| { | |
| "paper_id": "W89-0218", | |
| "header": { | |
| "generated_with": "S2ORC 1.0.0", | |
| "date_generated": "2023-01-19T03:45:00.838924Z" | |
| }, | |
| "title": "R ecognition of Com binatory Categorial Grammars and Linear Indexed Grammars", | |
| "authors": [ | |
| { | |
| "first": "K", | |
| "middle": [], | |
| "last": "Vijay-Shanker", | |
| "suffix": "", | |
| "affiliation": { | |
| "laboratory": "", | |
| "institution": "CIS University of Delaware Delaware", | |
| "location": { | |
| "postCode": "19716", | |
| "region": "DE" | |
| } | |
| }, | |
| "email": "" | |
| }, | |
| { | |
| "first": "David", | |
| "middle": [ | |
| "J" | |
| ], | |
| "last": "Weir", | |
| "suffix": "", | |
| "affiliation": {}, | |
| "email": "" | |
| } | |
| ], | |
| "year": "", | |
| "venue": null, | |
| "identifiers": {}, | |
| "abstract": "1A bottom-up finite state tree autom aton reads a tree bottom-up. The state that the autom aton associates with each node that it visits will depend on the states associated with the children of the node. 2 We consider LIG that correspond to the Chomsky normal form for CFG although we do not prove that all LIG have an equivalent grammar in this form. A discussion of the recognition algorithm for LIG in this form is sufficient to enable us to adapt it to give a recognition algorithm for CCG, which is the primary purpose of this paper.", | |
| "pdf_parse": { | |
| "paper_id": "W89-0218", | |
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| "abstract": [ | |
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| "text": "1A bottom-up finite state tree autom aton reads a tree bottom-up. The state that the autom aton associates with each node that it visits will depend on the states associated with the children of the node. 2 We consider LIG that correspond to the Chomsky normal form for CFG although we do not prove that all LIG have an equivalent grammar in this form. A discussion of the recognition algorithm for LIG in this form is sufficient to enable us to adapt it to give a recognition algorithm for CCG, which is the primary purpose of this paper.", | |
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| "section": "Abstract", | |
| "sec_num": null | |
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| ], | |
| "body_text": [ | |
| { | |
| "text": "In recent papers [14, 15, 3] we have shown that Combinatory Categorial Grammars (CCG), Head Gram mars (HG), Linear Indexed Gramm ars (LIG), and Tree Adjoining Grammars (TAG) are weakly equiv alent; i.e., they generate the same class of string languages. Although it is known that there are polynomial-time recognition algorithms for HG and TAG [7, 11] , there are no known polynomial-time recognition algorithms that work directly with CCG or LIG. In this paper we present polynomial time recognition algorithms for CCG and LIG that resemble the CKY algorithm for Context-Free Gram m ars (CFG ) [4, 16] .", | |
| "cite_spans": [ | |
| { | |
| "start": 17, | |
| "end": 21, | |
| "text": "[14,", | |
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| "start": 22, | |
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| "text": "3]", | |
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| "start": 344, | |
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| "start": 348, | |
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| "text": "11]", | |
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| "start": 595, | |
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| "text": "[4,", | |
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| "start": 599, | |
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| "text": "16]", | |
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| "section": "Introduction", | |
| "sec_num": "1" | |
| }, | |
| { | |
| "text": "The tree sets derived by a CFG can be recognized by finite state tree autom ata [10] 1. This is reflected in CFL bottom -up recognition algorithms such as the CKY algorithm. Intermediate configurations of the recognizer can be encoded by the states of these finite state autom ata (the nonterminal symbols of the gram m ar). The similarity of TAG, CCG, and LIG can be seen from the fact that the tree sets derived by these formalisms can be recognized by pushdown (rather than finite state) based tree autom ata. We give recognition algorithms for these formalisms by extending the CKY algorithm so that intermediate configurations are encoded using stacks. In [6] a chart parser for CCG is given where copies of stacks (derived categories) are stored explicitly in each chart entry. In Section 4 we show that storing stacks in this way leads to exponential run-time. In the algorithm we present here the stack is encoded by storing its top element together with information about where the remainder of the stack can be found. Thus, we avoid the need for multiple copies of parts of the same stack through the sharing of common substacks. This reduces the number of possible elements in each entry in the chart and results in a polynomial time algorithm since the time complexity is related to the num ber of elements in each chart entry. It is not necessary to derive separate algorithms for CCG, LIG, and TAG. In proving that these formalisms are equivalent, we developed constructions that map grammars between the different for malisms. We can make use of these constructions to adapt an algorithm for one formalism into an algorithm for another. First we present a discussion of the recognition algorithm for LIG in Section 22.", | |
| "cite_spans": [ | |
| { | |
| "start": 80, | |
| "end": 84, | |
| "text": "[10]", | |
| "ref_id": "BIBREF9" | |
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| { | |
| "start": 661, | |
| "end": 664, | |
| "text": "[6]", | |
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| "section": "Introduction", | |
| "sec_num": "1" | |
| }, | |
| { | |
| "text": "We present the LIG recognition algorithm first since it appeares to be the clearest example involving the use of the notion of stacks in derivations. In Section 3 we give an informal description of how to map a CCG to an equivalent LIG. Based on this relationship we adapt the recognition algorithm for LIG to one for CCG.", | |
| "cite_spans": [], | |
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| "eq_spans": [], | |
| "section": "Introduction", | |
| "sec_num": "1" | |
| }, | |
| { | |
| "text": "An Indexed Gram m ar [l] can be viewed as a CFG in which each nonterminal is associated with a stack of symbols. In addition to rewriting nonterminals, productions can have the effect of pushing or popping symbols on top of the stacks that are associated with each nonterminal. A LIG [2] is an Indexed G ram m ar in which the stack associated with the nonterminal of the LHS of each production can only be associated with one of the occurrences of nonterminals on the RHS of the production.", | |
| "cite_spans": [ | |
| { | |
| "start": 284, | |
| "end": 287, | |
| "text": "[2]", | |
| "ref_id": "BIBREF1" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Linear Indexed Gramm ars", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "Em pty stacks are associated with other occurrences of nonterminals on the RHS of the production. We write A[--] (or A [--7 ] ) to denote the nonterminal A associated with an arbitrary stack (or an arbitrary stack whose top symbol is 7 ). A nonterminal A with an empty stack is written", | |
| "cite_spans": [ | |
| { | |
| "start": 119, | |
| "end": 125, | |
| "text": "[--7 ]", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Linear Indexed Gramm ars", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "D e fin itio n 2.1 A LIG, G, is denoted by (V>/, Vj, V>, 5, P) where V'v is a finite set of nonterminals, V j is a finite set of terminals, Vj is a finite set of indices (stack symbols), S 6 Vn is the start symbol, and P is a finite set of productions, having one of the following forms. ", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Linear Indexed Gramm ars", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "In each of these two cases we say that A, is the d is tin g u is h e d child of A in the derivation.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": ".An[]T2", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "\u2022 If A[] a 6 P then r l A [}T 2 = > r ia r 2", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": ".An[]T2", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "The language generated by a LIG, G, N ote th a t tuples, as defined above, assum e the presence of a t least tw o stack sym bols. We m ust also consider tw o o th e r cases in which a n o n term in al is associated w ith eith er a stack of a single elem ent, or w ith the e m p ty stack . Suppose th a t A is associated w ith a stack containing only th e single sym bol 7 . T h is case will be rep resen ted using tuples of the form (A , 7 , A',p, < 7) ( w-\" indicates th a t an e m p ty stack is associated w ith A '). W hen an em p ty stack is associated w ith A we will use the tuple (A , -, -) . In discussing the general case for tuples we will use the form (A , 7 , A ', 7 ', p, < 7) with Wre now describe how each e n try Lij is filled. As th e algorithm proceeds, th e gap betw een i and j increases u n til it spans th e e n tire in p u t. T h e in p u t, <zi. . . a n , is accepted if (S , , -) E L\\ n.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": ".An[]T2", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "L (G ) = { w | S[] ==>\u2022 w }.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": ".An[]T2", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "New en tries are a d d ed to th e a rra y elem ents according to th e p ro d u ctio n s of th e g ra m m a r as follows. Step la. For each production A(--7] A, 7 , A2, 72, k + 1 ,; ) }", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 157, | |
| "end": 180, | |
| "text": "A, 7 , A2, 72, k + 1 ,;", | |
| "ref_id": "FIGREF0" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": ".An[]T2", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "-A i[]A 2[--] if (A i, -, , -) \u20ac Li'k and (A 2, 72, A3 , 7 3 , p, q) \u20ac Lk+i,j then Lij := Lij U { (", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "T h e p ro d u c tio n A[\u00bb7] -+ A i[]", | |
| "sec_num": "1." | |
| }, | |
| { | |
| "text": "Step lb. For each production A 71, A3 ,7 3 ,p, q) \u20ac Li,!, and (A2) -, -, -, - ", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 31, | |
| "end": 77, | |
| "text": "71, A3 ,7 3 ,p, q) \u20ac Li,!, and (A2) -, -, -, -", | |
| "ref_id": "FIGREF0" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "T h e p ro d u c tio n A[\u00bb7] -+ A i[]", | |
| "sec_num": "1." | |
| }, | |
| { | |
| "text": "[--7] -* > Ai[--]A2[] if (Ai,", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "T h e p ro d u c tio n A[\u00bb7] -+ A i[]", | |
| "sec_num": "1." | |
| }, | |
| { | |
| "text": ") \u20ac Lk+ij then Li j 1 --Li j U {(A ,7>A i,7i>*i^')}", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "T h e p ro d u c tio n A[\u00bb7] -+ A i[]", | |
| "sec_num": "1." | |
| }, | |
| { | |
| "text": "Step 2a. For these rules we say th a t (x/y) is th e p rim ary category and y th e secondary category. For these rules (x/y) is the prim ary category and ( .. )|2 . . . |n*n) the secondary category.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "T h e p ro d u c tio n A[\u00bb7] -+ A i[]", | |
| "sec_num": "1." | |
| }, | |
| { | |
| "text": "R estrictions can be associated w ith the use of each co m b in ato ry rule in R. T hese restrictions take the form of c o n stra in ts on th e in sta n tia tio n s of variables in th e rules. 1 . T h e leftm ost n o n term in al ( t a r g e t c a t e g o r y ) of the p rim ary category can be restric te d to be in a given subset of Vjv.", | |
| "cite_spans": [ | |
| { | |
| "start": 193, | |
| "end": 194, | |
| "text": "1", | |
| "ref_id": "BIBREF0" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "C om binatory Categorial Grammars", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "T h e catego ry to which y is in sta n tia te d can be restricted to be in a given finite subset of C ( V \\) . In th e p resen t discussion of C C G recognition we m ake the following assu m p tio n s concerning the form of th e g ra m m a r. 1 . In o rd er to sim plify our p re se n ta tio n we assum e th a t th e categories are parenthesis-free. The algorithm that we p re se n t can be adapted in a straightforw ard way to handle parenthesized cate gories and this m ore general algorithm is given in [1 2 ].", | |
| "cite_spans": [ | |
| { | |
| "start": 243, | |
| "end": 244, | |
| "text": "1", | |
| "ref_id": "BIBREF0" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "2.", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "2. We will assum e th a t th e function / does not assign categories to th e e m p ty strin g . This is co n sisten t w ith th e linguistic use of C C G a lth o u g h we have n o t show n t h a t th is is a norm al form for C C G .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "2.", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "In this section, we describe th e relatio n sh ip betw een LIG ", | |
| "cite_spans": [ | |
| { | |
| "start": 59, | |
| "end": 62, | |
| "text": "LIG", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "The LIG/CCG Relationship", | |
| "sec_num": "3.1" | |
| }, | |
| { | |
| "text": "We begin by considering the function, / , which assigns categories to each elem ent of V j-Suppose th a t c E f(a) w here c G C(Vh) and a G Vt -We should include th e production A[a] -* a where c -A a in P . For each co m b in ato ry rule in R w'e m ay include a n um ber of productions in P . From the definition of C C G it follows th a t the length of all secondary categories in th e rules R is bounded by som e c o n sta n t. T herefore th ere are a finite num ber of possible ground in sta n tia tio n s of the secondary category in each rule. T h u s we can rem ove variables in secondary categories by expanding the num ber of rules in R. T he rules th a t result will involve a secondary category c G C(Vjv) and a prim ary category of the form x/ A or x \\A where A 6 Vyv is the ta rg e t category of c. T he rule m ay also place a restriction on the value of the ta rg e t category of x. In the case of the prim ary categories of th e com binatory rules th ere is no bound on th eir length and we can n o t rem ove the variable th a t will be bound to the unbo u n d ed p a rt of the category (th e variable x above). T herefore the rules contain a single variable and are linear w ith respect to this variable; i.e., it ap p ears once on either side of the rule.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "177-", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "It is stra ig h tfo rw ard to convert co m binatory rules in this form in to corresponding LIG productions.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "177-", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "We illu s tra te how this can be done w ith an exam ple. Suppose we have the following com binatory rule.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "177-", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "x/A A / B \\ C \\ B -x/B \\C \\B w here th e ta rg e t categ o ry of x m ust be eith er C or D. T his is converted into the following two p ro d u ctio n s in P.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "177-", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "N otice th a t these LIG p ro d u ctio n s do not correspond precisely to our earlier definition. We are pushing and popping m ore th a t one sym bol on the stack and we have not associated em p ty stacks w ith all b u t one of th e RHS n o n term in als. A lthough this clearly does not affect weak generative pow er, as we will see in th e next section, it will require a m odification to th e recognition algorithm given earlier for LIG.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "C [ -/ B \\ C \\ B ] -C [ -M ] A [ / B \\ C \\ B ] D [ -/ B \\ C \\ B ] -D[-/A] A { / B \\ C \\ B ]", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "In o rd er to produce a C C G recognition algorithm we extend th e LIG recognition algorithm given in Section 2.2. From th e previous section it should be clear th a t the C C G and LIG algorithm s will be very sim ilar. T h erefo re we do n o t p resen t a detailed description of th e C C G alg o rith m . We use an a rray , C , w ith n 2 elem en ts, C tJ for 1 < t < j < n. T h e tuples in th e a rra y will have a slightly different ", | |
| "cite_spans": [], | |
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| "eq_spans": [], | |
| "section": ".2 R e c o g n it io n o f C C G", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "T h e recognition algorithm s given here have polynom ial-tim e com plexity because each array elem ent (e -g-\u00bb LX y J in LIG recognition) contains a polynom ial num ber of tuples (w ith respect to the difference betw een j and i). T hese tuples encode the top sym bol of the stack (or top sym bols of the category) to g e th e r w ith an in dication of w here the next p a rt of the stack (category) can be found. If we had sto re d th e e n tire stack in th e a rra y elem ents5, then each a rra y e n try could include exponentially m any elem ents. T h e recognition com plexity would then be exponential.", | |
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| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Im portance of Linearity", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "It is in te restin g to consider why it is not necessary to store the entire stack in the a rra y elem ents. A , 7 , .4', 7 ', p, q) 6 Lij. T his indicates the existence of a tuple, say (A ', 7 ', A\", 7 \", r, s) , in LP yq. It is crucial to note th a t when we are adding the first tuple to LX < J we are not concerned about th a t has th e form x c o n j x -\u25ba x w here th e variable x can be any category. W ith this schem a we have to check th e id e n tity of tw o derived categories. T his results in the loss of independence am ong paths in d eriv atio n trees. In [13] we have discussed th e notion of in d ep en d en t p a th s in derivation trees w ith respect to a range of g ra m m a tic a l form alism s. We have show n [12] th a t w hen C C G are extended with this c o o rd in a tio n schem a th e recognition problem becom es N P -com plete.", | |
| "cite_spans": [ | |
| { | |
| "start": 570, | |
| "end": 574, | |
| "text": "[13]", | |
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| "start": 731, | |
| "end": 735, | |
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| { | |
| "start": 109, | |
| "end": 132, | |
| "text": "A , 7 , .4', 7 ', p, q)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 182, | |
| "end": 211, | |
| "text": "say (A ', 7 ', A\", 7 \", r, s)", | |
| "ref_id": null | |
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| ], | |
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| "section": "Im portance of Linearity", | |
| "sec_num": "4" | |
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| { | |
| "text": "We have p resen ted a general schem e for polynom ial-tim e recognition of languages generated by a class of g ra m m a tic a l form alism s th a t are m ore pow erful th a n C F G . T his class of form alism s, which includes LIG , C C G , a n d T A G , derives m ore com plex trees th a n C F G due th e use of an additional sta c k -m a n ip u la tin g m echanism . Using c o n stru ctio n s given in [15, 3] , we have described how a recog n ition a lg o rith m p rese n ted for LIG can be a ", | |
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| "start": 404, | |
| "end": 408, | |
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| "end": 411, | |
| "text": "3]", | |
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| "section": "C onclusion", | |
| "sec_num": "5" | |
| }, | |
| { | |
| "text": "Intemational Parsing Workshop '89", | |
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| "text": "International Parsing Workshop '89", | |
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| { | |
| "text": "Note that / can assign categories to the em pty string, e, though, to our knowledge, this feature has not been employed in the linguistic applications of CCG.4There is no type-raising rule although its effect can be achieved to a limited extent since / can assign type-raised categories to lexical items.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "", | |
| "sec_num": null | |
| } | |
| ], | |
| "back_matter": [ | |
| { | |
| "text": "An e n try in C will be a six-tuple of the form ( A ,a ,/3 ,7 ,p , q) w here A E V /y ,,a ,(3 E ( { / A K v V and one of th e tw o cases applies. or 2 < 1(a) < s(G) -1, l((3) = s(G ) -1, 7 E { / , \\ }Viv , 1 < p < q < n 0 < 1(a) < 23(G ) -2, (3 = \u20ac, 7 = p = q -An e n try (A, a, (3,~/,p,q) is placed in C t,j when\u2022 If /3 = \u20ac and 7 -p -q --then A a a, .. . a 7.", | |
| "cite_spans": [ | |
| { | |
| "start": 279, | |
| "end": 289, | |
| "text": "(3,~/,p,q)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "annex", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "\u2022 If (3 \u00a3 e th en for som e a ' E ( { / , \\ }V/v)*, A a '/3 a a , . . . a ; and A a '/?7 =^> a \" . . . a 7.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "G", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "T h e step s of th e alg o rith m th a t apply for exam ples of forw ard application and forw ard composition are as follows.For each k betw een i and j, we look for (B, a , /?, 7 , p, < 7) E C,,* and (A , -) E C*+liJ w here B is a possible ta rg e t category of x and th e strin g (3a has /A as a suffix. If we find these tuples th en do th e following. ", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "G G", | |
| "sec_num": null | |
| } | |
| ], | |
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| "FIGREF0": { | |
| "type_str": "figure", | |
| "text": "A \\ , . . . , A n 6 Vn a^d flG { e } U V j. The relation => is defined as follows where a \u20ac V f and T i , T 2 are strings of nonterminals with G associated stacks. \u2022 If A[--7 ] -A i [ ] ...A t[ \" ] ...A n[] \u20ac P then T iA [q 7 ]T2 = > T 1 A1 [ ] ...A ,[ a ] ...A \" [ ] T If", | |
| "num": null, | |
| "uris": null | |
| }, | |
| "FIGREF1": { | |
| "type_str": "figure", | |
| "text": "the u n d e rsta n d in g th a t: A' G VN or 7 , 7 ' \u00a3 V/ or and p, q are integer betw een 1 and n or T h e alg o rith m can be u n d e rsto o d by verifying th a t at each step the following in v arian t holds. P r o p o s i t i o n 2.1 (A , 7 , A', 7', p, q) \u00a3 LX y J if and only if one of the following holds. If -y' ^ -th e n A[7 ] = > a , . . . a p_ i A'[ }aq+\\ . . . a; and A'[ol~i'\\ ===> ap .. . aq for som e a E Vf w here A ' is a distinguished descendent of A. N ote th a t this implies th a t for a l l 0 e Vf, A[j3~f] a l-. . . a p_ iA /[/3]a,+ 1 ...aj. T h u s, for (3 = 0 7 ', A [aYf] =^> a,-. . . a p_ i A '[a 7 /] a ,+ i .. .aj which implies A [a 7 ;7 ] =^=> a , .. .a j. If 7 7 = -^ A' th en A[7 ] ==> a,-.. .a3 and A '[] ap .. .aq. If A' = -th e n A[] =^=> a t-.. .aj.", | |
| "num": null, | |
| "uris": null | |
| }, | |
| "FIGREF2": { | |
| "type_str": "figure", | |
| "text": "A 2[-] is used while filling th e a rra y elem ent Lij as follows. For every k w here i < k < j, check th e previously com pleted a rra y elem ents L itk a n d L k+\\,j for (Ai,*-,and som e (A 2, 72, A3,7 3 ,P , <?), respectively. If these entries are found add (A , 7 , A2, 72? k + 1, j ) to Lij. If 72 = 73 = ^3 = P = q = ~ we Place ( A ,7 ,A 2, -, f c + l , j ) in Lij. From these en tries in L iyk and Ifc+i.j we know by P ro p o sitio n 2.1 th a t Ax[] =^=> a t-. . . a fc Recognition of LIG and .4.2[a] ==> a,k+i .. .a ; for some a E V}-. T h u s, Afcry] ==> a, \u2022. \u2022a: . T he p ro d u ctio n A[**-y] -Ai[*-]A2[] is handled sim ilarly. 2. Suppose A[-*] -* > A i[].42[--7 ] is a p roduction. W hen filling LtyJ we m ust check w hether the tuple (Ai,is in Lx and ( A2, 7 , A3, 73, p, q) is in L k+lyJ for som e k betw een i and j . If we do find these tuples th en we check in L v<q for some (A 3, 73, A4, 74, r, s). In this case we add (A , 73, A4, 74, r, s) to L{j. If 73 =th en the stack associated w ith A3 is empty, 74 = A 4 = r = s = -, and we add the tu p le ( A, r , 5) to L{yJ. T h e above steps can be related to P ro p o sitio n 2.1 as follows. (a) If 73 5* -th en for som e a \u20ac V /, A4[q74J =^> ar .. . a 3 a sub d eriv atio n of .43(0:7473] =^=> av ...aq a sub d eriv atio n of A2[c*74737] ==> a * + i . . . a j . C om bining this w ith A i[] ==> a , . . . a t we have A [q7473] ===> a, . . . a ; . (b) If 73 = -th en A3[] av ...aq is a su b derivation of A2[7 ] ==>\u2022 ^k+1 -..aj. C o m b in in g w ith Ai [ ] ==> a,-.. .a * , we get A[] = = > < Z j...a j. P ro d u c tio n s of th e form A[-*] -1 \u25ba Ai[-*7 ]A2[] are handled sim ilarly. 3 . S uppose A[] -a is a p ro d u ctio n . T his is used by th e algorithm in th e initializatio n of th e array L. If th e term in a l sym bol a is th e sam e as th e ith sym bol in th e in p u t strin g , i.e., a = a ,, then we include (A , -in th e a rra y elem ent Z ,tl. 2 .2 C o m p l e t e A lg o r it h m For i := 1 to n do Li.i := {(>1, I A []-\u00bb a,} For i := n to 1 do For j := j to n do For k := i to j -1 do", | |
| "num": null, | |
| "uris": null | |
| }, | |
| "FIGREF3": { | |
| "type_str": "figure", | |
| "text": "For each production A[-] -* A i[]A 2[--7] if (A 2 , 7 i A3, 7 3 , p, q) \u20ac \u00a3*+i,;> (A3 , 7 3 , A4 , 74, r, s) \u20ac \u00a3 Pl?, and ( 2 b. For each production A[--] -* \u2022 Ai[--7]A2[] if (Ai, 7 , A3 , 73, p, ?) 6 (A3, 73, A4, 74, r , 5) G Ip,}, and (A2) -, -, , -) 6 \u00a3*+1,;then Lij := Lij U { (A, 73, A4 , 74, r, s) } ny a rra y elem ent, say Z j j , is a set of tuples of the form ( A, 7 , A', 7 ', p, q) where p and q are either integers betw een i and j, or i = j = T he num ber of possible values for A, A', 7 , and 7' are each bounded by a c o n sta n t. T h u s the num ber of tuples in LXJ is at m ost 0((j -t)2). For a fixed value of i,j,k, steps l a and lb will a tte m p t to place at m ost 0((ji)2) tuples in L{j. Before adding m y tuple to Lij we first check w hether the tuple is already present in th a t a rra y elem ent. This can be done in c o n sta n t tim e on a RAM by assum ing th a t each a rra y elem ent LX tJ is itself an (i -f 1) x (j 4-1 ) array. A tu p le of the form( A, 7 , A', 7', p, q)will be in the (p,q)th elem ent of LX i] and a tuple of the form(A , -, -, -, -, -)will be in the (i + l,j + l)th elem ent of Lxj. T hus these steps take at most 0({j ~ 0 2) tim e -Sim ilarly, for a fixed value of i, j , and fc, steps 2a and 2b can add at m ost 0((j -i)2) d istin ct tuples. How ever, in these steps 0((j -i)4) not necessarily distinct tuples m ay be considered.T h ere are 0((j -i) 4) such tuples because the integers p,q,r,s can tak e values in the range betw een i and j. T h u s steps 2a and 2b m ay each tak e 0((j -i) 4) tim e for a fixed value of i,j,k. Since we have th ree in itial loops for i, j , and k, the tim e com plexity of the algorithm is 0 ( n 7) where the length of the in p u t is n.", | |
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| }, | |
| "FIGREF4": { | |
| "type_str": "figure", | |
| "text": "is an extension of C lassical C ateg o rial G ra m m a rs in which b o th function com position and fu nction a p p licatio n are allowed. In ad d itio n , forw ard and backw ard slashes are used to place conditions concerning th e relative ordering of a d jacen t categories th a t are to be com bined. C C G , G , is denoted by ( V j, V)v, 5 , / , R) where Vj is a finite set of term in als (lexical item s), V)v is a finite set of no n term in als (ato m ic categories), 5 is a distinguished m em ber of Vjv, / is a function th a t m aps elem ents of Vj U {e} to finite subsets of C(Vj\\r), th e set of c a te g o rie s,3 w here C(V}v) is th e sm allest set such th a t Vjv C C(V^) an d c i , c 2 G C(Vjv) im plies ( c i / c 2), ( c i \\ c 2) \u20ac C(VN), R is a finite set of c o m b in a to ry rules. T h e re are four ty p es of c o m b in a to ry rules involving variables x,y,z,z\\,... over C(V)y) and where It \u20ac { \\ > / } 4-", | |
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| }, | |
| "FIGREF5": { | |
| "type_str": "figure", | |
| "text": "generalized backw ard com position for some n > 1:(\u2022 -* (2/|l-l )|2 \u2022 \u2022 -ln*n) ( A y ) -' (\u2022 \u2022 .(x|i*i)|2 \u2022 \u2022 -ln*n)", | |
| "num": null, | |
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| }, | |
| "FIGREF6": { | |
| "type_str": "figure", | |
| "text": "erivations in a C C G , G = ( V j, Vyv, 5 , / , R), involve the use of th e c o m b in ato ry rules in R. Let = > G be defined as follows, w here T i , T 2 \u20ac [C{VN) u VT )m and c , c i , c 2 \u20ac C(VN). \u2022 If R co n tain s a c o m b in ato ry rule th a t has CiC2 -c as an in stan ce then T i c T 2 ==> T iG ic 2T If c 6 / ( a ) for som e a 6 Vt U{ c } an d c \u00a3 C(V)v) then T i c T 2 =\u25ba T i a T 2 GT h e strin g languages g e n e ra te d by a C C G , G , L(G) = { it; | 5 w \\ w \u20ac Vf }.", | |
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| }, | |
| "FIGREF7": { | |
| "type_str": "figure", | |
| "text": "an d C C G by discussing how we can construct from any C C G a w eakly equivalent LIG . T h e weak equivalence of LIG an d C C G w as established in [15]. T h e p u rp o se of this section is to show how a C C G recognition alg o rith m can be derived from th e a lg o rith m given ab o v e for LIG. G iven a C C G , G = ( V j, V\\r, 5 , / , R ) , we c o n stru c t an equivalent LIG , G ' = (V j, V)v, VjvU{/, \\} , S ,P )\u00bb as follows. E ach c ateg o ry in c 6 C(V]v) can be rep resen ted in G ' as a n o n term in a l an d associated stack A [a] w here A is th e ta rg e t c ateg o ry of c an d a \u20ac ({/\u00bb\\}V )v)* suck A a = c. N ote th a t we are assum ing th a t categories are parenthesis-free.", | |
| "num": null, | |
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| }, | |
| "FIGREF8": { | |
| "type_str": "figure", | |
| "text": "form from th o se of th e LIG alg o rith m . T his is because each d erivation step m ay depend on m ore th an one sym bol o f th e c a te g o ry (sta c k ). T h e n u m b er of such sym bols is b ounded by th e g ram m ar and is equal to th e n u m b er of sym bols in th e longest secondary category. We define th is bo u n d for a C C G , G = (V j, V}v, 5 , / , R ) as follows. Let 1(c) = k if c \u20ac ( { / A }^j v ) fc-Let 5(G ) be th e m axim um 1(c) of any c ateg o ry c G C(V}v) such th a t c can be th e secondary category of a c o m b in a to ry rule in R.As in th e LIG a lg o rith m we do not sto re th e en tire category explicitly. H ow ever, ra th e r th a n storing only th e to p sym bol locally, as in th e LIG alg o rith m , we sto re som e bo u n d ed n u m b er of sym bols locally to g e th e r w ith a in d icatio n of w here in C th e rem ain d er of th e categ o ry can be found. T his m odification is needed since a t each ste p in th e recognition alg o rith m we m ay have to exam ine the to p s(G) sym bols of a category. W ith o u t this extension we would be required to tra c e th ro u g h c(G) entries in C in order to exam ine th e to p c(G) sym bols of a categ o ry and th e a lg o rith m 's tim e com plexity would increase.", | |
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| }, | |
| "FIGREF9": { | |
| "type_str": "figure", | |
| "text": "how th e second tu p le cam e to be pu t in L p<q. T his is because the p roductions in LIG (com binatory rules in C C G ) are linear w ith respect to th eir u n b ounded stacks (categories). Hence th e derivations from different no n term in als and th eir associated stacks (categories) are independent of each other. InIndexed G ra m m a rs, p ro ductions can have the form A[-*7 ] -* > A i [--] A 2 [*\u2022]. In such productions there is no single distinguished child th a t inherits th e u n bounded stack from the no n term in al in the LHS of the pro d u ctio n . In a b o tto m -u p recognition algorithm the id en tity of the en tire stacks associated w ith A \\and A 2 has to be verified. T his nullifies any ad v an tag e from the sharing of stacks since we would have to exam ine the com plete stacks. A sim ilar situ a tio n arises in th e case of coordination schem a used to handle c e rtain form s of co o rd in atio n in D utch. A co o rdination schem a has been used by Steedm an[9]", | |
| "num": null, | |
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| }, | |
| "FIGREF10": { | |
| "type_str": "figure", | |
| "text": "d a p te d to give an alg o rith m for C C G . T hese are th e first polynom ial recognition a lg o rith m s th a t w ork directly w ith these form alism s. T his a p p ro ach can also be used to yield T A G recognition a lg o rith m , a lth o u g h th e TA G alg o rith m is not discussed in this p a p e r. A sim ilar a p p ro a c h has been in d ep en d en tly ta k e n by Lang [5] who p resen ts a E arley parser for T A G th a t a p p e a rs to be very closely related to the alg o rith m s p resen ted here. 5In the chart parser for CCG given by Pareschi and Steedman [6] the entire category is stored explicitly in each chart entry.", | |
| "num": null, | |
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| }, | |
| "TABREF0": { | |
| "num": null, | |
| "content": "<table><tr><td>CK Y alg o rith m for C F G recognition each a rray elem ent L t<J contains th a t subset of th e nonterm inal</td></tr><tr><td>sym bols th a t can derive th e su b strin g ax.. .a ; . In our algorithm the elem ents stored in Lij will encode</td></tr><tr><td>those n o n term in als and associated stacks th a t can derive th e strin g a, .. . a^.</td></tr><tr><td>In o rd er to o b tain a polynom ial algorithm we m ust encode the stacks efficiently. W ith each</td></tr><tr><td>n o n term in a l we store only the top of its associated stack and an indication of th e elem ent in L</td></tr><tr><td>where th e next p a rt of the stack can be found. T his is achieved by storing sets of tuples of the form</td></tr><tr><td>(.4 , 7 , A ' , 7 ',p, q) in the a rra y elem ents. Roughly speaking, a tuple (A , 7 , A', 7 ', p, < 7) is stored in I tiJ</td></tr><tr><td>w hen A [q7 /7 ] = > a, .. .aj and A/[q;7 /] -^ ap . . ,aq where q is a string of stack sym bols and A is</td></tr><tr><td>the unique distinguished descendent of A in the derivation of a</td></tr></table>", | |
| "type_str": "table", | |
| "text": "In considering th e recognition of LIG, we assum e th a t the underlying CFG is in C hom sky Norm al Form ; i.e., eith er tw o n o nterm inals (w ith their stack s) or a single term in al can a p p e ar on the RHS of a rule. A lthough we have not confirm ed w hether this yields a norm al form , a recognition algorithm for LIG in this form of LIG is sufficient to enable us to develop a recognition algorithm for C C G . We use an a rra y L consisting of n2 elem ents where th e strin g to be recognized is a x .. .a n . In the case of the", | |
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