G r e g e r s K o c h
D a t a l o g i s k I n s t i t u t K ø b e n h a v n s U n i v e r s i t e t S i g u r d s g a d e 41
D K - 2 0 0 0 K ø b e n h a v n N
N A T U R A L L A N G U A G E P R O G R A M M I N G
1. A i m s
T h i s is a s t u d y o f a u t o m a t e d p r o g r a m m i n g . We a r e a i m i n g at t h e s t e p w i s e d e v e l o p m e n t of a p r o g r a m m i n g e n v i r o n m e m t c o n s i s t i n g of an a u t o m a t i c t r a n s l a t i o n s y s t e m to t r a n s l a te t e x t s in n a t u r a l l a n g u a g e (e.g. s o f t w a r e r e q u i r e m e n t s p e c i f i c a t i o n s ) i n t o c e r t a i n l o g i c a l f o r m u l a e a c c o r d i n g
to s o m e s e m a n t i c th eo ry , as w e l l as i n to e x e c u t a b l e p r o g r a m s . T h e o n l y s e m a n t i c t h e o r i e s c o n s i d e r e d h e r e ar e l o g i c a l
t h e o r i e s , w h e t h e r t h e y be r e l a t e d to t h e l a m b d a c a l c u l u s or to th e p r e d i c a t e c a l c u l u s , a n d w e s h a l l h e n c e f o r t h talk a b o u t th e l e v e l of l o g i c a l r e p r e s e n t a t i o n (rather t h an s e m a n t i c r e p r e s e n t a t i o n ) .
T h e p a r a d i g m of t h e m e t h o d m a y b e g u e s s e d f r o m t h e f i g u r e 1. T h e h o r i s o n t a l a x i s d i s p l a y s t h e g a p b e t w e e n the h u m a n u s e r a n d t h e c o m p u t e r (that is th e s o - c a l l e d m a n - m a c h i n e c o m m u n i c a t i o n p r o b l e m ) . T h e v e r t i c a l a x i s i n d i c a t e s the level of a b s t r a c t i o n , f r o m l o w l e v e l s o f a b s t r a c t i o n up to h i g h e r ones. T h e a r e a s of n a t u r a l la ng u a g e , p r o g r a m m i n g language, a n d t h e p r e d i c a t e c a l c u l u s a r e i n d i c a t e d w i t h an o v e r l a p
F i g u r e 1
2. D e v e l o p i n g n a t u r a l s u b l a n g u a g e s
S t a r t i n g w i t h w h a t e v e r s e m a n t i c t h e o r i e s ar e a v a i l a b l e in th e l i t e r a t u r e the a i m h e r e is to d e v e l o p st e p by s t ep a c o m p u t a t i o n a l s u b l a n g u a g e of n a t u r a l la ng ua ge . T h i s d e v e l o p m e n t is p e r f o r m e d o n th e b a s i s of c a r e f u l l y s e l e c t e d e x a m p l e s (and it w i l l be i l l u s t r a t e d by e x a m p l e s ) . Th e t e x t u a l f o r m u l a t i o n o f t h e r e q u i r e m e n t s p e c i f i c a t i o n w i l l t h us be t r a n s l a t e d a u t o m a t i c a l l y in to a l o g i c a l r e p r e s e n t a t i o n .
T h e f i g u r e 2 sh o w s th e s a me t h i n g in a d i a g r a m m a t i c form. H e r e a r e i n d i c a t e d th e d o c u m e n t s o c c u r i n g d u r i n g the p r o c e s s , for i n s t a n c e
R e q u i r e m e n t S p e c i f i c a t i o n
a n d a f e w p r o c e s s e s to be d i s c u s s e d , for i n s t a n c e
T r a n s l a t e
S e v e r a l a s p e c t s o f th e f i g u r e 2 w i l l be c o m m e n t e d up on in t h e r e s t o f th e pape r.
3. R e v e r s e t r a n s l a t i o n
G e n e r a t i n g i.e. r e v e r s e t r a n s l a t i o n b a c k into n a t u r a l l a ng ua ge , m a k e s it p o s s i b l e to c h e c k th e q u a l i t y of the t e x t u a l t r a n s l a t i o n p r o c e s s .
It is a c h a r a c t e r i s t i c p r o p e r t y of lo g i c p r o g r a m m i n g l a n g u a g e s t h a t (at l e a s t in p r i n c i p l e ) t h e u s e r m a y a p p l y the sa me p r o g r a m for t r a n s l a t i o n a n d for g e n e r a t i o n .
4. L o g i c a l a l t e r n a t i v e s
T h e p a r t i c u l a r c h o i c e o f l o g i c a l r e p r e s e n t a t i o n is e s s e n t i al ly a r b i t r a r y , b u t th e s c i e n t i f i c l i t e r a t u r e h e a v i l y
F i g u r e 2
s u p p o r t s two k i n d s of r e p r e s e n t a t i o n s . H e r e w e c h o s e a logic g r a m m a r (like [4]). A n i n t e r e s t i n g a l t e r n a t i v e is a i n t e n s i o n a l g r a m m a r S la P H LI QA l [3 ,7,14,16] .
A l s o in a m o r e n a r r o w c o n t e x t of p a r s i n g (instead of full t r a n s l a t i o n ) th e f o l l o w i n g m e t h o d is p r o b a b l y p r e f e r a b l e to t h a t o f th e U p p s a l a C h a r t P a r s e r [17] as far as m o d i f i a bi l i t y , e x t e n s i b i l i t y , p o r t a b i l i t y , a n d e x p e r i m e n t a t i o n w i t h r e g a r d to g r a m m a t i c a l d e s c r i p t i o n s ar e c o n c e r n e d .
T h e s t e p w i s e d e v e l o p m e n t of th e co m p u t a t i o n a l s u b l a n g u a g e o f o r d i n a r y E n g l i s h u s e d a n u m b e r of c a r e f u l l y s e l e c t e d
b e n c h m a r k p r o b l e m s .
5. A n e x a m p l e
T h e f i r s t b e n c h m a r k p r o b l e m is s i m p l e e n o u g h to a l l o w a f a i r l y t h o r o u g h p r e s e n t a t i o n .
By th e e x a m p l e o f th e f a l l i b l e G r e e k we a l s o d e m o n s t r a t e th e i d ea of e x p r e s s i n g a g r a m m a r as a lo gi c p r o g ra m.
T h e p r o b l e m c o n s i s t s in h a v i n g the c o m p u t e r r e s p o n d in a s e n s i b l e w a y to th e f o l l o w i n g o r d i n a r y E n g l i s h t e x t (or n a t u r a l l a n g u a g e r e q u i r e m e n t s p e c i f i c a t i o n ) ;
T u r i n g is human. S o c r a t e s is human. S o c r a t e s is Gree k.
e v e r y h u m a n is f a l l i b l e .
w h i c h h u m a n is G r e e k a n d is f a l l i b l e ?
i.e. th e p r o g r a m s h o u l d a n s w e r th e q u e r y in the last s e n t e n c e .
W e s e l e c t th e f o l l o w i n g s i m p l e c o n t e x t - f r e e g r a m m a r < S e n t e n c e >
< T e r m >
< N o u n p h r a s e > < R e l a t i v e c l a u s e > < V e r b p h r a s e > < D e t e r m i n e r > < N o u n >
< T r a n s i t i v e v e r b > < P r o p e r n a m e >
= < T e r m > < V e r b p h r a s e >
= < P r o p e r n a m e > I < D e t e r m i n e r x N o u n p h r a s e > = < N o u n > I < N o u n x R e l a t i v e c l a u s e >
= t h a t < V e r b p h r a s e >
= <Transitiveverb><frerm> I <^Verbphrase>and ^erbphrase> = e v e r y I w h i c h
= h u m a n = is I i s n ' t
:= T u r i n g | S o c r a t e s I G r e e k I f a l l i b l e I human
If we just w a n t a p a r s e r (to a c c e p t or r e j e c t t h e i n p u t sentence) w e m a y s i m p l y c h a n g e th e g r a m m a r i n t o t h e l o g i c p r o g r a m o f f i g u r e 3, w h e r e t h e v a r i a b l e s f u n c t i o n as p o i n t ers to th e input string.
To so l v e o u r p r o b l e m o f t h e f a l l i b l e G r e e k w e n e e d a s o m e w h a t m o r e s o p h i s t i c a t e d t r a n s l a t i o n o f t h e a c c e p t e d i n p u t s e nt en ce , as s h o w n in f i g u r e 4. F o r t h e s a ke o f c l a r i t y w e h a v e h e r e o m i t t e d t h e v a r i a b l e s f u n c t i o n i n g as p o i n t e r s
(as in f i gu re 3). We h a v e o n l y g i v e n t h e v a r i a b l e s d e s i g n a t i n g the focus a n d r e s u l t ( s ) .
T r a n s l a t e :
S e nt en ce (x ,z ) if T e rm (x ,y ) & V e r b p h r a s e ( y , z ) . Term(x,y) if P r o p e r n a m e ( x , y ) .
Term(x,z) if D e t e r m i n e r ( x , y ) & N o u n p h r a s e ( y , z ) . N o u n p h r a s e ( x , y ) if N o u n ( x , y ) .
N o u n p h r a s e ( x , z ) if N o u n ( x , y ) S R e l a t i v e c l a u s e ( y ,z ) . R e l a t i v e c l a u s e ( x , z ) i f C h e c k ( t h a t , x , y ) s V e r b p h r a s e ( y ,z) V e r b p h r a s e ( x , z ) i f T r a n s i t i v e v e r b ( x , y ) & T e r m ( y , z ) .
V e r b p h r a s e ( x , w ) if V e r b p h r a s e ( x , y ) S C h e c k ( a n d , y , z ) & V e r b p h r a s e (z , w ) .
D e t e r m i n e r ( x , y ) i f C h e c k ( e v e r y , x , y ) . D e t e r m i n e r ( x , y ) i f C h e c k ( w h i c h , x , y ) . N o u n ( x , y ) i f C h e c k ( h u m a n , x , y ) .
T r a n s i t i v e v e r b ( x , y ) i f C h e c k ( x , y )
T r a n s i t i v e v e r b ( x , y ) i f C h e c k ( i s n 't , x , y ) . P r o p e r n a m e ( x , y ) i f C h e c k ( T u r i n g , x , y ) . P r o p e r n a m e ( x , y ) i f C h e c k ( S o c r a t e s , x , y ) . P r o p e r n a m e ( x , y ) i f C h e c k ( G r e e k , x , y ) . P r o p e r n a m e ( x , y ) i f C h e c k ( f a l l i b l e , x , y ) . P r o p e r n a m e ( x , y ) i f C h e c k ( h u m a n ) , x , y ) .
S e n t e n c e (z) if T e r m ( x , z1 ,z)&Verbphrase ( x , z 1 ) . T e r m ( x , z , z ) i f P r o p e r n a m e ( x ) .
T e r m ( x , z 1 , z ) if D e t e r m i n e r ( x , z 2 ,z 1 , z )& N o u n p h r a s e ( x ,z 2 ) N o u n p h r a s e ( x , z ) i f N o u n ( x , z ) .
N o u n p h r a s e ( x , z 1 & z 2 ) i f N o u n ( x , z 1 )S R e l a t i v e c l a u s e ( x , z 2 ) R e l a t i v e c l a u s e ( x , z ) i f C h e c k ( t h a t ) S V e r b p h r a s e ( x , z ) . V e r b p h r a s e ( x , z ) i f T r a n s i t i v e v e r b ( x , y ,z 1 ) & T e r m ( y ,z 1 ,z) V e r b p h r a s e ( x , z 1 & z 2 ) i f V e r b p h r a s e ( x , z 1 )& C he ck (a nd )
& V e r b p h r a s e ( x , z 2 ).
D e t e r m i n e r (x, z 1, z2 , Vx[ z1=^z2 ]) if C h e c k ( e v e r y ) . D e t e r m i n e r ( x , z 1 , z 2 ,W h i c h ( x , z 1& z 2 )) if C h e c k ( w h i c h ) . N o u n ( x , I s ( x , H u m a n ) ) if C h e c k ( h u m a n ) .
T r a n s i t i v e v e r b ( x , y , I s ( x , y ) ) if C h e c k ( i s ) . T r a n s i t i v e v e r b (x,y,~*Is (x,y) ) if C h e c k (isn't). P r o p e r n a m e ( T u r i n g ) i f C h e c k ( T u r i n g ) .
P r o p e r n a m e ( S o c r a t e ^ ( i f C h e c k ( S o c r a t e s ). P r o p e r n a m e ( G r e e k ) if C h e c k ( G r e e k ) .
P r o p e r n a m e ( F a l l i b l e ) i f C h e c k ( f a l l i b l e ) . P r o p e r n a m e ( H u m a n ) i f C h e c k ( h u m a n ) .
F i g u r e 4. T r a n s l a t e :
T h e c o m p u t a t i o n a l p r o c e s s i n g of th e s e c o n d a n d the s e c o n d la st s e n t e n c e s a r e d i s p l a y e d in t h e f i g u r e s 5 a n d 6,
r e s p e c t i v e l y .
E x a m p l e
S o c r a t e s IS
S e n t e n c e (z)
h u m a n
= Is(x,y)
= I s ( S o c r a t e s , Human)
F i g u r e 5
E x a m p l e
e v e r y h u m a n
S e n t e n c e ( z )
IS f a l l i b l e
= Vx [ I s (x. Human) "^Is (x,Fallible) ]
[image:8.595.81.511.117.731.2]Is (Turing, H u m a n ) . Is (Socrates, H u m a n ) . Is (Socrates, G r e e k ) .
Vx [ Is (x. Human) =^Is (x. F a ll ib le ) ] .
W h i c h (x. Is (x,Greek) & Is (x,Falli bl e) &Is (x,Human) )
F i g u r e 7. T r a n s l a t i o n ou tp ut :
Th e f u r t h e r t r a n s f o r m a t i o n o f f o r m u l a e in t h e l o g i c a l r e p r e s e n t a t i o n d e p e n d s h e a v i l y u p o n th e p a r t i c u l a r c h o i c e of logic p r o g r a m m i n g la ng u a g e . In c a s e o f a d i a l e c t of P r o l o g t h e t r a n s f o r m a t i o n s h o u l d c o n s t i t u t e a n o r m a l i z a t i o n i n to c o n j u n c t i v e n o r m a l f o r m (or c l a u s a l f o r m ) . H e r e the q u e s t i o n is r a i s e d w h e t h e r o r n o t t h e s u b l a n g u a g e a c t u a l l y w i l l g e n e r a t e l o g i c a l f o r m u l a e in a c l a u s e l f o r m
t h a t ar e d e f i n i t e (Horn c l a u s e s ) . (An i n t e r e s t i n g p r o b l e m w o u l d b e to c h a r a c t e r i z e t h e s u b l a n g u a g e s s a t i s f y i n g t h i s r e q u i r e m e n t )
In the fi rs t b e n c h m a r k p r o b l e m w e g e t t h e n o r m a l i z e d f o r m u l a e of figu re 8.
C l a u s a l f o r m ( c o n j u n c t i v e n o r m a l f o r m ) : I s ( T u r i n g , H u m a n ) .
I s ( S o c r a t e s ,H u m a n ) . I s ( S o c r a t e s , G r e e k ) .
I s ( x , F a l l i b l e ) i f I s ( x , H u m a n ) .
Print(x) if I s ( x , G r e e k ) & I s ( x , F a l l i b l e ) ( x , H u m a n ) .
F i g u r e 8.
6. N o r m a l i z a t i o n
7. V e r i f y i n g h a n d w r i t t e n p r o g r a m s
The l o g i c a l r e p r e s e n t a t i o n m a y be u s e d in t h e c o n t e x t of
v e r i f y i n g h a n d w r i t t e n lo gi c p r o g r a m s , as i n d i c a t e d in f i g u r e 2 As an e x a m p l e of v e r i f i c a t i o n w e h a v e t h e s e c o n d b e n c h m a r k p r o b l e m w h i c h is a v a r i a n t of th e A l p i n e C l u b p r o b l e m f r o m the a r t i f i c i a l i n t e l l i g e n c e l i t e r a t u r e [15].
8. N a t u r a l l a n g u a g e p r o g r a m m i n g
c o n s i d e r e d as y e t a n o t h e r le ve l (a fifth) in the c o n t e x t of th e f o ur l e v e l s d e a l t w i t h in th e E U R O T R A p r o j e c t [13], A c t u a l l y it is an o b v i o u s p o s s i b i l i t y to e x t r a c t th i s k i nd of i n f o r m a t i o n f r o m t h e third, s o - c a l l e d l o g i c o - s e m a n t i c
le v e l a n d b u i l d th e r e c o m m e n d e d l o g i c a l r e p r e s e n t a t i o n s f r o m t h a t i n f o r m a t i o n . U n f o r t u n a t e l y , I t e n d to be v e r y p e s s i m i s t i c as to w h e t h e r th i s t a s k w i l l a c t u a l l y be r e a l i s e d b y t h e E U R O T R A p a r t i c i p a n t s .
It is r e l a t i v e l y e a s y to s u p p l e m e n t t h is a u t o m a t e d p r o g r a m m i n g s y s t e m w i t h s p e c i f i c r u l e s c o n c e r n i n g the p r o b l e m d o m a i n to m a k e it a k n o w l e d g e - b a s e d sy stem, w h e t h e r the r u l e s ar e f o r m u l a t e d in a n o t a t i o n a k i n to th e c h o s e n l o g i c a l r e p r e s e n t a t i o n , or in the f o r m of a d d i t i o n a l te xt s in n a t u r a l l a ng ua ge .
So t h i s s y s t e m m a y c o n s t i t u t e the k e r n e l of a k n o w l e d g e b a s e d a u t o m a t e d p r o g r a m m i n g sy stem, w h e r e fr am e i n f o r m a t i o n s p e c i f i c to th e u n i v e r s e of d i s c o u r s e ar e to be added.
T h i s k i n d o f p r o g r a m m i n g m a y v e r y w e l l be t e r m e d " p r o g r a m m i n g in n a t u r a l l a n g u a g e " or " n a t u r a l l a n g u a g e p r o g r a m m i n g " ( h e n c e t h e t i t l e o f t h is p a p e r ) .
T h e t h i r d b e n c h m a r k p r o b l e m is a p r o t o t y p i c a l d a t a b a s e q u e r y l a n g u a g e p r o b l e m [19]. T h e f o u r t h b e n c h m a r k p r o b l e m is a small c o m p u t e r a i d e d d e s i g n p r o b l e m in a r c h i t e c t u r a l design. T h e s e b e n c h m a r k p r o b l e m s f a ll i n t o th e c l a s s o f n a t u r a l l a n g u a g e p r o g r a m m i n g . F u r t h e r d e t a i l s m a y b e f o u n d in th e r e p o r t [12].
9. S t a t u s r e m a r k s
T h e s y s t e m h e r e w a s w r i t t e n for E n g l i s h to d e v e l o p an E n g l i s h c o m p u t a t i o n a l s u b l a n g u a g e . A n o b v i o u s a l t e r n a t i v e c o u l d be to d e v e l o p a s i m i l a r s y s t e m in s o me (or every) S c a n d i n a v i a n l a n g u a g e (and it m i g h t i n c l u d e the s u r f a c e s t r u c t u r e s o f th e l o g i c p r o g r a m m i n g l a n g u a g e ) . A r e l a t e d e x p e r i m e n t u s i n g th e s a me m e t h o d in th e a u t o m a t e d t r a n s l a t i o n f r o m J a p a n e s e w i l l be r e p o r t e d o n (in [2]).
W h e n d e v e l o p i n g th e a p p r o p r i a t e n a t u r a l s u b l a n g u a g e c e r t a i n d i f f i c u l t i e s s h o w e d u p in c o n n e c t i o n w i t h p r o n o u n s . T h e y m a y be e x e m p l i f i e d b y th e s e n t e n c e :
A m a n t a k e s an a p p l e a n d he e a t s it.
The d i f f i c u l t i e s c o n c e r n e d th e s c o p e r u l e s o f th e q u a n t i f i c a t i o n a n d t h e y c o u l d c e r t a i n l y be o v e r c o m e b y e x t e n d i n g th e l o g i c a l co nn e c t i v e s in t o t w o - d i m e n s i o n a l o p e r a t o r s in a
s y s t e m a t i c m a n n e r [12].
As far as the p l u r a l of n o u n s a n d q u a n t i f i c a t i o n a r e c o n c e r n e d , t h e y w e r e n e e d e d in t h e f o u r t h b e n c h m a r k p r o b l e m . T h e r e s e e m s to be e s s e n t i a l l y s i x d i f f e r e n t w a y s to e x t e n d th e c o m p u t a t i o na l s u b l a n g u a g e w i t h q u a n t i f i c a t i o n , as i l l u s t r a t e d in f i g u r e 9,
T h e r e is a n e e d for g a i n i n g e x p e r i e n c e w i t h t h e u s e o f s y s t e m s like th is one. V i r t u a l l y n o t h i n g is a v a i l a b l e in th e s c i e n t i f i c l i t e r a t u r e .
S u c h a s y s t e m s h o u l d n o t be c o n s i d e r e d c o m p l e t e l y t r i v i a l to use, a l t h o u g h its p o t e n t i a l i t y fo r p o p u l a r i z a t i o n s h o u l d be r e c o g n i s e d (V ir tu al ly e v e r y b o d y w h o is n o t a n a n a l p h a b e t m i g h t le a r n to use i t ) .
D e c i s i o n s : 1 :
2;
3
1
S h o u l d th e t r a n s l a t i o n be o n e - p a s s or t w o - p a s s (many-pass) ? S h o u l d th e r e s u l t be e x p r e s s e d in h i g h e r o r d e r f u n c t i o n s
(or c a r d i n a l i t y ) ?
S h o u l d t h e r e be tw o t r u t h - v a l u e s or t h r e e (many) ?
Pass es
2:
H i g h e r - o r d e r
3:
V a l u e s
1 N o 2 (here)
1 N o 3
-1 Yes 2 ( I n t e n s i o n a l g r a m m a r s a n d
1 Yes 3 L e x i c a l - F u n c t i o n a l G r am ma rs )
2 Ye s 2 (here)
2 Ye s 3 (Logic gr am m a r s ) .
[image:12.595.68.534.489.761.2]A f e w r e l a t e d c o n t r i b u t i o n s f r o m m y i n s t i t u t e are i n c l u d e d in t h e l i st t h o u g h n o t r e f e r r e d to in t h e paper.
10. R e f e r e n c e s
[1] F. Als et al: Compiling in Prolog
(in Danish), DIKU report 83/16, Institute of Datalogy. Copenhagen University, 1983.
[2] A. Bernth: Logics applied to the translation of Japanese (in Danish), these proceedings.
[3] W.Bronnenberg et al.; The question answering system PHLIQA1, in L. Bole (ed.) Natural communication with computers Vol. 2, 1980.
[4] A. Colmerauer: An interesting subset of natural language, in ClarJc and Tarnlund (eds.) Logic programming, 1982. [5] N.D.Jones and A.Mycroft: Stepwise development of denota
tional semantics for Prolog, DIKU report 83/1, Institute of Datalogy, Copenhagen University, 1983. [6] P.H. Jørgensen and G. Koch: Two hew methods of natural
language database queries (in Danish), Proc. NordDATA Conf., Copenhagen 1981, 2, 227-232.
[7] G.^^^ch: Experimental formalization of Danish . Institute of Datalogy, Copenhagen University, 1979. DIKU report 79/19 (in Danish).
[8] G. Koch: A Prolog way of representing natural language fragments, DIKU report 80/16, Institute of Datalogy, Copenhagen University, 1980.
[9] G. Koch: A problem oriented software development method in computational linguistics (in Danish), Proc. De Nordi- slce Datalingvistikdagene, E. Lien (red.),
Trondheim University, 1981, 47-63.
[10] G. Koch: Grammars and predicate calculus, DIKU report 81/16, Institute of Datalogy, Copenhagen University, 1981.
[11] G. Koch and K.B. Larsen: Logical prototyping in system development (in Danish), Proc. NordDATA Conf.,Göteborg,
1982, 1, 270-273.
[12] G. Koch: Stepwise development of logic programmed software development methods, DIKU report 83/5, Institute of Datalogy, Copenhagen University, 1983.
[13] B. Maegaard and H.Ruus; Multilingual syntax and morphology for machine translation, in K. Hyldgaard-Jensen and
B.Maegaard (eds.): Machine translation and comptutational lexicography, Copenhagen 1982, 26-34.
[14] R. Montague; Formal philosophy, 1974.
[15] N.J.Nilsson; Principles of artificial intelligence, Springer-Verlag 1982.
[16] P.S.Olsen; Computational philosophy, future DIKU report, [17]
[18]
[19] J.E.'uilman:Principles of database systems, 1980. Institute of Datalogy, Copenhagen University, 1984.,0 V
Sågvall-Hein: A parser for Swedish, Uppsala University, UCDL-R-83-2.