LAWS FOR DEFAULT UNIFICATION... In the examples, we freely combine default a n d non- default information prefixed by I' in template definitions.. Evans, Roger & Gazdar, Gerald 1989 Infe
Trang 1D 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
e-mall : gosse@riksnl.uucp
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
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
A m a j o r p r o b l e m with a t t e m p t s to define a n
o p e r a t i o n s u c h a s d e f a u l t u n i f i c a t i o n for complex feature s t r u c t u r e s Is t h a t t h e r e are a t least two ways to t h i n k a b o u t this operation It
c a n b e defined a s a n o p e r a t i o n w h i c h Is like
o r d i n a r y unification, with the exception t h a t In case o f a unification failure, t h e v a l u e of the
n o n - d e f a u l t f e a t u r e s t r u c t u r e t a k e s
p r e c e d e n c e (Kaplan 1987, S h i e b e r 1986a)
A n o t h e r option Is n o t to rely o n u n i f i c a t i o n failure, b u t to r e m o v e d e f a u l t i n f o r m a t i o n
a b o u t a f e a t u r e f a l r e a d y if t h e n o n - d e f a u l t
f e a t u r e s t r u c t u r e c o n s t r a i n s the c o n t e n t s o f f
in some way This view u n d e r l i e s m o s t of the
d e f a u l t m e c h a n i s m s u s e d in G P S G 1 The
1 Actually, in GPSG b o t h n o t i o n s of
d e f a u l t u n i f i c a t i o n a r e u s e d In S h l e b e r ' s (1986a) formulation of the o f the Foot Feature Principle, for e x a m p l e , t h e o p e r a t i o n a d d
c o n s e r v a t i v e l y (which n o r m a l l y relies on
u n i f i c a t i o n failure) is r e s t r i c t e d to f e a t u r e s
t h a t a r e f r e e (i.e u n i n s t a n t l a t e d a n d n o t covarying with some o t h e r feature)
Trang 2d 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 feature 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 <p> : ¢x (where a is a t o m i c or NIL or TOP)
NIL
T O P
a a • A (the s e t of atoms)
<p> : 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*
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 / < p > 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 ~ <p>: a i f D / < p > is defined 3 a n d
D/<p> ~ 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}
D / < I p > is d e f i n e d iff D/,<l> a n d D ' / < p > a r e defined, where D'= D/<I>
Trang 3(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 , < P l P 2 > : a i n S :
< p l > e E "->VP3EE : < P 3 P 2 > : u i n S
l~ ~ E I , E2 in S:
< p l P 2 > E E2, < p l > E E 1 >
~ P 3 6 E1 : <p3P2 > E E 2
c V E in S, there is n o <p> e E,
such that <pl> is re~ll,ed in S
d V E in S, there is no <p> e E such that
<p> : 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 < p r > 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
T O P ~B S = S
c < p > : 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
= <p>: a ^ S, o t h e r w i s e
where E ' i s {<p>~ E I D ( S ) ~ E ' a n d p'e E'}
u { <p>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}
e ( ¥ A ~ ) ( B ~ = $, f f y A ~ = T O P ,
= (W (B ¢) A (X ~B ¢}, otherwise
Trang 4T 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 <p> : 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) [ < p l > < p n > ] / / 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
[ < p l p l > <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 ¢ = ¢}
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
1 T h i s s h o u l d n o t be c o n f u s e d with t h e (invalid) s t a t e m e n t t h a t ¥ (9 (X (9 ~ } = X (9 (V ( 9 ¢ )
Trang 5(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 >
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
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
The n o t a t i o n J u s t i n t r o d u c e d i n c r e a s e s t h e expressive power of s t a n d a r d m e t h o d s for the description of f e a t u r e s t r u c t u r e s a n d c a n be
u s e d for a n e l e g a n t t r e a t m e n t of s e v e r a l linguisUc p h e n o m e n a
Trang 6NON-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) :
I <cat arg case> ffi dat )
gedenken (to c ~ n m e m ~ a t e )
! <cat arg case> = gen )
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,
170
Trang 7only 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}
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
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