| { | |
| "paper_id": "W98-0120", | |
| "header": { | |
| "generated_with": "S2ORC 1.0.0", | |
| "date_generated": "2023-01-19T06:04:52.041528Z" | |
| }, | |
| "title": "A hierarchy of local TDGs", | |
| "authors": [ | |
| { | |
| "first": "Laura", | |
| "middle": [], | |
| "last": "Kallmeyer", | |
| "suffix": "", | |
| "affiliation": {}, | |
| "email": "" | |
| } | |
| ], | |
| "year": "", | |
| "venue": null, | |
| "identifiers": {}, | |
| "abstract": "\u00fc ni versit\u00e4t T\u00fcbingen Seminar f\u00fcr Sprachwissenschaft Wilhelmstr. 113 D-72074 T\u00fcbingen, Germany lk\u00a9sfs.nphil.uni-tuebingen.de l lntrod uction '.\\.tany recent vacidllL~ of Tree Adjoining Grammars (TAG) allow an underspecilication of the parent rela.tion bet ween nodes in a t ree, i.e. they do not deaJ l'<ith fully spec1fied tre~ as it is the ca.se with TAGs. Such TAG variants are for ex'1Il1plc Descnpt1011 Tree Grammurs (DTG) (fuunbow. Vijay-Shanker and Weir 1995),", | |
| "pdf_parse": { | |
| "paper_id": "W98-0120", | |
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| "abstract": [ | |
| { | |
| "text": "\u00fc ni versit\u00e4t T\u00fcbingen Seminar f\u00fcr Sprachwissenschaft Wilhelmstr. 113 D-72074 T\u00fcbingen, Germany lk\u00a9sfs.nphil.uni-tuebingen.de l lntrod uction '.\\.tany recent vacidllL~ of Tree Adjoining Grammars (TAG) allow an underspecilication of the parent rela.tion bet ween nodes in a t ree, i.e. they do not deaJ l'<ith fully spec1fied tre~ as it is the ca.se with TAGs. Such TAG variants are for ex'1Il1plc Descnpt1011 Tree Grammurs (DTG) (fuunbow. Vijay-Shanker and Weir 1995),", | |
| "cite_spans": [], | |
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| "section": "Abstract", | |
| "sec_num": null | |
| } | |
| ], | |
| "body_text": [ | |
| { | |
| "text": "TAG variant. local TDG, is an extension of TAG geuecatiug ti:ee tl~scriptions. Local TDGs even allow an underspec11icat1on of the dominance relation betwe\u20acn node names and thereby provide the poosibility to generate underspecified representations for structuraJ ambiguities such as quantifier scope ambiguities.", | |
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| "text": "This abstract dcals with formal properties of local TDGs. A hierarchy of local TDGs is established together with a pumping lemma for local TDGs of a. certa.in rank. With this pumping lemma one ca.n prove that the class of local TDGs of a. certain rank n contains the Jruiguage L, := {a.f \u2022 \u2022 \u2022 a~ 1 k ~ O} iff i :::; 211.", | |
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| "text": "Loca.I TDGs, proposed in (Kallmeyer 1997) , consist of trce descriptions, so-callcd e/ementary descriptions, and a s pecific start d escripiion. These tree descriptions are negatian and disjunction free formulas in a quantifier-free first order logic. This logic allows the description of relations between node names k 1 , k2 such as parent reia.tion 1 i.e. im mediate dominante) k 1 <J k1, <lominance (reflexive transitive closure of the parent relation) k1 <J\" k2, linear precedence k1 -.: k2 and equality k1 ::::: k\u2022i. Furthermore, nodes are supposed to be labelled by terminals or by 76 atomic feature structuces. The \\abeling function is d~noted by o, and for a node name Je, 6(k) ~ t sigmfies that k has a terminal labe\\ t, and a.(o(k)) R: u signifies that Je is la.belled by a. feature structuce containing the attribute value pair (a, v).", | |
| "cite_spans": [ | |
| { | |
| "start": 25, | |
| "end": 41, | |
| "text": "(Kallmeyer 1997)", | |
| "ref_id": "BIBREF0" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Local TDGs", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "T'r e\u20ac des cri p t ion s in a local TD G are o f a. certai.n form , r ou gh l y spe a.k.i n g t hey consis t of ful! y specified (sub)tree descriptions that are connected by dominance relations. 1", | |
| "cite_spans": [], | |
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| "section": "Local TDGs", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "In ll.ll elementary description 1/J, some of the node names are marked (thooe in the set K\"); this is important for the derivation of descriptions. A sample local ~DG is shown in Fig. 1 Fig. 1 tary or start description, the so-called derivation description of this deri vation step ( first locali ty condition), 4. for ea.ch marked name k\" in 1.1: with a parent, there must be a strong dominance k 1 <:J* k 2 in <) 1 such that k2 ~ k-. is added and the snbdescription between k\" and the next marked or minimal name dominating k.;. must be dorninated by k 1 (second locality condition), 5. and the result rfi2 must be maximally underspecifled.", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 179, | |
| "end": 185, | |
| "text": "Fig. 1", | |
| "ref_id": "FIGREF2" | |
| }, | |
| { | |
| "start": 186, | |
| "end": 192, | |
| "text": "Fig. 1", | |
| "ref_id": "FIGREF2" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Local TDGs", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "As the first condition shows, marked names are comparable to foot nodes in an auxiliary tree in a TAG since they specify those pans of an e!ementary description 1/1 that must be connected to a derived description 4i whcn adding 1/J to <P in a derivatiou step.", | |
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| "section": "Local TDGs", | |
| "sec_num": "2" | |
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| "text": "The second condition describes a kind of substitution. Only !eaf names in the old description can become equivalent to names that do not dominate other ma.rked narnes.", | |
| "cite_spans": [], | |
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| "section": "Local TDGs", | |
| "sec_num": "2" | |
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| "text": "Conditions 3. and 4. express the locality of the derivations. All names in the old description that are chosen for nev; equivalences must bc part of the derivation description, and furthermore a subdescription between two minimal or marked narnes must be \"inserted\" into R strong dominance where the domina.ted narne is pa.rt of the derivation description. These conditions can be compared to the Jocality restriction of the derivation in a. set-loClll multicomponent TAG (MG-TAG) (Weir 1988) . In fa.ct, for each set-loca1 MC-TAG, an equivalent locaJ TDG can be constructed (KaJlmeyer 1998a). However, local TDGs are more powerful than set-locaJ MC-TAGs because the locality condition restricts only the derivation of descriptions but not the wa.y a minimal structure for a derived description is obtained. This locality constitutes a crucial difference between local TDGs and DTGs since derivations in DTGs are non-local. Each subtree of a d-tree that is a.dded in a derivation step to a derived d-tree ' \"T can be inserted into any of the d-edges in T", | |
| "cite_spans": [ | |
| { | |
| "start": 481, | |
| "end": 492, | |
| "text": "(Weir 1988)", | |
| "ref_id": "BIBREF11" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Local TDGs", | |
| "sec_num": "2" | |
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| "text": "If a marked name has no parent, then an underspecification of the dominance relation ca.n occur in the result of a derivation step (see (KaJlmeyer 1998b , Kallmeyer 1998a ). In this paper, such cases are not considered, and for the examples mentioned here, the fifth condition is of no consequence. ", | |
| "cite_spans": [ | |
| { | |
| "start": 136, | |
| "end": 152, | |
| "text": "(KaJlmeyer 1998b", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 153, | |
| "end": 170, | |
| "text": ", Kallmeyer 1998a", | |
| "ref_id": "BIBREF1" | |
| } | |
| ], | |
| "ref_spans": [], | |
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| "section": "Local TDGs", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "= lfis /\\ !/Jz /\\ k1 ~ ku /\\ k2 ~ k11 /\\ k.i ~ k23 /\\ k3 <:]\" k15-", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
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| "section": "Local TDGs", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "A local TDG generates a set of descriptions. Each of these descriptions denotes infinitely many trees. The trees in the tree langu.age of a local TDG are those trees that are \"minimal\" for one of the derived de5criptions. A minimal tree of a description efJ is a. tree 1 satisfying 4> in such a way that 1. all parent rela.tions in 1 are described in l/J, and 2. if two different node names in </> denote the sarne node in /, then these two na.mes neither have both a parent in lfi nor have both a daughter in r;i.", | |
| "cite_spans": [], | |
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| "section": "77", | |
| "sec_num": null | |
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| "text": "The first condition makes sure that everything in / is described in l/J, and with the second condition no parent relation in the tree is described more than once in I/>.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
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| "section": "77", | |
| "sec_num": null | |
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| { | |
| "text": "For the local TDG in Fig. 1 for exarnple, only those descriptions have a minimal tree that are derived by adding t/J 1 in tbe last derivation step.", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 21, | |
| "end": 27, | |
| "text": "Fig. 1", | |
| "ref_id": "FIGREF2" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "77", | |
| "sec_num": null | |
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| { | |
| "text": "The string lan9uage of a local TDG Gis the set of all strings yielded by the trees in the tree language of G.", | |
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| "section": "77", | |
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| "text": "TDGs allow \"multicomponent\" derivations and a uniform complementation opera.tion similar to subsertion in DTGs. F\\J.rthermore, they provide underspecified representations for scope ambiguities (Kallmeyer 1998b ) since they a.llow the genera.tion of descriptions with underspecified dominance relations.", | |
| "cite_spans": [ | |
| { | |
| "start": 193, | |
| "end": 209, | |
| "text": "(Kallmeyer 1998b", | |
| "ref_id": "BIBREF2" | |
| } | |
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| "ref_spans": [], | |
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| "section": "77", | |
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| "text": "For a given TAG, an equivalent loc.al TDG with at most one marked name per elementary de.5cription can be easily constructed. Obviously, the extra power of local TDGs io contrast to TAGs arises Erom the possibility of marking more than one node oame in an elementary description. In Fig. 1 for example, 1/11 and 1/>z both contain two ma.rked names. Tbe language generated by this local TDG is no TAL. This suggests the de.finition of a hierarchy of local TDGs depending on the maximal number of ma.rked node narnes in an elemeutary description.", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 283, | |
| "end": 289, | |
| "text": "Fig. 1", | |
| "ref_id": "FIGREF2" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Rank of a local TDG", | |
| "sec_num": "3" | |
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| "text": "Two kinds of marked names can be distinguished: marked names where the part of the description dominating this name can be put somewhere uin between\" on the one hand (e.g. k 11 and k23in1/J2 in Fig. 1) , and on the other hand marked node names tha.t must be identified with a leaf name ( e.g. k 3 and k~ in Tji2 in Fig. 2) . Since there is a. similarity between foot nodes of auxiliary trees in TAGs and the first kind of marked node names, these are caJled adju.nctianmarked (a-markedl. For similar reasons. t.he second Start descriptio11: <f>S = .\\: t <J\" k, /\\ k2 <l kJ /\\ kJ <J\" k-1 f\\ k,_ < ks /\\ cat (.i(ki) .? k1 Roughly speaking, in a derivation step , for each s-marked name in the new elementary description, there is one substring added to the yield of the descri ption, and for each a-ma.rked name, two substrings are added (e.g. a 1 oz for k3 in Fig. 2, a 1 a2 and 01as for k11 in Fig. 1 and a3a4 and a~as for k 23 in Fig . I) . Therefore, a-marked names count twice as much as s-marked names for the rank of a local TDG:", | |
| "cite_spans": [ | |
| { | |
| "start": 607, | |
| "end": 614, | |
| "text": "(.i(ki)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [ | |
| { | |
| "start": 194, | |
| "end": 202, | |
| "text": "Fig. 1)", | |
| "ref_id": "FIGREF2" | |
| }, | |
| { | |
| "start": 316, | |
| "end": 323, | |
| "text": "Fig. 2)", | |
| "ref_id": "FIGREF3" | |
| }, | |
| { | |
| "start": 860, | |
| "end": 874, | |
| "text": "Fig. 2, a 1 a2", | |
| "ref_id": "FIGREF2" | |
| }, | |
| { | |
| "start": 895, | |
| "end": 910, | |
| "text": "Fig. 1 and a3a4", | |
| "ref_id": "FIGREF2" | |
| }, | |
| { | |
| "start": 932, | |
| "end": 940, | |
| "text": "Fig . I)", | |
| "ref_id": null | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Rank of a local TDG", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "s l. T, k2 s \"' 1 1 T2 1o, Ti \u2022s TJ \"\u2022 T2 \u2022o s \"\"", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Rank of a local TDG", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "a local TDG G is of rank n iff n = ma:i: { i 1 there is an elementary iJ; in G such that i is twice the number of a-marked names in 1/1 plus the number of s-marked names in 1/J}.", | |
| "cite_spans": [], | |
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| "section": "Rank of a local TDG", | |
| "sec_num": "3" | |
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| { | |
| "text": "For a given locaJ TDG it is always possible to find a weak]y equivalent local TDG with ane more s-marked name per elementary description . Therefore, the cla.ss of languages generated by local TDGs of rank i forms a subset of the dass of languages generated by local TDGs of rank i + l for i ~ O.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Rank of a local TDG", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "As shown in (Kallmeyer 1998a) , the cla.sses of local TDLs of rank 0 and 1 are equal, they a.re exactly the context-free languages. The dass of local TDLs of rank 2 contaius all TALs.", | |
| "cite_spans": [ | |
| { | |
| "start": 12, | |
| "end": 29, | |
| "text": "(Kallmeyer 1998a)", | |
| "ref_id": "BIBREF1" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Rank of a local TDG", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "The idea of the pumping Jemma for local TDGs of a certain rank n is similar to the one leading to the pumping lemma for TALs in (Vijay-Shanker 1987) .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "A pumping lemma", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "As shown in (Kallmeyer 1997) , the derivation process in a locaJ TDG can be described by a contextfree grammar GcF\u2022 For GcF, the pumping lemma for context-free languages holds. This means that in a derivation tree {of GcF) from a certain tree height oo, there is a subtree '\"f that can be iterated . For the corresponding local TDG, this signifies that an elercentary 1/J can be added twice such that: before adding 1/J again we have the following situation for a string w yielded by the old descrip-", | |
| "cite_spans": [ | |
| { | |
| "start": 12, | |
| "end": 28, | |
| "text": "(Kallmeyer 1997)", | |
| "ref_id": "BIBREF0" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "A pumping lemma", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "tion: U/ = %10V1 \u2022 \u2022 \u2022 l:1m-l VmXlm where Xli E T\", v 1 \u2022 \u2022 \u2022", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "A pumping lemma", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "Vm is the string yielded by the subdescription derived from .,P (ordered by linear precedence). As a next derivatiou step, 1/J is added again. If the grammar is of rank n, then by addiog !JJ. the string w can be split by inserting at most n new strings. Before the next adding of 1/J (corresponding to another iteration) takes place, these substrings will be expanded", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "A pumping lemma", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "to substrings U>1, \u2022 \u2022 \u2022 , Wn with 101 \u2022 \u2022 \u2022 Wn = v1 \u2022 \u2022 \u2022 Vm.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "A pumping lemma", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "These w; may be split into several words ( with other words in betweea) but the order of the letters is as", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "A pumping lemma", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "in v 1 \u2022 \u2022 \u2022tim .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "A pumping lemma", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "If this is repeated k times, k ~ l, then one ends up with a word contaning the letters of :r 1 ; = x10 \u2022\u2022\u2022 1\"1m and k occurrences of all symbols of w 1 \u2022 \u2022 \u2022 w\" that are for each of these occurrences (from left to right) ordered as in W1 \u2022 \u2022 \u2022 Wn. In the last steps (after the iterations of the derivation subtree -y) , the symbols of some string x~ Er\u2022 are added .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "A pumping lemma", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "Therefore the pumping lemma is as fo llows: for each ward u: in the string language of a lacal TDG of rank n with lwl great.er than some constant cc :", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "A pumping lemma", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "after rcmoving thc Letters of some words x1 and x 2 from w, the resulting ward has the form wi \u2022 \u2022 \u2022wn. Then for each k there is a ward uPl in the language containing also the letters af x 1 and x 2 , such that : if these letters are removed from w!kl, the result \u00fcP> is a word that can be obtained by taking k occurrences With the pumping lemma, it can be easily shown", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "A pumping lemma", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "that for i > 2n , L, = {ai\" \u2022 \u2022 \u2022 ai 1 m ~ O} does not", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "A pumping lemma", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "satisfy the pumping Jemma for TDGs of rank n and therefore cannot be generated by a local TDG of rank n .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "A pumping lemma", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "Consequently, for all n ~ 1, the string languages of TDGs of rank n form a proper subset of the string languages generated by TDGs of rank n + 1.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "A pumping lemma", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "In this paper, the rank of a local TDG was delined based on the number of marked names in the elementary descriptions of the grammar. Two kinds of marked names are distingui.shed, namely s-marked and a-marked names. Since derivations in local TDGs can be described by a context-free grammar, the pumping lemma for conteict-free grammars can be applied to the derivation trees of a local TDG . Thi.s lea.ds to the proof of a pumping lemma for local TDGs of a certain rank n. Roughly said, according to this pumping lemma, in a. derivation step, for each s-marked name in the new elementary clescription, one substring is a.dded, and for each a-marked name, two substrings a.re added. With this pumping lemma one can show that for n ~ 1 the languages generated by local TDGs of rank n form a proper subsel of languages generated by Jocal TDGs of rank n+ 1.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Conclusion", | |
| "sec_num": "5" | |
| }, | |
| { | |
| "text": "Some of the conditions bolding f\u00fcr de.criptinus \u2022 local TDG are ler~ a.side here. For a forma.! defuiitio: 0~local TDGs see(Kallmeyer 1998a).", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
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| "section": "", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Tl.tes~ two chan.cterizations are riot exdusive, for examples of node oa.mes tbat are both a-marked and smarked see(K&llmeyer 1998a).", | |
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| "back_matter": [], | |
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| "FIGREF0": { | |
| "text": "Fig. 1for e:xample, a. derivation step l/Js ;i5 q, 1 is possible with </>1", | |
| "num": null, | |
| "type_str": "figure", | |
| "uris": null | |
| }, | |
| "FIGREF1": { | |
| "text": ") :::: 5 f\\ cat(J(k2)) ::::: T1 /\\ cat(J(k3)):::: T2 /\\ cat(6(k4)) ;:::: Ta/\\ \u00f6(ks):::: l Elementary descriptions:1Jl1 = ka <J\" kr /\\ k~ <l ks f, ks <l\" k~ f\\ k9 <l kioA cat(J(k\u00f6)) :::: S /\\ \u2022 \u2022 \u2022 !/12 = l.:11 <J\" k12 /\\ k12 <l kJJ A k12 <l ki< /\\ k12 <l kir /\\ l.:13 -( k1~ /\\ kH -< ku /\\ k 14 <l\" kll f 1 .. .. .. /\\ cot(6(k11 )) :::: S /\\ cot(J(k12)) :::: S /\\ .. . . .. /\\ J(k26) :::: 07 /\\ 6(.\\:27) :::: llg J\\\" 1 = {ks,k10},K.; 2 = {k11,k23} Graphical represeotations: (marked names with asterisk) r/'s 1P1", | |
| "num": null, | |
| "type_str": "figure", | |
| "uris": null | |
| }, | |
| "FIGREF2": { | |
| "text": "Local TDG for {a)a2'a;'a4oga5a7a~ 10 ~ n} with two a-rnarked names in each elementary de-", | |
| "num": null, | |
| "type_str": "figure", | |
| "uris": null | |
| }, | |
| "FIGREF3": { | |
| "text": "Local TDG for { a)a~a3a~ 1 0 :::; n} wi th two s-rnarked names in each elernentary description 78 kind of marked names are called substitution-marked (s-marked}.~", | |
| "num": null, | |
| "type_str": "figure", | |
| "uris": null | |
| }, | |
| "FIGREF4": { | |
| "text": "of w1 \u2022 \u2022 \u2022 u:n and then, starting with c, taking (in arbitrary order) always the left letter of one of these k words as the next letter in wlk l. Furthermore, \u00fcl \"l still contains as substrings one occurrence of each of the words w1 , \u2022 \u2022 \u2022, w\" (in this order) . For the language L2n :== {ai\" \u2022 \u2022 \u2022 a2'\" 10 ~ m} for example the lemma for rank n holds with ca = 2n -1, Xi = :r2 = t : if w = a\\ \u2022 \u2022 \u2022 a2:,, then w 1 = a;i_ 1 a~.", | |
| "num": null, | |
| "type_str": "figure", | |
| "uris": null | |
| }, | |
| "TABREF0": { | |
| "content": "<table><tr><td>\u2022</td><td>.1..</td><td>h</td><td>l</td><td>2</td></tr><tr><td colspan=\"5\">co~bmed to obt:w1 a new de:icription r/>,_. Roughly</td></tr><tr><td colspan=\"5\">sa.id, 4'l can be v1ewed as a conjunction of ef> 1 , 1/i and new formula.s k :::::: k' or lc <l\u2022 lc' where k is a. name</td></tr><tr><td colspan=\"5\">from efi 1 and k' a name from f/l. This derivation step</td></tr><tr><td colspan=\"3\">must besuch tha.t</td><td/><td/></tr><tr><td/><td colspan=\"4\">1. for a. node name k-1> in 1/J, there is a. new equiv-</td></tr><tr><td/><td/><td colspan=\"3\">aleuce iff either k-4> is IIL8lked or k-4> is minimal</td></tr><tr><td/><td/><td colspan=\"3\">(dominated by uo other name, e.g. kG in iti 1 and k11 in t/J-J. in</td></tr></table>", | |
| "html": null, | |
| "num": null, | |
| "type_str": "table", | |
| "text": "(in the graphical repre-semat1ons, same of the node names are om.itted for reasons of rea.da.bility). Conjuncts such a.s k <l\u2022 k tree descriptions. In each derivation step, a der~ved </11 and_ an elementary de5cription 1\" are" | |
| } | |
| } | |
| } | |
| } |