Computational interpretation of propositional impredicativity in univalent foundations

Establish a computational interpretation of the propositional impredicativity axioms for univalent foundations, specifically the axioms enabling propositional resizing, so that these impredicative principles can be justified within a computationally meaningful framework for HoTT/UF.

Background

The paper adopts a predicative approach in univalent foundations by not assuming Voevodsky’s propositional resizing axioms. The authors emphasize philosophical and technical reasons for this choice, including model-theoretic and proof-theoretic considerations. A central motivation is the lack of a known computational interpretation of propositional impredicativity in HoTT/UF, which would justify adding such axioms without sacrificing constructive validity.

Propositional resizing (an impredicative principle) allows propositions in higher universes to be treated as small, simplifying constructions like arbitrary unions in powersets. In the absence of a computational interpretation, the authors avoid these axioms and instead develop domain theory predicatively using type universes and equivalences to manage size issues.