I n d e f i n i t e Reasoning
with
D e f i n i t e Rules
L . T h o r n e M c C a r t y and R o n v a n der M e y d e n
Computer Science Department
Rutgers University
New Brunswick, NJ 08903, USA
[email protected], [email protected]
Abstract
In t h i s p a p e r we present a n o v e l e x p l a n a t i o n
o f t h e source o f i n d e f i n i t e i n f o r m a t i o n i n c o m m o n sense r e a s o n i n g :
Indefinite information
arises f r o m r e p o r t s a b o u t t h e w o r l d expressed
i n t e r m s o f c o n c e p t s t h a t have been d e f i n e d usi n g only definite rules. A d o p t i n g this p o i n t of
view, we show t h a t first-order logic is insuffic i e n t l y expressive t o h a n d l e i m p o r t a n t e x a m ples of c o m m o n sense r e a s o n i n g .
As a r e m edy, we p r o p o s e t h e use of c i r c u m s c r i b e d defi n i t e rules, a n d we then investigate the p r o o f
t h e o r y o f t h i s m o r e expressive f r a m e w o r k . W e
consider t w o a p p r o a c h e s :
F i r s t , prototypical
proofs, a s p e c i a l t y p e of p r o o f by i n d u c t i o n ,
w h i c h yields a sound p r o o f theory. Second, we
describe cases in w h i c h t h e r e exists a decision
procedure f o r a n s w e r i n g queries, a p a r t i c u l a r l y
s i g n i f i c a n t r e s u l t because i t shows t h a t i t i s possible t o have d e c i d a b l e q u e r y processing i n circ u m s c r i b e d t h e o r i e s t h a t are n o t e q u i v a l e n t t o
any first order theory.
1
Introduction
C o m m o n sense r e a s o n i n g has a bias t o w a r d s definite
information, as m a n y researchers have n o t e d .
Thus
J o h n s o n - L a i r d [1983] has a r g u e d t h a t h u m a n b e i n g s
d r a w inferences i n p a r t i c u l a r " m e n t a l m o d e l s ' ' , a n d
Levesque [1986] has u r g e d , for s i m i l a r reasons, t h e s t u d y
o f " v i v i d k n o w l e d g e ' ' i n a r t i f i c i a l i n t e l l i g e n c e . T h i s bias
t o w a r d s d e f i n i t e i n f o r m a t i o n has also p e r m e a t e d c o m p u t e r science p r o p e r . T h e f i e l d o f logic p r o g r a m m i n g , f o r
e x a m p l e , is based on a subset of f i r s t - o r d e r logic consisti n g e n t i r e l y o f d e f i n i t e clauses. I n d e e d , M a k o w s k y [1985]
has a r g u e d t h a t " H o r n f o r m u l a s m a t t e r " i n c o m p u t e r
science precisely because t h e y have a u n i q u e m i n i m a l
model.
If t h i s is so, t h e n h o w does indefinite information arise
i n c o m m o n sense r e a s o n i n g ? I n t h i s p a p e r , w e a d o p t a
novel p o i n t of view on this question, a n d explore the consequences. We suggest (as a G e d a n k e n e x p e r i m e n t ) t h a t
c o m m o n sense r e a s o n i n g is based e x c l u s i v e l y on definite
rules, a n d t h a t i n d e f i n i t e i n f o r m a t i o n arises f r o m t h e a p p l i c a t i o n o f these d e f i n i t e rules t o d e f i n i t e f a c t s i n s i t u a -
890
Logic Programming
tions where c o m m u n i c a t i o n is almost always incomplete.
A n i n d i v i d u a l w h o receives r e p o r t s a b o u t t h e w o r l d does
n o t u s u a l l y k n o w t h e d e f i n i t e f a c t s u p o n w h i c h these rep o r t s are b a s e d , a n d f o r such a n i n d i v i d u a l t h e w o r l d i s
a ( c h a o t i c a l l y ! ) i n d e f i n i t e p l a c e . O u t o f necessity, a n i n d i v i d u a l i n t h i s s i t u a t i o n does i n d e f i n i t e r e a s o n i n g w i t h
d e f i n i t e rules.
If we adopt this p o i n t of view, however, it turns out
t h a t the standard way of representing indefinite inform a t i o n in a r t i f i c i a l i n t e l l i g e n c e - i.e., by a first-order
language capable of asserting disjunctive and existent i a l f a c t s - is i n a d e q u a t e . We show in S e c t i o n 3 t h a t
f i r s t - o r d e r l o g i c c a n n o t express a s i m p l e f o r m o f i n d e f i n i t e i n f o r m a t i o n t h a t seems essential f o r c o m m o n
sense r e a s o n i n g . I n s t e a d , we suggest in S e c t i o n 4 t h a t
an adequate representation of indefinite i n f o r m a t i o n
r e q u i r e s t h e use of circumscription [ M c C a r t h y , 1980;
M c C a r t h y , 1986]. B u t c i r c u m s c r i p t i o n i s a second-order
f o r m a l i s m . I s i t p l a u s i b l e t o p r o p o s e such a c o m p l e x
l o g i c a l l a n g u a g e f o r such a m u n d a n e p u r p o s e as c o m m o n sense r e a s o n i n g ? W e b e l i e v e t h e answer is: Yes.
T h e p r i n c i p a l t e c h n i c a l c o n t r i b u t i o n o f t h e present p a per, in Sections 5 a n d 6, is to suggest t w o new t e c h n i q u e s for c o m p u t i n g t h e inferences t h a t f o l l o w f r o m a
circumscribed definite rulebase. These techniques a p p l y
i n special cases, o f course, b u t t h e y seem p r o m i s i n g for
various practical applications.
In S e c t i o n 5, we d e v e l o p t h e n o t i o n of a prototypical proof, w h i c h is a s p e c i a l t y p e of i n d u c t i v e p r o o f . If
a q u e r y is e n t a i l e d by a c i r c u m s c r i b e d r u l e b a s e , t h e n
a p r o t o t y p i c a l p r o o f is guaranteed to exist, a n d if a
p r o t o t y p i c a l p r o o f succeeds, t h e n a n i n d u c t i o n schema
can be a u t o m a t i c a l l y generated.
Furthermore, an ind u c t i o n s c h e m a i n o u r s y s t e m has v e r y s i m p l e a n d a p pealing proof-theoretic properties. T h i s means t h a t we
c a n search s y s t e m a t i c a l l y f o r i n d u c t i v e p r o o f s , a t least
i n c e r t a i n s p e c i a l cases t h a t seem t o b e o f c o n s i d e r a b l e
p r a c t i c a l i m p o r t a n c e . These t e c h n i q u e s c a n n o t give u s
a c o m p l e t e p r o o f p r o c e d u r e , o f course. B u t i n S e c t i o n 6 ,
we show t h a t an a c t u a l decision procedure exists f o r cert a i n o t h e r special cases t h a t seem t o b e o f p r a c t i c a l i m p o r t a n c e . P r e v i o u s w o r k o n c i r c u m s c r i p t i o n has t r i e d t o
find conditions under w h i c h a circumscribed theory "collapses" to a first-order t h e o r y [ L i f s c h i t z , 1985], so t h a t
s t a n d a r d theorem-provers for first-order logic could then
b e a p p l i e d . H o w e v e r , these c o n d i t i o n s are e x t r e m e l y re-
s t r i c t i v e , a n d recent w o r k b y K o l a i t i s a n d P a p a d i m i t r i o u
[1990] has s h o w n t h a t t h e existence o f a n e q u i v a l e n t f i r s t o r d e r t h e o r y i s i t s e l f a n u n d e c i d a b l e p r o p e r t y . O u r results i n S e c t i o n 6 i m p r o v e u p o n t h i s p r e v i o u s w o r k i n
t w o w a y s : ( i ) w e show t h a t c o m p l e t e q u e r y - a n s w e r i n g i s
possible even f o r t h e o r i e s w h o s e c i r c u m s c r i p t i o n i s n o t
first-order; a n d (ii) we p r o v i d e a f u l l decision procedure
i n such cases r a t h e r t h a n a s e m i - d e c i d a b l e p r o o f t h e o r y .
T h e o v e r a l l f r a m e w o r k i n w h i c h w e have c o n d u c t e d
t h i s research i s p r e s e n t e d i n S e c t i o n 2 , w h i c h i s based o n
p r e v i o u s l y p u b l i s h e d w o r k [ M c C a r t y , 1988a; M c C a r t y ,
1 9 8 8 b ; B o n n e r et al., 1989]. In S e c t i o n 2, we define p r e cisely w h a t w e m e a n b y a " d e f i n i t e " r u l e , a n d w e a r g u e
t h a t o u r l o g i c s h o u l d b e intuitionistic, r a t h e r t h a n class i c a l , since i n t u i t i o n i s t i c l o g i c a d m i t s a l a r g e r class of
rules t h a t have " d e f i n i t e " p r o p e r t i e s .
Since o u r basic
l a n g u a g e i s i n t u i t i o n i s t i c , t h o u g h , a n o t h e r n o v e l aspect
of our work is t h a t we then w r i t e the circumscription
a x i o m as a sentence in second-order intuitionistic logic.
A l t h o u g h we state some of the properties of i n t u i t i o n i s t i c c i r c u m s c r i p t i o n i n S e c t i o n 4 , a m o r e d e t a i l e d discussion w i l l b e p r e s e n t e d i n a f o r t h c o m i n g p a p e r [ M c C a r t y ,
1991].
2
Definite Rules
H o r n clauses p r o v i d e t h e s t a n d a r d e x a m p l e s o f d e f i n i t e
rules, f o r a s i m p l e r e a s o n . E v e r y set o f p o s i t i v e H o r n
clauses has a u n i q u e m i n i m a l H e r b r a n d i n t e r p r e t a t i o n
[van E m d e n a n d K o w a l s k i , 1976; A p t a n d v a n E m d e n ,
1982], o r , e q u i v a l e n t l y , an initial model [ G o g u e n a n d M e seguer, 1986; M a k o w s k y , 1985]. As a r e s u l t , a set of posi t i v e H o r n clauses, R, has b o t h t h e disjunctive property
a n d t h e existential property: A d i s j u n c t i o n of a t o m i c f o r mulae,
is entailed by R if and only if
R
or
R
B, and an existentially quantified atomic formula,
is entailed by R if and only if
for
some g r o u n d s u b s t i t u t i o n
C l o s e l y r e l a t e d is a p r o o f t h e o r e t i c p r o p e r t y , t h e existence of linear proofs. I n d e e d ,
it is f a i r to say t h a t it is t h e procedural interpretation
of a declarative semantics, embodied in SLD-resolution
[ L l o y d , 1987], t h a t m a k e s H o r n - c l a u s e logic such a p o w e r f u l t o o l f o r l o g i c p r o g r a m m i n g , f o r d e d u c t i v e databases
and for knowledge representation in general.
I t i s w e l l k n o w n t h a t these u s e f u l p r o p e r t i e s are lost
i f w e m o v e b e y o n d t h e H o r n - c l a u s e subset o f f i r s t - o r d e r
(classical) l o g i c . H o w e v e r , we c a n preserve these p r o p e r ties f o r a l a r g e r class of rules if we use first-order intuitionistic logic. C o n s i d e r t h e f o l l o w i n g e x a m p l e :
(1)
G a b b a y , 1985; M c C a r t y , 1988a; M c C a r t y , 1 9 8 8 b ; M i l l e r ,
1989: H a l l n a s a n d S c h r o e d e r - H e i s t e r , 1988; B o n n e r ct a/.,
1989J, a n d h a v e been s h o w n t o b e u s e f u l f o r h y p o t h e t i c a l
r e a s o n i n g [ B o n n e r , 1988], f o r l e g a l r e a s o n i n g [ M c C a r t y ,
1989], f o r m o d u l a r l o g i c p r o g r a m m i n g [ M i l l e r , 1989]. a n d
f o r n a t u r a l l a n g u a g e u n d e r s t a n d i n g [Pareschi, 1988].
N o t e t h a t a set o f r u l e s i n t h e f o r m ( 1 ) , i n t e r p r e t e d
classically, w o u l d g i v e u s f u l l f i r s t - o r d e r l o g i c . H o w e v e r ,
i n t e r p r e t e d i n t u i t i o n i s t i c a l l y , these r u l e s g i v e us a p r o p e r
subset o f f i r s t - o r d e r l o c k w i t h i n t e r e s t i n g s e m a n t i c p r o p erties [ M c C a r t y , 1 9 8 8 a ] . 1 F i r s t , a set o f i n t u i t i o n i s t i c e m b e d d e d i m p l i c a t i o n s R has a final Kripke model, w h i c h
we denote by K * . T h i s means: G i v e n any K r i p k e m o d e l
K of R, t h e r e exists a u n i q u e h o m o m o r p h i s m
from K
i n t o K * . S e c o n d , K * i s generic f o r H o r n - c l a u s e queries.
T h i s m e a n s : A H o r n clause q u e r y
is entailed by R if
and only if
i s t r u e i n K * . T h i r d , K * has a unique minimal substatc, d e n o t e d by
. T a k e n t o g e t h e r , these
results i m p l y t h a t R has t h e d i s j u n c t i v e a n d e x i s t e n t i a l p r o p e r t i e s f o r H o r n clause q u e r i e s , a s t r a i g h t f o r w a r d
g e n e r a l i z a t i o n o f t h e f a c t t h a t a set o f (classical) p o s i t i v e
H o r n clauses has t h e d i s j u n c t i v e a n d e x i s t e n t i a l p r o p e r ties f o r a t o m i c queries. For t h i s r e a s o n , i t i s a p p r o p r i a t e
t o refer t o rules i n t h e f o r m (1) a s " d e f i n i t e " rules.
T h e p r o o f theory for i n t u i t i o n i s t i c embedded implicat i o n s is also a s t r a i g h t f o r w a r d g e n e r a l i z a t i o n of t h e p r o o f
t h e o r y f o r classical H o r n - c l a u s e l o g i c [ M c C a r t y , 1988b].
C o n s i d e r a rulebase c o n s i s t i n g o f (1) p l u s t h e f o l l o w i n g
t w o rules:
(2)
(3)
Assume a database of assertions:
'Container (p)' and
* Heated ( p ) ' , a n d consider the query:
SterileContainer
O u r p r o o f procedure begins by constructi n g an S L D - r e f u t a t i o n tree in an initial tableau, To, a n d
proceeds u n t i l i t has r e d u c e d t h e o r i g i n a l q u e r y t o t h e
goal:
Note that
this goal is a universally quantified i m p l i c a t i o n . At this
p o i n t , t h e p r o o f p r o c e d u r e : ( i ) replaces t h e v a r i a b l e y
w i t h a s p e c i a l c o n s t a n t ' ! y 1 ' a n d ( i i ) c o n s t r u c t s a n auxiliary tableau, T 1 , w i t h a g o a l ' D e a d ( ! y i ) ' a n d a d a t a b a s e
c o n s i s t i n g o f t h e assertions ' B u g ( ! y 1 ) ' a n d ' I n s i d e ( ! y i , p ) \
T h i s g o a l succeeds, u s i n g rules ( 2 ) a n d ( 3 ) , w h i c h means
t h a t t h e u n i v e r s a l l y q u a n t i f i e d i m p l i c a t i o n also succeeds.
T h u s t h e o v e r a l l p r o o f succeeds, w i t h a d e f i n i t e answer
substitution
For a d d i t i o n a l examples of
such p r o o f s , see [ M c C a r t y , 1 9 8 8 b ] .
3
T h i s e x p r e s s i o n s h o u l d be r e a d : "If every b u g y i n s i d e
c o n t a i n e r x is a d e a d b u g , t h e n x is a s t e r i l e c o n t a i n e r / '
R e c a l l t h a t a p o s i t i v e H o r n clause i s a n i m p l i c a t i o n w i t h
a n a t o m i c conclusion a n d w i t h a n antecedent consisting
o f a c o n j u n c t i o n o f a t o m i c f o r m u l a e . B u t r u l e (1) has a
H o r n clause " e m b e d d e d " i n i t s a n t e c e d e n t . W e c a l l such
rules embedded implications [ M c C a r t y , 1988a; M c C a r t y ,
1 9 8 8 b ] . I n t u i t i o n i s t i c e m b e d d e d i m p l i c a t i o n s have been
s t u d i e d b y several researchers [ G a b b a y a n d R e y l e , 1984;
Indefinite Information
For c o m m o n sense r e a s o n i n g , d e f i n i t e rules b y themselves
appear to be insufficient. Consider the following examp l e , discussed b y M o o r e [1982]. T h e r e are t h r e e b l o c k s :
B l o c k ( a ) , B l o c k ( b ) , B l o c k ( c ) ; a n d t w o colors: Red and
Green.
T h e blocks are stacked in the f o l l o w i n g way:
O n ( a , b ) , O n ( b , c ) ; a n d t h e t o p a n d b o t t o m b l o c k s are
1
T h e standard semantics for i n t u i t i o n i s t i c logic is K r i p k e
semantics. See [ K r i p k e , 1965; F i t t i n g , 1969].
McCarty a n d van der Meyden
891
p a i n t e d green a n d r e d , r e s p e c t i v e l y : G r e e n ( a ) , R e d ( c ) .
We do not know the color of the m i d d l e block, b u t we
k n o w t h a t every block is p a i n t e d either red or green:
(4)
W e w a n t t o k n o w i f t h e r e i s s o m e t h i n g g r e e n o n somet h i n g r e d , a s i t u a t i o n t h a t c o u l d be represented by the
following predicate:
(5)
I n t u i t i v e l y , t h e answer s h o u l d b e : Yes. E i t h e r ' b ' i s a r e d
b l o c k , i n w h i c h case ' a ' a n d ' b ' c o n s t i t u t e a g r e e n - o n - r e d
p a i r , o r else ' b ' i s a green b l o c k , i n w h i c h case ' b ' a n d ' c '
constitute a green-on-red pair.
H o w s h o u l d w e represent i n d e f i n i t e i n f o r m a t i o n o f t h i s
sort? O n e a p p r o a c h i s t o a d d t w o n e w t y p e s o f rules t o
our language:
(6)
(7)
We c a l l these rules disjunctive a n d existential assertions,
respectively. T h e a d d i t i o n of disjunctive and existential
assertions to a set of e m b e d d e d i m p l i c a t i o n s is e q u i v a l e n t t o f u l l f i r s t - o r d e r i n t u i t i o n i s t i c l o g i c , o f course, b u t
there are reasons t o w r i t e a rulebase i n t h i s special f o r m .
A l t h o u g h w e n o l o n g e r have t h e p r o p e r t i e s o f d e f i n i t e
rules discussed i n S e c t i o n 2 , i t m a k e s sense t o s t a y a s
close t o these p r o p e r t i e s a s p o s s i b l e . W e t h u s show i n
[ M c C a r t y , 1990] t h a t a set o f rules i n t h e f o r m ( 1 ) , (6)
a n d (7) has a f i n a l K r i p k e m o d e l , I * , a l t h o u g h I * does
n o t , i n g e n e r a l , have a u n i q u e m i n i m a l s u b s t a t e . W e also
i n v e s t i g a t e i n [ M c C a r t y , 1990] a n e x t e n s i o n o f t h e l i n e a r
p r o o f p r o c e d u r e i n [ M c C a r t y , 1988b] w h i c h uses a l i m i t e d
" d i s j u n c t i v e s p l i t t i i i g " o p e r a t i o n whenever it encounters
a rule in the f o r m (6), a n d we show t h a t this p r o o f p r o cedure i s c o m p l e t e f o r i n t u i t i o n i s t i c ( b u t n o t classical)
l o g i c . These r e s u l t s are r e l a t e d t o recent w o r k o n disjunctive logic programming [ M i n k e r a n d R a j a s e k a r , 1990;
L o v e l a n d , 1987; L o v e l a n d , 1988].
B u t d o r u l e s i n t h e f o r m ( 6 ) a n d (7) r e a l l y c a p t u r e
o u r c o m m o n sense r e a s o n i n g w i t h i n d e f i n i t e i n f o r m a t i o n ?
Suppose t h e f o l l o w i n g d e f i n i t i o n i s p a r t o f t h e ' b l o c k s
world':
(8)
(9)
M o r e o v e r , t h e r e is no set of f i r s t - o r d e r rules t h a t gives us
t h e i n t u i t i v e l y c o r r e c t r e s u l t i n t h i s case, since t r a n s i t i v e
closure c a n n o t b e d e f i n e d i n f i r s t - o r d e r l o g i c [ A h o a n d
U l l m a n , 1979]. I s t h e r e a n a l t e r n a t i v e ?
4
Circumscription
I n t h i s s e c t i o n , w e consider a r a d i c a l l y d i f f e r e n t a p p r o a c h
to the representation of indefinite i n f o r m a t i o n . Assume
t h a t t h e r e exists a set of definite rules t h a t c a n be a p p l i e d
to a w o r l d of definite f a c t s . A s s u m e also t h a t someone
else has o b s e r v e d t h e w o r l d , a p p l i e d t h e r u l e s , a n d rep o r t e d some o f these d e f i n i t e c o n c l u s i o n s . O u r j o b i s t o
m a k e inferences a b o u t t h e a c t u a l s t a t e o f t h e w o r l d , even
t h o u g h w e h a v e n o t observed i t d i r e c t l y . For e x a m p l e ,
w e m i g h t b e t o l d t h a t block ' a ' i s above block ' b \ using
t h e d e f i n i t i o n o f ' A b o v e ' i n rules ( 8 ) - ( 9 ) , a n d w e m i g h t
want to k n o w whether there is s o m e t h i n g on 'b' T h e
w o r l d c o u l d b e i n i n f i n i t e l y m a n y different states, w i t h
infinitely m a n y different configurations of ' O n ' facts, all
supporting the conclusion 'Above(a,b)' Our i n f o r m a t i o n a b o u t t h e w o r l d i s t h u s h i g h l y indefinite.
Intui t i v e l y , h o w e v e r , w e o u g h t t o b e able t o c o n c l u d e t h a t
is t r u e .
T h e f o r m a l m a c h i n e r y w e need f o r t h i s a p p r o a c h i s
p r o v i d e d b y M c C a r t h y ' s t h e o r y o f circumscription [ M c C a r t h y , 1980; M c C a r t h y , 1986]. Since w e are w o r k i n g
w i t h i n t u i t i o n i s t i c l o g i c , h o w e v e r , we need to use an intuitionistic v e r s i o n o f t h e c i r c u m s c r i p t i o n a x i o m .
Let
R be a f i n i t e set of e m b e d d e d i m p l i c a t i o n s , a n d let P
be a t u p l e consisting of the "def i n e d p r e d i c a t e s " t h a t a p p e a r o n t h e l e f t - h a n d sides o f
the rules in R. L e t R ( P ) denote the c o n j u n c t i o n of the
rules i n R , w i t h t h e p r e d i c a t e s y m b o l s i n P t r e a t e d a s
free p a r a m e t e r s , a n d let R ( X ) b e t h e same a s R ( P ) b u t
w i t h the predicate constants
replaced
by predicate variables
Definition 4.1:
T h e circumscription axiom i s t h e
f o l l o w i n g sentence i n second o r d e r i n t u i t i o n i s t i c
logic:2
a n d suppose w e h a v e been t o l d t h a t ' A b o v e ( a , b ) ' i s t r u e
according to this definition. Is there something on ' b ' ?
I n t u i t i v e l y , t h e answer s h o u l d b e : Yes. B u t i t i s s t a n d a r d
p r a c t i c e [ K o w a l s k i , 1979] t o w r i t e t h e ' o n l y i f ' h a l f o f
definition (8)~(9) as follows:
Above(x,y) => O n ( x , y) V
(3z)[Qn(x, z)
(10)
A A b o v e ( z , y)]
A n d u s i n g ( 1 0 ) , t h e f o r m a l answer t o o u r q u e r y is: N o . I t
is straightforward to verify that the final K r i p k e model
f o r rules ( 8 ) - ( 1 0 ) i n c l u d e s a s u b s t a t e t h a t c o n t a i n s a n
i n f i n i t e sequence o f ' A b o v e ' r e l a t i o n s , a n d n o r e l a t i o n i n
t h e f o r m ' O n ( w , b ) \ For e x a m p l e :
Above(a, b),
On(a,
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c1),
Above(c1,b),
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W e d e n o t e t h i s expression b y C t r c u m ( R ( P ) ; P ) , a n d
w e refer t o i t a s " t h e c i r c u m s c r i p t i o n o f P i n R ( P ) . "
T h i s a x i o m has t h e same i n t u i t i v e m e a n i n g t h a t i t has i n
classical l o g i c : I t states t h a t t h e e x t e n s i o n s o f t h e p r e d i cates in P are as s m a l l as p o s s i b l e , g i v e n t h e c o n s t r a i n t
2
We define second-order i n t u i t i o n i s t i c logic precisely in
[ M c C a r t y , 1990], following standard accounts such as [Troelstra and van Dalen, 1988].
that R ( P ) must be true. Since the logic is intuitionistic, however, the axiom minimizes extensions at every
substate of every Kripke model that satisfies R. 3
If R consists of rules ( 8 ) - ( 9 ) , then the circumscription
o f ' A b o v e ' in R forces 'Above' to be the transitive closure
of ' O n ' [McCarthy, 1980; Lifschitz, 1985], and this entails
the following implication:
(11)
We thus have a solution to the problem posed at the
beginning of this section.
How might we compute such inferences, in general?
We discuss this question in [McCarty, 1990], and we
summarize our results briefly here. Let us formulate the
general query problem as follows:
(12)
where Q and R are embedded implications, and
is an
implication in the f o r m (11) w i t h a positive disjunctive
or existential conclusion and an antecedent consisting of
a conjunction of atomic formulae. Our approach is to
construct a final Kripke model for
under various assumptions about Q and R, and then
to show that this final Kripke model is generic for the
query
In the present paper, we w i l l only consider the
case in which R is a set of Horn clauses, but we w i l l
analyze the general case of embedded implications in a
forthcoming paper [McCarty, 199l]. For Horn clauses,
the construction of the final Kripke model is simple, and
is related to recent results of Kolaitis and Papamitriou
[1990]. First, let K* be the final Kripke model for R
itself, as defined in Section 2. Now let B be the set of
base predicates in R, that is, the predicates that do not
appear in P, and let b be the Herbrand base constructed
using B alone. Define:
where K * ( s ) denotes the set of substates s' in K* such
that
Intuitively, C* consists of the set of least
fixpoints of the Horn clauses in ft applied to all possible
combinations of ground atomic formulae that are constructible using the predicates in B. We then have the
following result:
T h e o r e m 4 . 2 : Let R be a set of Horn clauses, and
let Q be a set of embedded implications. Let C be
the largest subset of C* that satisfies Q. Then C is
a final Kripke model for
Circum(7t(P);P).
Since we can also show that final Kripke models are
generic for queries in the f o r m
we can solve the query
problem (12) by showing that
is true in C.
In the remainder of this paper, we w i l l investigate
two ways to do this using certain additional assumptions about the f o r m of Q and R. As an illustration of
our techniques, we w i l l work w i t h a single example that
combines the two examples in Section 3. Let R be the
following set of rules:
ChristmasBlock
(13)
3
Note that we have written the circumscription axiom as
a second-order universally-quantified embedded implication.
Alternative versions, using negation and second-order existential quantification, which are equivalent in classical logic,
would not be equivalent intuitionistically.
(14)
(15)
(16)
(17)
Intuitively, rules (13)-(14) define the concept of a
'ChristmasBlock', and rules (15)-(17) define the concept
of a stack of 'ChristmasBlocks'. Suppose we are told
that there exists a stack of 'ChristmasBlocks' in which
block 'a' is above block 'b' and furthermore that ' a '
and ' b ' are painted green and red, respectively. Does it
follow that there is something green on something red?
Intuitively, the answer should be: Yes. Formally, we
can pose this question by circumscribing the predicates
'ChristmasBlock,' ' O n C B ' and ' A b o v e C B ' in rules (13)(17), adding rule (5) to Q, and then asking whether the
following implication is entailed:
(18)
We w i l l show how to solve this problem in the following
two sections of the paper, using two different methods.
5
Prototypical Proofs
In this section, we consider a class of inductive proofs in
which the induction schema takes the f o r m of an intuitionistic embedded implication. We can think of these
proofs as having two parts: The first part is a prototypical proof, and it is guaranteed to exist whenever the
query
is entailed by
Circum(R(P);P).
The second
part involves the proof of an embedded implication w i t h
an embedded second-order universal quantifier, and it is
conjectured to exist whenever the prototypical proof succeeds. A l t h o u g h second-order intuitionistic logic is incomplete, in general, the fragment of second-order logic
that we use to state the induction schema happens to
have a complete proof procedure. This means that it
is possible to automate the search for a solution to our
sample problem, and to certain other similar problems.
First, note that rules (13)-(15) in our sample problem are nonrecursive Horn clauses. For such rules, the
solution is the same in intuitionistic logic as it is in classical logic [Reiter, 1982; Lifschitz, 1985]. Let Comp(7t)
denote Clark's Predicate Completion [Clark, 1978]. We
then have the following result, which is proven in [McCarty, 1990]:
T h e o r e m 5 . 1 : Let R be a set of nonrecursive Horn
clauses. Then
Circum(R(P);P)
is equivalent to
Comp(R).
The remaining rules in our sample problem have a
simple f o r m , suggesting the following:
D e f i n i t i o n 5.2: R is a linear recursive definition of
the predicate A if it consists of:
1. A Horn clause w i t h ' A ( X ) ' on the left-hand side
and a conjunction of nonrecursive predicates on
the right-hand side, and
2. A Horn clause that is linear recursive in A,
McCarty a n d van der Meyden
893
'Green(a)' and 'Red(b)' in its data base, and w i t h
'GreenOnRed' as its goal. Our first step is to show, if
possible, that this proof succeeds using the prototypical definition in (19). However, the reader can easily verify that there exists an SLD-refutation proof of
'GreenOnRed' in this tableau using rules (5) and (19),
and using Comp(R) applied to rule (15). Moreover, from
an inspection of this proof, it is apparent that there also
exists a proof of the following universally quantified i m plication:
(21)
Let us call this implication
Then
is the following universally quantified i m plication:
(22)
and if we can prove (22) we w i l l also have a proof of
our original query (18). Therefore, using the induction
schema in Definition 5.3, we t r y to prove
This goal is an implication w i t h a second-order
universal quantifier, so we create a new tableau,
we add
to the data base, and we t r y to prove
in T1.
Let us write out each of these schemata in detail,
is the following implication:
(23)
and
is equivalent to the following implication:
To prove (24), we instantiate x and z to the special constants
' and
we add the right-hand side of (24)
to the data base of T 1 , and we try to prove the left-hand
side of (24). The remaining details are somewhat tedious, but the reader should be able to check that this
proof does in fact go through. The main point to note is
that the proof now uses Comp(R) applied to rules (13)
and (14), which yields a disjunctive assertion. We thus
need to apply the "disjunctive s p l i t t i n g " operation discussed in Section 3. When this final step succeeds, however, the inductive proof itself succeeds, and the query
is shown to be true.
The justification for our approach can be found in the
following two theorems, which are proven in [McCarty,
1990] using our results on final Kripke models. In the
statement of these theorems, S(A) denotes the set of
all induction schemata for A that can be constructed
using Definition 5.3, and V(A) denotes the prototypical
definition of A given by Definition 5.2. B o t h theorems
apply to the situation in which R is a linear recursive
definition, Q is a set of embedded implications, and
is
an implication w i t h a positive disjunctive or existential
conclusion and an antecedent consisting of a conjunction
of atomic formulae.
T h e o r e m 5.4:
894
Logic Programming
T h e o r e m 5.5:
Theorem 5.4 tells us that inductive proofs are sound
but not necessarily complete, while Theorem 5.5 tells us
that prototypical proofs are complete but not necessarily
sound. Together, these theorems sanction the strategy
we illustrated in our example: T r y to find a prototypical
proof first, and then use this proof to suggest a suitable
induction schema.
Our sample problem is still relatively simple, but we
have constructed proofs of this sort for more difficult
problems.
In particular, we have applied our techniques to prove various properties of P R O L O G programs [Kanamori and F u j i t a , 1986; Elkan and McAllester, 1988]. For example, let ' A p p e n d ( l , m , n ) ' be defined
as usual. Let 'Reverse(r,*)' be defined as follows:
Reverse(nil, nil)
We can then show that
[Reverse(x,y)
Rever8e(y,x)]' is entailed by the circumscription of ' A p pend' and 'Reverse' in this rulebase. For the details of
this proof, see [McCarty, 1990].
6
A Decision Procedure
We have shown in Theorem 5.4 that a certain class of i n duction schemata provides a sound inference procedure
for circumscribed definite rules. Furthermore, the structure of these schemata allows us to use a complete proof
theory for second-order embedded implications in the i n ductive step of the proof. This raises the question: Is the
resulting proof theory for the circumscribed rulebase itself complete? We show in this section that it is not. In
fact there can be no such proof theory, since the query
problem w i l l be shown to be not even semi-decidable.
Nevertheless, we w i l l demonstrate that one can still find
interesting classes of decidable queries. Our results are
significant since they show that, even in cases where the
circumscription of a theory is not first-order equivalent,
it is possible to decide certain broad classes of queries.
We refer the reader to [van der Meyden, 1990] for proofs
of the results in this section.
For the remainder of this section we restrict our attention to rules R which contain only positive Horn clauses
w i t h o u t function symbols, i.e., all programs are D A T A L O G programs [Chandra and Harel, 1982]. Furthermore, we require that there be no repeated variables in
the heads of rules. 4 We consider the following restricted
formulation of the query problem:
(25)
where D is a set of ground base and defined atoms, and
is a closed positive existential formula in the base and
defined predicates, i.e., a formula constructed using only
the operators
and
The proof of the following
result is by an encoding of the containment problem for
context-free languages:
4
This last restriction is made to simplify the presentation
only; our results still hold if it is removed.
T h e o r e m 6 . 1 ; For arbitrary D A T A L O G rules R,
sets of ground atoms D and positive existential
queries
the problem D
Circum(7l(P);P)
is undecidable.
We w i l l now show that the complement of the query
problem is recursively enumerable. Define an expansion
of a defined a t o m A(x) by R to be any set E(x) of base
atoms such that either
1. E = { B 1 ( x ; b ) , . . . , B n ( x ; b ) } for some tuple b of
new constants, and for some rule
in R such that all the B i are base predicates, or
2.
for
some tuple b of new constants, and for some rule
in R such that all the B i are base predicates, all the
A j are defined predicates, and each Ej(x;y) is an
expansion of Aj(x; y) by R.
Also, if a is a tuple of constants, then we say that E(a)
is an expansion of A (a) by R if E(x) is an expansion
of A(x) by R. We write
ExpandR(A)
for the set of
expansions of A by R.. If D is a set of ground atoms in
both base and defined predicates, then an expansion of
D by R is any set of ground atoms obtained f r o m D by
replacing each defined a t o m A
D by an expansion of
A by R.
For example, let R consist of the rules (13)-(17). Then
the set:
is an expansion of D = {Green(a), AboveCB(a, b),
Red(b)}.
The following proposition shows that we may restrict
attention to a denumerable set of models of a particular
form when answering queries:
Lemma 6.2 shows that the problem of deciding that
a query is not entailed is semi-decidable, since it suffices to find a single expansion of D in which the
query fails. It follows f r o m this and Theorem 6.1 that
is not a recursively enumerable set.
Thus the techniques of Section 5 can only provide sufficient conditions for answering queries.
In spite of these undecidability results, there exist
broad classes of queries for which the query problem can
be shown to be decidable, even in cases when the circumscription of R is not equivalent to a first-order theory.
Define a query to be basic if it contains occurrences of
base predicates only.
McCarty a n d van der M e y d e n
895
decidable for sets of atoms D and queries containing the predicate P as well as the base predicates.
These results are all consequences of a theorem of
Courcelle [1990] concerning graph grammars. In general, we retain decidability when we permit in the query
any predicate to which the rulebase gives a definition expressible in monadic second-order logic. Unfortunately,
it can be shown to be undecidable whether a rulebase
defines a predicate expressible in monadic second-order
logic, so we must be content w i t h enumerating special
cases, as above, if we wish to go beyond the class of
basic queries.
7
Conclusion
We have presented two approaches to answering queries
in the presence of indefinite information, both of which
are able to handle the "stack of Christmas blocks" example. The reader may have wondered why, since the
method of Section 6 is a decision procedure, one would
bother w i t h the prototypical proofs of Section 5? The
reason is that the the decision procedure works for a
smaller class of rules than the prototypical proofs. On
the other hand, the price paid by the prototypical proofs
for their ability to deal w i t h a potentially larger set of
examples is logical incompleteness. It would be interesting to determine the extent to which this incompleteness
corresponds to the incompleteness of human reasoning
when faced w i t h indefinite information of comparable
logical complexity.
References
[McCarty, 1988a] L.T. McCarty. Clausal intuitionistic
logic. I. Fixed-point semantics. Journal of Logic Programming, 5 ( 1 ) : 1 - 3 1 , 1988.
[McCarty, 1988b] L.T. McCarty. Clausal intuitionistic
logic. I I . Tableau proof procedures. Journal of Logic
Programming, 5(2):93-132, 1988.
[McCarty, 1990] L.T. McCarty. C o m p u t i n g w i t h prototypes. Technical Report L R P - T R - 2 2 , Computer Science Department, Rutgers University, 1990. A preliminary version of this paper was presented at the
Bar Ran Symposium on the Foundations of Artificial
Intelligence, Ramat Gan, Israel, June 1989.
[McCarty, 1991] L.T. McCarty. Circumscribing embedded implications. In A. Nerode et al., editors, Proceedings, First International Workshop on Logic Programming and Non-Monotonic Reasoning, page (forthcoming). M I T Press, 1991.
[van der Meyden, 1990] R. van der Meyden.
Recursively indefinite databases. In S. Abiteboul and P.C.
Kanellakis, editors, Proceedings of the Third International Conference on Database Theory, pages 364-378.
Springer LNCS No. 470, 1990.
N o t e : These references are truncated because of space
limitations. A f u l l list of references is available from the
authors on request.
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