Generalized Decidability via Brouwer Trees

2026-02-11

AI Summary

This work addresses the coarseness of traditional decidability classifications in constructive mathematics by introducing Brouwer ordinals within homotopy type theory to define a refined notion of α-decidability. The resulting hierarchy of decidability precisely corresponds to ordinal indices and unifies, as well as generalizes, classical decidability and semi-decidability. We establish that α-decidable propositions are closed under conjunction and, in certain cases, under disjunction. Furthermore, we provide upper bounds on the decidability of countable intersections and unions, as well as iterated quantifiers—e.g., showing ω²-decidability in specific instances. All results have been formally verified in Cubical Agda.

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
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In the setting of constructive mathematics, we suggest and study a framework for decidability of properties, which allows for finer distinctions than just"decidable, semidecidable, or undecidable". We work in homotopy type theory and use Brouwer ordinals to specify the level of decidability of a property. In this framework, we express the property that a proposition is $\alpha$-decidable, for a Brouwer ordinal $\alpha$, and show that it generalizes decidability and semidecidability. Further generalizing known results, we show that $\alpha$-decidable propositions are closed under binary conjunction, and discuss for which $\alpha$ they are closed under binary disjunction. We prove that if each $P(i)$ is semidecidable, then the countable meet $\forall i\in \mathbb N. P(i)$ is $\omega^2$-decidable, and similar results for countable joins and iterated quantifiers. We also discuss the relationship with countable choice. All our results are formalized in Cubical Agda.

Problem

• decidability
• constructive mathematics
• Brouwer ordinals
• homotopy type theory
• semidecidability

Innovation

• generalized decidability
• Brouwer ordinals
• homotopy type theory
• constructive mathematics
• Cubical Agda