Ferenc G'ecseg and Magnus Steinby. Tree Automata. Akad'emiai Kiad'o, 1984.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... F . A tree transformation f T# is called total if dom(f) T# . 2.5 Relabeling tree transducers Now we define the relabeling tree transducers considered in this paper. The reader who wants to know more details about relabeling tree transducers is recommended to consult with [Eng75] [GS84], and [GS97] We should mention, our relabelings in [Eng75] were called finite state relebalings. 4 A top down relabeling is a tuple r = Q, #, #, Q # , R) where Q is a unary ranked alphabet, called the set of states; # and # are ranked alphabets, called the input and output ranked alphabet, ....

....are denoted by FBRel, DBRel, and BHom sa , respectively. Analogously, we have BHom sa DBRel FBRel BRel. 2.6 Properties of relabelings Now we recall some well known properties of the tree transformation classes introduced above. For more details the reader can consult with [Eng75] and [GS84]. Fact 1: The tree transformation classes induced by the top down and by the bottom up relabelings are the same, i.e. TRel = BRel. In what follows we will denote this class by Rel. Also FTRel = FBRel, which class will be denoted by FRel. Moreover, when speaking about these classes, we drop the ....

[Article contains additional citation context not shown here]

F. Gecseg, M. Steinby, Tree Automata, Akademiai Kiado, Budapest, 1984.

....t i , tn 1 ) If R n 1, then the i th cylindrification of R is the relation R n 1 defined by (t1 , t i 1 , t, t i , t n ) R(t1 , t i 1 , t i , t n ) We summarize here results on tree automata that we use. More details can be found in [9, 15]. Theorem 15 (Decidable Emptiness) The emptiness problem for tree automata is decidable. In fact, it can be decided in linear time in the size of the automaton. Theorem 16 (Closure Properties) Tree automata are closed under intersection, union, complementation, cylindrification, and projection. ....

F. Gecseg and M. Steinby. Tree Automata. Akademiai Kiado, 1984.

....feature trees (or multitrees) is equational iff it is recognizable. Proof Since J recognizability corresponds to the characterization by congruence relations, and Theorem 4. 1 covers the case of feature trees of depth 1, the proof can be done following the standard one for first order trees (cf. [GS84]) 2 10 7 Conclusion and Further Work The results of this paper together present a complete regular theory of feature trees. They offer a solution to the concrete practical problem of computing with types denoting sets of feature trees as described in the introduction. Now, it is interesting to ....

F. G'ecseg and M. Steinby. Tree Automata. Akad'emiai Kiad'o, Budapest, 1984. 12

....a e. 2 Remark 4. Mishra [30] uses a class of set constraints with a non standard interpretation over non empty path closed sets of finite trees to approximate the success set of a logic program. A set of trees is path closed if it can be recognized by a deterministic top down tree automaton [20]. Set constraints over non empty path closed sets also have the properties 1. and 2. above. Due to the non standard interpretation, this holds even if n ary constructor terms are allowed on the left side of the inclusion. For example, the constraint f(x; y) f(a; a) f(b; b) has a greatest ....

F. G'ecseg and M. Steinby. Tree Automata. Akademiai Kiado, 1984.

.... we again obtain a double exponential upperbound, which is a significant improvement over the non elementary upperbound of the complexity of the decision procedure presented in [9] 2 Preliminaries We assume the reader is familiar with the basics of term rewriting ( 1, 7, 14] and tree automata ([4, 10]) We recall the following definitions from [9] We refer to the latter paper for motivation and examples. Let R be a TRS over a signature F . The sets of ground redexes, ground normal forms, and root stable forms of R are denoted by REDEXR , NFR , and RSR . Let R ffl be the TRS R[ fffl fflg ....

F. G'ecseg and M. Steinby. Tree Automata. Akad'emiai Kiad'o, Budapest, 1984.

....v0, v(m 1) #(v0) #(v(m 1) V (v) #(v) #. The automaton A accepts t if there is a run # of A on t such that #(#) is the set of # labeled finite trees accepted by A. # Finite tree automata are closed under Boolean operations and are decidable in polynomial time [58]. This notion of finite tree is too general for our purposes, since our treelike structures have always a bounded branching degree. Therefore, we introduce the following definition of at most k ary finite tree. A set D T k is an at most k ary tree domain if: T k , xi T k and j ....

F. Gecseg and M. Steinby. Tree Automata. Akademiai Kiado, Budapest, 1984.

....T is a finite subset of CS ) and the right hand side of each production in P is an expression over CS and N , of the same type as the left hand side. Obviously, the context free language L(G) generated by G is a set of expressions over CS (and it is also a regular tree language, see [GecSte]) The graph language 13 generated by G is val(L(G) val(e) L(G) Notethattype(val(L(G) type(S) By Val(CFG(CS) we denote the class of all graph languages generated by context free graph grammars over CS. By the results of [MezWri] it is the class of equational subsets of the ....

....28, each symbol b B has type (1, 1) in val(L(G) and has an arbitrary type with respect to h. We may assume that G is in normal form, i.e. that all its productions are c g or X #Z or X #Z,whereX,Y,Z # N and g (this is the usual normal form of regular tree grammars, see, e.g. [GecSte]) We have to construct a context free graph grammar G # = N # ,T # ,P # ,S # ) such that val(L(G # ) h(L) The idea of the construction is the same as in the proof of Theorem 28: each graph gr(w)forwhichh(w) is defined, is transformed into exp(gr(w) where is the unique decoration of ....

F.Gecseg, M.Steinby; Tree Automata,Akademiai Kiado, Budapest, 1984

....are non terminals and c, d are terminals. For every set variable X, we get a corresponding tree grammar describing the set X. This description is sufficiently explicit because there are standard algorithms for membership, emptiness, intersection and other properties of regular tree grammars[8]. 4.3 Solving set constraints with projections and intersections In this section, we give the transformations for solving systems of set constraints in variable expression form from [9] A set constraint E 1 E 2 is in variableexpression form if its right side E 2 is just a set variable. The ....

F. Gecseg and M. Steinby. Tree Automata. Akademiai Kiado, Budapest, 1984.

....that capture the e ects of extended forms. After simpli cation, program points whose associated nonterminals do not have a right side good form are identi ed as dead. All the production forms here are the same as or similar to those studied by many people. For example, standard forms are as in [17, 25, 11], copy forms are common in grammars, selector forms are rst seen in [25] and conditional forms have counterparts in [4, 20] Overall, the constraints and simpli cations rules here extend those by Jones and Muchnick [25] Notation. We use a set based language. It is based on SETL [55, 56] ....

F. Gecseg and M. Steinb. Tree Automata. Akademiai Kiado, Budapest, 1984.

....a right side good form are identi ed as dead. Appendix A gives a small example program together with the constructed grammar, a user query, and the simpli cation result. All the production forms here are the same as or similar to those studied by many people. For example, standard forms are as in [14, 22, 9], copy forms are common in grammars, selector forms are rst seen in [22] and conditional forms have counterparts in [3, 17] Overall, the constraints here extend those by Jones and Muchnick [22] Notation. We use a set based language. It is based on SETL [41, 42] extended with a xed point ....

F. Gecseg and M. Steinb. Tree Automata. Akademiai Kiado, Budapest, 1984.

....nil( cons(D; cons(D; nil( g. Applying each element to a given value, say, cons(2; cons(4; nil) yields f ; cons( cons( nil) g, of which cons( cons( nil) is the least upper bound. Formally, the grammars we use to represent liveness patterns are regular tree grammars [18], which allow bounded, and often precise, representations of unbounded structures [27,43,44,2,55,9,50] A regular tree grammar based liveness pattern, called liveness grammar in short, denoted G, is a quadruple hT ; N ; P; Si, where T is a set of terminal symbols including L, D, and all possible ....

....a set of terminal symbols including L, D, and all possible constructors c, N is a set of nonterminal symbols N , P is a set of productions of the form: N D; N L; or N c n (N 1 ; N n ) 6) and nonterminal S is the start symbol. So, our liveness grammars use general regular tree grammars [27,18,9] extended with the special constants D and L. A sentence generated by a liveness grammar is a liveness pattern. Let language LG be the set of sentences generated by G. The projection function that G represents, denoted [ G] is: G] x) tf (x) j 2 LG g (7) where t is the least upper bound ....

[Article contains additional citation context not shown here]

F. Gecseg and M. Steinb. Tree Automata. Akademiai Kiado, Budapest, 1984.

....obtain are valid in the general case (section 4.4) Finally, we extend the previous result to bottom up transducers (section 4.5) 2 Preliminaries Main definitions and results about tree transducers can be found in J.Engelfriet s papers ( 7] 8] 10] and in the book of F.Gecseg and M. Steinby [13]. In this section, we just give basic definitions and properties used in the paper. 2.1 Trees A ranked alphabet is a pair ( Sigma; ae) where Sigma is a finite alphabet and ae is a mapping from Sigma to IN. Usually, we will write Sigma for short. For any oe in Sigma, ae(oe) is called the rank ....

....forest with the tree transformation we consider and conversely. Thus, we inherit the good closure and decidability properties of recognizable forests. Property. The class REC of recognizable forests is effectively closed under union, intersection and complementation and emptiness is decidable [13]. Closure of T LAB under union. Let T 1 and T 2 be two relabelings. With T 1 and T 2 we associate the automata A 1 and A 2 (as defined before) So b T 1 [ b T 2 = f(t; u) t; u) 2 b T 1 or (t; u) 2 b T 2 g = f(t; u) t; u] 2 F (A 1 ) or [t; u] 2 F (A 2 )g = f(t; u) t; u] 2 F (A 1 ) ....

[Article contains additional citation context not shown here]

F. Gecseg and M. Steinby. Tree automata. Akademiai Kiado, Budapest. 1984.

....denotes the set f1; pg. A torsion from [p] to [q] is a mapping from [p] to [q] We denote it by q; 1) p) 2.2 Letter to Letter Top down Tree Transducers Main definitions and results can be found in J.Engelfriet s papers ( 7] 8] and in the book of F.Gecseg and M. Steinby [10]. We just give the definitions and properties used in this paper. Definitions. A top down tree transducer is a 5 tuple T = Sigma ; Delta; Q; I; R where Sigma and Delta are the ranked alphabets of respectively input and output symbols, Q is the finite set of states, I is the subset of Q ....

F. Gecseg and M. Steinby. Tree automata. Akademiai Kiado, Budapest. 1984.

....new classes is based on the decidability of emptiness for tree automata with equalities or disequalities on direct subterms [5] 2 Preliminaries In this section, we just recall definitions and properties used in the following. We refer the reader to [11] for tree rewriting systems and to [7] and [10] for tree transducers. A ranked alphabet is a pair ( Sigma; ae) where Sigma is a finite alphabet and ae is a mapping from Sigma to IN. Usually, we will write Sigma for short. For any oe of Sigma, ae(oe) is called the rank of oe. For any integer n, Sigma n denotes the subset of Sigma of ....

....and nondeleting letter to letter rewriting system (two trees being equivalent if and only if their irreducible forms are equal) for the so defined class of tree automata, denoted by REC , equivalence is decidable. 3. 1 The Class REC A good survey on classical tree automata can be found in [10]. An equivalence description on n elements is a partition of [n] Let j be an equivalence relation on T Sigma and let d be an equivalence description on [n] A tuple of terms (t i ) i2[n] satisfies the equivalence description d if and only if for any X 2 d, for any i and j 2 X, t i j t j , and ....

F. Gecseg and M. Steinby. Tree automata. Akademiai Kiado, Budapest. 1984.

No context found.

Ferenc G'ecseg and Magnus Steinby. Tree Automata. Akad'emiai Kiad'o, 1984.

No context found.

Ferenc Gecseg and Magnus Steinby. Tree Automata. Akademiai Kiado, Budapest, 1984.

No context found.

Ferenc Gecseg and Magnus Steinby. Tree Automata. Akademiai Kiado, Budapest, 1984.

No context found.

F. Gecseg and M. Steinby. Tree Automata. Akademiai Kiado, 1984.

No context found.

F. G'ecseg and M. Steinby. Tree Automata. Akad'emiai Kiad'o, Budapest, 1984.

No context found.

Ferenc Gecseg and Magnus Steinby. Tree Automata. Akademiai Kiado, Budapest, 1984.

No context found.

F. Gecseg and M. Steinby. Tree Automata. Akademiai Kiado, 1984.

No context found.

F. G'ecseg and M. Steinby. Tree Automata. Akad'emiai Kiad'o, 1984.

No context found.

F. G'ecseg and M. Steinby. Tree Automata. Akademiai Kiado, Budapest, 1984.

No context found.

Gecseg, F., Steinby, M.: Tree Automata. Akademiai Kiado, Budapest (1984)

No context found.

Ferenc Gecseg and Magnus Steinb. Tree Automata. Akademiai Kiado, Budapest, 1984.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC