Defaults in Unification Grammar

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D E F A U L T S I N U N I F I C A T I O N G R A M M A R Gosse B o u m a

Research Institute for Knowledge S y s t e m s Postbus 463, 6200 AL Maa.qtrlcht. The Netherlands

ABSTRACT

I n c o r p o r a t i o n o f d e f a u l t s in g r a m m a r f o r m a l i s m s is i m p o r t a n t for r e a s o n s o f l i n g u i s t i c a d e q u a c y a n d g r a m m a r o r g a n i z a t i o n . In this p a p e r we p r e s e n t a n algorithm for h a n d l i n g default information in unification g r a m m a r . T h e algorithm specifies a logical o p e r a t i o n o n f e a t u r e s t r u c t u r e s , merging with the n o n - d e f a u l t s t r u c t u r e only t h o s e p a r t s o f t h e d e f a u l t f e a t u r e s t r u c t u r e which are n o t c o n s t r a i n e d b y the n o n - d e f a u l t s t r u c t u r e . We p r e s e n t v a r i o u s l i n g u i s t i c applications of default unification.

L INTRODUCTION

MOTIVATION. T h e r e a two, n o t quite u n r e l a t e d , r e a s o n s f o r i n c o r p o r a t i n g d e f a u l t s m e c h a n i s m s into a linguistic formalism. First, l i n g u i s t s h a v e o f t e n a r g u e d t h a t c e r t a i n p h e n o m e n a are described m o s t n a t u r a l l y with the u s e of r u l e s or o t h e r formal devices t h a t m a k e u s e o f a n o t i o n o f d e f a u l t (see, for instance, G a z d a r 1987). The s e c o n d r e a s o n is t h a t t h e u s e o f d e f a u l t s s i m p l i f i e s t h e d e v e l o p m e n t o f large a n d complex g r a m m a r s , in particular, the d e v e l o p m e n t of lexicons for s u c h g r a m m a r s (Evans & G a z d a r 1988). T h e l a t t e r s u g g e s t s t h a t the u s e of d e f a u l t s Is o f p a r t i c u l a r r e l e v a n c e for t h o s e b r a n d s o f Unification G r a m m a r (UG) t h a t are lexicalist, t h a t is, in which the lexicon is the main s o u r c e o f g r a m m a t i c a l i n f o r m a t i o n ( s u c h a s C a t e g o r i a l U n i f i c a t i o n G r a m m a r (Uskorelt 1986, C a l d e r et al. 1988) a n d H e a d - d r i v e n P h r a s e S t r u c t u r e G r a m m a r (Pollard & Sag

1987)).

We p r o p o s e a m e t h o d f o r i n c o r p o r a t i n g d e f a u l t s Into UG, in s u c h a w a y t h a t It b o t h e x t e n d s t h e linguistic a d e q u a c y o f UG a n d s u p p o r t s the f o r m u l a t i o n of rules, t e m p l a t e s a n d lexical e n t r i e s for m a n y unification-based theories. In the next section, we define default unification, a logical o p e r a t i o n on f e a t u r e s t r u c t u r e s . It is defined for a language. FM/, ~, w h i c h is in m a n y r e s p e c t s i d e n t i c a l to t h e language FML as defined in Kasper & R o u n d s ( 1 9 8 6 ) . N e x t , w e c o m e to l i n g u i s t i c applications o f default unification. A linguistic n o t a t i o n is introduced, w h i c h can be u s e d to describe a n u m b e r of linguistically interesting

p h e n o m e n a , s u c h a s f e a t u r e p e r c o l a t i o n . coordination, a n d m a n y a s p e c t s of inflectional morphology. Furthermore. it can be u s e d in the s e n s e o f F l l c g l n g e r et al. (1985) to define e x c e p t i o n s to r u l e s , n o n - m o n o t o n i c specialization o f t e m p l a t e s or i r r e g u l a r lexlcal entries.

BACKGROUND. T h e r e a r e s e v e r a l p r o p o s a l s w h i c h hint at the possibility of a d d i n g default m e c h a n i s m s to the linguistic formalisms a n d theories Just m e n t i o n e d . The fact t h a t GPSG ( G a z d a r et al., 1985) m a k e s h e a v y u s e of defaults, h a s led to some r e s e a r c h c o n c e r n i n g t h e c o m p a t i b i l i t y of GPSG with a f o r m a l i s m s u c h PATR-II (Shieber 1986a) a n d c o n c e r n i n g the logical n a t u r e o f the m e c h a n i s m s u s e d in GPSG (Evans 1987). S h i e b e r (1986a) proposes a n operation . r i d conservatively, which a d d s i n f o r m a t i o n of a f e a t u r e s t r u c t u r e A to a f e a t u r e s t r u c t u r e B, in a s f a r a s t h i s information is not in conflict with i n f o r m a t i o n In B. Suggestions for similar o p e r a t i o n s can be f o u n d in Shivber (1986b:59-61) (the overwrite option of PATR-II) a n d Kaplan (1987) (priority union). Fllckinger et al. (1985) a r g u e for the i n c o r p o r a t i o n o f d e f a u l t i n h e r i t a n c e m e c h a n i s m s In UG a s a n a l t e r n a t i v e for the t e m p l a t e s y s t e m of PATR-II.

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d i s t i n c t i o n b e t w e e n t h e two a p p r o a c h e s is e s p e c i a l l y r e l e v a n t f o r r e e n t r a n t f e a t u r e values.

The definition p r e s e n t e d in the n e x t section is defined a s a n o p e r a t i o n o n a r b i t r a r y f e a t u r e structures, a n d t h u s it is more general t h a n the o p e r a t i o n s o d d conservatively or overwrite, in which o n l y one s e n t e n c e a t a t i m e (say, <X 0 h e a d > = <X 1 h e a d > o r <subject case> = nominative] is a d d e d to a f e a t u r e description. An o b v i o u s a d v a n t a g e o f o u r a p p r o a c h is t h a t overwriting a s t r u c t u r e F with 1 ~ is equivalent to adding F a s default information to F'. Default u n i f i c a t i o n , a s d e f i n e d below, follows t h e a p p r o a c h in w h i c h d e f a u l t i n f o r m a t i o n is r e m o v e d if it is c o n s t r a i n e d in the n o n - d e f a u l t s t r u c t u r e . T h i s decision is to a c e r t a i n e x t e n t l i n g u i s t i c a l l y m o t i v a t e d (see s e c t i o n 3), b u t p e r h a p s m o r e i m p o r t a n t is t h e fact t h a t we w a n t e d to avoid t h e following p r o b l e m . F o r a r b i t r a r y f e a t u r e s t r u c t u r e s , t h e r e is n o t a l w a y s a u n i q u e w a y to resolve a unification conflict, n o r is it n e c e s s a r i l y t h e c a s e t h a t one s o l u t i o n s u b s u m e s o t h e r solutions. C o n s i d e r for i n s t a n c e the e x a m p l e s in (I).

(1) default n o n - d e f a u l t

a <f>ffia <f> = <g> <g> = b

b. <f> = <g> < f > f a <g> ffi b

To resolve the conflict, in Ca), either one of the e q u a t i o n s could b e r e m o v e d . In (b), e i t h e r the fact t h a t <g> = b or the r e e n t r a n c y could b e r e m o v e d (in b o t h cases, this would remove the inpllcit fact t h a t <f> = b). An a p p r o a c h w h i c h o n l y t r i e s t o r e m o v e t h e s o u r c e s o f a unification conflict, will t h u s b e forced to m a k e a r b i t r a r y d e c i s i o n s a b o u t t h e o u t c o m e o f t h e d e f a u l t unification p r o c e d u r e . At l e a s t for the p u r p o s e s o f g r a m m a r development, this s e e m s to be a n u n d e s i r a b l e situation 1.

2 . D E S C R I P T I O N O F T H E A L G O R I T H M

THE LANGUAGE FML*. D e f a u l t u n i f i c a t i o n is d e f i n e d in t e r m s of a f o r m a l l a n g u a g e for feature s t r u c t u r e s , b a s e d on K a s p e r & R o u n d s ' (1986) l a n g u a g e FML. FML* d o e s not c o n t a i n d i s j u n c t i o n , h o w e v e r , a n d f u r t h e r m o r e , e q u a t i o n s o f t h e f o r m / : f ( w h e r e ¢~ is a n a r b i t r a r y f o r m u l a ) a r e r e p l a c e d b y e q u a t i o n s

1 However, in E v a n s ' (1987) v e r s i o n o f F e a t u r e Specification Defaults, it is s i m p l y allowed t h a t a c a t e g o r y d e s c r i p t i o n h a s m o r e t h a n o n e ' s t a b l e e x p a n s i o n ' .

of the form  : ¢x (where a is a t o m i c or NIL or TOP).

(2) ~ ~ FML*

NIL T O P

a a • A (the s e t of atoms)

: a p e L* (L the s e t of labels) a n d a • A u {TOP,NIL} [<pl>,..,<pn>] e a c h P i e L*

¢ ^ ¥ ¢,¥ • FML*

W e a s s u m e that feature structures are r e p r e s e n t e d a s directed acycllc g r a p h s (dags). T h e d e n o t a t i o n D(¢) o f a f o r m u l a ¢ is t h e m i n i m a l e l e m e n t w,r.t, s u b s u m p t i o n 2 in t h e s e t of d a g s t h a t satisfy it. The conditions u n d e r w h i c h a d a g D s a t i s f i e s a f o r m u l a o f FML* (where D / is t h e d a g t h a t is f o u n d if we follow t h e p a t h p t h r o u g h t h e d a g D) a r e a s follows :

(3) S~-WmTZCS Or FML ° a D ~ NIL a l w a y s b. D ~ TOP n e v e r c D ~ a i f D f a

d D ~ : a i f D / is defined 3 a n d D/ ~ a,

e. D J = ¢ ^ X f f D ~ b a n d D R

£ D ~ [<pl>,..,<pn>] if the values of all Pl ( I _< I < n) are equivalent. NORMAL FORM REQUIREMENTS. D e f a u l t u n i f i c a t i o n s h o u l d b e a s e m a n t i c a l l y well- b e h a v e d o p e r a t i o n , t h a t is, the r e s u l t of t h i s o p e r a t i o n s h o u l d d e p e n d o n l y o n t h e d e n o t a t i o n o f t h e f o r m u l a ' s involved. S i n c e d e f a u l t u n i f i c a t i o n is a n o n - m o n o t o n i c o p e r a t i o n , h o w e v e r , in w h i c h p a r t s o f t h e d e f a u l t i n f o r m a t i o n m a y d i s a p p e a r , a n d since t h e r e a r e in g e n e r a l m a n y f o r m u l a e d e n o t i n g t h e s a m e d a g , e s t a b l i s h i n g t h i s is n o t completely trivial. In particular, we m u s t m a k e s u r e t h a t t h e f o r m u l a w h i c h p r o v i d e s t h e d e f a u l t i n f o r m a t i o n is in t h e following n o r m a l form:

2 A d a g D s u b s u m e s a d a g D' if the s e t of f o r m u l a e s a t i s f y i n g D' c o n t a i n s t h e s e t of f o r m u l a e s a t i s f y i n g D (Eisele & DSrre. 1988: 287}.

3 D/<l> is defined iff I e Dom(D).

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(4) FML" N o r m a l F o r m

A f o r m u l a S i s in FML* NFiff: a V E / n S , : a i n S :

e E "->VP3EE : : u i n S l~ ~ E I , E2 in S:

E E2, E E 1 --> ~ P 3 6 E1 : <p3P2 > E E 2 c. V E in S, there is n o e E,

such that <pl> is re~ll,ed in S.

d V E in S, there is no e E such that : a (a e A) is in S.

(5) B

A p a t h <pl> is r e a l i z e d in S lff is defined in D(@ (l,r E L) (cf. Elsele & D0n-e,

1988 : 288).

For every f o r m u l a S in FML*, there is a f o r m u l a S' in FML* NF. w h i c h is e q u i v a l e n t to it w.r.t u n i f i c a t i o n , t h a t is, for w h i c h t h e following holds:

(6) ~/7. e FML*: S ^ 7. ~ TOP ¢~ S' ^ 7. ~ TOP Note t h a t t h i s d o e s n o t i m p l y t h a t S a n d S' h a v e t h e s a m e d e n o t a t i o n . T h e two f o r m u l a e below, f o r e x a m p l e , a r e e q u i v a l e n t w.r.t. unification, y e t denote different d a g s :

(7) a. <:f> : a ^ [<f>,<g>] b. < f > : a ^ < g > : a

For c o n d i t i o n s (4a,b), it is e a s y to see t h a t (6) h o l d s (it follows, for i n s t a n c e , f r o m t h e e q u i v a l e n c e l a w s (21} a n d (22) in K a s p e r & R o u n d s , 1986: 261). Condition (4c} c a n b e m e t b y r e p l a c i n g e v e r y o c c u r e n c e o f a n equivalence c l a s s [<pl>,..,<pn>] in a f o r m u l a S b y a c o n j u n c t i o n o f e q u i v a l e n c e s [<p11>,..,<pnl>] for every <pi/> (1 < i < n} realized in D(S}. F o r example, if L = {f,g), (Sb} is the NF of (Sa).

{8) a [<f>,<g>]^ <ff>:NiL

b. [<ff>,<gf>] ^ [<fg>,<gg>] ^ ~ : NIL C o n d i t i o n (4d} c a n b e m e t b y e l i m i n a t i n g e q u i v a l e n c e c l a s s e s o f p a t h s l e a d i n g to a n a t o m i c value. T h u s , (To) is the NF of (7a). Note t h a t t h e effect of (4c,d) is t h a t t h e v a l u e o f e v e r y p a t h w h i c h is m e m b e r o f s o m e equivalence class is NIL.

A default f o r m u l a h a s to b e in FML" NF for two r e a s o n s . F i r s t , all i n f o r m a t i o n w h i c h is implicit in a f o r m u l a , s h o u l d b e r e p r e s e n t e d explicitly, so we c a n c h e c k e a s i l y w h i c h p a r t s of a f o r m u l a n e e d t o b e r e m o v e d to avoid p o t e n t i a l u n i f i c a t i o n conflicts w i t h t h e n o n - default f o r m u l a . . T h i s Is g u a r a n t e e d b y (4a,b). Second, all r e e n t r a n t p a t h s s h o u l d h a v e NIL a s value. This is g u a r a n t e e d b y (4c,d) a n d m a k e s it possible to r e p l a c e a n e q u i v a l e n c e c l a s s b y a w e a k e r s e t o f e q u a t / o n s , in w h i c h a r b i t r a r y long e x t e n s i o n s o f t h e old p a t h n a m e s m a y o c c u r (if s o m e p a t h would h a v e a v a l u e o t h e r t h a n NIL, c e r t a i n e x t e n s i o n s c o u l d lead to i n c o n s i s t e n t results}.

LAWS FOR DEFAULT UNIFICATION. D e f a u l t

u n i f i c a t i o n is a n o p e r a t i o n w h i c h t a k e s two f o r m u l a s a s a r g u m e n t s , r e p r e s e n t i n g d e f a u l t a n d n o n - d e f a u l t i n f o r m a t / o n respectively. T h e d a g d e n o t e d b y t h e r e s u l t a n t f o r m u l a is s u b s u m e d b y t h a t o f t h e n o n - d e f a u l t a r g u m e n t , b u t not n e c e s s a r i l y b y t h a t o f t h e default a r g u m e n t .

T h e l a w s for d e f a u l t u n i f i c a t i o n (defined a s Default ~ Non-default = Result, w h e r e Default is in FMLS-NF] a r e listed below.

(9) D~AULTUNa~C.ATSOm :

a _{S e NIL} _{= S} S e T O P = T O P NIL (B S =~b T O P ~B S = S

b. a ~ S = S

S ~ a = a

c. : a ~ S =S, ffD(S)I=<P'> : a , p' a p r e f l x o f p , a e A. = ~, ifD(S} I = <pp'> :a.

= ~ , ff 3 p ' E E : D ( O ) I = E a n d p ' is a prefix of p.

= : a ^ S, o t h e r w i s e .

cL E

G)

S

=

F~E//~

where E ' i s {~ E I D ( S ) ~ E ' a n d p'e E'} u { e E I D(S) ~ <p'> : a} (p' a prefix of p, a e A) a n d Z is {<p'> l D(S) l = <pp'> :a. and p ~ E}.

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T h i s definition of d e f a u l t unification r e m o v e s all d e f a u l t i n f o r m a t i o n w h i c h m i g h t lead to a u n i f i c a t i o n c o n f l i c t . F u r t h e r m o r e , it is designed in s u c h a w a y t h a t the o r d e r in which i n f o r m a t i o n is r e m o v e d is irrelevant (note t h a t o t h e r w i s e the s e c o n d c a s e in (9e) would b e invalid). T h e first two c a s e s of (9c) a r e n e e d e d to r e m o v e all s e n t e n c e s : a, which refer to a p a t h w h i c h is b l o c k e d o r w h i c h c a n n o t receive a n a t o m i c v a l u e in ¢. The third c a s e in (9c) is needed for situations s u c h a s (I0). (I0) (<fg> : a ^ <h g> : b) (9 [<f>, <h>]

In (9d), we first r e m o v e from a n e q u i v a l e n c e c l a s s all p a t h s w h i c h h a v e a prefix t h a t is a l r e a d y in a n e q u i v a l e n c e c l a s s or w h i c h h a s a n a t o m i c value. The r e s u l t of this s t e p is E-E'. Next, we modify the equivalence class, so t h a t it allows e x c e p t i o n s (i.e. t h e posslbtlity of n o n - u n i f i a b l e v a l u e s ) for all p a t h s w h i c h a r e e x t e n s i o n s of p a t h s in E-E' a n d a r e defined in ¢. We c a n t h i n k o f m o d i f i e d e q u i v a l e n c e c l a s s e s a s a b b r e v i a t i o n s f o r a s e t o f (unmodified) equivalence classes:

(11) [ .... ] / / Z = ¢ , w h e r e ~ is t h e c o n j u n c t i o n of all e q u i v a l e n c e c l a s s e s [ .... <pnpl>]. s u c h t h a t pl is n o t defined in Z, b u t p r is in z, for s o m e l,r e L

An e x a m p l e s h o u l d m a k e this clearer: (12) [<f>,<g>,<h>l ( 9 ( < g > : a A < f g > : b ) =

l<f>,<h> l//{<g>} A (<f> : a ^ <fg> : b). The r e s u l t of default unification in this case is t h a t one e l e m e n t ( <g> } is r e m o v e d f r o m the d e f a u l t e q u i v a l e n c e c l a s s s i n c e it is c o n s t r a i n e d in b y the n o n - d e f a u l t information. F u r t h e r m o r e , t h e e q u i v a l e n c e is modified, s o t h a t it allows for exceptions for the p a t h s

<fg>

a n d <h g>. A p p l y i n g t h e r u l e in (I I), a n d a s s u m i n g t h a t L = {f,g,h}, we conclude t h a t (13) [<f>,<h> ]//{<g>} =

[<ff>, <hf>] A [<fh>, <h h> ].

N o t e t h a t t h e r e p l a c e m e n t o f m o d i f i e d e q u i v a l e n c e c l a s s e s b y o r d i n a r y e q u i v a l e n c e c l a s s e s is a l w a y s possible, a n d t h u s the r e s u l t of (9el) is equivalent to a formula in FML*. Finally. (ge) s a y s t h a t . given a c o n s i s t e n t d e f a u l t f o r m u l a , t h e o r d e r in w h i c h d e f a u l t i n f o r m a t i o n is a d d e d to t h e n o n - d e f a u l t

f o r m u l a is u n i m p o r t a n t . 1 (This d o e s not hold for i n c o n s i s t e n t d e f a u l t f o r m u l a e , h o w e v e r , since d e f a u l t unification with t h e individual conJuncts might filter out enough information to m a k e the resultant formula a consistent extension of the non-defauR formula, whereas T O P O ¢ = ¢}.

T h e monotonlclty properties of default unification are listed below {where < is subsumption}:

(14} a , ~ X ^ ,

(but n o t X ~ X ^ * )

b. X-<X ' ~ (X ^ @ ~ 0 C ' ^ ~ ) ( b u t n o t ¢ s ¢ ' ~ (g ^ ¢ ) <_ ( X ^ ¢ ' ) ) (14a) s a y s t h a t d e f a u l t unification is m o n t o n l c a d d i t i o n o f i n f o r m a t i o n to t h e n o n - d e f a u l t information. (14b) s a y s t h a t the f u n c t i o n a s a whole is m o n o t o n i c o n l y w.r.t, t h e d e f a u l t a r g u m e n t : a d d i n g m o r e d e f a u l t i n f o r m a t i o n leads to e x t e n s i o n s of the result. Adding n o n - d e f a u l t i n f o r m a t i o n is n o n - m o n o t o n i c . however, a s t h i s m i g h t c a u s e m o r e of t h e d e f a u l t i n f o r m a t i o n to g e t r e m o v e d o r overwritten.

T h e laws in (9) prove t h a t f o r m u l a e containing t h e ( 9 - o p e r a t o r c a n a l w a y s b e r e d u c e d to s t a n d a r d f o r m u l a e of FML*. T h i s implies t h a t f o r m u l a e u s i n g t h e ( 9 - o p e r a t o r c a n still be i n t e r p r e t e d a s d e n o t i n g dags. F u r t h e r m o r e , it follows t h a t addition of d e f a u l t unification to a u n i f i c a t i o n - b a s e d f o r m a l i s m s h o u l d b e s e e n o n l y a s a w a y to i n c r e a s e t h e e x p r e s s i v e power o f tools u s e d in defining t h e g r a m m a r (and t h u s . a c c o r d i n g to D6rre et al. (1990) d e f a u l t u n i f i c a t i o n w o u l d b e a n 'off line' e x t e n s i o n o f t h e f o r m a l i s m , t h a t is, its effects c a n be c o m p u t e d a t compile time).

A NOTE ON I M P L E M E N T A T I O N . W e h a v e i m p l e m e n t e d d e f a u l t u n i f i c a t i o n in Prolog. F e a t u r e s t r u c t u r e s a r e r e p r e s e n t e d b y o p e n e n d e d lists ( c o n t a i n i n g e l e m e n t s o f t h e f o r m

label=Value ),

a t o m s a n d v a r i a b l e s to r e p r e s e n t c o m p l e x f e a t u r e s t r u c t u r e s , a t o m i c v a l u e s a n d r e e n t r a n c i e s r e s p e c t i v e l y (see G a z d a r & Mellish, 1989). T h i s i m p l e m e n t a t i o n h a s the a d v a n t a g e t h a t it is c o r r e s p o n d s to FML* NF.

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(15) a. If=X, gfXl Y]

b.

[ f = a , g = a

I _Y]

c. [f=[h=a I X l ] , g = [ h f a I X I ] I_Y]

d [f=[h=a I Xl,g=[h=._Z IX1] I Y ]

If we unify (15a) with [[=al_Yl]. we get (15b), in which the value of g h a s been u p d a t e d a s well T h u s , the r e q u i r e m e n t s of (4a,b) are always met, a n d f u r t h e r m o r e , the r e e n t r a n c y a s s u c h between f a n d g is n o longer visible (condition 4c). If we unify (I 5a) with U'=[h=a I X 2 ) I Y3], we get (15c), in which the variable X h a s been replaced b y X 1 , which c a n be i n t e r p r e t e d a s ranging over all p a t h s t h a t are realized b u t not defined u n d e r f ( c o n d l t l o n (4d)). Note also t h a t this r e p r e s e n t a t i o n h a s t h e advantage t h a t we c a n d e f i n e a r e e n t r a n c y for all r e a l i z e d features, w i t h o u t having to specify the set o f possible f e a t u r e s o r e x p a n d i n g the v a l u e o f f into a list c o n t a i n i n g all t h e s e features. If we

d e f a u l t u n i f y (15a) w i t h

[f=[hffial_X2II_X,3]

a s

n o n - d e f a u l t i n f o r m a t i o n , for i n s t a n c e , t h e r e s u l t is r e p r e s e n t a b l e a s (15d). T h e r e e n t r a n c y for all undefined features u n d e r f is r e p r e s e n t e d b y X1. The c o n s t a n t NIL of FML* is r e p r e s e n t e d a s a Prolog variable ( _Z in this case). T h u s , the seemingly s p a c e c o n s u m i n g p r o c e d u r e of bringing a formula into FML* NF a n d t r a n s f o r m i n g t h e o u t p u t of (9d) into FML* is a v o i d e d c o m p l e t e l y . T h e a c t u a l d e f a u l t unification p r o c e d u r e is a modified version o f the merge operation defined in D6rre & Elsele

(1986).

3 . L I N G U I S T I C A P P L I C A T I O N S

Default unification c a n be u s e d to e x t e n d the s t a n d a r d PATR-II ( S h i e b e r e t al.. 1983) m e t h o d s for defining feature s t r u c t u r e s . In the examples, we freely combine default a n d non- default information (prefixed by I') in template definitions.

(16) a. D E T : ( l<cat arg> ffi N t<cat val> ffi NP <cat dir> = right <cat arg> = <cat val> <cat val n u m > = sg

<cat val case> = nom ). b. NP: ( <cat> = n o u n

<bar> ffi2 ). c. N : ( <cat> = n o u n

<bar> =1 ).

(16) d e s c r i b e s a f r a g m e n t o f C a t e g o r l a l Unification G r a m m a r (Uszkorelt. 1986, Calder et al. 1988. Bouma. 1988). T h e corresponding feature s t r u c t u r e for a definition s u c h as (16a)

is d e t e r m i n e d a s follows: first, all d e f a u l t information a n d all n o n - d e f a u l t information is u n i f i e d s e p a r a t e l y , w h i c h r e s u l t s in two f e a t u r e - s t r u c t u r e s (17a,b). The r e s u l t i n g two f e a t u r e s t r u c t u r e s a r e m e r g e d b y m e a n s of default unification (I 7c).

(]7)

] ]

c a s e = nora a. | d i r = r i g h t

t - a r g = < 1 >

b.

El t°" 'II

vaJ = b a r =

c a t

=

[ c a t = : ]

L a r g b a r =

c.

m l

cat

ffi

m

r°., = ]

l b a r

= 2

v a l ffi { 1 } / n u m

L c a s e d i r ffi r i g h t

r,,,,, = 2r,,]

I b a r

a r g ffi { 1 } / n u m

- L e a s e

m

m m

In (17c) the equivalence <cat val> = <cat an3> h a d to b e r e p l a c e d b y a w e a k e r s e t of e q u i v a l e n c e s , w h i c h h o l d s for all f e a t u r e s u n d e r val o r arg. e x c e p t c a t a n d bar. We r e p r e s e n t this b y u s i n g []-bracketed indices, i n s t e a d of <> a n d b y m a r k i n g the a t t r i b u t e s which are exceptions in ix)/([ i t a l i c . .

TWo things are worth noticing. First of all, the unificaUon of n o n - d e f a u l t information prior to merging it with the n o n - d e f a u l t information, g u a r a n t e e s t h a t all default i n f o r m a t i o n m u s t be u n i f i a b l e , a n d t h u s it e l i m i n a t e s the p o s s i b i l i t y o f i n h e r i t a n c e c o n f l i c t s i n s i d e t e m p l a t e definitions. Second, the d i s t i n c t i o n b e t w e e n d e f a u l t a n d n o n - d e f a u l t information is r e l e v a n t o n l y in d e f i n i t i o n s , n o t in t h e c o r r e s p o n d i n g f e a t u r e s t r u c t u r e s . This m a k e s the u s e o f the T - o p e r a t o r completely local: if a definlUon c o n t a i n s a template, we can replace this t e m p l a t e b y t h e c o r r e s p o n d i n g f e a t u r e s t r u c t u r e a n d we do not need to w o r r y a b o u t the fact t h a t this t e m p l a t e might c o n t a i n the T-operator.

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NON-MONOTONIC INHERITANCE OF INFORMATION IN TEMPLATES. T h e u s e o f d e f a u l t u n i f i c a t i o n e n a b l e s u s to u s e t e m p l a t e s even in t h o s e c a s e s w h e r e n o t all t h e i n f o r m a t i o n in t h e t e m p l a t e Is c o m p a t i b l e with t h e i n f o r m a t i o n already p r e s e n t in the definition.

G e r m a n t r a n s i t i v e v e r b s n o r m a l l y t a k e a n accusative NP a s a r g u m e n t b u t there are some v e r b s w h i c h t a k e a dative o r genitive NP a s a r g u m e n t . This Is e a s i l y a c c o u n t e d for b y defining t h e c a s e o f t h e a r g u m e n t o f t h e s e v e r b s a n d l n h e r i t t n g all o t h e r I n f o r m a t i o n from the template ~ r .

(]8) a. "IV: ( <cat val> = V P <cat arg> ffi NP <cat arg case> = a c c ).

b. he]fen (Whelp) :

( TV

I <cat arg case> ffi dat ).

gedenken (to c ~ n m e m ~ a t e )

( TV

! <cat arg case> = gen ).

SPECIALIZATION OF REENTRANCIES. A n i m p o r t a n t function of default unification is t h a t It allows u s to define e x c e p t i o n s to the fact t h a t two r e e n t r a n t f e a t u r e s t r u c t u r e s a l w a y s h a v e to d e n o t e e x a c t l y t h e s a m e f e a t u r e s t r u c t u r e s . T h e r e Is a w i d e c l a s s o f l i n g u i s t i c c o n s t r u c t i o n s w h i c h s e e m s to r e q u i r e s u c h m e c h a n i s m s .

Specifiers in CUG c a n be defined a s f u n c t o r s w h i c h t a k e a c o n s t i t u e n t of c a t e g o r y C a s a r g u m e n t , a n d r e t u r n a c o n s t i t u e n t of category C, with the exception t h a t one or more specific feature v a l u e s a r e c h a n g e d (see Bach, ] 9 8 3 , B o u m a , ]988). E x a m p l e s o f s u c h categories are d e t e r m i n e r s (see (]6a)), c o m p l e m e n t i z e r s a n d auxiliaries.

(]9) a. that : ( <cat y a h = <cat arg> <cat arg> = S

<cat arg vform> = fin 1<cat a r g comp> = none l<cat val comp> = that ).

b. will : ( <cat val> = <cat arg> <cat rag> = VP <cat val> = V P

1<cat arg vform> ffi bse l<cat val vform> ffi fin ).

Note t h a t t h e e q u a t i o n <cat val> = <cat arg>

will c a u s e all a d d i t i o n a l f e a t u r e s o n t h e a r g u m e n t w h i c h are n o t explicitly m e n t i o n e d In the n o n - d e f a u l t p a r t of t h e definition to percolate u p to the value.

Next, consider coordination of NPs.

(20) X0 --> X] X2Xo

<X2 cat> ffi conJ ¢X0> ffi <XI> ¢Y,O> ffi ~ <g0 cat> = n p <X 2 wform> ffi a n d k X 0 num> ffi plu I<X 1 n u m > =NIL ! <X2 num> ffi NIL).

{20) could be u s e d a s a rule for c o n j u n c t i o n of NPs in UG. It r e q u i r e s i d e n t i t y b e t w e e n t h e m o t h e r a n d t h e two c o o r d i n a t e d e l e m e n t s . However, r e q u i r i n g t h a t t h e t h r e e n o d e s be unifiable w o u l d be to strict. The n u m b e r of a conjoined NP Is a l w a y s p l u r a l a n d d o e s n o t depend on the n u m b e r of the coordinated NPs. F u r t h e r m o r e , t h e n u m b e r o f two c o o r d i n a t e d e l e m e n t s n e e d n o t be identical. T h e n o n - default information in (20) t a k e s care of this. The effect of this s t a t e m e n t Is t h a t a d d i n g the default informaUon <X0> = <XI> a n d <gO > ffi <X3> will result in a feature s t r u c t u r e in which

XO, X 1 a n d X 3 are unified, except for t h e i r values for <num>. We are n o t interested in the ruan-values of the conJuncts, so t h e y are set to N/L {which should be interpreted a s in section 2). The hum -value of the result is always p/u.

INFLECTIONAL MORPHOLOGY. W h e n s e e n from a CUG perspective, the categories of inflectional affixes a r e c o m p a r a b l e to t h o s e o f specifiers. The plural suffix -s for forming plural n o u n s can, for i n s t a n c e , be e n c o d e d a s a f u n c t i o n from (regular) s i n g u l a r n o u n s into Identical, b u t plural, n o u n s . Thus. we get the following categorization:

(21) - s : ( <cat val> = <cat arg> <cat arg cat> ffi n o u n <cat arg class> = regular l<cat arg n u m > ffi sg l<cat val Hum> = plu ).

Again, all a d d i t i o n a l i n f o r m a t i o n p r e s e n t on the a r g u m e n t w h i c h Is n o t m e n t i o n e d in the n o n - d e f a u l t p a r t o f t h e d e f i n i t i o n , Is percolated u p to the value automatically.

I, EXICAL D E F A U L T S . T h e l e x i c a l f e a t u r e specification d e f a u l t s of GPSG c a n a l s o be i n c o r p o r a t e d . C e r t a i n i n f o r m a t i o n h o l d s for m o s t lexlcal i t e m s o f a c e r t a i n category, b u t n o t for p h r a s e s o f t h l s c a t e g o r y . A u n i f l c l a t l o n - b a s e d g r a m m a r t h a t i n c l u d e s a morphological c o m p o n e n t (see, for i n s t a n c e , Calder, 1989 a n d Evans & Gazdar, 1989), would p r o b a b l y list o n l y (regular) r o o t f o r m s a s lexlcal items. For regular n o u n s , for instance,

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only the

singular

form

would be listed in the lexicon. S u c h i n f o r m a t i o n can be a d d e d to lexicon d e f i n i t i o n s b y m e a n s o f a lexlcal default rule:

{22) v. N ==> ( 3SG <class> = regular}

b. (x~v ffi N.

s h e e p = ( N

<mum> =NIL <class> = irregular}. The i n t e r p r e t a t i o n o f A ==> B is a s follows: If the definition D o f a lexical item is unifiable with A, t h a n e x t e n d D to B ( B D. T h u s , the lexlcal e n t r y cow would be e x t e n d e d with all t h e i n f o r m a t i o n in the d e f a u l t r u l e above, w h e r e a s the lexical e n t r y for sheep would only be e x t e n d e d w i t h t h e i n f o r m a t i o n t h a t <person> = 3. Note t h a t a d d i n g t h e d e f a u l t information to the template for N directly, a n d t h e n overwriting it in the irregular cases is not a feasible alternative, a s this would force us to distinguish b e t w e e n the template N if u s e d to describe n o u n s a n d the template N if u s e d in complex categories s u c h as NP/N or N / N (i.e. for d e t e r m i n e r s o r adjectives it is not typically the case t h a t they c o m b i n e only with r e g u l a r and singular nouns).

& C O N C L U S I O N S

We h a v e p r e s e n t e d a g e n e r a l definition for d e f a u l t unification. The fact t h a t It does not f o c u s one the r e s o l u t i o n of f e a t u r e conflicts alone, m a k e s it p o s s i b l e to define d e f a u l t u n i f i c a t i o n a s a n o p e r a t i o n o n f e a t u r e s t r u c t u r e s , r a t h e r t h a n a s a n operation adding o n e e q u a t i o n at a tlme to a given f e a t u r e d e s c r i p t i o n . T h i s g e n e r a l i z a t i o n m a k e s it possible to give a u n i f o r m t r e a t m e n t of s u c h t h i n g s a s a d d i n g d e f a u l t I n f o r m a t i o n to a t e m p l a t e , overwriting o f f e a t u r e v a l u e s a n d lexical d e f a u l t r u l e s . We believe t h a t the examples in section 3 d e m o n s t r a t e t h a t this is a useful e x t e n s i o n o f UG, as it s u p p o r t s the definition o f exceptions, the formulation more a d e q u a t e theories o f f e a t u r e percolation, a n d the e x t e n s i o n of UG with a m o r p h o l o g i c a l c o m p o n e n t .

R E F E R E N C E S

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in C a t e g o r l a l U n i f i c a t i o n G r a m m a r , Lingu/st/cs, vo126, 21-46.

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Calder, Jo; Klein, Ewan & Zeevat, H e n k 1988 U n i f i c a t i o n C a t e g o r i a l G r a m m a r : a concise, e x t e n d a b l e g r a m m a r for n a t u r a l l a n g u a g e p r o c e s s i n g . Proceedings of Collng 1988, H u n g a r i a n A c a d e m y o f Sciences, Budapest, 83-86.

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Eisele, A n d r e a s & D6rre, J o c h e n 1 9 8 6 A L e x l c a l - F u n c t l o n a l G r a m m a r S y s t e m in

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E v a n s , R o g e r 1 9 8 7 T o w a r d s a F o r m a l specification of D e f a u l t s in GPSG. In E. Klein & J . van B e n t h e m (eds.), Categories, Polymorphlsm and Unification. University of E d i n b u r g h , E d i n b u r g h / University of Amsterdam, Amsterdam, 73-93.

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