| { | |
| "paper_id": "C82-1012", | |
| "header": { | |
| "generated_with": "S2ORC 1.0.0", | |
| "date_generated": "2023-01-19T13:13:06.963111Z" | |
| }, | |
| "title": "", | |
| "authors": [ | |
| { | |
| "first": "Werner", | |
| "middle": [], | |
| "last": "Dilger", | |
| "suffix": "", | |
| "affiliation": { | |
| "laboratory": "", | |
| "institution": "Universit~t Kaiserslautern Fachbereich Informatik", | |
| "location": { | |
| "postCode": "D-6750", | |
| "settlement": "Kaiserslautern", | |
| "country": "FR Germany" | |
| } | |
| }, | |
| "email": "" | |
| } | |
| ], | |
| "year": "", | |
| "venue": null, | |
| "identifiers": {}, | |
| "abstract": "Tree directed grammars as a special kind of translation grammars are defined. It is shown that a loop-free tree directed grammar can be transformed into an equivalent top-down tree transducer, and from this fact it follows that given an arbitrary context-free language as input, a tree directed grammar produces an output language which is at most context-sensitive.", | |
| "pdf_parse": { | |
| "paper_id": "C82-1012", | |
| "_pdf_hash": "", | |
| "abstract": [ | |
| { | |
| "text": "Tree directed grammars as a special kind of translation grammars are defined. It is shown that a loop-free tree directed grammar can be transformed into an equivalent top-down tree transducer, and from this fact it follows that given an arbitrary context-free language as input, a tree directed grammar produces an output language which is at most context-sensitive.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Abstract", | |
| "sec_num": null | |
| } | |
| ], | |
| "body_text": [ | |
| { | |
| "text": "Within the natural language information system PLIDIS [6] a semantic processor was implemented for the translation of syntactically analyzed sentences into expressions of a predicate calculus-orlented internal representation language. This semantic processor was designed according to a translation grammar defined by Wulz [8] , which is similar to the transformation grammar introduced by Chomsky [3] . The operations on trees which are defined in thetransformation grammar, i.e. deletion, insertion, and transposition of subtrees, are also available in the Wulz grammar. Therefore it can be assumed that it is equivalent to the transformation grammar with regard to the input/output-relation.", | |
| "cite_spans": [ | |
| { | |
| "start": 54, | |
| "end": 57, | |
| "text": "[6]", | |
| "ref_id": "BIBREF5" | |
| }, | |
| { | |
| "start": 323, | |
| "end": 326, | |
| "text": "[8]", | |
| "ref_id": "BIBREF7" | |
| }, | |
| { | |
| "start": 398, | |
| "end": 401, | |
| "text": "[3]", | |
| "ref_id": "BIBREF2" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "INTRODUCTION", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "But when the Wulz grammar was realized within PLiDIS for a section of German, only of a few of its possibilities was made use. No real transformation was prescribed by the PLIDIS translation rules, they only checked the parse tree and produced an output separated from this tree. Thus, what was realized in the PLIDIS translation rules can be better described by another kind of translation grammar, namely the tree directed grammar (TDG). When we investigate the TDGs and their relation to tree transducers it turns out that they are less powerful than transformation grammars.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "INTRODUCTION", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "We define trees in the manner of [2] and [7] as mappings from tree domains (special subsets of N*, where N is the set of natural numbers) into an alphabet Z and call them therefore trees \"over\" Z. We assume for the rest of the paper that Z is ranked. Because trees are flnlte mappings it is convenient to identify a tree with its graph. If u is an element of a tree domain, a 6 ~, and t(u) = a, then the pair <u,a> is called a n0dz of t.", | |
| "cite_spans": [ | |
| { | |
| "start": 33, | |
| "end": 36, | |
| "text": "[2]", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 41, | |
| "end": 44, | |
| "text": "[7]", | |
| "ref_id": "BIBREF6" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "TREE DIRECTED GRA~tMARS", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Let T be any set of trees over E. A TDG G T for T is a quadruple", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "TREE DIRECTED GRA~tMARS", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "G T = (~,A,n,~)", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "TREE DIRECTED GRA~tMARS", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "where ~ is the alphabet of terminals of G m, ~ is the set of productions of G T, and u E E U 4. It follows fr6m this definition that the elements of E play the role of nonterminals in G T. When they are used for this purpose in the productions, they are enclosed in brackets, so we get from E the set", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "TREE DIRECTED GRA~tMARS", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "[~] = {[~]la ~ ~}", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "TREE DIRECTED GRA~tMARS", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "The elements of ~ are further used in the structural condition parts of the productions. There we should be able to distinguish between different occurrences of ~he same symbol in a tree. In order to represent such distinctions, the symbols are provided with indices, so we get from E the set EIN D = U {aila E E} iEIND for some index set IND (in general a subset of N).", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "TREE DIRECTED GRA~tMARS", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "with a q ~, e e (4 U [EIND])*, and sc is a structural condition which contains the symbol a I .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Now a production p E ~ is a triple ([al],SO,~)", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "In order to explain the application of a production we have to define the structural conditions. Assume, x E E and X = {Xl,X2,...}.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Now a production p E ~ is a triple ([al],SO,~)", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Then the set of structural indiuid~als is SI = EIN D U X There are four two-place predicates defined on SI, namely DOM (\"dominates immediately\"), DOM* (\"dominates\"), LFT (\"is immediately left from\"), and LFT * (\"is left from\"). Atomic structural conditions", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Now a production p E ~ is a triple ([al],SO,~)", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "are TRUE, FALSE, P(~,~)", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Now a production p E ~ is a triple ([al],SO,~)", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "where P is one of the four predicates above and ~,~ E SI.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Now a production p E ~ is a triple ([al],SO,~)", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "A ~t~uct~ral ~ondltZon is then an atomic structural condition or a The structural conditions are similar to the local constraints of Joshi and Levy [5] , and it can be shown that both are equivalent with regard to their ability to describe relations on the set of nodes of a tree. ~L-HAREHARE with regard to the tree of the example above.", | |
| "cite_spans": [ | |
| { | |
| "start": 148, | |
| "end": 151, | |
| "text": "[5]", | |
| "ref_id": "BIBREF4" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Now a production p E ~ is a triple ([al],SO,~)", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "A top-dow~ IAZ\u00a2 t~n~da~zr (TDTT) (cf. There are some obvious similarities between TDGs and TDTTs. It is easy to see that not every TDTT can be transformed into an equivalent TDG, because the TDTTs have the states as an additional means to direct derivations. In some cases the derivation can be directed by appropriate structural conditions in the same way as it is done by states, but i~ is easy to construct examples where this is impossible. On the other hand, each TDG can be transformed into an equivalent TDTT. The main step of this transformation is to put together some of the productions so that the resulting productions satisfy the condition that all symbols of the structural condition part except a I are situated below the symbolu I in each tree, where a I correspond~ to the first component of the @roduction. E ) The first and the second production satisfy the condition, the third one does not, because the nodes assigned to x I and x~ are above that one assigned to c I in each tree which satisfies the \u00a7tructural condition. But we ca~ put together the second and the third production and get a new one:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "TOP-DOWN TREE TRANSDUCERS i", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "([bl],DOM(bl,C 1) ^ LFT(el,c I) ^ LFT(bl,X 2) ^ DOM*(x2,el), H[el]E)", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "TOP-DOWN TREE TRANSDUCERS i", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "NOW this production is \"better\" than the third above, but it does not yet satisfy our condition. Therefore we put it together with the first one and get it performs the same derivation as the four productions above. The productions resulting from this transformation process are all proceeding downward in a tree. Each of them can be transformed into a TDTT of its own and finally these single TDTTs are composed to one TDTT which is equivalent to the TDG.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "TOP-DOWN TREE TRANSDUCERS i", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "The transformation process sketched above can be made only if the TDG is loop-free. That means that each node of a tree is passed during a derivation in TDG at most once. Now we can adopt the result of Baker [I] about top-down tree transductions. It states that the family of the images of recognizable sets of trees (e.g. the set of derivation trees of a context-free grammar) under a top-down transduction is properly contained in the family of deterministic context-sensltive languages. In other words, the result of t~e translation of the set of derivation trees of a context-free grammar by a TDG is at most a deterministic contextsensitive language.", | |
| "cite_spans": [ | |
| { | |
| "start": 208, | |
| "end": 211, | |
| "text": "[I]", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "TOP-DOWN TREE TRANSDUCERS i", | |
| "sec_num": null | |
| } | |
| ], | |
| "back_matter": [], | |
| "bib_entries": { | |
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| }, | |
| "ref_entries": { | |
| "FIGREF0": { | |
| "text": "So e.g. the set {<(),a>,<(O),b>,<(1),d>,<(2),a>,<(O,O),e>,<(O,1),c>, <(2,O),d>,<(2,1),b>,<(2,2),e>,<(O,1,0),e>,<(2,1,O),c>, <(2,1,1),d>,<(2,1,0,O),d>} represents the tree of fig. I.", | |
| "type_str": "figure", | |
| "num": null, | |
| "uris": null | |
| }, | |
| "FIGREF1": { | |
| "text": "Boolean expression over the set of atomic structural conditions. For example, if ~ = {a,b,c,d,e}, IND = {1,2}, then the following expressions are structural conditions: I. DOM(al,b I) 2. DOM(bl,X I) ^ LFT(el,x I) 3. DOM(Xl,C I) ^ LFT*(Xl,X 2) ^ DOM*(x2,e I) 4. DOM(al,b I) ^ (NDOM*(bl,e I) v LFT(bl,dl)) The semantics of a structural condition is defined in the usual way by an interpreting function from the condition into a semantic domain. Here, the trees of T are semantic domains. The four predicates DOM, ...,LFT \u2022 are always interpreted in the same way, and this interpretation should be obvious. The main part of the interpretation is the assignment of the structural individuals to the nodes of a tree, which is called the ,od~ ~Zgnment. A mapping of the individuals of a structural condition into the set of nodes of a tree is a node assignment, if it obeys the following restrictions: If a 6 E, then an individual ~ (i 6 IND) should be assigned to a node with label a, whereas the individuals e~ and e. (i ~ j) should be assigned to different nodes with the ~ same 3 label u. An individual x~ 6 X can be assigned to an arbitrary node. A tree t ~atZsfig~ a structural condition sc if there exists a node assignment such that sc holds for the assigned nodes of t under the assumed interpretation of the four predicates and the usual interpretation of the Boolean operators. The reader is invited to check, how the tree of the example above satisfies the structural conditions I. -4.", | |
| "type_str": "figure", | |
| "num": null, | |
| "uris": null | |
| }, | |
| "FIGREF2": { | |
| "text": "Assume, p =([ul],sc,~) is a production of G m. Then the structural individual u I m~st occur in sc. Assume further that y = y1[ai]Y2 where yl,y 2 E (~ u [ZTw_]) ~, i E IND, and there is a node assignment which m~ps e~ on a nod~U<u,a> in tree t and t satisfies sc in such a way that ~I ~s mapped on <u,a> as well, then p can be applied to y:YI[ui]Y2 ~ y1~Y2 GT,tSome of the individuals of X occurring in e may be replaced by the node assignment for sc by individuals of [~T,,~]. In this way derivations in G T with regard %o a tree taze def!~d. If a derivation stops with a word y e ~, y can be regarded as a translation of t. Assume e.g. we are given the following four productions: ([al],DOM(a1~bl),[bl][bl ]) ([bl],DOM(bl,X I) ^ LFT(el,xl),H[Xl]) ([Cl],DOM(Xl,Cl) ^ LFT(Xl,X 2) ^ DOM*(x2,el),[el]E) ([el],TRUE,AR) By means of these productions we can perform the derivation [al] P---[bl][b 1] v--H[Cl][bl] P--H[el]E[bl ] P---HARE[b 1]", | |
| "type_str": "figure", | |
| "num": null, | |
| "uris": null | |
| }, | |
| "FIGREF3": { | |
| "text": "[4]) is a transducing automaton which proceeds top-down from the root to the leaves in a tree and in each step yields an output. It is defined as a quintuple M = (Q,?-,A,qo,R) where ~ and ~ are defined as before, Q is a finite set of states, q E Q is the initial state and R is a finite set of rules of thefSrm q(u(~1...Tk)) ---> ylq1(Til)Y2q2(~i2 ) -.. Ynqn(Tin)Yn+ I with n,k z O; I ~ i. < k for I < j ~ n; q'q1'''''qn E Q, ~ E T, Yl'\" \"-'Yn+1 E &*. 3 k is the rank of ~ and the T 4 are variables over T. when SUCh a rule is applied to a tree ~ at a node with label u, the variables ~ are replaced by those subtrees of t whose roots are immediately dof~inated by the node with label a. Assume e.g. we are given the TDTT M = ({qo,ql }, {a,b,c,d,e}, {A,E,H,R},qo,R) with R = { qo(a(\u00a2lT2T3)) --~ Hq1(T3)qo(T1)Hq I(T3)qO(T 1), qo(b(TiT2 )) --> qo(\u00a22), qo(C(~1)) --~ E, ql (a(\u00a21\u00a22T3)) ---> qI(\u00a23 )' q1(e) --~ AR } M performs on the tree cf the example above the derivation qc (a (b (ec (e)) da (db (c (d) d) e) ) ) k--Hql (a(db(c(d)d)e))qc(b(ec(e)))Hq1(a(db(c(d)d)e))qo(biec(e))) ~---Hql (e) qo (b (ec (e)) ) Hql (a (db (c (d) d) e) ) qc (b (ec (e)) ) ~--HARqo (b (ec (e)) ) Hql (a (db (c (d) d) e) ) qo (b (ec (e)) ) P---HARqo (c (e)) Hql (a [db (c (d) d) e) ) qo (b (ec (e)) ) P--HAREHql(a(db(c(d)d)e))qo(b(ec(e))) ~---HAREHARE TDGs AND TDTTs", | |
| "type_str": "figure", | |
| "num": null, | |
| "uris": null | |
| }, | |
| "FIGREF4": { | |
| "text": "Take e.g. the productions ([al],DOM(al,bl),[bl][bl]) ([bl],DOM(bl,X I) ^ LFT(el,xl),H[Xl]) ([Cl],DOM(Xl,C I) a LFT(xl,x 2) ^ DOM*(x2,el),[el]", | |
| "type_str": "figure", | |
| "num": null, | |
| "uris": null | |
| }, | |
| "FIGREF5": { | |
| "text": "[al],DOM(al,b I) ^ DOM(bl,c I) ^ LFT(el,c I) ^ LFT(bl,X 2) ^ DOM*(x2,el),H[el]EH[el]E) This production is acceptable and together with the production ([el],TRUE,AR)", | |
| "type_str": "figure", | |
| "num": null, | |
| "uris": null | |
| } | |
| } | |
| } | |
| } |