Bottom-Up and Top-Down Tree Series Transformations

We generalize bottom-up tree transducers and top-down tree transducers to the concept of bottom-up tree series transducer and top-down tree series transducer, respectively, by allowing formal tree series as output rather than trees, where a formal tree series is a mapping from output trees to some semiring. We associate two semantics with a tree series transducer: a mapping which transforms trees into tree series (for short: tree to tree series transformation or t-ts transformation), and a mapping which transforms tree series into tree series (for short: tree series transformation or ts-ts transformation). We show that the standard case of tree transducers is reobtained by choosing the boolean semiring under the t-ts semantics. Also, for each of the two types of tree series transducers and for both types of semantics, we prove a characterization which generalizes in a straightforward way the corresponding characterization result for the underlying tree transducer class. More precisely, we prove that polynomial bottom-up tree series transducers can be characterized by the composition of nite state relabeling tree series transducers and homo-morphism tree series transducers. Moreover, we prove that the total deterministic top-down tree series transducers can be characterized by the composition of homomorphism tree series transducers and linear total deterministic top-down tree series transducers.