Incomparability Results for Classes of Polynomial Tree Series Transformations

We consider (subclasses of) polynomial bottom-up and top-down tree series transducers over a partially ordered semiring A = (A,⊕, ,0,1, ), and we compare the classes of tree-to-tree-series and o-tree-to-tree-series transformations computed by such transducers. Our main result states the following. If, for some a ∈ A with 1 a, the semiring A is a weak a-growth semiring and either (i) the semiring A is additively idempotent and x, y ∈ {polynomial, deterministic, total, deterministic and total, homomorphism}, or (ii) 1 ≺ 1⊕ 1 and x, y ∈ {deterministic, deterministic and total, homomorphism}, then the statements x BOT(A) on y BOT(A) and x BOT(A) on y TOP(A) hold. Therein x BOT(A) for mod ∈ {ε, o} denotes the class of mod-tree-to-tree-series transformations computed by bottom-up tree series transducers, which have property x, over the semiring A (the class y TOP(A) is de ned similarly for top-down tree series transducers). Besides, on denotes incomparability with respect to set inclusion.