The Equivalence of Bottom-Up and Top-Down Tree-to-Graph Transducers

Abstract

We introduce the bottom-up tree-to-graph transducer, which is very similar to the usual (total deterministic) bottom-up tree transducer except that it translates trees into hypergraphs rather than trees, using hypergraph substitution instead of tree substitution. If every output hypergraph of the transducer is a jungle, i.e., a hypergraph that can be unfolded into a tree, then the tree-to-graph transducer is said to be tree-generating and naturally defines a tree-to-tree translation. We prove that bottom-up tree-to-graph transducers define the same treeto-tree translations as the previously introduced top-down tree-to-graph transducers. This is in contrast with the well-known incomparability of the usual bottom-up and top-down tree transducers.] 1998Academic Press

A bottom-up characterization of deterministic top-down tree transducers with regular look-ahead

“…On the other hand, (nondeterministic) bottom-up and top-down tree automata recognize the same class of tree languages [15,16], linear and nondeleting bottom-up and top-down tree transducers compute the same class of tree transformations [5], and tree-generating bottom-up and top-down tree-to-graph transducers are equivalent [12].…”

“…One is the bottom-up processing, which starts at the leaves of a tree s and ends up in the root of s. The other one is the top-down processing, when the process starts at the root of s and proceeds towards the leaves of s. Therefore the literature distinguishes between bottom-up (or: frontier-to-root) and top-down (or: root-to-frontier) tree automata, tree transducers, and tree-to-graph transducers [17][18][19]5,6,1,15,16,12].…”

Formal Tree Series

“…We do not consider generalizations of the IO-substitution. Bozapalidis [11], Engelfriet, Fülöp, Vogler [19] and Fülöp, Vogler [23] consider these generalizations to formal tree series.…”

“…Kuich [39] generalized a restricted form of top-down tree transducers to tree series transducers which map formal tree series into formal tree series. Engelfriet, Fülöp, Vogler [19] and Fülöp, Vogler [23] generalized this approach and defined bottom-up and top-down tree series transducers as generalization of frontier-to-root and root-to-frontier tree transducers in the sense of Gécseg, Steinby [24,25]. In this section we only consider the case of top-down tree series transducers.…”

“…Formal tree series were introduced by Berstel, Reutenauer [5], and then extensively studied by Bozapalidis [7,8,9,10,11], Bozapalidis, Rahonis [12], Kuich [38,39,40,41,43], Engelfriet, Fülöp, Vogler [19] and Fülöp, Vogler [23].…”