators. This provides a syntax-independent representation of simple type theories. We describe multisorted second-order presentations, such as the presentation of the simply-typed $\lambda$-calculus, and their clone-theoretic algebras; free algebras on clones abstractly describe the syntax of simple type theories quotiented by equations such as $\beta$- and $\eta$-equality. We give a construction of free algebras and derive a corresponding induction principle, which facilitates syntax-independent proofs of properties such as adequacy and normalization for simple type theories. Working only with clones avoids some of the complexities inherent in presheaf-based frameworks for abstract syntax.