Input Products for Weighted Extended Top-Down Tree Transducers

Abstract

A weighted tree transformation is a function τ : TΣ×TΔ → A where TΣ and TΔ are the sets of trees over the ranked alphabets Σ and Δ, respectively, and A is the domain of a semiring. The input and output product of τ with tree series ϕ: TΣ → A and ϕ: TΔ → A are the weighted tree transformations ϕ Δ τ and τ Δ Ψ, respectively, which are defined by (ϕ ◃ τ)(t, u) = ϕ(t) ċ τ (t, u) and (τ ▹ ψ)(t, u) = τ (t, u) ċ ψ(u) for every t ∈ TΣ and u ∈ TΔ. In this contribution, input and output products of weighted tree transformations computed by weighted extended top-down tree transducers (wxtt) with recognizable tree series are considered. The classical approach is presented and used to solve the simple cases. It is shown that input products can be computed in three successively more difficult scenarios: nondeleting wxtt, wxtt over idempotent semirings, and weighted top-down tree transducers over rings.