Compositions of n tree transducers

Abstract

Top-down tree transductions, introduced as models of syntax-directed translations and transformational grammars in [12] and [11], are not closed under relational composition. However, closure under relational composition is not always needed; for example, in establishing closure properties of surface sets.(Surface sets are tree-languages obtained as ranges of transductions.) One such closure property, whether the image of a surface set under a transduction is a surface set, remained open. In this paper, we show that transductions need not preserve surface sets. In fact, we exhibit a hierarchy of tree languages obtained by successive transductions. We do not have a good proof that the hierarchy inclusions are proper, but there are strong reasons for so suspecting. As is customary in tree automata papers, we spend some effort on notation. This time, we present a list of first-order axioms for plane (ordered) trees. The Gorn-Brainerd-Doner representation of trees as prefix-closed sets of sequences ([5],[2],[3]) really is a representation in the sense that any (well-founded) abstract tree satisfying our axioms is isomorphic to a prefix-closed set. We then adopt Rosen's notation [10] for trees in subsequent definitions. In our opinion, however, a universally acceptable notation remains to be discovered.