[Paper Review] Impredicative Encodings of (Higher) Inductive Types

Steve Awodey, Jonas Frey+1 more|arXiv (Cornell University)|Feb 8, 2018

Logic, programming, and type systems36 references3 citations

TL;DR

This paper presents a type-theoretic method to refine impredicative encodings of inductive and higher inductive types in dependent type theory, restoring $η$-rules and enabling dependent elimination by internalizing parametricity and dinaturality using identity types. The key contribution is a provable, internalized $η$-equality for impredicative encodings, which enables full dependent eliminators for types like natural numbers and higher inductive types such as the circle $ℂ^1$ and 1-truncation.

ABSTRACT

Postulating an impredicative universe in dependent type theory allows System F style encodings of finitary inductive types, but these fail to satisfy the relevant η-equalities and consequently do not admit dependent eliminators. To recover η and dependent elimination, we present a method to construct refinements of these impredicative encodings, using ideas from homotopy type theory. We then extend our method to construct impredicative encodings of some higher inductive types, such as 1-truncation and the unit circle S1.

Motivation & Objective

• To address the longstanding problem of impredicative encodings in type theory failing to satisfy $η$-rules and dependent elimination.
• To restore $η$-equalities and dependent eliminators for finitary inductive types such as natural numbers and binary sums.
• To extend the method to higher inductive types, including 1-truncation and the unit circle $ℂ^1$, using homotopy type theory principles.
• To provide a fully internalized, type-theoretic refinement of impredicative encodings, avoiding reliance on external models or semantics.
• To establish consistency and model-theoretic foundations for the refined encodings using realizability and groupoidal models.

Proposed method

• Refine impredicative encodings using identity types to internalize parametricity and dinaturality, ensuring propositional $η$-equalities.
• Apply a type-theoretic version of the Yoneda lemma to recover $η$-rules for non-recursive types like $A + B$.
• Construct initial algebras via impredicative encodings in a way that supports dependent elimination for inductive types.
• Use the notion of $0$-types and $n$-types from homotopy type theory to generalize the method to $1$-types and higher inductive types.
• Leverage the uniqueness of identity proofs (UIP) at the $0$-type level to derive dependent eliminators from propositional $η$-equalities.
• Formalize the encodings in a type theory with an impredicative universe $Σ$ closed under small sums and identity types, enabling internal reasoning about coherence and universality.

Experimental results

Research questions

• RQ1 Can impredicative encodings of finitary inductive types be refined to satisfy $η$-rules and dependent elimination within type theory?
• RQ2 How can parametricity and dinaturality be internalized in type theory to restore $η$-equalities without relying on external models?
• RQ3 Can the method be extended to higher inductive types such as $1$-truncation and the unit circle $ℂ^1$?
• RQ4 What coherence conditions are required to generalize the encoding method to $n$-types for $n > 1$, and are they tractable?
• RQ5 What is the consistency strength and model-theoretic justification for the refined encodings in realizability and groupoidal models?

Key findings

• The refined impredicative encodings satisfy propositional $η$-equalities, which are sufficient to derive dependent eliminators as shown in [AGS12, AGS17].
• The encoding of the natural numbers $ℕ$ as $∀ X. olinebreak (X \to X) \to X \to X$ is refined to support dependent elimination via identity types.
• The binary sum $A + B$ is encoded impredicatively and refined using a Yoneda-style argument to recover its $η$-rule.
• The method successfully encodes the $1$-truncation of a type and the unit circle $ℂ^1$ as higher inductive types using impredicative constructions.
• The encodings are stable under truncation levels: for example, the $ℕ$ encoding remains a $0$-type even when constructed in a $1$-type context.
• A realizability model with an impredicative universe closed under sums and identity types validates the system, and groupoidal models support $1$-types.