Match weak interaction preorder with Lévy–Longo tree preorder

Establish that the weak head variant of interaction preorder for the untyped λ-calculus (defined using weak head reduction that does not reduce under abstractions) coincides with the Lévy–Longo tree preorder; specifically, prove that two terms are related by the weak interaction preorder if and only if their Lévy–Longo trees are ordered/equal accordingly.

Background

The paper in the paper focuses on head reduction, with interaction equivalence characterized via Böhm trees. The authors suggest analogous development for weak head reduction (also called lazy reduction), where Lévy–Longo trees provide the corresponding semantic representation.

They conjecture that the weak interaction preorder aligns exactly with Lévy–Longo tree preorder, but note that crafting a weak version of their interaction Böhm-out technique may be required, as classical separation results for weak head reduction are not directly available without extensions.

References

Weak Head. Our study focuses on the paradigmatic case of head reduction, but it could be adapted to weak head reduction (sometimes called lazy reduction [Abramsky and Ong 1993]). It is folklore that the Lévy-Longo tree preorder (a weak variant of Böhm trees) matches the weak head type preorder. We conjecture that the Levy-Longo tree preorder matches exactly the weak head variant of our interaction preorder.

— Interaction Equivalence (2409.18709 - Accattoli et al., 27 Sep 2024) in Section 10, Future Work (Weak Head)