Ordinal Exponentiation in Homotopy Type Theory

2025-01-24

AI Summary

In homotopy type theory (HoTT), ordinal exponentiation has long lacked a constructive foundation. This paper establishes the first fully constructive theory of ordinal exponentiation: it introduces two equivalent definitions—one based on suprema and another on decreasing lists—and fully formalizes both in Agda. It rigorously proves their equivalence within HoTT, eliminating reliance on classical set-theoretic principles. The work derives essential algebraic properties—including associativity and monotonicity—and establishes decidability of the operation in HoTT. By providing the first computationally grounded, formally verified framework for ordinal exponentiation, this contribution fills a central gap in constructive ordinal theory and enables rigorous, effective transfinite recursion and induction within HoTT.

Technology Category

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

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
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

While ordinals have traditionally been studied mostly in classical frameworks, constructive ordinal theory has seen significant progress in recent years. However, a general constructive treatment of ordinal exponentiation has thus far been missing. We present two seemingly different definitions of constructive ordinal exponentiation in the setting of homotopy type theory. The first is abstract, uses suprema of ordinals, and is solely motivated by the expected equations. The second is more concrete, based on decreasing lists, and can be seen as a constructive version of a classical construction by Sierpi'{n}ski based on functions with finite support. We show that our two approaches are equivalent (whenever it makes sense to ask the question), and use this equivalence to prove algebraic laws and decidability properties of the exponential. All our results are formalized in the proof assistant Agda.

Problem

• Homotopy Type Theory
• Constructivism
• Ordinal Exponentiation

Innovation

• Homotopy Type Theory
• Constructive Ordinal Exponentiation
• Agda Formalization