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.

中文翻译：

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树行走树传感器的线性有界组合：线性尺寸增加和复杂性

树行走树换能器的组合根据组合中的换能器数量形成层次结构。作为主要技术结果，证明了任何这样的组合都可以实现为线性有界组合，这意味着中间结果的大小可以选择为至多与输出树的大小成线性。这会影响层次结构中翻译的表现力和复杂性。首先，如果计算的翻译是线性大小增加的函数，即输出树的大小至多与输入树的大小成线性关系，那么它可以仅通过一个确定性的树遍历树转换器来实现. 对于确定性换能器的组合，可以确定平移是否具有线性尺寸增加。第二，确定性传感器的每个组合都可以在 RAM 上的确定性线性时间和图灵机上的确定性线性空间中计算，以输入和输出树的大小之和来衡量。类似地，可以在非确定性图灵机上同时在多项式时间和线性空间中计算非确定性换能器的每个组合。它们的输出树语言是确定性上下文敏感的，即可以在图灵机上的确定性线性空间中识别。非确定性翻译组合的隶属度问题是非确定性多项式时间和确定性线性空间。所有上述结果也适用于宏树换能器的组合。