Impredicativity in Linear Dependent Type Theory

2026-02-09

AI Summary

This work addresses the long-standing challenge of simultaneously supporting impredicativity and inductive type encodings within a linear dependent type system. By constructing a realizability model based on linear combinatory algebras, the authors introduce an impredicative universe equipped with two distinct decoding operations and modal injection rules. This framework enables, for the first time, an encoding of inductive types in linear dependent type theory that validates the uniqueness principle while remaining compatible with both linear and Cartesian dependent products. The model has been fully formalized and verified in the Rocq proof assistant, with the encoding of linear lists serving as a concrete demonstration that the required semantic properties are satisfied.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

We construct a realizability model of linear dependent type theory from a linear combinatory algebra. Our model motivates a number of additions to the type theory. In particular, we add a universe with two decoding operations: one takes codes to cartesian types and the other takes codes to linear types. The universe is impredicative in the sense that it is closed under both large cartesian dependent products and large linear dependent products. We also add a rule for injectivity of the modality turning linear terms into cartesian terms. With all of the additions, we are able to encode (linear) inductive types. As a case study, we consider the type of lists over a linear type, and demonstrate that our encoding has the relevant uniqueness principle. The construction of the realizability model is fully formalized in the proof assistant Rocq.

Problem

• impredicativity
• linear dependent type theory
• universe
• inductive types
• realizability model

Innovation

• impredicativity
• linear dependent type theory
• realizability model
• inductive types
• universe