Generalizing Brouwer trees beyond N^N to represent general second-order functionals

Determine whether there exists a useful generalization of Brouwer trees that can represent general second-order functionals F : ({a : A} P(a)) → ({b : B} Q(b)) without restricting the domain A to an initial segment of the natural numbers N, and, if such a generalization exists, construct it and characterize its properties and limitations.

Background

The paper contrasts dialogue trees used to represent continuous functionals F : N^N → N with Brouwer trees, noting that Brouwer trees query inputs in the fixed order 0, 1, 2, … and hence do not label nodes explicitly with questions. Dialogue trees, by contrast, allow arbitrary question orders and can more flexibly model interaction.

When moving to the general dependently typed setting of functionals F : ({a : A} P a) → ({b : B} Q b), the authors point out that it is unclear how to adapt Brouwer’s fixed-order querying scheme. The difficulty arises because enforcing a fixed numeric order on queries appears to require restricting A to an initial segment of N, which is incompatible with arbitrary types A and dependent position families P.