## Macro Tree Transducers, Attribute Grammars, and MSO Definable Tree Translations (1998)

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Venue: | Inform. and Comput |

Citations: | 49 - 20 self |

### BibTeX

@ARTICLE{Engelfriet98macrotree,

author = {Joost Engelfriet and Sebastian Maneth},

title = {Macro Tree Transducers, Attribute Grammars, and MSO Definable Tree Translations},

journal = {Inform. and Comput},

year = {1998},

volume = {154},

pages = {34--91}

}

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### Abstract

A characterization is given of the class of tree translations definable in monadic second order logic (MSO), in terms of macro tree transducers. The first main result is that the MSO definable tree translations are exactly those tree translations realized by macro tree transducers (MTTs) with regular look-ahead that are single use restricted. For this the single use restriction known from attribute grammars is generalized to MTTs. Since MTTs are closed under regular look-ahead, this implies that every MSO definable tree translation can be realized by an MTT. The second main result is that the class of MSO definable tree translations can also be obtained by restricting MTTs with regular look-ahead to be finite copying, i.e., to require that each input subtree is processed only a bounded number of times. The single use restriction is a rather strong, static restriction on the rules of an MTT, whereas the finite copying restriction is a more liberal, dynamic restriction on the ...

### Citations

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Citation Context ... the normal form of a (with respect to )) and is denoted by nf(); a). In the remainder of this section we recall some basic notions concerning trees, tree translations, and tree languages (see, e.g., =-=[GS84]-=-). 2.1 Ranked Alphabets and Trees A set \Sigma together with a mapping rank \Sigma : \Sigma ! N is called a ranked set. For ks0, \Sigma (k) is the set foe 2 \Sigma j rank \Sigma (oe) = kg; we also wri... |

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Citation Context ... s]][[oe / s 0 ]] = t[[oe / s[[oe / s 0 ]]]] and if oe 0 6= oe then t[[oe / s]][[oe 0 / s 0 ]] = t[[oe 0 / s 0 ; oe / s[[oe 0 / s 0 ]]]], and similar for the general case (cf. Sections 3.4 and 3.7 of =-=[Cou83]-=-). The notion of second order substitution is closely related to that of a tree homomorphism; associativity of second order substitution corresponds to the fact that tree homomorphisms are closed unde... |

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Citation Context ...he class of translations realized by sur ATTs with look-ahead is precisely the class of translations realized by sur MTTs with regular lookahead. Unlike ATTs, MTTs are closed under regular look-ahead =-=[EV85]-=-. Hence, every MSO definable tree translation can be realized by an MTT. Let us discuss this result in more detail. There is a close relationship between the states of an MTT and the synthesized attri... |

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Citation Context ...cro tree transducers [Eng80, CF82, EV85] and attributed tree transducers [Ful81, EF81, CF82] (an extensive survey of these two models of syntax-directed semantics is presented in the recent monograph =-=[FV98]-=-). Moreover, for macro tree transducers we introduce the notion of state sequence (Definition 3.7), 5 and for attributed tree transducers we recall from [BE98] the notion of an attributed relabeling (... |

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Citation Context ...REL, and T R -REL is its obvious extension with regular look-ahead. If we denote by B-REL the class of total relabelings in DBQREL, then it is easy to show that B-REL ` T R -REL (cf. Lemma 2.10(3) of =-=[Eng77]-=-). 13 In [BE98] a class of relabelings which is based on attribute grammars was considered. We will show in Section 4 that these attributed relabelings have the same power as top-down relabelings with... |

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Citation Context ...ger class of transducers for which the translations they realize are MSO definable. This class is obtained by restricting the MTTs to be finite copying. The notion of finite copying was introduced in =-=[AU71]-=- for generalized syntax-directed translation schemes, which are closely related to top-down tree transducers. For top-down tree transducers it was investigated in [ERS80]. Intuitively, an MTT is finit... |

57 |
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Citation Context ...ms of ATTs. Since it is not possible for an ATT to compute a label according to values of other attributes, we need to add the ability to evaluate expressions in an algebra W over finite domains (cf. =-=[CF82]-=-). An ordinary ATT A can be used to compute the new label for each node as a W -expression (in some attribute a 0 ). For an input tree s the relabeled tree t is obtained by replacing the label of each... |

51 |
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Citation Context ...restriction is a more liberal, dynamic restriction on the derivations of an MTT. 1 Introduction Formulas in monadic second order logic (MSO) can be used to define functions from graphs to graphs (cf. =-=[Cou94]-=-), called MSO (graph) transductions. MSO transductions have nice properties, comparable to those of finite state transductions on strings. In particular, they are closed under composition and they pre... |

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Citation Context ...s defined in terms of the outside attributes of oe (i.e., we assume Bochmann Normal Form [Boc76]). Note also that our definition of ATTs in Definition 3.9 is different from the original definition in =-=[Ful81]-=-. There, for every inherited attribute b, the right-hand side of the (hb; 1i; root)-rule is restricted to trees over \Delta. In the appendix of [Gie88] this difference was pointed out and the term ful... |

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Citation Context ...3.9 is different from the original definition in [Ful81]. There, for every inherited attribute b, the right-hand side of the (hb; 1i; root)-rule is restricted to trees over \Delta. In the appendix of =-=[Gie88]-=- this difference was pointed out and the term full attributed tree transducer was used to refer to the transducers of Definition 3.9. In what follows let A = (Syn; Inh; \Sigma; \Delta; root; a 0 ; R) ... |

48 | Some open questions and recent results on tree transducers and tree languages - Engelfriet - 1980 |

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Citation Context ...IO macro grammars, which can be seen as MTTs without input and with strings in the right-hand sides of the rules, Fischer proves a nondeleting ("argument-preserving") normal form in Theorem =-=3.1.10 of [Fis68]-=-. Our proof will be essentially the same, but it will need regular look-ahead to preserve the determinism of the MTT. Given an arbitrary MTT R , how can we construct a nondeleting MTT R M 0 which real... |

41 |
Context-free graph grammars
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Citation Context ...te a monadic tree of size n into the full binary tree of size 2 n \Gamma 1). As discussed in the Introduction, it is known from [EvO97, CE95] that the class of contextfree graph languages (see, e.g., =-=[Eng97]-=-) can be obtained by applying MSO graph transductions to regular tree languages. Hence, the class of tree languages generated by contextfree graph grammars equals the class MSOTT (REGT ) of MSO defina... |

39 | Semantic evaluation from left to right - Bochmann - 1976 |

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Citation Context ...nite copying was introduced in [AU71] for generalized syntax-directed translation schemes, which are closely related to top-down tree transducers. For top-down tree transducers it was investigated in =-=[ERS80]-=-. Intuitively, an MTT is finite copying, if each input subtree and each parameter is copied only a bounded number of times. In contrast to the single use restriction, finite copying is a dynamic restr... |

32 | Hyperedge Replacement: Grammars and Languages, volume 643 - Habel |

32 | Benefits of tree transducers for optimizing functional programs - Kühnemann - 1530 |

32 |
Fundamental Properties of Infinite Trees, Theoret
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Citation Context ...itution) second-order substitution is associative, i.e., that t _ s _ s$ =t _ s _ s$ and if _${_ then t _ s _$ s$ =t _$ s$, _ s _$ s$ , and similarly for the general case (cf. Sections 3.4 and 3.7 of =-=[Cou83]-=-). The notion of second-order substitution is closely related to that of a tree homomorphism; associativity of second-order substitution corresponds to the fact that tree homomorphisms are closed unde... |

30 | A comparison of tree transductions defined by monadic second order logic and by attribute grammars
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Citation Context ... the input and output graphs to be (node-labeled, ordered) trees, then we obtain a function from trees to trees, i.e., a tree translation. In This work was supported by the EC TMR Network GETGRATS. 1 =-=[BE98]-=- the class of MSO definable tree translations was investigated and it was proved that it equals the class of tree translations realized by attributed tree transducers (ATTs) with look-ahead which are ... |

30 | A logical characterization of the sets of hypergraphs defined by hyperedge replacement grammars - Courcelle, Engelfriet - 1995 |

29 |
Increasing modularity and language-independency in automatically generated compilers
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Citation Context ...are a variation of attribute grammars in which all attribute values are trees (see [Ful81, FV98]), and the sur property is a well-known restriction on the rules of an attribute grammar (introduced in =-=[Gan83]-=-): each attribute is used at most once. Like for attribute grammars, the class of translations realized by sur ATTs is closed under composition, which does not hold for unrestricted ATTs (see also [Ku... |

29 |
On multiple context-free grammars. Theoretical Computer Science 88(2):191–229
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Citation Context ...ges this equivalence is proved in Section 4 of [vV96], and also in [Wei92] using different formalisms. It is also easy to see (cf. [vV96]) that this is the class of multiple context-free languages of =-=[SMFK91]-=-. These equivalences are discussed in more detail in Section 6 of [Eng97] (where the tree grammars of [Rao97] are called multiple regular tree grammars). Instead of taking regular tree languages as in... |

26 |
The string generating power of context-free hypergraph grammars
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- 1991
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Citation Context ...guages generated by contextfree graph grammars equals the class MSOTS (REGT ) of MSO definable tree-to-string translations applied to regular tree languages. As a corollary we get a result known from =-=[EH91]-=- (cf. Section 6 of [Eng97]). 50 Corollary 7.8 yT fc (REGT ) is the class of string languages generated by (HR or NR) context-free graph grammars. It is easy to see, cf. Example 1(6, yield) of [BE98], ... |

24 |
Three hierarchies of transducers
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- 1982
(Show Context)
Citation Context ...Eng97]. Note also that yT (REGT ) is properly included in yMTT (REGT ); as an example, the language f(a n b) 2 n j ns1g is obviously in yMTT (REGT ), but not in yT (REGT ) as shown in Theorem 3.16 of =-=[Eng82]-=-. Weak Finite Copying Our definition of finite copying for MTT R s is a generalization of the one for top-down tree transducers. It does not distinguish states which contribute to the output from thos... |

22 | Node replacement graph grammars - Engelfriet, Rozenberg |

22 |
Berechnungsstärken von Teilklassen primitiv-rekursiver Programmschemata
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- 1997
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Citation Context ...83]): each attribute is used at most once. Like for attribute grammars, the class of translations realized by sur ATTs is closed under composition, which does not hold for unrestricted ATTs (see also =-=[Kuh97]-=-). This closure property remains when look-ahead is added. The look-ahead of an ATT can be understood as a preprocessing attribute grammar, all attributes of which are finite-valued and which merely r... |

21 |
High level tree transducers and iterated pushdown tree transducers
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- 1988
(Show Context)
Citation Context ...anslations we use the subscript mon to denote the restriction of X to translations with monadic output alphabets. Note that an alternative proof of pMTT R mon = yT R can be found in Theorem 8.7(a) of =-=[EV88]-=-. Lemma 7.6 pMTT R mon = yT R and pMTT R fc;mon = yT R fc . Proof. Let M be a T R and let M 0 be an MTT R mon . We may assume that q 0 does not appear in any right-hand side of a rule of M . Furthermo... |

18 | Oostrom, Logical description of context-free graphlanguages - Engelfriet, van - 1997 |

17 |
Rational tree relations
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Citation Context ...t T R sur = T R fc . 2 If we apply the classes in Theorem 7.4 to REGT, then we obtain the following corollary. Corollary 7.5 MSOTT dir (REGT ) = ATT os;sur (REGT ) = T ssur (REGT ) = T fc (REGT ). In =-=[Rao97]-=- a special type of tree grammar is investigated which generates tuples of trees. It is straightforward to see that these grammars correspond to sur ATTs which have synthesized attributes only (cf. [vV... |

16 |
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Citation Context ...mber of parameters, whereas a synthesized attribute may depend on any number of inherited attributes (depending on the input subtrees). For a particular subclass of ATT, called absolutely noncircular =-=[KW76]-=-, the set of inherited attributes on which a synthesized attribute depends is fixed for every input symbol. Then, M can be constructed straightforwardly [CF82]. For a general ATT A an MTT M can be con... |

15 | Tree transducers and syntax-directed semantics - Engelfriet - 1981 |

15 |
The translation power of top-down tree-to-graph transducers
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Citation Context ...ch that hq; si(y 1 ; : : : ; y m ) ) M t. Note that, for s 2 T \Sigma ,sM (s) = M q 0 (s). The q-translations of trees in T \Sigma can be characterized inductively as follows. Lemma 3.4 (Lemma 4.8 of =-=[EV94]-=-) Let M = (Q; P; \Sigma; \Delta; q 0 ; R; h) be an MTT R . For every q 2 Q, oe 2 \Sigma (k) , ks0, and s 1 ; : : : ; s k 2 T \Sigma , M q (oe(s 1 ; : : : ; s k )) = rhs(q; oe; hh(s 1 ); : : : ; h(s k ... |

15 |
Tree transducers with external functions. Theoret
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Citation Context ... this paper we always mean noncircular ATTs when referring to ATTs. If an ATT is noncircular, then the derivation relation on any tree root(s), s 2 T \Sigma , is confluent and terminating (see, e.g., =-=[FHVV93]-=-). Thus, every attribute instance hc; vi 2 hAtt; Occ(root(s))i has a unique normal form nf() A;root(s) ; hc; vi) 2 T \Delta ; intuitively, this is the value of the attribute c at node v. Let us now de... |

14 | Linear context-free rewriting systems and deterministic tree-walking transducers
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(Show Context)
Citation Context ...c (REGT ) is the class of tree languages which they generate, i.e., ATT os;sur (REGT ). For the corresponding classes of yield languages this equivalence is proved in Section 4 of [vV96], and also in =-=[Wei92]-=- using different formalisms. It is also easy to see (cf. [vV96]) that this is the class of multiple context-free languages of [SMFK91]. These equivalences are discussed in more detail in Section 6 of ... |

9 |
A characterization of attributed tree transformations by a subclass of macro tree transducers. Theory Comput
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Citation Context ...the oe-rules of M , then this construction only works if each occurrence has the same trees t 1 ; : : : ; t m as parameters (or if different parameters will be deleted during the derivation of M ; in =-=[FV97]-=- a characterization of ATT in terms of MTTs is given which is based on this observation). If M is ssuri, then there is at most one occurrence of hq; x i i in the oe-rules of M and hence we can constru... |

8 | Tree languages generated by context-free graph grammars
- Engelfriet, Maneth
- 2000
(Show Context)
Citation Context ...in [Dre97]. In fact, the results of [Dre97] can be used to obtain, in a more direct way, the above characterization by MTTs of the tree languages generated by context-free graph grammars, as shown in =-=[EM]-=-. This paper is structured as follows. Section 2 contains basic notions concerning trees, tree translations, and tree languages. In Section 3 we recall the notions of macro tree transducer and attribu... |

7 | A characterization of the sets of hypertrees generated by hyperedge-replacement graph grammars
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(Show Context)
Citation Context ...or HR and NR). By our results this is the class of output languages of sur (or, equivalently, finite-copying) MTTs taking regular tree languages as input. A related result has recently been proved in =-=[Dre97]-=-. In fact, the results of [Dre97] can be used to obtain, in a more direct way, the above characterization by MTTs of the tree languages generated by context-free graph grammars, as shown in [EM]. This... |

7 | Attributs sémantiques et schémas de programmes - Franchi-Zannettacci - 1982 |

7 |
Context-free graph grammars, in: Handbook of Formal Languages
- Engelfriet
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(Show Context)
Citation Context ...anslate a monadic tree of size n into the full binary tree of size 2 n &1). As discussed in the Introduction, it is known from [EvO97, CE95] that the class of context-free graph languages (see, e.g., =-=[Eng97]-=-) can be obtained by applying MSO graph transductions to regular tree languages. Hence, the class of tree languages generated by context-free graph grammars equals the class MSOTT (REGT) of MSO defina... |

7 | The generating power of total deterministic tree transducers
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(Show Context)
Citation Context ...ing the regular tree language T 7 of all trees over 7 as input. For a class X of tree transducers we denote by OUT(X ) the class of output tree languages generated by X. It is shown in Theorem 4.7 of =-=[Man98]-=- that every regular tree language is the output tree language of a ``semi-relabeling.'' Since every semi-relabeling is a strongly sur top-down tree transducer (cf. the discussion in the beginning of S... |

6 | The formal power of one-visit attribute grammars - Engelfriet, File - 1981 |

5 |
On the generating power of deterministic tree transducers
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(Show Context)
Citation Context ...taking the regular tree language T \Sigma of all trees over \Sigma as input. For a class X of tree transducers we denote by OUT (X) the class of output tree languages generated by X. It is known from =-=[Man96]-=- that every regular tree language is the output tree language of a finite copying top-down tree transducer with input copying bound 1. Hence, REGT ` OUT (T fc ). It follows that for each class X which... |

4 | Generalized context-free grammars
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(Show Context)
Citation Context ...at sts M (i 1 ; i) \Delta \Delta \Delta sts M (i n ; i) is a permutation of the elements of hhQ; fp i gii that appear insM q0 (s[ui / p i ]), i.e., of the elements in sts M (s; ui). 2 In Lemma 5.3 of =-=[vV96]-=- it is proved that yT fc (REGT ) ` yT ssur (REGT ). The idea is to use as states of the ssur top-down tree transducer M 0 the state sequences of the finite copying top-down tree transducer M (with a b... |

3 |
Attributed tree transducers cannot induce all deterministic bottom-up tree transformations
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(Show Context)
Citation Context ...can be understood as a preprocessing attribute grammar, all attributes of which are finite-valued and which merely relabels each node of the input tree. Note that ATTs are not closed under look-ahead =-=[FV95]-=-, and in particular that not every translation realized by a sur ATT with look-ahead can be realized by an ATT without look-ahead. In this paper we want to characterize the class of MSO definable tree... |

1 | Hyperedge replacement graph - Drewes, Kreowski, et al. - 1997 |

1 |
Handbook of Graph Grammars and Computing by Graph Transformation
- grammars
- 1997
(Show Context)
Citation Context ...for HR and NR). By our results this is the class of output languages of sur (or, equivalently, finitecopying) MTTs taking regular tree languages as input. A related result has recently been proved in =-=[Dre97]-=-. In fact, the results of [Dre97] can be used to obtain, in a more direct way, the above characterization by MTTs of the tree languages generated by context-free graph grammars, as shown in [EM]. This... |