Equivalence of almost '!'-depth 1 λ-transducers with MSO relabeling and MSOT-S2

Establish that the class of tree-to-tree functions computed by almost '!'-depth 1 λ-transducers, when preprocessed by a monadic second-order (MSO) relabeling, is exactly the class MSOT-S2. Concretely, prove the expressive-power equality between the composition of an almost '!'-depth 1 λ-transducer with an MSO relabeling and the class MSOT-S2 of monadic second-order tree transductions with sharing (as denoted by the authors).

Background

The paper studies the expressive power of (almost) affine λ-transducers for tree-to-tree functions and connects them to tree-walking transducers. It establishes that purely affine λ-transducers correspond to reversible tree-walking transducers, while almost purely affine λ-transducers correspond to tree-walking transducers. With MSO relabeling as preprocessing, purely affine λ-transducers capture MSOT, and almost purely affine λ-transducers capture MSOT-S.

In the perspectives section, the authors propose relaxing the affine constraint by bounding the nesting depth of the exponential modality '!' in the λ-calculus, defining a '!'-depth n system. They conjecture, for n=1, a precise expressive-power equivalence: the composition of an almost '!'-depth 1 λ-transducer with an MSO relabeling equals the class MSOT-S2.