Non-Derivability Results in Polymorphic Dependent Type Theory

2026-03-04

AI Summary

This study investigates the derivability of inductive, coinductive, and quotient types equipped with induction principles within the second-order dependent type theory λP2. By employing categorical semantics and constructing countermodels, it establishes for the first time that parametric quotient types and strong coinduction principles are not definable in pure λP2. The work further demonstrates that functional extensionality plays a crucial role in the derivability of induction principles. Specifically, in an extension of λP2 featuring Σ-types and identity types with uniqueness of identity proofs—but lacking functional extensionality—neither the induction principle for natural numbers nor the coinduction principle for stream types is provable. These results precisely delineate the minimal theoretical extensions required to support strong induction and coinduction.

Technology Category

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

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs

Abstract

In the pure Calculus of Constructions (CC) one can define data types and function over these, and there is a powerful higher order logic to reason over these functions and data types. This is due to the combination of impredicativity and dependent types, and most of these features can already be observed in polymorphic (second order) dependent type theory $λ$P2. The impredicative encoding of data types (in $λ$P2 or CC) is powerful but not fully satisfactory: for example, the induction principle is not provable. As a matter of fact, it can be shown that induction is not provable for whatever possible representation of data types. In a recent paper, Awodey, Frey and Speight show that in an extension of $λ$P2 with Sigma-types, identity types with uniqueness of identity proofs and function extensionality, it is possible to define data types for which the induction principle is provable. More recently it has been shown that in this extension of $λ$P2, also quotient types can be defined with the proper induction principle, and, using quotient types, coinductive types can be defined with the proper coinduction principle. This leaves various questions open: Are quotient types with induction principle not definable in the original $λ$P2? And how about coinductive types, is it impossible to get a strong coinduction principle in $λ$P2? Looking at it from the other side: which of the extensions used are really needed to make induction and coinduction work? In this paper, we contribute partial answers to these questions: parametric quotient types are not definable in $λ$P2 and the well-known definable stream type does not have a coinduction principle. For the latter question we show that, if we just extend $λ$P2 with Sigma-types and identity types with uniqueness of identity proofs, we still cannot prove an induction principle for the natural numbers. So function extensionality is crucial in making induction provable. We show these results by studying models of $λ$P2 where the types representing these principles are empty, so these models act as counter models to the derivability of the principles.

Problem

• induction principle
• coinduction principle
• quotient types
• dependent type theory
• Calculus of Constructions

Innovation

• dependent type theory
• induction principle
• coinduction
• function extensionality
• counter models