Tree transductions and families of tree languages

Abstract

Interest in the study of sets of trees, tree languages, has led to the definition of finite automata which accept trees [2,11] and transducers which map trees into other trees [7,9,10]. These generalized machines may read treesfinite automata which accept trees [2,11] and of transducers which map trees into other trees [7,9,10]. These generalized machines may read trees either “top-down” (from the root toward the leaves) or “bottom-up” (from the leaves toward the root). Here it is shown that both the class of top-down transductions and the class of bottom-up transductions can be characterized in terms of two restricted classes of tree transductions. From these ductions and the class of bottom-up transductions can be characterized in terms of two restricted classes of tree transductions. From these characterizations, it is shown that the composition of any n bottom-up transductions can be realized by the composition of n+1 top-down transductions, and similarly, the composition of any n top-down transductions can be realized by the composition of n+1 bottom-up transductions. Next, we study the families of tree languages which can be obtained from the recognizable sets (sets accepted by finite tree automata) by the composition of n top-down or bottom-up transductions, n>0. The yield operation, which concatenates the leaves of a tree from left to right to form a of string, languages from the hierarchy of families of tree languages. It is shown that each family of string languages in this hierarchy is properly contained in the family of context-sensitive languages.