LP rules in unification grammar

La ur i C a r l s o n

U n i v e r s i t y of H e l s i n k i

R e s e a r c h U n i t for C o m p u t a t i o n a l L i n g u i s t i c s

LP R U L E S IN U N I F I C A T I O N G R A M M A R

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

-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 N x V * .

u => w i f f u = x Ay a n d w = xUy f o r s o m e A -> U in P, x y in V * .

w in L( A) if f A = > * w. L( G) = L ( S ) .

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.

1.2. 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)

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:

G* = (N',T',P*,Cat), T* = F + C + = ), (, 1, v , f ai 1.

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

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.

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).

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 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

-w o r d s , 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 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 ( V a r ) to L ( C a t ) .

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 ( x y ) = s ( x ) s ( y ) .

A s u b s t i t u t i o n w h i c h is o n e - o n e in L ( V a r ) 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.

A 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 a r ) 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 a x i m a l l y g e n e r a l u n i f i e r (mgu) for A , B , or s = m g u ( A , B ) , if u ( s ( A B ) ) = 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 n g schema.

m g u ( A , B ) = u ( A , B , 0 ) , w h e r e

-u(c,c,0) = Oy c i n C

u ( v , B , 0 ) = < v , B > if V in L ( V a r ) d o e s n o t o c c u r in B.

u ( ( f l = c l . . . f n = c n ) , ( f l = d l . . . f n = d n ) , 0 )

u ( c l , d l , u ( ( f 2 = c 2 . . . f n = c n ), ( f 2 = d 2 . . . f n = d n ) , 0 )

u(A,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 w i t h B iff m g u ( 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) • A s u b s u m e s B iff m g u ( 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 = x Ay a n d w = s ( x Wy ) w h e r e B - > W == p for s o m e p r o d u c t i o n p i n P, x y i n V * and s u b s t i t u t i o n s = m g u ( A , B ) .

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.

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.

3R-E x a m p l e UCFG;

( c a t = S num=vO) - > ( c a t = A n u m=v l ) ( c a t = B n um=v l ) ( c a t = C num=vl )

( c a t = v l num=( num=v2) ) - > ( c a t = v l num=v2) ( c a t = v l num=0)

( c a t = A

nuiff=0)

( c a t = B

num=0)

( c a t = C

nuiD=0)

T h i 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 e

1.3. 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 B l f ••• 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 re la ti on , we p r o c e e d as fo ll o w s . Let L be a

-s u b -s e t of V * . W e d e f i n e S a t ( L , < ) as t h e s e t of t h o s e w o r d s w i n L t h a t a r e n o t of f o r m x By Az w h e r e A < B. If w is in S a t ( 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 = x A y , w = x U y , 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-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 , A i - > a j f o r 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 .

A U I D L P g r a m m a r is a p a i r H = ( G r < ) 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 * . S a t ( L , < ) = (w i n 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 = x Ay a n d w = s ( x Wy ) 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 i n P, x y i n V * a n d s u b s t i t u t i o n s = m g u ( A , B ) .

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 g e n e r a l i z e d to su bs um pt io n.

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 r y s y m b o l s .)

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 = x Ay a n d w = s ( x W y ) 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 i n P, xy i n V * and s u b s t i t u t i o n s = m g u ( 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 ( t y p e = s u b ) , 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 p r o p e r l y u n i f i e s w i t h the pa ir of c a te go ri es .

4-A s s 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 - > x C ' . y B ' 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 t o B ' A * " . It is m a t c h e d 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 t o s ( B * A * ) " .

W h e n e v e r a n e w i t e m i = s ( A -> x C ' . y z ) is to be c o m b i n e d f r o m i t e m s j = A - > x . y C ' 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 o n t h e c o n s t r a i n t l i s t o f i o r j o r i s o f f o r m " B < C t o C * B ' " , B* i n y

and B < C a LP r u l e . ) 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 y pairs. (The idea is a d a p t e d f r om K a r t t u n e n (this volume).)

-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 .

(

1

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 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 ( A B ) .

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 ( A B ) 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 , s i n c e ( t y p e = s u b ) = c ( t y p e = m a i n ) ,

a p p l i c a t i o n of NP < V ( t y p e = s u b ) to t h e p a i r V ( t y p e = 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 ( t y p e = m a i n ) .

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 x y s u b s u m e s AB t h e n x y s u b s u m e s

CD.

Or e q u i v a l e n t l y .

(5) X < y if x y s u b s u m e s AB, e l s e x y 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

6-(6) X < 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 x y s u b s u m e s ( ( x c a t ) = V) ( y c a t )

= N P )) t h e n x y s u b s u m e s ( ( x t y p e ) = s u b )

(8) A s s u m e x > y. T h e n if x y s u b s u m e s ( ( x c a t ) = NP) (y c a t )

= V ) ) th en x y s u b s u m e s ( ( x t y p e ) = m a i n )

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 . )

T h e n e x t e x a m p l e is a r u l e of f u n c t i o n a l w o r d o r d e r i n t e r p r e t a t i o n .

(9) X < y if x y s u b s u m e s ( x = ( y s u b j e c t ) ) e l s e x y

s u b s u m e s ( ( x f u n c t i o n ) = 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

-c o n d i t i o n s .

R e f e r e n c e s

A h o , A. a n d J. U l l m a n ( 1 9 7 2 ) , T h e T h e o r y o f P a r s i n g , T r a n s l a t i o n , and C o m p i l i n g . V o l u m e 1 : P a r s i n g . P r e n t i c e - H a l l .

B a r t o n , E. (1985), " T h 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 P r o c e e d i n g s o f t h e 2 3 r d A n n u a l M e e t i n g o f t h e 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 ( 1 9 8 5 ) , G e n e r a l i z e d P h r a s e S t r u c t u r e 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), I n t r o d u c t i o n t o A u t o m a t a T h e o r y , L a n g u a g e s , and C o m p u t a t i o n . A d d i s o n - W e s l e y .

S h i e b e r , S. ( 1 9 8 4 ) , " D i r e c t P a r s i n g o f I D / L P G r a m m a r s " ,

L i n g u i s t i c s and P h i l o s o p h y 7 , 13 5- 15 4.