Deriving Induction Principles for Inductive and Higher Inductive Types via Parametricity

Conceitos Básicos

Parametricity, expressed in terms of the category-theoretic concept of cohesion, can be used to derive induction principles for inductive and higher inductive types from their recursors, avoiding the need to explicitly specify complex coherence conditions.

Resumo

The content discusses the use of parametricity, formulated in terms of cohesion, to derive induction principles for inductive and higher inductive types.

The key insights are:

1. Parametricity can be understood as an essentially modal aspect, connected to the concept of cohesion in category theory. This allows for a more axiomatic and synthetic treatment of parametricity, compared to previous analytic approaches.

2. By formalizing the notion of "graph types" which capture the idea of a predicate or relation as a type family, the author is able to prove a substitution lemma that serves as the basis for deriving induction principles.

3. The author demonstrates this approach by deriving induction principles for the natural numbers (N) and the circle (S1) higher inductive type, showing how the complexity of coherence conditions can be managed using parametricity.

4. The formalization is carried out in Agda, with careful attention paid to ensuring computational soundness and canonicity of the resulting definitions and theorems.

The overall contribution is a unifying framework for understanding and applying parametricity, with applications in solving problems around the complexity of coherence conditions in homotopy type theory and related fields.

Estatísticas

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Citações

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Principais Insights Extraídos De

Parametricity via Cohesion

by C.B.... às arxiv.org 04-08-2024

https://arxiv.org/pdf/2404.03825.pdf

Perguntas Mais Profundas

How might this approach to parametricity via cohesion be extended to handle more complex higher inductive types, inductive-inductive types, or even dependent type theories with richer type-forming operations

This approach to parametricity via cohesion can be extended to handle more complex higher inductive types, inductive-inductive types, or even dependent type theories with richer type-forming operations by incorporating additional modalities and axioms that capture the specific structure and coherence requirements of these types. For higher inductive types, such as the circle S1 with non-trivial identifications, the framework can be expanded to include modalities that account for higher-dimensional coherence conditions and constructors. By defining appropriate adjoint functors and modalities, the framework can capture the essential properties of these types and derive parametricity theorems for them. Similarly, for inductive-inductive types, where types are mutually defined, the framework can be adapted to handle the interplay between different constructors and their coherence requirements. By introducing modalities that reflect the interdependencies between the components of inductive-inductive types, the framework can provide a systematic way to reason about parametricity in these settings.

What are the limitations of this approach, and are there any fundamental obstacles to generalizing it further or applying it to other settings

While the approach to parametricity via cohesion offers a promising way to internalize parametric reasoning in type theory, there are limitations and challenges to consider. One fundamental obstacle is the complexity that arises when dealing with higher-dimensional coherence conditions, especially in the context of higher inductive types. As the types become more intricate and involve non-trivial identifications or higher-dimensional structures, the modalities and axioms needed to capture these properties may become increasingly intricate and difficult to formalize. Additionally, the framework may face challenges in handling certain exotic or non-standard type-forming operations that do not fit neatly into the existing categorical semantics of cohesion. Generalizing the framework to encompass a wide range of type theories with diverse features and operations may require a deep understanding of the underlying category theory and modal logic, as well as innovative approaches to model complex coherence conditions effectively.

Beyond the derivation of induction principles, what other applications might this framework of parametricity via cohesion have in type theory, homotopy theory, or programming language theory

Beyond the derivation of induction principles, the framework of parametricity via cohesion has various applications in type theory, homotopy theory, and programming language theory. One key application is in the analysis and verification of program modules, where parametricity theorems can ensure the uniformity and modularity of module structures. By using the framework to reason about parametricity internally in dependent type theories, programmers can establish strong guarantees about the behavior and interactions of modules within a system. Additionally, in homotopy theory, the framework can be utilized to study higher-dimensional coherence conditions and derive induction principles for higher inductive types, enabling a deeper understanding of the topological and geometric aspects of type theory. Moreover, in constructive mathematics, the framework can aid in developing formal proofs and reasoning about parametricity in a rigorous and systematic manner, leading to more reliable and verifiable software systems.