Stratified Type Theory: A Type Theory with Stratified Typing Judgements and Floating Function Types

Core Concepts

Stratified Type Theory (StraTT) is a type theory that stratifies typing judgements rather than universes, and includes a separate nondependent function type with a floating domain to overcome the limitations of strict stratification.

Abstract

The paper introduces Stratified Type Theory (StraTT), a type theory that takes a different approach to preventing logical inconsistency compared to the typical universe hierarchy. Instead of stratifying universes, StraTT stratifies the typing judgements themselves.
The key features of StraTT are:

Stratified dependent function types: The domain type must be at a strictly lower level than the codomain type.
Displacement: Global definitions can be uniformly displaced to higher levels, providing a simple form of level polymorphism.
Floating nondependent function types: These have a domain type that "floats" to match the level of the overall function type, overcoming limitations of strict stratification.

The authors prove logical consistency for the subsystem subStraTT, which includes only stratified dependent functions and displacement. They also prove type safety for the full StraTT, including floating nondependent functions, though consistency of the full system remains an open problem.
The paper also includes several examples demonstrating the expressivity of StraTT, such as defining decidable types, Leibniz equality, and dependent pairs. Finally, the authors describe a prototype implementation that extends StraTT with datatypes and type inference.

Stats

None.

Quotes

None.

Key Insights Distilled From

Stratified Type Theory

by Jonathan Cha... at arxiv.org 04-09-2024

https://arxiv.org/pdf/2309.12164.pdf

Deeper Inquiries

How might the authors approach proving consistency of the full StraTT system, including floating nondependent function types

To prove the consistency of the full StraTT system, including floating nondependent function types, the authors could consider several approaches. One potential strategy could involve further refining the logical relation used in the consistency proof to accommodate the interaction between floating functions and cumulativity. This may involve developing a more intricate inductive–recursive definition that accounts for the behavior of floating nondependent functions and their impact on the overall consistency of the system. Additionally, the authors could explore the possibility of introducing new constraints or rules that govern the behavior of floating functions to ensure that they do not introduce inconsistencies into the system. By carefully analyzing the specific challenges posed by floating nondependent functions and their interaction with other features of StraTT, the authors can work towards establishing the consistency of the full system.

What are the potential tradeoffs between the design of StraTT and existing proof assistant systems in terms of working with universe levels

The design of StraTT introduces novel concepts such as stratified typing judgments and displacement, offering an alternative approach to handling type universes and dependent functions. In comparison to existing proof assistant systems, StraTT's stratification technique provides a different perspective on managing universe levels and ensuring consistency within the type system. One potential tradeoff of StraTT's design is the complexity introduced by the stratification and displacement mechanisms, which may require users to have a deeper understanding of the system's internal workings. This could potentially lead to a steeper learning curve for users unfamiliar with these concepts. On the other hand, StraTT's approach may offer more flexibility and control over universe levels, allowing for a more nuanced and precise handling of type dependencies compared to traditional proof assistants.

Could the stratification and displacement techniques used in StraTT be applied to other type theories beyond dependent type systems

The stratification and displacement techniques used in StraTT could potentially be applied to other type theories beyond dependent type systems. These techniques offer a structured way to manage the levels of typing judgments and ensure consistency within the type system. By incorporating similar stratification principles and displacement mechanisms, other type theories could potentially benefit from enhanced control over type dependencies and universe levels. This could lead to more robust and expressive type systems that prevent logical inconsistencies and enable more sophisticated type checking and inference capabilities. Overall, the concepts introduced in StraTT have the potential to inspire new approaches to type theory design in a broader context.