On Tree Transducers for Partial Functions

In [4, Theorem X7.11 it is shown that if a partial function can be realized by a (nondeterministic) finite state a-transducer, then it can also be realized by a (deterministic) generalized bimachine (introduced in [l 11). Thus the generalized bimachines compute precisely all partial functions which are realizable by a-transducers. It can easily be shown that instead of a generalized bimachine (which roughly cbonsists of two deterministic sequential machines working in opposite directions on the input string) one can also take a deterministic gsm with regular look-ahead, i.e. a deterministic generalized sequential machine whose moves are determined also in the basis of whether or not the rest of the input string belongs to certain regular languages (cf. [3]). In this note we generalize this result to tree traxducers [ 12,9,8,2,5,6] and show that for tree transducers the role of the generalized bimachine is played by the deterministic top-down tree transducer with regular look-ahl:ad (abbreviated to dt’-fst), introduced in [6]. In particular we prove that if ai partial function of trees can be realized by any finite sequence of (bottom-up OI top-down) nondeterministic tree transducers, then it can also be realized by Q dt’-fst. This implies that every string function obtwined fron a sequence of tree transducers by considering yieids, is a generalized syntax-dlirected translation, 2s defined in [I]. it sezms to be open whether the composition of n + 1 tree transdlucers is more powerful than that of n tree transducers (for ranges this is conjectured in [8]). The above result shows that it is not true for partial functions. Thus, examples to prove the hierarchy proper are necessarily relations which are not hnctions. We adopt the notation and terminology of [5] and [6$. We recall that both finite state tree transducers and the tree transformations they realize, are called fst. The composition of relations R 1 and R2 is