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Linear-bounded composition of tree-walking tree transducers: linear size increase and complexity

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Abstract

Compositions of tree-walking tree transducers form a hierarchy with respect to the number of transducers in the composition. As main technical result it is proved that any such composition can be realized as a linear-bounded composition, which means that the sizes of the intermediate results can be chosen to be at most linear in the size of the output tree. This has consequences for the expressiveness and complexity of the translations in the hierarchy. First, if the computed translation is a function of linear size increase, i.e., the size of the output tree is at most linear in the size of the input tree, then it can be realized by just one, deterministic, tree-walking tree transducer. For compositions of deterministic transducers it is decidable whether or not the translation is of linear size increase. Second, every composition of deterministic transducers can be computed in deterministic linear time on a RAM and in deterministic linear space on a Turing machine, measured in the sum of the sizes of the input and output tree. Similarly, every composition of nondeterministic transducers can be computed in simultaneous polynomial time and linear space on a nondeterministic Turing machine. Their output tree languages are deterministic context-sensitive, i.e., can be recognized in deterministic linear space on a Turing machine. The membership problem for compositions of nondeterministic translations is nondeterministic polynomial time and deterministic linear space. All the above results also hold for compositions of macro tree transducers. The membership problem for the composition of a nondeterministic and a deterministic tree-walking tree translation (for a nondeterministic IO macro tree translation) is log-space reducible to a context-free language, whereas the membership problem for the composition of a deterministic and a nondeterministic tree-walking tree translation (for a nondeterministic OI macro tree translation) is possibly NP-complete.

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Notes

  1. The name “tree-walking tree transducer” was introduced in [26]. The adjective “tree-walking” stands for the fact that the transducer walks on the input tree (just as the tree-walking automaton of [2]). The tt is the generalization to trees of the two-way finite-state string transducer, which walks on its input string in both directions and produces the output string one-way from left to right. Note that “tree-walking” and “two-way” alliterate.

  2. That is as opposed to a “backward deterministic” context-free grammar in which distinct rules have distinct right-hand sides, see, e.g., [26]. A forward deterministic context-free grammar that generates a string is also called a “straight-line” context-free grammar.

  3. The tt M is circular if there exist \(t\in T_\varSigma \), \(u\in \mathcal{N}(t)\), \(q\in Q\), and \(s\in T_\varDelta ({\text {Con}}(t))\) such that \(\langle q,u \rangle \Rightarrow ^*_{M,t} s\) and \(\langle q,u \rangle \) occurs in s. Thus, M is noncircular if and only if \(G_{M,t}\) is nonrecursive for every \(t\in T_\varSigma \), which implies that \(L(G_{M,t})\) is finite. Note that a total deterministic tt is noncircular if and only if for every \(t\in T_\varSigma \), \(u\in \mathcal{N}(t)\), and \(q\in Q\) there exists \(s\in T_\varDelta \) such that \(\langle q,u \rangle \Rightarrow ^*_{M,t} s\). It can be shown that for every finitary tt there is an equivalent noncircular tt, but that will not be needed in this paper.

  4. In [28], \({\textsf {dTT}} \) and \({\textsf {dTT}} ^\ell \) are denoted by \({\textsf {dTT}} ^{{{\textsc {mso}}}}\) and \({\textsf {dTT}} \), respectively.

  5. We note that an alternative proof is by Lemma 26 (in Sect. 6) and [34, Theorem 7.4] (see also [65, Section 5]). For the reader familiar with mso translations, see [14], we note that it is proved in [29, Section 4] that \({\textsf {dTT}} ^{\mathrm {s}}_\mathrm {rel}\) is the class of mso (tree) relabelings, and that REGT, which is the class of mso definable tree languages, is closed under inverse mso (tree) transductions by [14, Corollary 7.12].

  6. To be precise, the regular sub-test \(T({\text {mark}}(T^\bullet _\varSigma ))\).

  7. For the definition of \(\alpha (u)\) see Sect. 3.

  8. By mistake, [31, Theorem 35] is stated for \(n\ge 1\) only. It also holds for \(n=0\) by [31, Lemma 34 and Theorem 31].

  9. Recall from Sect. 2 that the rank of a node is the rank of its label, i.e., the number of its children.

  10. To be precise, \(|t|\le (2\cdot |t|_0-1)+|t|_1\) where \(|t|_0\) and \(|t|_1\) are the number of leaves and monadic nodes of t, respectively.

  11. There are several such computations, but they all have the same unique derivation tree in \(L(G^\mathrm {der}_{M,t})\). The definition of productivity clearly does not depend on the particular choice of the derivation.

  12. We do not know whether Theorem 44 holds for nondeterministic tt’s, i.e., whether it is decidable for a composition of nondeterministic tt’s whether or not it realizes a translation in \({\textsf {LSIF}} \).

  13. A “visible” pebble can be observed by the transducer during its entire life time (as usual for pebbles), whereas an “invisible” pebble p cannot be observed during the life time of a pebble \(p'\) of which the life time is nested within the one of p; thus, such a pebble \(p'\) “hides” the pebble p.

  14. Note that there is a straightforward one-to-one correspondence between the leftmost derivations of G and \(G'\), and between their derivation trees. Since G is \(\varepsilon \)-free, the derivation trees have the same height.

  15. Note that a node of t has the same label and child number in t and \(\#(t,s)\), except when it has child number 1 in \(\#(t,s)\) in which case it has child number 0 or 1 in t, depending on whether or not its parent in \(\#(t,s)\) has label \(\#\).

  16. The path language of a tree \(s\in T_\varOmega \) consists of all strings in \(\varOmega ^*\) that are obtained by walking along a path from the root of s to one of its leaves, writing down the labels of the nodes of that path from left to right.

  17. In fact, \(\mathrm{enc}_\varSigma \) can even be computed without pushdown: for every forest \(f\in F_\varSigma \), \(\mathrm{enc}_\varSigma (f)\) can be obtained from f by removing all left-brackets, changing each right-bracket into e, and adding one e at the end.

  18. It can be shown that the nondeterministic version of Lemma 73 also holds, but we will not do that here.

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Acknowledgements

We are grateful to the reviewers for their constructive comments.

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  1. LIACS, Leiden University, Leiden, The Netherlands

    Joost Engelfriet

  2. Google Japan G.K., Tokyo, Japan

    Kazuhiro Inaba

  3. Department of Mathematics and Informatics, Universität Bremen, P.O. Box 330 440, 28334, Bremen, Germany

    Sebastian Maneth

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Engelfriet, J., Inaba, K. & Maneth, S. Linear-bounded composition of tree-walking tree transducers: linear size increase and complexity. Acta Informatica 58, 95–152 (2021). https://doi.org/10.1007/s00236-019-00360-8

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