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Domain Theory in Constructive and Predicative Univalent Foundations

Authors Tom de Jong , Martín Hötzel Escardó



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Author Details

Tom de Jong
  • University of Birmingham, UK
Martín Hötzel Escardó
  • University of Birmingham, UK

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Tom de Jong and Martín Hötzel Escardó. Domain Theory in Constructive and Predicative Univalent Foundations. In 29th EACSL Annual Conference on Computer Science Logic (CSL 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 183, pp. 28:1-28:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.CSL.2021.28

Abstract

We develop domain theory in constructive univalent foundations without Voevodsky’s resizing axioms. In previous work in this direction, we constructed the Scott model of PCF and proved its computational adequacy, based on directed complete posets (dcpos). Here we further consider algebraic and continuous dcpos, and construct Scott’s D_∞ model of the untyped λ-calculus. A common approach to deal with size issues in a predicative foundation is to work with information systems or abstract bases or formal topologies rather than dcpos, and approximable relations rather than Scott continuous functions. Here we instead accept that dcpos may be large and work with type universes to account for this. For instance, in the Scott model of PCF, the dcpos have carriers in the second universe U₁ and suprema of directed families with indexing type in the first universe U₀. Seeing a poset as a category in the usual way, we can say that these dcpos are large, but locally small, and have small filtered colimits. In the case of algebraic dcpos, in order to deal with size issues, we proceed mimicking the definition of accessible category. With such a definition, our construction of Scott’s D_∞ again gives a large, locally small, algebraic dcpo with small directed suprema.

Subject Classification

ACM Subject Classification
  • Theory of computation → Constructive mathematics
  • Theory of computation → Type theory
Keywords
  • domain theory
  • constructivity
  • predicativity
  • univalent foundations

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