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arXiv · 2601.23082

A Complete Finitary Refinement Type System for Scott-Open Properties

Colin Riba·Adam Donadille
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Abstract

We are interested in proving input-output properties of functions that handle infinite data such as streams or non-wellfounded trees. We provide a finitary refinement type system which is (sound and) complete for Scott-open properties defined in a fixpoint-like logic. Working on top of Abramsky's Domain Theory in Logical Form, we build from the well-known fact that the Scott domains interpreting recursive types are spectral spaces. The usual symmetry between Scott-open and compact-saturated sets is reflected in logical polarities: positive formulae allow for least fixpoints and define Scott-open sets, while negative formulae allow for greatest fixpoints and define compact-saturated sets. A realizability implication with the expected (contra)variance on polarities allows for non-trivial input-output properties to be formulated as positive formulae on function types.

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Colin Riba, Adam Donadille. 2026-01-30. A Complete Finitary Refinement Type System for Scott-Open Properties. https://arxiv.org/abs/2601.23082

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