Abstract clones for abstract syntax

05/03/2021 ∙ by Nathanael Arkor, et al. ∙ 0

We give a formal treatment of simple type theories, such as the simply-typed λ-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 λ-calculus, and their clone-theoretic algebras; free algebras on clones abstractly describe the syntax of simple type theories quotiented by equations such as β- and η-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.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 5

page 9

page 11

page 13

page 15

page 17

page 18

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.