Bounded First-Class Universe Levels in Dependent Type Theory

2025-02-27

AI Summary

In dependent type theory, universes serve only as syntactic markers and cannot participate in computation or express constraints. Method: We introduce the first explicit type system supporting bounded first-class universes, embedding universes as ordinary terms into syntax and reduction, enabling universe polymorphism and universe constraints; the system is parameterized over arbitrary well-founded universe hierarchies. Results: It satisfies subject reduction, type safety, logical consistency, and canonicity. All metatheoretic properties are mechanically verified in Lean. Our key contribution is the first fully formalized explicit syntax and typing rules for constrained first-class universes—overcoming the traditional non-computational treatment of universes—and establishing a new foundation for higher-order, proof-relevant type systems that simultaneously ensures expressive power and formal reliability.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Decentralized Web and Fediverse systems
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

In dependent type theory, being able to refer to a type universe as a term itself increases its expressive power, but requires mechanisms in place to prevent Girard's paradox from introducing logical inconsistency in the presence of type-in-type. The simplest mechanism is a hierarchy of universes indexed by a sequence of levels, typically the naturals. To improve reusability of definitions, they can be made level polymorphic, abstracting over level variables and adding a notion of level expressions. For even more expressive power, level expressions can be made first-class as terms themselves, and level polymorphism is subsumed by dependent functions quantifying over levels. Furthermore, bounded level polymorphism provides more expressivity by being able to explicitly state constraints on level variables. While semantics for first-class levels with constraints are known, syntax and typing rules have not been explicitly written down. Yet pinning down a well-behaved syntax is not trivial; there exist prior type theories with bounded level polymorphism that fail to satisfy subject reduction. In this work, we design an explicit syntax for a type theory with bounded first-class levels, parametrized over arbitrary well-founded sets of levels. We prove the metatheoretic properties of subject reduction, type safety, consistency, and canonicity, entirely mechanized from syntax to semantics in Lean.

Problem

• Design syntax for type theory with bounded first-class levels.
• Ensure type safety and consistency in dependent type theory.
• Mechanize proofs of metatheoretic properties in Lean.

Innovation

• Introduces bounded first-class universe levels
• Uses level polymorphism for definition reusability
• Mechanizes syntax to semantics in Lean