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Forward and Backward Application of Symbolic Tree Transducers

We consider symbolic tree automata (sta) and symbolic tree transducers (stt). We characterize s-recognizable tree languages (which are the tree languages recognizable by sta) in terms of (classical) recognizable tree languages and relabelings. We prove that sta and the recently introduced variable tree automata are incomparable with respect to their recognition power. We define symbolic regular tree grammars and characterize s-regular tree languages in terms of regular tree languages and relabelings. As a consequence, we obtain that s-recognizable tree languages are the same as s-regular tree languages. We show that the syntactic composition of two stt computes the composition of the tree transformations computed by each stt, provided that (1) the first one is deterministic or the second one is linear and (2) the first one is total or the second is nondeleting. We consider forward application and backward application of stt and prove that the backward application of an stt to any s-recognizable tree language yields an s-recognizable tree language. We give a linear stt of which the range is not an s-recognizable tree language. We show that the forward application of simple and linear stt preserves s-recognizability. As a corollary, we obtain that the type checking problem of simple and linear stt and the inverse type checking problem of arbitrary stt is decidable.