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Forward and backward application of symbolic tree transducers

Abstract

We consider symbolic tree automata (sta) and symbolic regular tree grammars and their corresponding classes of tree languages: s-recognizable tree languages and s-regular tree languages. We prove that the following three classes are equal: the class of s-recognizable tree languages, the class of s-regular tree languages, and the class of images of classical recognizable tree languages under tree relabelings. Moreover, the sta and the recently introduced variable tree automata are incomparable with respect to their recognition power. Also, we consider symbolic tree transducers (stt) and prove the following theorems. 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 one is nondeleting. Backward application of an stt to any s-recognizable tree language yields an s-recognizable tree language. There is a linear stt of which the range is not an s-recognizable tree language. Forward application of simple and linear stt preserves s-recognizability. A restricted version of both the type checking problem of simple and linear stt, and the inverse type checking problem of arbitrary stt is decidable. Since we deal with trees over infinite alphabets, we require appropriate conditions on sta and stt such that all the proofs are constructive.

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Notes

  1. 1.

    We mean that for every \(\varphi \in \Phi \) there is an explicit algorithm which, for every \( a\in U\) as input, computes \(\varphi (a)\).

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Acknowledgments

The authors would like to thank the referees for their work and valuable suggestions, which improved the quality of the paper.

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Correspondence to Zoltán Fülöp.

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Research of this author was financially supported by the programs TÁMOP-4.2.1/B-09/1/KONV-2010-0005 and TÁMOP-4.2.2.A-11/1/KONV-2012-0073 of the Hungarian National Development Agency.

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Fülöp, Z., Vogler, H. Forward and backward application of symbolic tree transducers. Acta Informatica 51, 297–325 (2014). https://doi.org/10.1007/s00236-014-0197-7

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