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| "paper_id": "W85-0104", | |
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| "generated_with": "S2ORC 1.0.0", | |
| "date_generated": "2023-01-19T06:13:34.803877Z" | |
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| "title": "", | |
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| "abstract": "", | |
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| "paper_id": "W85-0104", | |
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| "abstract": [], | |
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| { | |
| "text": "T h e n the d e f i n i t i o n of U C F G c a n be a p p r o a c h e d as fo llows.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "G = (N,T,P,S) w h e r e N = L(G') fo r t h e f o l l o w i n g CF G:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "D e f i n i t i o n of u n i f i c a t i o n ba se d C F g r a m m a r (UCFG)", | |
| "sec_num": "1.2." | |
| }, | |
| { | |
| "text": "G* = (N',T',P*,Cat), T* = F + C + = ), (, 1, v , f ai 1.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "D e f i n i t i o n of u n i f i c a t i o n ba se d C F g r a m m a r (UCFG)", | |
| "sec_num": "1.2." | |
| }, | |
| { | |
| "text": "P' = C a t -> f a i l C a t -> Var^ V a r -> v; V a r l C a t -> c, c in C", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "D e f i n i t i o n of u n i f i c a t i o n ba se d C F g r a m m a r (UCFG)", | |
| "sec_num": "1.2." | |
| }, | |
| { | |
| "text": "C a t -> ( f l = C a t f 2 = C a t ... f n = C a t ) , w h e r e fl,f2,...,fn = F P a f i n i t e s u b s e t of N x V * , S in N.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "D e f i n i t i o n of u n i f i c a t i o n ba se d C F g r a m m a r (UCFG)", | |
| "sec_num": "1.2." | |
| }, | |
| { | |
| "text": "T h e f o r e m o s t d i f f e r e n c e h e r e is t h a t t h e s e t of n o n t e r m i n a l s y m b o l s of a U C F G is i n f i n i t e (it is t h e l a n g u a g e of a n o t h e r c o n t e x t free grammar).", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "D e f i n i t i o n of u n i f i c a t i o n ba se d C F g r a m m a r (UCFG)", | |
| "sec_num": "1.2." | |
| }, | |
| { | |
| "text": "A l t h o u g h o n l y a f i n i t e n u m b e r of n o n t e r m i n a l s c a n o c c u r in r u l e s , a r u l e c a n m a t c h ( u n i f y w i t h ) a n i n f i n i t y of d i f f e r e n t a U C F G r u l e c a n h a v e a n i n f i n i t e n u m b e r of r u l e i n s t a n c e s . T h e m a i n d i f f e r e n c e s c o m p a r e d to C F a r e t h a t t h e m a t c h i n g r e l a t i o n b e t w e e n A a n d t h e l e f t h a n d s i d e of p is n o t i d e n t i t y b u t u n i f i c a t i o n (a r u l e c a n m a t c h a n i n f i n i t e n u m b e r of n o n t e r m i n a l s c o n s i s t e n t w i t h it) a n d t h a t t h e e x p a n s i o n of A w i t h p c a n i n s t a n t i a t e (rewrite) n o n t e r m i n a l s in u o u t s i d e A. = mgu(A,B) .", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 768, | |
| "end": 778, | |
| "text": "= mgu(A,B)", | |
| "ref_id": null | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "D e f i n i t i o n of u n i f i c a t i o n ba se d C F g r a m m a r (UCFG)", | |
| "sec_num": "1.2." | |
| }, | |
| { | |
| "text": "u ( v , B , 0 ) = < v , B > if V in L ( V", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "D e f i n i t i o n of u n i f i c a t i o n ba se d C F g r a m m a r (UCFG)", | |
| "sec_num": "1.2." | |
| }, | |
| { | |
| "text": "g e n e r a l i z e d to su bs um pt io n. T h i s g r a m m a r l e f t g e n e r a t e s the s e n t e n c e s J u n g e n si nd J u n g e n and d a s s J u n g e n J u n g e n si nd (but no t the i l l i c i t orders). T h e s t a n d a r d i n t e r p r e t a t i o n of a u n i f i c a t i o n LP r u l e A < B c a n be r e s t a t e d as fo l l o w s . Or e q u i v a l e n t l y . ", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "T h i s d e f i n i t i o n c o m b i n e s t h e U C F G a n d I D L P y i e l d r e l a t i o n s in t h e s t r a i g h t f o r w a r d way. LP r u l e s a r e u s e d as l o c a l t e s t s c h e c k i n g a p e r m u t a t i o n of t h e R H S of a r u l e w h e n t h e r u l e is a p p l i e d . C a t e g o r y i d e n t i t y a s t h e m a t c h i n g r e l a t i o n is", | |
| "sec_num": null | |
| } | |
| ], | |
| "back_matter": [], | |
| "bib_entries": {}, | |
| "ref_entries": { | |
| "FIGREF0": { | |
| "text": "T h e p u r p o s e of t h i s p a p e r is t o c o n s i d e r e x t e n s i o n of u n i f i c a t i o n b a s e d c o n t e x t f r e e g r a m m a r w i t h (a s u i t a b l e g e n e r a l i z a t i o n of) the ID LP f o r m a l i s m of G a z d a r et al. (1985). W e d e f i n e u n i f i c a t i o n b a s e d c o n t e x t f r e e g r a m m a r (UCFG) as a g e n e r a l i z a t i o n of C F grammar. T h e s h a r e d p r o p e r t y is the C F forrr of p r o d u c t i o n s (a s i n g l e n o n t e r m i n a l o n t h e L H S of a ru le). T h i s a l l o w s u s e of a p p r o p r i a t e l y m o d i f i e d C F G p a r s i n g a l g o r i t h m s . T h e H U G g r a m m a r f o r m a l i s m of L a ur i K a r t t u n e n (paper p r e s e n t e d in this co nf er en ce ) c a n be r e g a r d e d as an i n s t a n c e of UCFG. T h e s t r u c t u r e of the p r e s e n t p a pe r is as f o l l o w s . In s e c t i o n 1, d e f i n i t i o n s a r e g i v e n to r e l e v a n t t y p e s of g r a m m a r . S e c t i o n 2 d i s c u s s e s g e n e r a l i z a t i o n of I D L P g r a m m a r to u n i f i c a t i o n . P p a r s i n g p r o b l e m b o u n d up w i t h the g e n e r a l i z a t i o n is d e s c r i b e d ir s e c t i o n 2.2, w h i c h c l o s e s w i t h a d e f i n i t i o n of u n i f i c a t i o n IDLI grammar. D i r e c t p a r s i n g of U I D L P g r a m m a r is d i s c u s s e d in sectior 2.3. T h e l a s t s e c t i o n 2.4 p r o p o s e s a g e n e r a l i z a t i o n of the n o t i o n of a u n i f i c a t i o n LP r u l e , a i m e d to a v o i d t h e c o s t of p a r s i n g f u l l U I D L P g r a m m a r w h i l e r e t a i n i n g e n o u g h e x p r e s s i v e p o w e r f o r t h e s t a t e m e n t o f c o m m o n t y p e s o f w o r d o r d e r c o n s t r a i n t s . 1. D e f i n i t i o n s S t a n d a r d n o t a t i o n s of f o r m a l g r a m m a r t h e o r y a r e u s e d (see e.g H o p c r o f t a n d U l l m a n (1979) f o r d e f i n i t i o n s ) , in, +, a n d i f f are u s e d fo r s e t t h e o r e t i c m e m b e r s h i p , u n i o n a n d d e f i n i t i o n a l -3S-LP rules in unification grammar Lauri Carlson Proceedings of NODALIDA 1985, pages 35-48 e q u i v a l e n c e , r e s p e c t i v e l y . 1.1. D e f i n i t i o n of s t a n d a r d CF g r a m m a r W e r e c a p i t u l a t e the d e f i n i t i o n of s t a n d a r d CF g r a m m a r to s e r v e as a p o i n t of c o mp ar is on . G = <N,T,P,S>, N, T d i s j o i n t and finite, P a f i n i t e s u b s e t of NxV*. u => w i f f u = xAy a n d w = xUy f o r s o m e A -> U in P, xy in V*. w in L(A) if f A =>* w. L(G) = L(S).", | |
| "type_str": "figure", | |
| "uris": null, | |
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| }, | |
| "FIGREF1": { | |
| "text": "n o n t e r m i n a l s in t h e c o u r s e of d i f f e r e n t d e r i v a t i o n s . In o t h e r", | |
| "type_str": "figure", | |
| "uris": null, | |
| "num": null | |
| }, | |
| "FIGREF2": { | |
| "text": "T o d e f i n e t h e y i e l d r e l a t i o n , w e n e e d to i n t r o d u c e a n u m b e r of a u x i l i a r y concepts. 1.2.1. S u b s t i t u t i o n s An s u b s t i t u t i o n is a p a r t i a l f u n c t i o n s f r om L(Var) to L(Cat). A s u b s t i t u t i o n s is e x t e n d e d to V * b y t h e c l a u s e s(b s t i t u t i o n w h i c h is o n e -o n e in L(Var) is a c a l l e d a r e n a m i n g of v a r i a b l e s . W e d e f i n e a n o t a t i o n a l e q u i v a l e n c e r e l a t i o n = = a m o n g a l p h a b e t i c v a r i a n t s so t h a t x==y if x = s(y ) for a r e n a m i n g of v a r i a b l e s s. = = is e x t e n d e d to p r o d u c t i o n s in t h e o b v i o u s way: p = A -> U == B -> W iff AU == BW.", | |
| "type_str": "figure", | |
| "uris": null, | |
| "num": null | |
| }, | |
| "FIGREF3": { | |
| "text": "s u b s t i t u t i o n s w i t h v a l u e s in L (V ar) c a n be i t e r a t e d .T o d e f i n e e v e n t u a l v a l u e s of s, w e d e f i n e s * ( x ) so t h a t S*(x) = X if s(x) is u n d e f i n e d or s'^(x) = x for s o m e n>0; e l s e s*(x) = S * ( s ( x ) ) . T h e s m a l l e s t s u b s t i t u t i o n is t h e e m p t y s u b s t i t u t i o n 0. T h e i n c o n s i s t e n t s u b s t i t u t i o n 1 is s u c h that l(x ) = f a i l for any x. 1.2.2. U n i f i c a t i o n A u n i f i e r for c a t e g o r i e s A,B is a s u b s t i t u t i o n s s.t. s(A) = s(B). s is a m axim ally g e n e r a l u n i f i e r (mgu) for A,B, or s = mgu(A,B), if u(s(AB)) = u(AB) f o r a n y u n i f i e r u for A,B. It c a n be d e t e r m i n e d up to a l p h a b e t i c v a r i a n c e u s i n g the f o l l o w i", | |
| "type_str": "figure", | |
| "uris": null, | |
| "num": null | |
| }, | |
| "FIGREF4": { | |
| "text": "a r ) d o e s n o t o c c u r in B. B,s) = s + u ( s * ( A ) , s * ( B ) ,0) o t h e r w i s e u( A, B, s) = 1. A u n i f i e s with B iff mgu(A,B) is not 1. T h e u n i f i c a t i o n of A and B is mgu (A,B)' (A) = mgu (A,B)' (B) \u2022 A subsumes B iff mgu(A,B)(A) = = A. T h a t is, u n i f i c a t i o n of B i n t o A d o e s n o t c h a n g e A. 1.2.3. D e r i v a b i l i t y W i t h thes e b a s i c c o nc ep ts , we can d e f i n e the U C F G y i e l d r e l a t i o n as f o ll ow s. u => w iff u = xAy a n d w = s(xWy) w h e r e B -> W == p for s o m e p r o d u c t i o n p in P, xy in V* and s u b s t i t u t i o n s = mgu(A,B).", | |
| "type_str": "figure", | |
| "uris": null, | |
| "num": null | |
| }, | |
| "FIGREF5": { | |
| "text": "T h e == r e l a t i o n in the d e f i n i t i o n a l l o w s r e n a m i n g of v a r i a b l e s b e t w e e n r e p e a t e d a p p l i c a t i o n s of a r u l e . W e h a v e c h o s e n to r e n a m e the p r o d u c t i o n p. E q u i v a l e n t l y , we c o u l d r e n a m e u. s U C F G g e n e r a t e s the n o n -C F l a n g u a g e1.3. C F I D L P g r a m m a r S t a n d a r d r e w r i t i n g r u l e s c o n t a i n d o m i n a n c e a n d p r e c e d e n c e i n f o r m a t i o n c o n n e c t e d together. T h e idea of IDLP g r a m m a r s is to s e p a r a t e d o m i n a n c e f r o m p r e c e d e n c e . T h e r e a r e t w o t y p e s of r u l e s ; ID r u l e s of f o r m A Blf \u2022\u2022\u2022 f Bri w h e r e A is a n o n t e r m i n a l a n d B l , . . . Bn f o r m a m u l t i s e t (set w i t h p o s s i b l e re pe ti ti on s) of s y m b o l s in V, and LP r u l e s of fo rm A < B w h e r e A and B are n o n t e r m i n a l s in N. A C F I D L P g r a m m a r H c a n be d e f i n e d as a p a i r (G,<) w h e r e G is a s t a n d a r d C F g r a m m a r and < is a s t r i c t p a r t i a l o r d e r i n g in NxN. F o r s i m p l i c i t y , w e f i x a s t a n d a r d o r d e r i n g of t h e R H S ' s of ID r u l e s w h i c h c o n t a i n s < as a s u b s e t . H e n c e f o r t h w e a s s u m e ID r u l e s are n o r m a l i z e d into s t a n d a r d order. To d e f i n e the y i e l d r e l a t i o n , we p r o c e e d as f o l l o w s . Le t L be a", | |
| "type_str": "figure", | |
| "uris": null, | |
| "num": null | |
| }, | |
| "FIGREF6": { | |
| "text": "s u b s e t of V*. W e d e f i n e Sat(L ,<) as t h e s e t of t h o s e w o r d s w in L t h a t a r e n o t of f o r m xByAz w h e r e A < B. If w is in Sat(V*r<) w e s a y t h a t w s a t i s f i e s <. A p r o d u c t i o n p s a t i s f i e s < iff its R H S s a t i s f i e s <. W e d e f i n e a r e l a t i o n -> in V* so t h a t x -> y iff y is a p e r m u t a t i o n of x and y s a t i s f i e s <. T h e n u => w in H P st.t W -> U. (G,<) i f f u = xAy, w = xUy, a n d A -> W is in T h e n o v e l t y he re is th at a g i v e n p r o d u c t i o n a c t u a l l y d e f i n e s a set of o r d e r e d p r o d u c t i o n s o b t a i n e d by p e r m u t i n g its ri gh t ha nd side in LP a c c e p t a b l e ways. T h u s a n y I D L P g r a m m a r h a s an e q u i v a l e n t o r d e r e d g r a m m a r . C o n v e r s e l y , a n y o r d e r e d G h a s a t r i v i a l e q u i v a l e n t I L D P G . T h e c o n v e r s i o n s a f f e c t the n o n t e r m i n a l v o c a b u l a r y and t h e r e w i t h the p a r s e trees g e n e r a t e d by the grammars. In G a z d a r et al. (1 98 5: 49 ) t h e s t r o n g g e n e r a t i v e c a p a c i t y of IDLP g r a m m a r s is c h a r a c t e r i z e d in t e rm s of the E C P O ( E q u i v a l e n t C o n s t a n t P a r t i a l O r d e r i n g ) p r o p e r t y . In an I D L P G w i t h f i x e d n o n t e r m i n a l v o c a b u l a r y , if A < B in t h e R H S of o n e p r o d u c t i o n , A < B in t h e R H S of a l l p r o d u c t i o n s . If a n d o n l y if a o r d e r e d g r a m m a r h a s t h i s p r o p e r t y , it c a n be r e w r i t t e n as an I D L P g r a m m a r w i t h o u t c h a n g i n g the n o n t e r m i n a l v o c a b u l a r y . S i n c e it is e a s y to m o v e f r o m t h e e x t e n d e d v o c a b u l a r y of a c o v e r i n g IDLP g r a m m a r to the o r i g i n a l v o c a b u l a r y of the o r d e r e d g r a m m a r in any g i v e n d e r i v a t i o n (cf. A h o and U 1 l m a n 7 2 :275), the E C P O p r o p e r t y is of s l i g h t p r a c t i c a l interest. T h i s is a l l the m o r e t r u e in U C F G , w h e r e t h e f e a t u r e c o m p o s i t i o n of t h e t o p c a t e g o r y p r o v i d e s a m o r e f l e x i b l e d e s c r i p t i o n of s e n t e n c e s t r u c t u r e than d e r i v a t i o n history. -4D-40 Proceedings of NODALIDA 1985 1.3.2. P a r s i n g of IDLP g r a m m a r T h i s s e c t i o n a s s u m e s s t a n d a r d n o t i o n s of E a r l e y p a r s i n g (see e.g. A h o a n d O i l m a n 1972). B a r t o n (1985) sh o w s p a r s i n g of IDLP g r a m m a r s N P c o m p l e t e in the w o r s t c a s e (w h e n < is e m p t y ) . T h e b a s i c f a c t is t h a t t h e n u m b e r n! of p e r m u t a t i o n s of a s t r i n g of l e n g t h n g r o w s e x p o n e n t i a l l y w i t h n. T h e c o m b i n a t o r i a l e x p l o s i o n a r i s e s in c a s e s of l e x i c a l a m b i gu it y, w h e r e (say) a s t r i n g of f o r m a l...a n c a n be p a r s e d in n! w a y s b y an I D L P g r a m m a r H = {G,<) w i t h G = (S -> A l...A n , Ai -> aj for a l l i , j ) a n d < e m p t y . T h o u g h e x p o n e n t i a l in t h e w o r s t c a s e , d i r e c t p a r s i n g of I D L P g r a m m a r s c a n s h o w s a v i n g s c o m p a r e d to p a r s i n g c o r r e s p o n d i n g o r d e r e d g r a m m a r s . T h e s o u r c e of t h e s a v i n g s is t h a t t h e n u m b e r of i t e m s in t h e p a r s e t a b l e c a n be k e p t s m a l l b y k e e p i n g t o g e t h e r i t e m s t h a t a r e r e p r e s e n t e d s e p a r a t e l y in t h e o r d e r e d g r a m m a r . In t h e w o r s t c a s e , t h i s g e t s t h e n u m b e r of i t e m s d o w n f r o m 0 ( /G /! ) to 0 ( 2^^/ ) (one i t e m p e r s u b s e t of R H S s y m b o l s i n s t e a d of o n e per permutation). A n E a r l e y p a r s e i t e m A -> x.y is < -> to i t e m A -> z.w iff x < -> z a n d y < -> w. T o s i m p l i f y t h e i d e n t i f i c a t i o n of e q u i v a l e n t items, th ey c a n be n o r m a l i z e d to a fixe d a l p h a b e t i c order. T h e t e s t yB < -> By i n v o l v e s c h e c k i n g L P -a c c e p t a b i 1 i ty of By T h i s can be d o n e by l o o p i n g t h r o u g h a l l LP r u l e s and p a i r s B,C, C in y. S i n c e t h e r e s u l t of t h e t e s t d e p e n d s o n l y o n G, it c a n be pr e c o m p u t e d . 2. U I D L P g r a m m a r 2.1. D e f i n i t i o n A n a l o g y w i t h the CF c a se s u g g e s t s the f o l l o w i n g d e f i n i t i o n s . _^1_ 41 Proceedings of NODALIDA 1985 A U I D L P g r a m m a r is a p a i r H = (Gr<) w h e r e G is a U C F G a n d < is a s t r i c t p a r t i a l orde r on V. L e t L be a s u b s e t of V*. Sat(L ,<) = (w in L: w is n o t of f o r m xCyDz w h e r e A < B a n d CD s u b s u m e s BA). If w is in S a t(V *,< ) we s a y t h a t w s a t i s f i e s <. A p r o d u c t i o n p s a t i s f i e s < iff it s RHS s a t i s f i e s <. x -> y iff y is a p e r m u t a t i o n of x and y s a t i s f i e s <. T h e n u => w in H = (G,<) iff u = xAy a n d w = s(xWy) w h e r e s(W) < -> U a n d B -> U == p fo r s o m e p r o d u c t i o n p in P, xy in V* a n d s u b s t i t u t i o n s", | |
| "type_str": "figure", | |
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| }, | |
| "FIGREF7": { | |
| "text": "2.2. N o n l o c a l s u b s u m p t i o n p r o b l e m It f o l l o w s f r o m t h e s e d e f i n i t i o n s t h a t t h e L P -a c c e p t a b l e p e r m u t a t i o n s of a n i n s t a n c e of a U I D r u l e c a n be a p r o p e r s u b s e t of the p e r m u t a t i o n s of the o r i g i n a l rule. Th e r e a s o n is th at in g e n e r a l , S a t ( L ,< ) is a s u b s e t of S a t ( s ( L ) , < ) b u t n o t v i c e versa. I n s t a n t i a t i o n c a n m a k e LP r u l e s a p p l i c a b l e w h i c h do not a p p l y to the u n i n s t a n t i a t e d rule. In ot he r words, in U I D L P g r am ma r, w o r d orde r c a n be s e n s i t i v e to c o n t e x t . F o r e x a m p l e , t h e o r d e r of a V a n d i t s c o m p l e m e n t s m a y d e p e n d on t h e c h a r a c t e r of t h e c l a u s e t h e y b e l o n g to ( G e r m a n , Finnish). T h i s c a n n o t be d e c i d e d l o c a l l y by l o o k i n g at the v e r b a n d its c o m p l e m e n t s . T h i s s i t u a t i o n is e x e m p l i f i e d in t h e f o l l o w i n g U I D L P g r a m m a r . (CF c a t e g o r y s y m b o l s i n d e x e d w i t h f e a t u r e e q u a t i o n s s e r v e a s s h o r t h a n d s f o r U C F G c a t e g o", | |
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| "text": "H o w e v e r , the sa me g r a m m a r p a r s e d b o t t o m up righ t to l e f t a c c e p t s the i l l i c i t s e n t e n c e d a s s J u n g e n si nd J u n g e n ; m i l a r p a r a d o x c a n be c o n s t r u c t e d b e t w e e n le ft and ri gh t top d o w n d e r i v a t i o n s o f a v a r i a n t g r a m m a r w i t h r u l e s Ss r e s u l t is u n d e s i r a b l e in t h a t it i m p o r t s a n o r d e r d e p e n d e n t , p r o c e d u r a l f e a t u r e to a n o t h e r w i s e d e c l a r a t i v e fo rm a l i s m . T h e L P -a c c e p t a b i 1 ity of a s e n t e n c e ca n d e p e n d on the o r d e r a p p l y i n g t h e r u l e s of g r a m m a r , s o t h a t c e r t a i n p a r s i n g s t r a t e g i e s c a n p a s s s e n t e n c e s w h i c h f a i l L P -r u l e s o n t h e su rf ac e. A s fa r as t h e d e f i n i t i o n of d e r i v a b i l i t y is c o n c e r n e d , w h a t w e s h o u l d h a v e s a i d to b e g i n w i t h is f a i r l y c l e a r . W e w a n t LP r u l e s to r e g u l a t e t h e o r d e r of s i s t e r c o n s t i t u e n t s at a l l s t a g e s of 43 Proceedings of NODALIDA 1985 d e r i v a t i o n (in p a r t i c u l a r , at the end of d e r i v a t i o n s ) . T h i s can be s t a t e d as a g l o b a l c o n d i t i o n o n d e r i v a t i o n s , or, as is a c t u a l l y d o n e in G P S G ( G a z d a r e t al. 1 9 8 5 : 4 6 ) , by i n t e r p r e t i n g L P c o n d i t i o n s a s n o d e a d m i s s i b i l i t y c o n d i t i o n s . I n t h i s in t e r p r e t a t i o n , an L P -r u l e A < B reads: (LP) A p a i r of n o d e s B*A* s u b s u m i n g BA c a n n o t a p p e a r as s i s t e r s in a d e r i v a t i o n tree. G i v e n t h a t L P -a c c e p t a b i 1 i t y is t a k e n c a r e of by ( L P ) , w e c a n s i m p l i f y the U I D L P G y i e l d r e l a t i o n as f o l l o w s : u => w in H = (G,<) iff u = xAy a n d w = s(xWy) w h e r e s(W) is a p e r m u t a t i o n of U a n d B -> U == p for s o m e p r o d u c t i o n p in P, xy in V* and s u b s t i t u t i o n s = mgu(A,B). 2.3. P a r s i n g of U I D L P g r a m m a r s O n e w a y to p a r s e U I D L P as r e v i s e d a b o v e is to s i m p l y a p p l y L P r u l e s in the c o m p l e t e d p a r s e so as to f i l t e r out L P -i n c o n s i s t e n t p a r s e s . T h i s is c o n c e p t u a l l y s t r a i g h t f o r w a r d b u t i n e f f i c i e n t . A l l p e r m u t a t i o n s w o u l d be c o n s i d e r e d o n l y to be d i s c a r d e d at t h e f i n a l step. W e s h o u l d b r i n g L P c o n s t r a i n t s to b e a r as s o o n as p o s s i b l e , i.e. a p p l y L P r u l e s a s g l o b a l c o n s t r a i n t s o n der i v a t i o n s . C o n s i d e r a g a i n t h e i l l i c i t p a r s e a b o v e . A l t h o u g h V a n d its c o m p l e m e n t NP do not s u b s u m e the LP r u l e NP < V(type = sub), th ey d o u n i f y w i t h t h e r u l e . (If t h e y d i d n o t u n i f y w i t h t h e r u l e , t h e y c o u l d n o t s u b s u m e it l a t e r e i t h e r . ) S u c h u n d e c i d e d a p p l i c a t i o n s of LP r u l e s s h o u l d r e m a i n a c t i v e u n t i l a d e c i s i o n c a n be m a d e . (A g a i n , t h e d e c i s i o n s h o u l d be d o n e as e a r l y as p o s s i b l e to c u t o f f p a r s e s . ) L e t us s a y t h a t in s u c h c a s e s , t h e LP r u l e properly u n ifies w i t h the pa ir of c a te go ri es . u m e B < C p r o p e r l y u n i f i e s w i t h B \\ C w h e n i t e m A -> xC'.yB'z is f o r m e d . W e n e e d to s a v e t h e a c t i v e c o n s t r a i n t w i t h t h e u n d e c i d e d i t e m in s u c h a w a y t h a t it w i l l be r e a p p l i e d to i n s t a n c e s of the o r i g i n a l u n d e c i d e d pair. T o d o so, w e a s s o c i a t e w i t h e a c h u n d e c i d e d i t e m w i t h a l i s t of c o n s t r a i n t s of the f o l l o w i n g kind. A c o n s t r a i n t c is of fo rm \"A < B to B'A*\". It is matched if A*B* s u b s u m e s AB, i r r e l e v a n t if A'B* d o e s n o t u n i f y w i t h AB, a n d a c t i v e o t h e r w i s e . If s is an s u b s t i t u t i o n , s(c) = \"A < B to s(B*A*)\". W h e n e v e r a n e w i t e m i = s(A -> xC'.yz) is to be c o m b i n e d f r o m i t e m s j = A -> x.yC'w a n d k = C\" -> u. u s i n g m g u s, w e c h e c k e a c h c o n s t r a i n t o n t h e c o n s t r a i n t l i s t (s (c ): c i s on the c o n s t r a i n t l i s t of i or j or is of form \"B < C to C*B'\", B* in y and B < C a LP r u le .) If c is m a t c h e d , r e j e c t t h e c o m b i n a t i o n . If c is i r r e l e v a n t , d e l e t e t h e c o n s t r a i n t f r o m t h e c o n s t r a i n t li s t . F i n a l l y , if t h e c o m b i n a t i o n is n o t r e j e c t e d , a s s i g n t h e r e m a i n i n g c o n s t r a i n t s as the c o n s t r a i n t l i s t of the n e w item. T h e a b o v e p r o c e d u r e is r a t h e r c u m b e r s o m e . W h a t is m o r e , it is d i f f i c u l t to find c a s e s in a c t u a l l a n g u a g e s w h e r e it is r e a l l y needed. T h e ty pe s of c o n t e x t s e n s i t i v e w o r d or d e r c o n s t r a i n t s I h a v e f o u n d a l l o w for a s i m p l e r f i x w h i c h is s k e t c h e d in t h e f o l l o w i n g section. 2.4. G e n e r a l i z e d LP r u l e s A U C F G c a t e g o r y c a n b e r e p r e s e n t e d b y a s e t o f f e a t u r e e q u a t i o n s , s p e c i f y i n g v a l u e s of f e a t u r e s i n s t a n t i a t e d in e a c h ca te go ry . C o n v e r s e l y , the set c o n s t i t u t e s the l e a s t s o l u t i o n of its r e p r e s e n t i n g e q u a t i o n s in the d o m a i n of U C F G c a t e go ri es . A p a i r of U C F G c a t e g o r i e s c a n be s i m i l a r y r e p r e s e n t e d as a hi g h e r or de r c a t e g o r y w i t h a t t r i b u t e s 0,1 for the m e m b e r s of the pa ir . F e a t u r e e q u a t i o n s c a n t h e n be u s e d to d e s c r i b e c a t e g o r ypairs. (The idea is a d a p t e d f r om K a r t t u n e n (this volume).)", | |
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| }, | |
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| "text": "If A'B* s u b s u m e s AB, t h e n A' < B*.T h e c o n t r a p o s i t i o n of this c o n s t r a i n t (given A* is not B*) is( 2 ) If A* > B*, th en not: A'B* s u b s u m e s AB. If t h e c o m p l e m e n t c(A B ) o f t h e s p e c i f i c a t i o n AB c a n b e e x p r e s s e d , t h e c o n t r a p o s i t i v e c o n s t r a i n t c a n b e f u r t h e r r e w r i t t e n as (3) If A* > B* t h e n A'B* u n i f i e s w i t h c(AB). T h e n (3) c o u l d be d i r e c t l y v e r i f i e d b y u n i f y i n g c(AB) i n t o A'B' w h e n e v e r t h e a n t e c e d e n t of (3) is t a k e n . T h i s s u f f i c e s to g u a r a n t e e s a t i s f a c t i o n of t h e c o n s e q u e n t of (3), in f a c t s t r e n g t h e n s it to s u b s um pt io n. In o u r e x a m p l e g r a m m a r l i c a t i o n of NP < V(type = sub) to t h e p a i r V(type = x) NP c o u l d s i m p l y i n s t a n t i a t e V into V (type=m ain). T h e s e o b s e r v a t i o n s s u g g e s t the f o l l o w i n g g e n e r a l i z a t i o n of the n o t i o n of a LP rule. (4) A s s u m e x > y. T h e n if xy s u b s u m e s AB t h e n xy s u b s u m e s CD.", | |
| "type_str": "figure", | |
| "uris": null, | |
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| }, | |
| "FIGREF10": { | |
| "text": "xy s u b s u m e s AB, e l s e xy s u b s u m e s CD. L e t us c o n s i d e r s o m e s p e c i a l c a s e s of t h i s g e n e r a l form. T h e o r i g i n a l r u l e A < B is o b t a i n e d as", | |
| "type_str": "figure", | |
| "uris": null, | |
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| }, | |
| "FIGREF11": { | |
| "text": "< y if xy s u b s u m e s AB e l s e xy s u b s u m e s f a i l . Our e x a m p l e of n o n l o c a l s u b s u m p t i o n is ta k e n c a r e of by the pair of r u l e s (7) A s s u m e x > y. T h e n if xy s u b s u m e s ((x cat) = V) (y cat) = NP)) t h e n xy s u b s u m e s ((x type) = sub) u m e x > y. T h e n if xy s u b s u m e s ((x cat) = NP) (y cat) = V)) th en xy s u b s u m e s ((x type) = main) T h e s e r u l e s i n s t a n t i a t e the m a i n and s u b o r d i n a t e c l a u s e f e a t u r e s at t h e t i m e w h e n t h e V P o r d e r is f i x e d . (An a b b r e v i a t i o n of c o m p l e m e n t a r y r u l e s s u c h a s ( 7 ) -( 8 ) i n t o o n e r u l e s e e m s a p p r o p r i a t e . ) if xy s u b s u m e s (x = (y s u b j e c t ) ) e l s e xy s u b s u m e s ((x function) = rheme) T h i s r u l e s a y s t h a t a s u b j e c t f o l l o w i n g t h e m a i n v e r b is rh em at ic . A s s u g g e s t e d b e f o r e , g e n e r a l i z e d L P r u l e s of f o r m (4) ( r e s p e c t i v e l y , (5)) c a n be i m p l e m e n t e d in p a r s i n g s o t h a t t h e c o n s e q u e n t (else) c o n d i t i o n of the r u l e is a c t u a l l y u n i f i e d in w h e n the or de r c o n d i t i o n of the r u l e is a p p l i c a b l e (violated). A n i m p l e m e n t a t i o n of g e n e r a l i z e d LP r u l e s of this kind into H U G is in progress. It s h o u l d be ke pt in m i n d th at g e n e r a l i z e d LP r u l e s do not s o l v e the n o n l o c a l s u b s u m p t i o n p r o b l e m . A t best, they m a k e it p o s s i b l e to a v o i d t h e p r o b l e m by a l l o w i n g s t a t e m e n t of s o m e c o n t e x t s e n s i t i v e w o r d o r d e r r u l e s w i t h o u t r e f e r e n c e to n o n l o c a l -a i-47 Proceedings of NODALIDA 1985", | |
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| "text": "e C o m p u t a t i o n a l D i f f i c u l t y of I D / L P Parsing\", in Proceedings of the 23rd Annual Meeting of the ACL, C h i c a g o . G a z d a r , G, E. K l e i n , G. P u l l u m a n d I. S a g (1985), G e n e r a lize d Phrase Structure Grcimmar. H a r v a r d U n i v e r s i t y Press. H o p c r o f t , J. a n d J. U l l m a n (1979), In tro d u c tio n to Automata Theory, Languages, and Computation. A d d i s o n -W e s l e y .", | |
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