A Complete Finitary Refinement Type System for Scott-Open Properties

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.

💡 Research Summary

The paper presents a finitary refinement type system that is both sound and complete for Scott‑open properties of programs manipulating infinite data structures such as streams and non‑well‑founded trees. The authors build on Abramsky’s “Domain Theory in Logical Form”, exploiting the fact that Scott domains interpreting recursive types are spectral spaces. In a spectral space, open sets and compact‑saturated sets are dual; the authors capture this duality by assigning a polarity to logical formulae. Positive formulae are closed under least fixed points (µ) and denote Scott‑open sets, whereas negative formulae are closed under greatest fixed points (ν) and denote compact‑saturated sets.

The core of the system is a three‑tiered logic L⁺, L⁻, and L± that annotates pure types. Formulae are built from propositional connectives, type‑specific modalities (e.g.,