Ferenc Gecseg and Magnus Steinby. Tree Languages. In: Grzegorz Rozenberg and Arto Salomaa, editors, Handbook of Formal Languages III, chapter 1, pages 1--68. Springer-Verlag, Heidelberg, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....subsets of V . The lowercase identi ers are rst order variables ranging over the elements of V . Notation X i (z) means that z is an element of the set X i . The predicate s i (x; y) means that hx; yi 2 s i holds in the graph G. For the precise de nition of monadic second order logic see e.g. [10], pp28. 9X0 ; Xk 1 : partit(X0 ; Xk 1 ) singl(X0 ; null) singl(X1 ; root) 8x 0 j k (X j (x) P j (x) 1) P j (x) P P j (x) 8y: s i (x; y) 0 l k hx j ;x l i2s X l (y) singl(X; y) 8z: X(z) y = z) partit(Y0 ; Yn 1) 8x: ....

Ferenc Gecseg and Magnus Steinby. Tree languages. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages. Vol. III: Beyond Words, chapter 1. Springer, 1997.

....1 ; A 2 ; W] A 3 ; B] Intuitively, on the right hand side of this production the lower right subsquare is refined one level further than the other three. Productions of this kind are, however, not essential, due to the following well known normal form result on regular tree grammars (see, e.g. [GS97]) Fact 7. For every regular tree grammar G one can effectively construct a regular tree grammar G = N; Sigma ; P; S) such that L(G ) L(G) and P contains only productions of the form A : f(A 1 ; A n ) where f 2 Sigma , rank Sigma (f) n, and A; A 1 ; A n 2 ....

....of a rule in R with the right hand side of that rule. The set Acc(ta) of accepted terms is given by Acc(ta) ft 2 T Sigma j t Gamma q for some q 2 Q f g. A finite bottom up tree automaton is a bottom up tree automaton with only finitely many states. Due to the following fact (see, e.g. [GS97]) finite bottom up tree automata are a useful tool if one aims at computability results concerning sets generated by regular tree grammars. 7 Fact 9. 1. A tree language T is regular if and only if T = Acc(ta) for some finite bottom up tree automaton ta. Moreover, there are effective ....

Ferenc G'ecseg and Magnus Steinby. Tree languages. In Grzegorz Rozenberg and Arto Salomaa, editors, Handbook of Formal Languages, volume 3, pages 1--68. Springer, Berlin, 1997.

....de nitions which capture the same constraints. This construction introduces a simple role each closed set of partial roles, similar to the construction showing the equivalence of deterministic and nondeterministic nite state automata [61] or deterministic and nondeterministic nite tree automata [34, 15]. Construction is complicated by the form of our slot constraints, but can be done by introducing additional roles that simulate slot constraint conjunction. The ability to perform conjunction of slot constraints is an easy consequence of the equivalence of slot constraints with the generalized ....

Ferenc Gecseg and Magnus Steinby. Tree languages. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages. Vol. III: Beyond Words, chapter 1. Springer, 1997.

....in compiler construction (for the latter, see the recent book by Fulop and Vogler [8] Since that time it has turned out that tree transducers are a useful tool for many other areas, too, and their properties and extensions have been studied by a variety of authors. For references see, e.g. [9, 8]. For the most part of this paper top down tree transducers are studied. As mentioned above, they can be seen as a generalisation of finite state string transducers (also called generalised sequential machines) to trees 1 . Like those, top down tree transducers are one way devices which process ....

Ferenc G'ecseg and Magnus Steinby. Tree languages. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages. Vol. III: Beyond Words, chapter 1, pages 1--68. Springer, 1997.

....the meaning of trees, but it does not say how these trees are to be generated. The main classes of tree generators considered in the following are regular and contextfree tree grammars, and top down tree transducers. These notions were first studied by Rounds and Thatcher in [31, 32, 33, 35] see [19, 20] for an introduction to the field and for many further references. In order to obtain the main results of this paper regular tree grammars and top down tree transducers will turn out to be sufficient. Context free tree grammars will only occur in examples showing possible generalisations. 3.1 ....

Ferenc G'ecseg and Magnus Steinby. Tree languages. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages. Vol. III: Beyond Words, chapter 1, pages 1--68. Springer, 1997.

....construction (for the latter, see the recent book by Fulop and Vogler [FV98] Since these times it has turned out that top down tree transducers are a useful tool for many other areas, too, and their properties and extensions have been studied by a variety of authors. For references see, e.g. [GS97,FV98]. As mentioned above, top down tree transducers are a generalisation of finitestate string transducers (also called generalised sequential machines) to trees 1 . Like these, top down tree transducers are one way machines which process their input in one direction, using a finite number of ....

Ferenc G'ecseg and Magnus Steinby. Tree languages. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages. Vol. III: Beyond Words, chapter 1, pages 1--68. Springer, 1997.

....display which is designed to visualize graphs will usually not be able to display the values obtained from an algebra over integer numbers. Presently, the main types of tree grammars, tree transducers, algebras, and displays that have been implemented are 1 ffl regular tree grammars (cf. [GS97]) ffl top down tree transductions (cf. e.g. Rou70, Tha70a, Eng75, GS84, GS97] and the YIELD transduction (cf. EV85] ffl algebras on truth values, strings, integer numbers, and terms (the free term algebra and the YIELD algebra) for chain code pictures (cf. MRW82] and turtle graphics ....

....will usually not be able to display the values obtained from an algebra over integer numbers. Presently, the main types of tree grammars, tree transducers, algebras, and displays that have been implemented are 1 ffl regular tree grammars (cf. GS97] ffl top down tree transductions (cf. e.g. [Rou70, Tha70a, Eng75, GS84, GS97]) and the YIELD transduction (cf. EV85] ffl algebras on truth values, strings, integer numbers, and terms (the free term algebra and the YIELD algebra) for chain code pictures (cf. MRW82] and turtle graphics [PL90] and for the type of pictures generated by collage grammars [HK91] and ....

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Ferenc G'ecseg and Magnus Steinby. Tree languages. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages. Vol. III: Beyond Words, chapter 1, pages 1--68. Springer, 1997.

....(n) instead of f . The set T Sigma of terms over Sigma is defined as usual, i.e. it is the smallest set such that f 2 T Sigma for every f (0) 2 Sigma and g[t 1 ; t n ] 2 T Sigma for every g (n) 2 Sigma (n 1) and all t 1 ; t n 2 T Sigma . A regular tree grammar (cf. [GS97]) is a tuple g = N; Sigma ; P; S) such that N is a finite set of nonterminals considered as symbols of rank 0, Sigma is a signature disjoint with N , P is a set of term rewrite rules of the form A t where A 2 N and t 2 T Sigma [N , and S 2 N is the start symbol. The rules in P are also ....

Ferenc G'ecseg and Magnus Steinby. Tree languages. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages. Vol. III: Beyond Words, chapter 1, pages 1--68. Springer, 1997.

.... a state transition is decided upon a character read from the input and a tuple of prior states (instead of exactly one prior state in the case of finite state automata) There are several types of tree automata and several ways to introduce them, e.g. consult Doner [12] G esceg and Steinby [20, 21] or Common et al. 10] We are particularly interested in finite bottom up tree automata and essentially follow the formalization of Schalkoff [59] or Miclet [50] Definition 2 (Bottom Up Tree Automata) A finite bottom up tree automata (or frontier toroot automata) is a quadruplet ( Sigma; Q; F; ....

Ferenc G'esceg and Magnus Steinby. Tree Languages. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume 3, pages 1--68. Springer, 1996.

....the class yR TBY of string languages which are yields of tree languages in R TBY is considered as an interesting special case. In particular, it is proved that yR TBY is a substitution closed full AFL. We assume the reader to be familiar with elementary tree language theory, see, e.g. [GS84, GS97]. 2 Preliminaries 2.1 Basic mathematical notation The set of all natural numbers (including 0) is denoted by N , N = N nf0g, and [n] f1; ng for every n 2 N . For a set S, S) denotes the powerset of S and jSj denotes its cardinality. The length of a finite sequence w is denoted by ....

....1 Delta Delta Delta s 0 k x k 1 Delta Delta Delta xm ] fl 0 s 0 ; which completes the proof. Next, some results concerning regular tree languages are recalled, starting with the definition of regular tree grammars. 3. 12 Definition (regular tree grammar and regular tree language, see [GS84, GS97]) A regular tree grammar is a tuple G = Sigma; Gamma; R; fl 0 ) where Sigma is a finite signature, Gamma is a finite signature of states (or nonterminals) of rank zero, Sigma Gamma = R Gamma Theta T Sigma ( Gamma) is a finite set of rewrite rules, and fl 0 2 Gamma is the initial ....

Ferenc G'ecseg and Magnus Steinby. Tree languages. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages. Vol. III: Beyond Words, chapter 1, pages 1--68. Springer, 1997.

....several other authors. In this paper, we will mainly consider the variant defined and investigated by Engelfriet in [Eng77] see also [Eng78, ERS80, FV89, SV95, GV96] called top down tree transducer with regular look ahead. For surveys on the theory of tree languages and tree transductions see [GS84, NP92, GS97]. Throughout the paper N denotes the set of natural numbers (including 0) and N = N nf0g. For n 2 N , n] denotes the set f1; ng. A sequence of length n 2 N over a set A is a mapping l : n] A. The length n of l is denoted by jlj, denotes the empty sequence (i.e. the one of length ....

....1 ; t n ] where t 0 contains exactly one occurrence of x 1 , then t R t 0 where t 0 = t 0 [ s 0 [ t 1 ; t n ] Next, the concepts of regular tree grammars and top down tree transducers with regular look ahead are recalled. 2. 1 Definition (regular tree grammar, cf. [GS97]) A regular tree grammar is a system g = N; Sigma; P; S) where N and Sigma are disjoint finite signatures, all symbols in N (the nonterminals) have rank 0, P N Theta T Sigma[N is a finite set of term rewrite rules (the productions) and S (the start symbol) is in N . The set of terms ....

Ferenc G'ecseg and Magnus Steinby. Tree languages. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages. Vol. III: Beyond Words, chapter 1, pages 1--68. Springer, 1997.

.... Recently, recursive neural networks have been proven to possess the computational power of at least Frontier toRoot Automata (Sperduti [15] Kuchler [16] This machine model for tree processing is known to be a generalization of the FSA concept for sequence processing (G esceg and Steinby [17] [18]) These results raise the question about the node complexity of recursive neural network implementations of FRAO. Can the previous methods and results be lifted from sequence to tree processing Furthermore, the consideration of recursive neural networks might also give new insights on the ....

....unfortunately, we conjecture to be NP complete. II. Background A. Tree Automata The computation model of tree automata is well understood and applied in several fields of computer science. There are several types of tree automata and several ways to introduce them (G esceg and Steinby [17] [18]) Here it is convenient to define an analogous extension to what is known as Mealy machines (Hopcroft and Ullman [24] in the case of sequence processing, i.e. automata that map trees to trees of the same shape. Definition 1: A (deterministic) frontier to root tree automaton 1 with output ....

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Ferenc G'esceg and Magnus Steinby, "Tree Languages", in Handbook of Formal Languages, G. Rozenberg and A. Salomaa, Eds., vol. 3, pp. 1--68. Springer, 1996.

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Ferenc Gecseg and Magnus Steinby. Tree Languages. In: Grzegorz Rozenberg and Arto Salomaa, editors, Handbook of Formal Languages III, chapter 1, pages 1--68. Springer-Verlag, Heidelberg, 1997.

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Ferenc Gcseg and Magnus Steinby. Tree languages. In Grzegorz Rozenberg and Arto Salomaa, editors, Handbook of Formal Languages: Beyond Words, volume 3, pages 1--68, Berlin, 1997. Springer

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