Generalized Universe Hierarchies and First-Class Universe Levels

02/27/2021
by   András Kovács, et al.
0

In type theories, universe hierarchies are commonly used to increase the expressive power of the theory while avoiding inconsistencies arising from size issues. There are numerous ways to specify universe hierarchies, and theories may differ in details of cumulativity, choice of universe levels, specification of type formers and eliminators, and available internal operations on levels. In the current work, we aim to provide a framework which covers a large part of the design space. First, we develop syntax and semantics for cumulative universe hierarchies, where levels may come from any set equipped with a transitive well-founded ordering. In the semantics, we show that induction-recursion can be used to model transfinite hierarchies, and also support lifting operations on type codes which strictly preserve type formers. Then, we consider a setup where universe levels are first-class types and subject to arbitrary internal reasoning. This generalizes the bounded polymorphism features of Coq and at the same time the internal level computations in Agda.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

04/08/2019

A General Framework for the Semantics of Type Theory

We propose an abstract notion of a type theory to unify the semantics of...
02/23/2021

Induction principles for type theories, internally to presheaf categories

We present new induction principles for the syntax of dependent type the...
11/12/2020

Functorial Semantics for Partial Theories

We provide a Lawvere-style definition for partial theories, extending th...
09/11/2020

A general definition of dependent type theories

We define a general class of dependent type theories, encompassing Marti...
02/24/2022

Strict universes for Grothendieck topoi

Hofmann and Streicher famously showed how to lift Grothendieck universes...
09/23/2020

Behavioral Specification Theories: an Algebraic Taxonomy

We develop a taxonomy of different behavioral specification theories and...
This week in AI

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