Domain theory in univalent foundations II: Continuous and algebraic domains

2024-07-09

arXiv.org

AI Summary

This work constructively establishes continuous and algebraic domain theory, overcoming reliance on the law of excluded middle, the axiom of choice, and universe resizing assumptions inherent in classical frameworks. Methodologically, it redefines continuity for dcpos within univalent type theory using categorical ideas of continuity, employs propositional truncation to ensure continuity is a decidable property, and rigorously constructs small-basis algebraic and continuous dcpos without size assumptions. Key contributions include: (i) the first formalization of core domain-theoretic concepts—such as algebraicity, continuity, and Scott-continuity—within a constructive, univalent setting free of choice and universe resizing; (ii) a characterization of small-basis dcpos as precisely those completely represented by small ideals; (iii) verification that the Scott $D_infty$ model forms an algebraic dcpo with small basis; and (iv) a fully constructive semantic foundation for the $lambda$-calculus. All results are formally verified in Agda.

Technology Category

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

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: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Vertical and domain-specific search

Abstract

We develop the theory of continuous and algebraic domains in constructive and predicative univalent foundations, building upon our earlier work on basic domain theory in this setting. That we work predicatively means that we do not assume Voevodsky's propositional resizing axioms. Our work is constructive in the sense that we do not rely on excluded middle or the axiom of (countable) choice. To deal with size issues and give a predicatively suitable definition of continuity of a dcpo, we follow Johnstone and Joyal's work on continuous categories. Adhering to the univalent perspective, we explicitly distinguish between data and property. To ensure that being continuous is a property of a dcpo, we turn to the propositional truncation, although we explain that some care is needed to avoid needing the axiom of choice. We also adapt the notion of a domain-theoretic basis to the predicative setting by imposing suitable smallness conditions, analogous to the categorical concept of an accessible category. All our running examples of continuous dcpos are then actually examples of dcpos with small bases which we show to be well behaved predicatively. In particular, such dcpos are exactly those presented by small ideals. As an application of the theory, we show that Scott's $D_infty$ model of the untyped $lambda$-calculus is an example of an algebraic dcpo with a small basis. Our work is formalised in the Agda proof assistant and its ability to infer universe levels has been invaluable for our purposes.

Problem

• Develops domain theory in constructive univalent foundations
• Addresses predicative continuity without propositional resizing
• Adapts basis concepts with smallness conditions for predicativity

Innovation

• Constructive predicative univalent foundations for domains
• Propositional truncation to ensure continuity property
• Small bases adaptation for predicative dcpos