## A comparison of pebble tree transducers with macro tree transducers (2003)

Venue: | Acta Informatica |

Citations: | 37 - 10 self |

### BibTeX

@ARTICLE{Engelfriet03acomparison,

author = {Joost Engelfriet and Sebastian Maneth},

title = {A comparison of pebble tree transducers with macro tree transducers},

journal = {Acta Informatica},

year = {2003},

volume = {39},

pages = {2003}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. The n-pebble tree transducer was recently proposed as a model for XML query languages. The four main results on deterministic transducers are: First, (1) the translation τ of an n-pebble tree transducer can be realized by a composition of n + 1 0-pebble tree transducers. Next, the pebble tree transducer is compared with the macro tree transducer, a well-known model for syntax-directed semantics, with decidable type checking. The 0-pebble tree transducer can be simulated by the macro tree transducer, which, by the first result, implies that (2) τ can be realized by an (n+1)-fold composition of macro tree transducers. Conversely, every macro tree transducer can be simulated by a composition of 0-pebble tree transducers. Together these simulations prove that (3) the composition closure of n-pebble tree transducers equals that of macro tree transducers (and that of 0-pebble tree transducers). Similar results hold in the nondeterministic case. Finally, (4) the output languages of deterministic n-pebble tree transducers form a hierarchy with respect to the number n of pebbles. 1

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Citation Context ...al intermediate tree, and then applying a YIELD mapping to that tree (realizing the second-order tree substitution inherent in M’s computation). In other words, DtMTT ⊆ DtT◦YIELD (Proposition 4.17 of =-=[CF82]-=-; cf. also Theorem 4.8 of [EV85]). Partialness and nondeterminism of an mtt can be handled by post-composing a total deterministic mtt with a corresponding top-down tree transducer (Corollary 6.12 of ... |

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Citation Context ... (see also [NSV01]). The macro tree transducer (mtt) is also a finite state device that translates trees into trees. It can be obtained by combining the top-down tree transducer and the macro grammar =-=[Fis68]-=-, i.e., the states of the top-down tree transducer may have parameters of type output tree, and thus computations can be spawned at non-leafsPebble and Macro Tree Transducers 3 nodes of the output tre... |

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Citation Context ...der is referred to [EM02b]. String automata that use pebbles in a stack-like fashion (which basically means that the pebbles have nested life times) were introduced in [GH96] and extended to trees in =-=[EH99]-=- (see also [NSV01]). The macro tree transducer (mtt) is also a finite state device that translates trees into trees. It can be obtained by combining the top-down tree transducer and the macro grammar ... |

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Citation Context ... our n-pebble tree transducer is there called an (n + 1)-pebble tree transducer. As observed in [MSV00], the pebble tree transducer can be obtained from the tree-walking automaton of [AU71] (see also =-=[ERS80]-=-) by adding pebbles and the ability to generate output trees rather than strings. We observe here that the deterministic pebble tree transducer without pebbles, i.e., the 0-ptt, is very closely relate... |

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Citation Context ...rs, but for more details the reader is referred to [EM02b]. String automata that use pebbles in a stack-like fashion (which basically means that the pebbles have nested life times) were introduced in =-=[GH96]-=- and extended to trees in [EH99] (see also [NSV01]). The macro tree transducer (mtt) is also a finite state device that translates trees into trees. It can be obtained by combining the top-down tree t... |

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Citation Context ..., ε, j〉 → σ(〈q, down1〉, . . . , 〈q, downk〉)s14 J. Engelfriet and S. Maneth It should be clear that indeed τMΣ = monΣ. Note that monΣ is an instance of a “regular insertion” (see, e.g., Section 2.3 of =-=[Eng82]-=-), which inserts strings (seen as monadic trees) of an arbitrary regular language Rσ above each symbol σ of an input tree. ⊓⊔ Size-to-Height Relationship of PTT Translations. In the next lemma we show... |

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Citation Context ...ct, by Subsection 3.2, 0-ptts are essentially attributed tree transducers; the addition of parameters to the attributes of the attributed tree transducer gives the macro attributed tree transducer of =-=[KV94]-=- which can be simulated by the composition of two attributed tree transducers (to be precise, the class of translations realized by macro attributed tree transducers equals the class of two-fold compo... |

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Citation Context ...ble tree translations can be characterized inside the class DPTT ∗ as those translations for which the number of different subtrees in the output tree is polynomial in the size of the input tree (cf. =-=[EM01]-=-, where the MSO definable tree translations are characterized inside the class DMTT as those translations for which the size of the output tree is linear in the size of the input tree). Acknowledgment... |

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Citation Context ...at for attribute grammars (see Subsection 3.2) these results are well known: top-down tree transducers can be simulated by attribute grammars [CF82] and so can YIELD mappings (shown in Theorem 1.3 of =-=[Eng81]-=-, without correctness proof, and in Corollary 6.24 of [FV98], using an indirect proof). Together with the above decomposition result this will allow us to prove the equality of the composition closure... |

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Citation Context ... by the use of parameters, we can remove all up instructions from a 0-ptt M, thus obtaining a stay-mtt that realizes the same translation as M. In fact, this result is already known. It was proved in =-=[EV86]-=- in the setting of transducers with storage. As discussed at the end of Subsection 3.3, 0-PTT = RT(Tree-walk) and hence 0-PTT ⊆ RT(P(Tree)). By Theorem 5.14, Corollary 5.21, and Theorem 4.18 of [EV86]... |

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Citation Context ...n closure many attractive properties are known; for instance: it has decidable type checking [EV85], the translations can be computed in linear time (in the sum of the sizes of input and output tree) =-=[Man02]-=-, and the output languages form a full AFL and have decidable emptiness and finiteness problems [DE98]. Before we discuss our results, let us consider the relationship of tree transducers to (XML base... |

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Citation Context ...than, the one for trees in this section. The one pebble is really needed: deterministic 0-pebble string transducers are closed under composition (because they are the two-way finite state transducers =-=[CJ77]-=-). Let us sketch the proof of this decomposition. Let M be an n-ptt, n ≥ 1. We want to discuss how to decompose M’s translation τM into the composition of a fixed total function EncPeb, realized by a ... |

7 | Output string languages of compositions of deterministic macro tree transducers
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Citation Context ...r n of pebbles, of the output languages of n-pebble tree transducers. The proof is based on the “mtt-hierarchy” of (string) output languages of n-fold compositions of mtts that was recently proved in =-=[EM02a]-=-. The result strengthens the hierarchy of translations of n-pebble tree transducers, which follows from an obvious size-to-height relationship for such translations (viz., the height of the output tre... |

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7 | Conditions for efficiency improvement by tree transducer composition - Voigtländer - 2002 |