on the relation between autoepistemic logic and ... - CiteSeerX
ON THE RELATION BETWEEN A U T O E P I S T E M I C LOGIC A N D C I R C U M S C R I P T I O N Preliminary Report K u r t Konolige* Artificial Intelligence Center and Center for the Study of Language and Information SRI International, Menlo Park, California 94025/USA Abstract C i r c u m s c r i p t i o n on the one hand and autoepist e m i c and default logics on the other seem to have q u i t e different characteristics as formal systems, w h i c h makes it d i f f i c u l t to compare t h e m as f o r m a l i z a t i o n s of defeasible connmonsense reasoning. In this paper we accomplish t w o tasks: (1) we extend the o r i g i n a l semantics of a u t o e p i s t e m i c logic to a language which i n cludes variables q u a n t i f i e d i n t o the context of the a u t o e p i s t e m i c o p e r a t o r , and (2) we show t h a t a certain class of autoepistemic theories in t h e extended language has a m i n i m a l - m o d e l sem a n t i c s corresponding t o c i r c u m s c r i p t i o n . W e conclude t h a t all of the first-order consequences of p a r a l l e l predicate c i r c u m s c r i p t i o n can be obt a i n e d f r o m this class of autoepistemic theories. T h e correspondence we c o n s t r u c t also sheds l i g h t o n the p r o b l e m a t i c t r e a t m e n t o f equality in circumscription.
1
Introduction
T h e r e l a t i o n s between the m a j o r n o n m o n o t o n i c logic form a l i s m s of AI — default logic, autoepistemic logic, and c i r c u m s c r i p t i o n — is of some i m p o r t a n c e , since all of these logics have been proposed as f o r m a l i s m s for various types of commonsense reasoning. T h e basic f o r m a l equivalence of d e f a u l t and a u t o e p i s t e m i c logic has already been s h o w n (see [ K o n o l i g e , 1987]), b u t the relation between c i r c u m s c r i p t i o n and default or autoepistemic logic r e m a i n s obscure. M o s t l y this is a consequence of the different f o u n d a t i o n s of these logics: c i r c u m s c r i p t i o n is based on a m i n i m a l - m o d e l semantics (see [Lifschitz, 1985]), w h i l e the others use more proof-theoretic techniques ( d e f a u l t logic [Reiter, 1980]) or an epistemic ope r a t o r ( a u t o e p i s t e m i c logic [ M o o r e , 1985]). In t r y i n g to express a u t o e p i s t e m i c or default, logic in c i r c u m s c r i p t i o n , researchers have f o u n d the basic probl e m to be t h a t a m i n i m a l - m o d e l or even prefered-model *This research was supported by the Office of Naval Research under Contract No. N00014-85-C-0251, by subcontract from Stanford University under the Defense Advanced Research Projects Administration under Contract No. N00039-84-C-0211, and by a gift from the System Development Foundation.
semantics s i m p l y does not have the c a p a b i l i t y of representing the requisite proof-theoretic or epistemic concepts (see [Shoham, 1987]). We agree w i t h this assessment, and say n o t h i n g further about it here. On the other h a n d , there have been several results on expressing c i r c u m s c r i p t i o n in default logic. These results are summarized in [ E t h e r i n g t o n , 1986]; they apply to the restricted case of predicate c i r c u m s c r i p t i o n w i t h no fixed predicates and w i t h a f i n i t e , fixed d o m a i n . From a model-theoretic p o i n t of view, the predicate c i r c u m s c r i p t i o n C i r c u m ( , 4 ; P; Z) of a first-order sentence A picks out those models of A in which the extension of the predicate P is m i n i m a l . T h e comparison is across models w i t h the same d o m a i n and denotation f u n c t i o n , but which m i g h t differ in the extensions of the predicates Z. A l l predicates other t h a n P and Z are fixed, t h a t is, cannot vary in a comparison of models. It was recently shown (see [de Kleer and Konolige, 1989]) t h a t fixed predicates are inessential in predicate c i r c u m s c r i p t i o n , t h a t is, there is a simple t r a n s l a t i o n f r o m any c i r c u m scription w i t h fixed predicates to one w i t h o u t . Hence fixed predicates no longer present an obstacle to representing circumscriptions in default or autoepistemic ogic. T h e problem of finite domains remains, however. In this paper we provide a solution to this p r o b l e m , by first extending autoepistemic logic to a language which allows q u a n t i f y i n g i n t o the epistemic operator, and then showing t h a t a certain class of autoepistemic theories, the M I N = theories, express all of the first-order consequences of predicate c i r c u m s c r i p t i o n .
2
S e m a n t i c s of Q u a n t i f y i n g - i n
A u t o e p i s t e m i c ( A E ) logic was defined by [Moore, 1985] as a f o r m a l account of an agent reasoning a b o u t her own beliefs. T h e agent's beliefs are assumed to be a set of sentences in some logical language augmented by a m o d a l operator L. As o r i g i n a l l y defined, and extended in [Konolige, 1987], its language does not p e r m i t variables quantified outside of a m o d a l operator to appear inside. In this section we f u r t h e r extend AE logic to deal with quantifying-in. 2.1
Logical preliminaries
We begin w i t h a language C for expressing self-belief, and introduce valuations of C. T h e t r e a t m e n t generally
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follows and extends [Konolige, 1987] Let be a f i r s t - o r d e r language w i t h e q u a l i t y and f u n c t i o n a l t e r m s . T h e n o r m a l f o r m a t i o n rules for for m u l a s of first-order languages h o l d . A sentence of is a f o r m u l a w i t h no free v a r i a b l e s ; an atom is a sentence of the f o r m P(t1, • • • , tn). We e x t e n d by a d d i n g a u n a r y m o d a l o p e r a t o r L; t h e e x t e n d e d l a n g u a g e is called can be defined recursively as c o n t a i n i n g all t h e f o r m a t i o n rules of plus the f o l l o w i n g : If
is a f o r m u l a of
t h e n so is
.
(1)
An expression is a modal atom. Sentences a n d a t o m s of are called ordinary. N o t e t h a t nestings such as are not a l l o w e d ; we consider o n l y a single level of nesting here. Because the a r g u m e n t of a m o d a l o p e r a t o r can c o n t a i n free v a r i a b l e s , there m a y be q u a n t i f y i n g i n t o the scope of a m o d a l a t o m , e.g., is a sentence. O f t e n we w i l l use a s u b s c r i p t " 0 " to i n d i c a t e a subset of o r d i n a r y sentences, e.g., Let u s r e s t r i c t ourselves for t h e m o m e n t t o m o d a l a t o m s w h i c h d o n o t c o n t a i n free v a r i a b l e s . F r o m the p o i n t o f view o f f i r s t - o r d e r v a l u a t i o n s , the m o d a l a t o m s are s i m p l y n i l a r y p r e d i c a t e s . O u r i n t e n d e d i n t e r p r e t a t i o n of these a t o m s is t h a t is an e l e m e n t of t h e b e l i e f set of t h e agent. So we w i l l consider v a l u a t i o n s of to be standard first-order valuations, w i t h the a d d i t i o n of a belief set T. T h e a t o m s are i n t e r p r e t e d as t r u e or false d e p e n d i n g on w h e t h e r is in T. To d i s t i n g u i s h these v a l u a t i o n s , we w i l l s o m e t i m e s call t h e m L valuations. T h e interaction of the interpretation of L w i t h firstorder v a l u a t i o n s is o f t e n a d e l i c a t e m a t t e r , a n d so a per spicuous t e r m i n o l o g y f o r t a l k i n g a b o u t L v a l u a t i o n s is necessary. I n p a r t i c u l a r , i t i s o f t e n useful t o d e c o u p l e the i n t e r p r e t a t i o n o f m o d a l a n d o r d i n a r y a t o m s . F i r s t order v a l u a t i o n s are b u i l t u p o n t h e t r u t h v a l u e s o f a t o m s : for o r d i n a r y a t o m s , t r u t h v a l u e s are g i v e n by a s t r u c t u r e (U,v,R), where v is a d e n o t a t i o n a l m a p p i n g f r o m t e r m s to elements of the universe U, a n d R. is a set of rela tions over U, one for each p r e d i c a t e . We w i l l refer to any such s t r u c t u r e as an ordinary index, a n d d e n o t e it w i t h t h e s y m b o l I. M o d a l a t o m s are g i v e n a t r u t h v a l u e by a belief set T, w h i c h is called a modal index. N o t e t h a t , because m o d a l o p e r a t o r s are n o t nested, o n l y t h e o r d i n a r y sentences o f t h e m o d a l i n d e x V are i m p o r t a n t . T h e t r u t h v a l u e o f any sentence i n can b e d e t e r m i n e d by the n o r m a l rules for f i r s t - o r d e r v a l u a t i o n s , g i v e n an o r d i n a r y and m o d a l i n d e x . W e w r i t e if a val u a t i o n ( / , T ) satisfies T h e valuation rule for m o d a l a t o m s can be w r i t t e n as if and only if
(2)
A v a l u a t i o n t h a t makes a every m e m b e r of a set of sentences t r u e is called a model of t h e set. A sentence t h a t is t r u e in every m e m b e r of a class of v a l u a t i o n s is called valid w i t h respect to t h e class. W e use C n ( X ) t o m e a n t h e f i r s t - o r d e r consequences o f a f i r s t - o r d e r set of sentences X. 2.2
Autoepistemic Extensions
I n [ K o n o l i g e , 1987], w e i n f o r m a l l y defined a n e x t e n s i o n of a set of sentences A as those consequences of A w h i c h
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Knowledge Representation
an agent, s h o u l d believe. T h e f o r m a l c o u n t e r p a r t is given
by: DEFINITION 2.1 the
Any set of sentences T which sa/ysfi.e.s
equation
is an a u t o e p i s t e m i c e x t e n s i o n of A. T h i s is a f i x e d - p o i n t e q u a t i o n for a b e l i e f set /', and is a c a n d i d a t e for t h e belief set of an ideal i n t r o s p e c t i v e agent w i t h premises .4. I t differs s l i g h t l y f r o m t h e o r i g i n a l d e f i n i t i o n o f [ K o n o l i g e , 1987] i n t h a t the m o d a l i n d e x consists of the o r d i n a r y p a r t ( t h e kernel of 7 1 ), t h i s suf fices because t h e l a n g u a g e of T does not i n c l u d e nested modal operators. N o t e t h a t we are c o n s i d e r i n g all m o d e l s of A in w h i c h t h e i n t e r p r e t a t i o n of is t h e b e l i e f set of t h e agent itself, t h a t is, t h e v a l u a t i o n s we consider all have a m o d a l i n d e x t h a t i s t h e b e l i e f set o f t h e a g e n t , f o l l o w i n g M o o r e , we c a l l such v a l u a t i o n s autoepistemic (or AE) v a l u a t i o n s . 2.3
Quantifying-in
W e w o u l d like t o e x t e n d t h e l a n g u a g e o f a u t o e p i s t e m i c l o g i c t o i n c l u d e v a r i a b l e s w h i c h are q u a n t i f i e d outside, t h e scope o f t h e m o d a l o p e r a t o r , b u t can a p p e a r inside, e.g., T h e p r o b l e m i s t h a t i t i s n o t o b v i o u s how t o e x t e n d t h e s e m a n t i c s o f t h e logic t o deal w i t h these " q u a n t i f i e d - i n " expressions. R e c a l l t h a t t h e b e l i e f set T is a set of sentences t h a t f o r m t h e beliefs of an agent.. To interpret we s i m p l y ask w h e t h e r t h e expression øis i n T . B u t w i t h t h e q u a n t i f i e d - i n l a n g u a g e , w e m u s t also b e able t o interpret w h e r e i s the proposition t h a t t h e i n d i v i d u a l x has t h e p r o p e r t y I n order t o c o n s t r u c t a p r o p o s i t i o n a l expression whose m e a n i n g is w e m u s t have some way o f r e f e r i n g t o i n d i v i d u a l s in the d o m a i n . T h e simplest, scheme f o r reference t o d o m a i n e l e m e n t s is to use t h e d e n o t a t i o n m a p v a l r e a d y present in the f i r s t - o r d e r m t e r p r e t i o n / . I n place o f the v a l u a t i o n rule f o r m o d a l a t o m s g i v e n a b o v e , w e use: iff
f o r some t e r m / such t h a t
(3)
T h a t is, we say t h a t (x) is believed if x has a n a m e / such is b e l i e v e d . G i v e n t h i s a d d i t i o n t o t h e v a l u a t i o n r u l e , w e can use t h e d e f i n i t i o n o f a u t o e p i s t e m i c e x t e n s i o n a b o v e for t h e case in w h i c h A c o n t a i n s q u a n t i f i e d - i n expressions. How ever, w e w i l l m a k e one t e c h n i c a l s t i p u l a t i o n t h a t w i l l b e useful i n l a t e r d e v e l o p m e n t s , t o i n s u r e t h a t there are e n o u g h " f r e e " names in t h e l a n g u a g e , The language contains a count ably infinite set of constants C which cannot be used in the premise set A of an extension.
(4)
A g i v e n i n d i v i d u a l x m a y h a v e n o n e or m a n y n a m e s in a m o d e l , a c i r c u m s t a n c e w h i c h leads to some i n t e r esting behavior of the fixed-point equation of Definition 2 . 1 . T o e x p l o r e some o f these, w e f i r s t m a k e t h e observa t i o n t h a t t h e revised d e f i n i t i o n o f s a t i s f i a b i l i t y f o r m o d a l a t o m s does n o t p e r t u r b t h e e x t e n s i o n s o f a n y set A t h a t c o n t a i n s n o q u a n t i f i e d - i n expressions.
21.801801 Let A = {Pa}. T h e r e is a single exten sion T of .4, with To - ( ' n ( P o ) . Therefore, we know t h a t LP a and ¬ L P b are in T. By the v a l u a t i o n rule for m o d a l a t o m s ( 3 ) , w i l l be true in (I,To) if there is some i n d i v i d u a l x such t h a t x — v(a). Ev ery i n t e r p r e t a t i o n / has some such i n d i v i d u a l , and hence true in every ( / , 7 b ) model of A, and hence in 7\
EXAMPLE
A n o t h e r i n t e r e s t i n g sentence contained in T is s : To see why this is so, let x be an i n d i v i d u a l w i t h , Since Pa is the only g r o u n d occurrence of the predicate in 7b, it must be the case t h a t LPx is false in any model (7,7b) of A } and hence s is an element of T. On the other h a n d , consider a s i m i l a r sentence .$' : It m i g h t be suspected t h a t s' is a m e m b e r of 71, b u t this is not the case. For a l t h o u g h x is the d e n o t a t i o n of/;, it may also be the d e n o t a t i o n of a in some first-order i n t e r p r e t a t i o n , and for (his i n t e r p r e t a t i o n , LPx w i l l be true. So s' w i l l n o t be true in all ( / , 7 b ) models of A T h i s e x a m p l e h i g h l i g h t s a curious s i t u a t i o n t h a t oc curs when knowledge of properties of i n d i v i d u a l s hinges on h a v i n g a name for t h a t i n d i v i d u a l : the epistemic op erator expresses knowledge of the intension of a t e r m . Let, us take P to be the p r o p e r t y of being rich, a to be the m a y o r , and b to be the former police chief. We have p r o o f t h a i the m a y o r is rich and no evidence t h a t the f o r m e r police chief is . These are statements a b o u t the intension of the terms a and /;, t h a t is, the m a y o r , whoever he is, is rich. On the other h a n d , the expression LPx when x is a q u a n t i h e d - i n variable says t h a t we k n o w Pc to be true for some intensional concept r whose d e n o t a t i o n is x. Now if we were to know t h a t a p a r t i c u l a r i n d i v i d u a l x is the former police chief, we s t i l l cannot, say t h a t we have no evidence t h a t x is rich, because x may also be the m a y o r . A n o t h e r consequence of the intensional nature of the epistemic o p e r a t o r is t h a t even t h o u g h a universal state m e n t m a y be t r u e in an AE v a l u a t i o n , its s u b s t i t u t i o n instances m a y n o t . Consider the v a l u a t i o n ( 7 , T ) , where and r = C n ( P a ) . We must have because if ' LPa is t r u e ; and if n o t , ¬Px is t r u e . However, the s u b s t i t u t i o n instance LPbV ¬Pb is not true in this v a l u a t i o n : LPb is n o t a m e m b e r of V, and Pb is t r u e in 7 because a—b. F i n a l l y , we n o t e t h a t the Barcan f o r m u l a every A E v a l u a t i o n , while the converse may he false. T h e reason for the latter is t h a t even t h o u g h every i n d i v i d u a l x has the p r o p e r t y P, some i n d i v i d u a l s m a y n o t be given a name in the AE v a l u a t i o n , and so LPx w i l l he false. T h i s scheme for e x t e n d i n g the semantics of L valu a t i o n s to the q u a n t i f i e d - i n case is s i m i l a r to t h a t pro posed for the s i m p l e epistemic operators in [Konolige, 1984]. It differs f r o m the approach of [Levesque, 1982, Levesque, 1987] in t h a t it is based on the intension of t e r m s r a t h e r t h a n their d e n o t a t i o n . Nevertheless there are m a n y p o i n t s of s i m i l a r i t y between the two approaches t h a t we have n o t i n v e s t i g a t e d .
3
MIN Theories
So far we have only looked at extensions of sets of firstorder sentences. T h e normal d e f i n i t i o n of extension (2.1) earned over to the extended language, w i t h o n l y a change in the valuation rule for m o d a l a t o m s , and a slight restriction on the constants a p p e a r i n g in the premise set,. We now examine a class of first-order sen tences t h a t we call M I N theories. A n y such theory has the f o r m (5) where W is a finite set of first-order sentences and the P, are a sequence of predicates. We w r i t e M ( W ; P1, .. . P n ) to indicate the M I N theory of W over the predicates P t . T h e idea behind M I N theories is to select AE v a l u a tions in which every i n d i v i d u a l not k n o w n to have the property Pi does not have this property, i.e., to m i n i m i z e the extension of each Pi. EXAMPLE
3.1
Let
W
=
{Fa},
and
let
S
-
Every v a l u a t i o n (I,S) w h i c h satisfies M ( I T ; P) makes Px true for x — v ( a ) , and false for every other x. T h u s , if we define T by
it is clear that 7b — S, and hence T is an extension of M ( I T ; P ) . In fact it is the only extension. Let IT - {Pu V P/;}, and as before let 5 = = x — a). Again we can show t h a t ev ery valuation (I, S) satisfying M(W,P) satisfies = x = a, and so S is the kernel of an AE extension of M ( W ' ; P ) . In this case the extension is not unique, there is another one whose kernel is Cn(Vx\Px,* = x. — h). T h e sentences w h i c h these t w o extension have in common are all first-order conse quences of Let W and let A g a i n we can show t h a t every v a l u a t i o n ( 7 , 5 ) sat isfying M ( I T ; P ) satisfies Vx.Pa: = z~a, and so S generates an AE extension 7'. B u t the choice of the constant a was a r b i t r a r y , and we can use any other constant in defining S. fience there are an infinite number of extensions of M ( W ; P ) ; the sen tences they have in common are the first-order con sequences of These examples are very suggestive of a correspon dence between M I N theories and the m i n i m a l models of W. However, there is one essential p o i n t of difference. A model / of W is m i n i m a l in P t if there is no other model with, the same universe and denotation function whose extensions of /\ are properly included in those of I. (Note t h a t we are assuming here t h a t the e x t e n sions of all predicates oilier t h a n the P ti can vary across compared models. In the next, subsection we consider the case of fixed predicates.) Comparisons are n o t m a d e between models w i t h different domains and different de n o t a t i o n functions, and hence choosing o n l y the m i n i m a l models of W w i l l not lead to any conclusions a b o u t the equality relation or the size of the d o m a i n not already apparent in W (this p o i n t lias been noted in [ E t h e r i n g t o n and Reifer, 1984]). On the other h a n d , this is n o t t r u e
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of the extensions of M I N theories: an extension can con tain conclusions not present in \V a b o u t e q u a l i t y anions, terms. For e x a m p l e , the set W = {¬Pa) a — b} does not have a = b a m o n g its f u s t - o r d e r consequences, yet. the single extension of M ( W , P) does. To get the cor respondence between M I N theories and m i n i m a l models correct, we need to fix the i n t e r p r e t a t i o n of e q u a l i t y in the former. 3.1
A second interesting property of parameter models is t h a t the Pminimal parameter models of a. finite set W are sufficient, for m i n i m a l e n t a i l m e n t , as we now show. 1 3.1 base language, set of sentences is true in all and only if it modeIs.
PROPOSITION
Fixing predicates
Proof. Suppose has a P - m i n i m a l count able model /. Let the constants C' be those members of C not mentioned by < We are free to construct a model V t h a t is the same as /, b u t in which all elements of / are denoted by one of C. V m u s t be m i n i m a l ; if it were n o t , then there i s another model / " w i t h the same d e n o t a t i o n f u n c t i o n and universe a,s I', b u t w i t h a smaller extension of P. We can con vert I" i n t o a model w i t h the same d e n o t a t i o n f u n c t i o n as /, which must then be less t h a n /, a contradiction. In the converse d i r e c t i o n , if has a m i n i m a l parameter m o d e l , it obviously has a m i n i m a l countable m o d e l .
We extend M I N theories by a d d i n g a set of predicates, the fixed predicates, to the o r i g i n a l d e f i n i t i o n . A M I N theory now has the f o r m
where the Q i are a sequence of predicates. We w r i t e M ( W ; P1, . . . P n ',Q1,..... Qm) to i n d i c a t e the M1N theory of W over the predicates P i w i t h Qj f i x e d . Note t h a t , to fix Q, b o t h Q and its n e g a t i o n ¬Q are m i n i m i z e d . I n general t h i s w i l l lead t o m u l t i p l e exten sions in w h i c h various c o m b i n a t i o n s of Qx and ¬Qx hold for each i n d i v i d u a l x. EXAMPLE
3.2
Let
W
=
The
M1N
the
ory M(W',P\Q) has t w o classes of extensions: one class contains {¬Pa,Qa}, w h i l e the other contains {¬Qa,Pa}. T h u s the m i n i m i z a t i o n of P does not force the acceptance of Qa in every e x t e n s i o n . T h e presence of a fixed Q a c t u a l l y creates an infi n i t e n u m b e r of extensions because of the presence of the countable set C of constants in L We w i l l consider extensions to be e q u i v a l e n t if they differ o n l y in sentences c o n t a i n i n g these constants; in this case, there are j u s t t w o n o n e q u i v a l e n t extensions. T h e e q u a l i t y predicate can be fixed, j u s t as any other predicate, and it is the class of M I N theories M ( W ; P\ — ) t h a t w e consider i n r e l a t i n g extensions t o m i n i m a l m o d els: call these M I N = theories. We now develop the re sult t h a t a first-order sentence is t r u e in the P - m i n i m a l models of W j u s t in case it is t r u e in every extension of
W(W;P-=). 3.2
Parameter models
A first-order i n t e r p r e t a t i o n in w h i c h every i n d i v i d u a l x is denoted by some t e r m is called a parameter i n t e r p r e t a t i o n . H e r b r a n d i n t e r p r e t a t i o n s are one t y p e o f parameter i n t e r p r e t a t i o n , in w h i c h every t e r m denotes itself. Pa rameter i n t e r p r e t a t i o n s are m o r e general t h a n H e r b r a n d i n t e r p r e t a t i o n s , since in the f o r m e r t w o t e r m s can refer to the same i n d i v i d u a l . Just as H e r b r a n d i n t e r p r e t a t i o n s are a sufficient se m a n t i c s for universal prenex sentences, so t o o p a r a m e ter i n t e r p r e t a t i o n s suffice for sets of first-order sentences. By "suffice" we mean t h a t any such set W has a model i f and o n l y i f i t has a p a r a m e t e r m o d e l . N o t e t h a t this s t a t e m e n t is n o t t r u e in general if W can c o n t a i n m e m bers of C: for e x a m p l e t h e set in w h i c h Pci is asserted for every constant has a model but no C-parameter model.
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Let he a sentence of. —C (the with out the constants C), and W a of the same language. Then P-minimal countable models of Wif is true in all P-minimal parameter
F i n a l l y , as the next p r o p o s i t i o n shows, parameter models are the only ones we need consider in f o r m i n g extensions o f M I N = theories. PROPosITION 3.2 Any interpreta11on ( / , T) is a modeI of a MIN= theory M(W; P; —) only if I is a parameter model. Proof. Suppose / isn't a parameter m o d e l . T h e n there is some element e of the universe of 1 such t h a t c is not denoted by any t e r m . T h u s b o t h -Lx = y and ¬ L x # y are t r u e for x=e, and this leads to a c o n t r a d i c t i o n in M(W;P;=). 3.3
The main theorem
For any P - m i n i n i a l parameter model / of W, we call the P-diagram of / the set of g r o u n d literals in P and = t h a t are t r u e in I. We first, show t h a t the d i a g r a m of / picks o u t a u n i q u e extension of M ( W ; P ; = ) , for w h i c h / is a model. 3.3 Let D be the diagram of a Pminimal parameter model I of W. Then (I, D) is a model of some extension of M ( W ; P; =).
PROPOSITION
Proof.
T h i s is a sketch of the proof. Let
We first show t h a t the r e s t r i c t i o n of S to g r o u n d P and e q u a l i t y literals is exactly the set D. N o t e t h a t D is complete w i t h respect to equal i t y l i t e r a l s : for all t e r m s a a n d 6, either a=b or a#b is in D, b u t n o t b o t h . F r o m the f i x i n g o f e q u a l i t y i n M ( W ; P ; = ) , a l l o f these are also contained in 5. D is also complete Note: we w i l l often use a single minimized predicate P in propositions in the rest of this paper; the extensions to multiple predicates is obvious.
4
Reasoning about Equality
In defining P - m i n i m a l interpretations, we have specified that two interpretations must have the same domain and denotation function in order to be comparable. T h i s corresponds to predicate circumscription w i t h a fixed interpretation of terms.
By Proposition 3.3 above, every P - m i n i m a l parameter model of W satisfies the kernel of some extension of M(W;P=). Conversely, by Proposition 3.4 above, the kernel of every extension of M(W;P;=) are the sentences true of some class of P-minirnal parameter models of W. Given the sufficiency of parameter models for m i n i m a l entailment (Proposition 3.1), we have the following theorem. THEOREM
3.5
The first-order sentences S true in every extension ofM(W; P ; = ) are exactly those sentence true in the P-minimal models of W.
This example is typical of the way in which c i r c u m scription handles equality: it does not allow any new conclusions about equality, because the denotations of terms are fixed across comparable models. However, it is also possible to compare models w i t h different denotation functions; the corresponding predicate circumscrip-
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t i o n allows f u n c t i o n s to vary, as well as predicates (see [ L i f s c h i t z , 1984]).
F r o m the above e x a m p l e , it seems t h a t a l l o w i n g terms to vary leads to the danger of unexpected identification of t e r m s , at least if we do n o t have axioms that e x p l i c i t l y say t h a t d i f f e r i n g t e r m s refer to different i n d i v i d u a l s We w o u l d like to t r e a t e q u a l i t y a m o n g terms somewhat in between the t w o extremes of fixed and v a r y i n g denota t i o n s : to r e m a i n agnostic a b o u t the equivalence of terms, b u t s t i l l be able to d r a w basic default conclusions. T h e M I N = theories, because of their relation to /■'m i n i m a l models, always leave the denotations of terms f i x e d , and so f a l l prey to the same problems w i t h equal i t y as c i r c u m s c r i p t i o n w i t h fixed terms. However, we can relax the r e s t r i c t i o n on denotations by using M1N theories, w i t h o u t the sentences f i x i n g equality.
References [Etherington and Reiter, 1984] R. E. Mercer D. W. Etherington and R. Reiter. On the adequacy of predicate circumscription for closed-world reason ing. In AAA! Workshop on Non-Mono!ante Reasoning. American Association for A r t i f i c i a l Intelligence, 1984. de Kleer and Konolige, 1989] Johan de Kleer and K u r t Konolige. E l i m i n a t i n g the fixed predicates from a cir cumscription. Artificial Intelligence, page to appear, 1989. [Etherington, 1986] David W i l l i a m E t h e r i n g t o n . Reasoning with Incomplete Information: Investigations of Non-Monotonia Reasoning. P h D thesis, University of British C o l u m b i a , Vancouver, British C o l u m b i a , 1986. [Imielinski, 1987] T. Imielinski. Results on translating defaults to circumscription. Artificial Intelligence, 32(1):131-146, 1987. [Konolige, 1984] K u r t Konolige. A Deduction Model of Belief and its Logics. P h D thesis, Stanford University, 1984. [Konolige, 1987] K u r t Konolige. On the relation be tween default logic and autoepistemic theories Artificial Intelligence, 35(3):343-382, 1987. [Levesque, 1982] Hector J. Levesque. A formal treat ment of incomplete knowledge bases. Technical Re port 614, Fairchild A r t i f i c i a l Intelligence Laboratory, Palo A l t o , C a l i f o r n i a , 1982. [Levesque, 1987] Hector J. Levesque. A l l 1 know; an abridged report. In Proceedings of the American Association of Artificial Intelligence. Seattle, W a s h i n g t o n , 1987.
M I N theories w i t h o u t e q u a l i t y f i x a t i o n are thus inter mediate between a fixed and v a r y i n g i n t e r p r e t a t i o n of equality, and seem to be the r i g h t level of v a r i a t i o n for commonsense reasoning in a b n o r m a l i t y theories.
5
Conclusion
We have extended the language and semantics of autoepistemic logic in a n a t u r a l way to the case of q u a n t i f i e d - i n variables. By l o o k i n g at a class of A10 the ories, the M I N = theories, we showed t h a t all of the firstorder consequences of predicate c i r c u m s c r i p t i o n could be expressed in a simple way in autoepistemic logic. T h i s is the first result on the r e l a t i o n s h i p of these two logics for the case of n o n f m i t e d o m a i n s . T h e results have been used to shed some l i g h t on the t r e a t m e n t of equality in commonsense reasoning.
[Lifschitz, 1984] V l a d i m i r Lifschitz. Some results on cir cumscription. In AAAl Workshop on Non-Monotonic Reasoning, 1984. [Lifschitz, 1985] V l a d i m i r Lifschitz. C o m p u t i n g circum scription. In Proceedings of the International Joint Conference on Artificial Intelligence, pages 121 127, Los Angeles, California, 1985. [McCarthy, 1986] John M c C a r t h y . A p p l i c a t i o n s of cir cumscription to formalizing commonsense knowledge. Artificial Intelligence, 28, 1986. [Moore, 1985] Robert C. Moore. Semantical considera tions on nonmonotonic logic. Artificial Intelligence, 25(1), 1985. [Perlis, 1986] Donald Perlis. On the consistency of com monsense reasoning. Computational Intelligence, 2, 1986. [Reiter, 1980] R a y m o n d Reiter. A logic for default rea soning. Artificial Intelligence, 13(1-2), 1980. [Shoham, 1987] Yoav Shoh a m . Reasoning about Change: Time and Causation from the Standpoint of Artificial Intelligence. M I T Press, Cambridge, Massachusetss, 1987.
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1
Introduction
T h e r e l a t i o n s between the m a j o r n o n m o n o t o n i c logic form a l i s m s of AI — default logic, autoepistemic logic, and c i r c u m s c r i p t i o n — is of some i m p o r t a n c e , since all of these logics have been proposed as f o r m a l i s m s for various types of commonsense reasoning. T h e basic f o r m a l equivalence of d e f a u l t and a u t o e p i s t e m i c logic has already been s h o w n (see [ K o n o l i g e , 1987]), b u t the relation between c i r c u m s c r i p t i o n and default or autoepistemic logic r e m a i n s obscure. M o s t l y this is a consequence of the different f o u n d a t i o n s of these logics: c i r c u m s c r i p t i o n is based on a m i n i m a l - m o d e l semantics (see [Lifschitz, 1985]), w h i l e the others use more proof-theoretic techniques ( d e f a u l t logic [Reiter, 1980]) or an epistemic ope r a t o r ( a u t o e p i s t e m i c logic [ M o o r e , 1985]). In t r y i n g to express a u t o e p i s t e m i c or default, logic in c i r c u m s c r i p t i o n , researchers have f o u n d the basic probl e m to be t h a t a m i n i m a l - m o d e l or even prefered-model *This research was supported by the Office of Naval Research under Contract No. N00014-85-C-0251, by subcontract from Stanford University under the Defense Advanced Research Projects Administration under Contract No. N00039-84-C-0211, and by a gift from the System Development Foundation.
semantics s i m p l y does not have the c a p a b i l i t y of representing the requisite proof-theoretic or epistemic concepts (see [Shoham, 1987]). We agree w i t h this assessment, and say n o t h i n g further about it here. On the other h a n d , there have been several results on expressing c i r c u m s c r i p t i o n in default logic. These results are summarized in [ E t h e r i n g t o n , 1986]; they apply to the restricted case of predicate c i r c u m s c r i p t i o n w i t h no fixed predicates and w i t h a f i n i t e , fixed d o m a i n . From a model-theoretic p o i n t of view, the predicate c i r c u m s c r i p t i o n C i r c u m ( , 4 ; P; Z) of a first-order sentence A picks out those models of A in which the extension of the predicate P is m i n i m a l . T h e comparison is across models w i t h the same d o m a i n and denotation f u n c t i o n , but which m i g h t differ in the extensions of the predicates Z. A l l predicates other t h a n P and Z are fixed, t h a t is, cannot vary in a comparison of models. It was recently shown (see [de Kleer and Konolige, 1989]) t h a t fixed predicates are inessential in predicate c i r c u m s c r i p t i o n , t h a t is, there is a simple t r a n s l a t i o n f r o m any c i r c u m scription w i t h fixed predicates to one w i t h o u t . Hence fixed predicates no longer present an obstacle to representing circumscriptions in default or autoepistemic ogic. T h e problem of finite domains remains, however. In this paper we provide a solution to this p r o b l e m , by first extending autoepistemic logic to a language which allows q u a n t i f y i n g i n t o the epistemic operator, and then showing t h a t a certain class of autoepistemic theories, the M I N = theories, express all of the first-order consequences of predicate c i r c u m s c r i p t i o n .
2
S e m a n t i c s of Q u a n t i f y i n g - i n
A u t o e p i s t e m i c ( A E ) logic was defined by [Moore, 1985] as a f o r m a l account of an agent reasoning a b o u t her own beliefs. T h e agent's beliefs are assumed to be a set of sentences in some logical language augmented by a m o d a l operator L. As o r i g i n a l l y defined, and extended in [Konolige, 1987], its language does not p e r m i t variables quantified outside of a m o d a l operator to appear inside. In this section we f u r t h e r extend AE logic to deal with quantifying-in. 2.1
Logical preliminaries
We begin w i t h a language C for expressing self-belief, and introduce valuations of C. T h e t r e a t m e n t generally
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1213
follows and extends [Konolige, 1987] Let be a f i r s t - o r d e r language w i t h e q u a l i t y and f u n c t i o n a l t e r m s . T h e n o r m a l f o r m a t i o n rules for for m u l a s of first-order languages h o l d . A sentence of is a f o r m u l a w i t h no free v a r i a b l e s ; an atom is a sentence of the f o r m P(t1, • • • , tn). We e x t e n d by a d d i n g a u n a r y m o d a l o p e r a t o r L; t h e e x t e n d e d l a n g u a g e is called can be defined recursively as c o n t a i n i n g all t h e f o r m a t i o n rules of plus the f o l l o w i n g : If
is a f o r m u l a of
t h e n so is
.
(1)
An expression is a modal atom. Sentences a n d a t o m s of are called ordinary. N o t e t h a t nestings such as are not a l l o w e d ; we consider o n l y a single level of nesting here. Because the a r g u m e n t of a m o d a l o p e r a t o r can c o n t a i n free v a r i a b l e s , there m a y be q u a n t i f y i n g i n t o the scope of a m o d a l a t o m , e.g., is a sentence. O f t e n we w i l l use a s u b s c r i p t " 0 " to i n d i c a t e a subset of o r d i n a r y sentences, e.g., Let u s r e s t r i c t ourselves for t h e m o m e n t t o m o d a l a t o m s w h i c h d o n o t c o n t a i n free v a r i a b l e s . F r o m the p o i n t o f view o f f i r s t - o r d e r v a l u a t i o n s , the m o d a l a t o m s are s i m p l y n i l a r y p r e d i c a t e s . O u r i n t e n d e d i n t e r p r e t a t i o n of these a t o m s is t h a t is an e l e m e n t of t h e b e l i e f set of t h e agent. So we w i l l consider v a l u a t i o n s of to be standard first-order valuations, w i t h the a d d i t i o n of a belief set T. T h e a t o m s are i n t e r p r e t e d as t r u e or false d e p e n d i n g on w h e t h e r is in T. To d i s t i n g u i s h these v a l u a t i o n s , we w i l l s o m e t i m e s call t h e m L valuations. T h e interaction of the interpretation of L w i t h firstorder v a l u a t i o n s is o f t e n a d e l i c a t e m a t t e r , a n d so a per spicuous t e r m i n o l o g y f o r t a l k i n g a b o u t L v a l u a t i o n s is necessary. I n p a r t i c u l a r , i t i s o f t e n useful t o d e c o u p l e the i n t e r p r e t a t i o n o f m o d a l a n d o r d i n a r y a t o m s . F i r s t order v a l u a t i o n s are b u i l t u p o n t h e t r u t h v a l u e s o f a t o m s : for o r d i n a r y a t o m s , t r u t h v a l u e s are g i v e n by a s t r u c t u r e (U,v,R), where v is a d e n o t a t i o n a l m a p p i n g f r o m t e r m s to elements of the universe U, a n d R. is a set of rela tions over U, one for each p r e d i c a t e . We w i l l refer to any such s t r u c t u r e as an ordinary index, a n d d e n o t e it w i t h t h e s y m b o l I. M o d a l a t o m s are g i v e n a t r u t h v a l u e by a belief set T, w h i c h is called a modal index. N o t e t h a t , because m o d a l o p e r a t o r s are n o t nested, o n l y t h e o r d i n a r y sentences o f t h e m o d a l i n d e x V are i m p o r t a n t . T h e t r u t h v a l u e o f any sentence i n can b e d e t e r m i n e d by the n o r m a l rules for f i r s t - o r d e r v a l u a t i o n s , g i v e n an o r d i n a r y and m o d a l i n d e x . W e w r i t e if a val u a t i o n ( / , T ) satisfies T h e valuation rule for m o d a l a t o m s can be w r i t t e n as if and only if
(2)
A v a l u a t i o n t h a t makes a every m e m b e r of a set of sentences t r u e is called a model of t h e set. A sentence t h a t is t r u e in every m e m b e r of a class of v a l u a t i o n s is called valid w i t h respect to t h e class. W e use C n ( X ) t o m e a n t h e f i r s t - o r d e r consequences o f a f i r s t - o r d e r set of sentences X. 2.2
Autoepistemic Extensions
I n [ K o n o l i g e , 1987], w e i n f o r m a l l y defined a n e x t e n s i o n of a set of sentences A as those consequences of A w h i c h
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Knowledge Representation
an agent, s h o u l d believe. T h e f o r m a l c o u n t e r p a r t is given
by: DEFINITION 2.1 the
Any set of sentences T which sa/ysfi.e.s
equation
is an a u t o e p i s t e m i c e x t e n s i o n of A. T h i s is a f i x e d - p o i n t e q u a t i o n for a b e l i e f set /', and is a c a n d i d a t e for t h e belief set of an ideal i n t r o s p e c t i v e agent w i t h premises .4. I t differs s l i g h t l y f r o m t h e o r i g i n a l d e f i n i t i o n o f [ K o n o l i g e , 1987] i n t h a t the m o d a l i n d e x consists of the o r d i n a r y p a r t ( t h e kernel of 7 1 ), t h i s suf fices because t h e l a n g u a g e of T does not i n c l u d e nested modal operators. N o t e t h a t we are c o n s i d e r i n g all m o d e l s of A in w h i c h t h e i n t e r p r e t a t i o n of is t h e b e l i e f set of t h e agent itself, t h a t is, t h e v a l u a t i o n s we consider all have a m o d a l i n d e x t h a t i s t h e b e l i e f set o f t h e a g e n t , f o l l o w i n g M o o r e , we c a l l such v a l u a t i o n s autoepistemic (or AE) v a l u a t i o n s . 2.3
Quantifying-in
W e w o u l d like t o e x t e n d t h e l a n g u a g e o f a u t o e p i s t e m i c l o g i c t o i n c l u d e v a r i a b l e s w h i c h are q u a n t i f i e d outside, t h e scope o f t h e m o d a l o p e r a t o r , b u t can a p p e a r inside, e.g., T h e p r o b l e m i s t h a t i t i s n o t o b v i o u s how t o e x t e n d t h e s e m a n t i c s o f t h e logic t o deal w i t h these " q u a n t i f i e d - i n " expressions. R e c a l l t h a t t h e b e l i e f set T is a set of sentences t h a t f o r m t h e beliefs of an agent.. To interpret we s i m p l y ask w h e t h e r t h e expression øis i n T . B u t w i t h t h e q u a n t i f i e d - i n l a n g u a g e , w e m u s t also b e able t o interpret w h e r e i s the proposition t h a t t h e i n d i v i d u a l x has t h e p r o p e r t y I n order t o c o n s t r u c t a p r o p o s i t i o n a l expression whose m e a n i n g is w e m u s t have some way o f r e f e r i n g t o i n d i v i d u a l s in the d o m a i n . T h e simplest, scheme f o r reference t o d o m a i n e l e m e n t s is to use t h e d e n o t a t i o n m a p v a l r e a d y present in the f i r s t - o r d e r m t e r p r e t i o n / . I n place o f the v a l u a t i o n rule f o r m o d a l a t o m s g i v e n a b o v e , w e use: iff
f o r some t e r m / such t h a t
(3)
T h a t is, we say t h a t (x) is believed if x has a n a m e / such is b e l i e v e d . G i v e n t h i s a d d i t i o n t o t h e v a l u a t i o n r u l e , w e can use t h e d e f i n i t i o n o f a u t o e p i s t e m i c e x t e n s i o n a b o v e for t h e case in w h i c h A c o n t a i n s q u a n t i f i e d - i n expressions. How ever, w e w i l l m a k e one t e c h n i c a l s t i p u l a t i o n t h a t w i l l b e useful i n l a t e r d e v e l o p m e n t s , t o i n s u r e t h a t there are e n o u g h " f r e e " names in t h e l a n g u a g e , The language contains a count ably infinite set of constants C which cannot be used in the premise set A of an extension.
(4)
A g i v e n i n d i v i d u a l x m a y h a v e n o n e or m a n y n a m e s in a m o d e l , a c i r c u m s t a n c e w h i c h leads to some i n t e r esting behavior of the fixed-point equation of Definition 2 . 1 . T o e x p l o r e some o f these, w e f i r s t m a k e t h e observa t i o n t h a t t h e revised d e f i n i t i o n o f s a t i s f i a b i l i t y f o r m o d a l a t o m s does n o t p e r t u r b t h e e x t e n s i o n s o f a n y set A t h a t c o n t a i n s n o q u a n t i f i e d - i n expressions.
21.801801 Let A = {Pa}. T h e r e is a single exten sion T of .4, with To - ( ' n ( P o ) . Therefore, we know t h a t LP a and ¬ L P b are in T. By the v a l u a t i o n rule for m o d a l a t o m s ( 3 ) , w i l l be true in (I,To) if there is some i n d i v i d u a l x such t h a t x — v(a). Ev ery i n t e r p r e t a t i o n / has some such i n d i v i d u a l , and hence true in every ( / , 7 b ) model of A, and hence in 7\
EXAMPLE
A n o t h e r i n t e r e s t i n g sentence contained in T is s : To see why this is so, let x be an i n d i v i d u a l w i t h , Since Pa is the only g r o u n d occurrence of the predicate in 7b, it must be the case t h a t LPx is false in any model (7,7b) of A } and hence s is an element of T. On the other h a n d , consider a s i m i l a r sentence .$' : It m i g h t be suspected t h a t s' is a m e m b e r of 71, b u t this is not the case. For a l t h o u g h x is the d e n o t a t i o n of/;, it may also be the d e n o t a t i o n of a in some first-order i n t e r p r e t a t i o n , and for (his i n t e r p r e t a t i o n , LPx w i l l be true. So s' w i l l n o t be true in all ( / , 7 b ) models of A T h i s e x a m p l e h i g h l i g h t s a curious s i t u a t i o n t h a t oc curs when knowledge of properties of i n d i v i d u a l s hinges on h a v i n g a name for t h a t i n d i v i d u a l : the epistemic op erator expresses knowledge of the intension of a t e r m . Let, us take P to be the p r o p e r t y of being rich, a to be the m a y o r , and b to be the former police chief. We have p r o o f t h a i the m a y o r is rich and no evidence t h a t the f o r m e r police chief is . These are statements a b o u t the intension of the terms a and /;, t h a t is, the m a y o r , whoever he is, is rich. On the other h a n d , the expression LPx when x is a q u a n t i h e d - i n variable says t h a t we k n o w Pc to be true for some intensional concept r whose d e n o t a t i o n is x. Now if we were to know t h a t a p a r t i c u l a r i n d i v i d u a l x is the former police chief, we s t i l l cannot, say t h a t we have no evidence t h a t x is rich, because x may also be the m a y o r . A n o t h e r consequence of the intensional nature of the epistemic o p e r a t o r is t h a t even t h o u g h a universal state m e n t m a y be t r u e in an AE v a l u a t i o n , its s u b s t i t u t i o n instances m a y n o t . Consider the v a l u a t i o n ( 7 , T ) , where and r = C n ( P a ) . We must have because if ' LPa is t r u e ; and if n o t , ¬Px is t r u e . However, the s u b s t i t u t i o n instance LPbV ¬Pb is not true in this v a l u a t i o n : LPb is n o t a m e m b e r of V, and Pb is t r u e in 7 because a—b. F i n a l l y , we n o t e t h a t the Barcan f o r m u l a every A E v a l u a t i o n , while the converse may he false. T h e reason for the latter is t h a t even t h o u g h every i n d i v i d u a l x has the p r o p e r t y P, some i n d i v i d u a l s m a y n o t be given a name in the AE v a l u a t i o n , and so LPx w i l l he false. T h i s scheme for e x t e n d i n g the semantics of L valu a t i o n s to the q u a n t i f i e d - i n case is s i m i l a r to t h a t pro posed for the s i m p l e epistemic operators in [Konolige, 1984]. It differs f r o m the approach of [Levesque, 1982, Levesque, 1987] in t h a t it is based on the intension of t e r m s r a t h e r t h a n their d e n o t a t i o n . Nevertheless there are m a n y p o i n t s of s i m i l a r i t y between the two approaches t h a t we have n o t i n v e s t i g a t e d .
3
MIN Theories
So far we have only looked at extensions of sets of firstorder sentences. T h e normal d e f i n i t i o n of extension (2.1) earned over to the extended language, w i t h o n l y a change in the valuation rule for m o d a l a t o m s , and a slight restriction on the constants a p p e a r i n g in the premise set,. We now examine a class of first-order sen tences t h a t we call M I N theories. A n y such theory has the f o r m (5) where W is a finite set of first-order sentences and the P, are a sequence of predicates. We w r i t e M ( W ; P1, .. . P n ) to indicate the M I N theory of W over the predicates P t . T h e idea behind M I N theories is to select AE v a l u a tions in which every i n d i v i d u a l not k n o w n to have the property Pi does not have this property, i.e., to m i n i m i z e the extension of each Pi. EXAMPLE
3.1
Let
W
=
{Fa},
and
let
S
-
Every v a l u a t i o n (I,S) w h i c h satisfies M ( I T ; P) makes Px true for x — v ( a ) , and false for every other x. T h u s , if we define T by
it is clear that 7b — S, and hence T is an extension of M ( I T ; P ) . In fact it is the only extension. Let IT - {Pu V P/;}, and as before let 5 = = x — a). Again we can show t h a t ev ery valuation (I, S) satisfying M(W,P) satisfies = x = a, and so S is the kernel of an AE extension of M ( W ' ; P ) . In this case the extension is not unique, there is another one whose kernel is Cn(Vx\Px,* = x. — h). T h e sentences w h i c h these t w o extension have in common are all first-order conse quences of Let W and let A g a i n we can show t h a t every v a l u a t i o n ( 7 , 5 ) sat isfying M ( I T ; P ) satisfies Vx.Pa: = z~a, and so S generates an AE extension 7'. B u t the choice of the constant a was a r b i t r a r y , and we can use any other constant in defining S. fience there are an infinite number of extensions of M ( W ; P ) ; the sen tences they have in common are the first-order con sequences of These examples are very suggestive of a correspon dence between M I N theories and the m i n i m a l models of W. However, there is one essential p o i n t of difference. A model / of W is m i n i m a l in P t if there is no other model with, the same universe and denotation function whose extensions of /\ are properly included in those of I. (Note t h a t we are assuming here t h a t the e x t e n sions of all predicates oilier t h a n the P ti can vary across compared models. In the next, subsection we consider the case of fixed predicates.) Comparisons are n o t m a d e between models w i t h different domains and different de n o t a t i o n functions, and hence choosing o n l y the m i n i m a l models of W w i l l not lead to any conclusions a b o u t the equality relation or the size of the d o m a i n not already apparent in W (this p o i n t lias been noted in [ E t h e r i n g t o n and Reifer, 1984]). On the other h a n d , this is n o t t r u e
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of the extensions of M I N theories: an extension can con tain conclusions not present in \V a b o u t e q u a l i t y anions, terms. For e x a m p l e , the set W = {¬Pa) a — b} does not have a = b a m o n g its f u s t - o r d e r consequences, yet. the single extension of M ( W , P) does. To get the cor respondence between M I N theories and m i n i m a l models correct, we need to fix the i n t e r p r e t a t i o n of e q u a l i t y in the former. 3.1
A second interesting property of parameter models is t h a t the Pminimal parameter models of a. finite set W are sufficient, for m i n i m a l e n t a i l m e n t , as we now show. 1 3.1 base language, set of sentences is true in all and only if it modeIs.
PROPOSITION
Fixing predicates
Proof. Suppose has a P - m i n i m a l count able model /. Let the constants C' be those members of C not mentioned by < We are free to construct a model V t h a t is the same as /, b u t in which all elements of / are denoted by one of C. V m u s t be m i n i m a l ; if it were n o t , then there i s another model / " w i t h the same d e n o t a t i o n f u n c t i o n and universe a,s I', b u t w i t h a smaller extension of P. We can con vert I" i n t o a model w i t h the same d e n o t a t i o n f u n c t i o n as /, which must then be less t h a n /, a contradiction. In the converse d i r e c t i o n , if has a m i n i m a l parameter m o d e l , it obviously has a m i n i m a l countable m o d e l .
We extend M I N theories by a d d i n g a set of predicates, the fixed predicates, to the o r i g i n a l d e f i n i t i o n . A M I N theory now has the f o r m
where the Q i are a sequence of predicates. We w r i t e M ( W ; P1, . . . P n ',Q1,..... Qm) to i n d i c a t e the M1N theory of W over the predicates P i w i t h Qj f i x e d . Note t h a t , to fix Q, b o t h Q and its n e g a t i o n ¬Q are m i n i m i z e d . I n general t h i s w i l l lead t o m u l t i p l e exten sions in w h i c h various c o m b i n a t i o n s of Qx and ¬Qx hold for each i n d i v i d u a l x. EXAMPLE
3.2
Let
W
=
The
M1N
the
ory M(W',P\Q) has t w o classes of extensions: one class contains {¬Pa,Qa}, w h i l e the other contains {¬Qa,Pa}. T h u s the m i n i m i z a t i o n of P does not force the acceptance of Qa in every e x t e n s i o n . T h e presence of a fixed Q a c t u a l l y creates an infi n i t e n u m b e r of extensions because of the presence of the countable set C of constants in L We w i l l consider extensions to be e q u i v a l e n t if they differ o n l y in sentences c o n t a i n i n g these constants; in this case, there are j u s t t w o n o n e q u i v a l e n t extensions. T h e e q u a l i t y predicate can be fixed, j u s t as any other predicate, and it is the class of M I N theories M ( W ; P\ — ) t h a t w e consider i n r e l a t i n g extensions t o m i n i m a l m o d els: call these M I N = theories. We now develop the re sult t h a t a first-order sentence is t r u e in the P - m i n i m a l models of W j u s t in case it is t r u e in every extension of
W(W;P-=). 3.2
Parameter models
A first-order i n t e r p r e t a t i o n in w h i c h every i n d i v i d u a l x is denoted by some t e r m is called a parameter i n t e r p r e t a t i o n . H e r b r a n d i n t e r p r e t a t i o n s are one t y p e o f parameter i n t e r p r e t a t i o n , in w h i c h every t e r m denotes itself. Pa rameter i n t e r p r e t a t i o n s are m o r e general t h a n H e r b r a n d i n t e r p r e t a t i o n s , since in the f o r m e r t w o t e r m s can refer to the same i n d i v i d u a l . Just as H e r b r a n d i n t e r p r e t a t i o n s are a sufficient se m a n t i c s for universal prenex sentences, so t o o p a r a m e ter i n t e r p r e t a t i o n s suffice for sets of first-order sentences. By "suffice" we mean t h a t any such set W has a model i f and o n l y i f i t has a p a r a m e t e r m o d e l . N o t e t h a t this s t a t e m e n t is n o t t r u e in general if W can c o n t a i n m e m bers of C: for e x a m p l e t h e set in w h i c h Pci is asserted for every constant has a model but no C-parameter model.
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Knowledge Representation
Let he a sentence of. —C (the with out the constants C), and W a of the same language. Then P-minimal countable models of Wif is true in all P-minimal parameter
F i n a l l y , as the next p r o p o s i t i o n shows, parameter models are the only ones we need consider in f o r m i n g extensions o f M I N = theories. PROPosITION 3.2 Any interpreta11on ( / , T) is a modeI of a MIN= theory M(W; P; —) only if I is a parameter model. Proof. Suppose / isn't a parameter m o d e l . T h e n there is some element e of the universe of 1 such t h a t c is not denoted by any t e r m . T h u s b o t h -Lx = y and ¬ L x # y are t r u e for x=e, and this leads to a c o n t r a d i c t i o n in M(W;P;=). 3.3
The main theorem
For any P - m i n i n i a l parameter model / of W, we call the P-diagram of / the set of g r o u n d literals in P and = t h a t are t r u e in I. We first, show t h a t the d i a g r a m of / picks o u t a u n i q u e extension of M ( W ; P ; = ) , for w h i c h / is a model. 3.3 Let D be the diagram of a Pminimal parameter model I of W. Then (I, D) is a model of some extension of M ( W ; P; =).
PROPOSITION
Proof.
T h i s is a sketch of the proof. Let
We first show t h a t the r e s t r i c t i o n of S to g r o u n d P and e q u a l i t y literals is exactly the set D. N o t e t h a t D is complete w i t h respect to equal i t y l i t e r a l s : for all t e r m s a a n d 6, either a=b or a#b is in D, b u t n o t b o t h . F r o m the f i x i n g o f e q u a l i t y i n M ( W ; P ; = ) , a l l o f these are also contained in 5. D is also complete Note: we w i l l often use a single minimized predicate P in propositions in the rest of this paper; the extensions to multiple predicates is obvious.
4
Reasoning about Equality
In defining P - m i n i m a l interpretations, we have specified that two interpretations must have the same domain and denotation function in order to be comparable. T h i s corresponds to predicate circumscription w i t h a fixed interpretation of terms.
By Proposition 3.3 above, every P - m i n i m a l parameter model of W satisfies the kernel of some extension of M(W;P=). Conversely, by Proposition 3.4 above, the kernel of every extension of M(W;P;=) are the sentences true of some class of P-minirnal parameter models of W. Given the sufficiency of parameter models for m i n i m a l entailment (Proposition 3.1), we have the following theorem. THEOREM
3.5
The first-order sentences S true in every extension ofM(W; P ; = ) are exactly those sentence true in the P-minimal models of W.
This example is typical of the way in which c i r c u m scription handles equality: it does not allow any new conclusions about equality, because the denotations of terms are fixed across comparable models. However, it is also possible to compare models w i t h different denotation functions; the corresponding predicate circumscrip-
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1217
t i o n allows f u n c t i o n s to vary, as well as predicates (see [ L i f s c h i t z , 1984]).
F r o m the above e x a m p l e , it seems t h a t a l l o w i n g terms to vary leads to the danger of unexpected identification of t e r m s , at least if we do n o t have axioms that e x p l i c i t l y say t h a t d i f f e r i n g t e r m s refer to different i n d i v i d u a l s We w o u l d like to t r e a t e q u a l i t y a m o n g terms somewhat in between the t w o extremes of fixed and v a r y i n g denota t i o n s : to r e m a i n agnostic a b o u t the equivalence of terms, b u t s t i l l be able to d r a w basic default conclusions. T h e M I N = theories, because of their relation to /■'m i n i m a l models, always leave the denotations of terms f i x e d , and so f a l l prey to the same problems w i t h equal i t y as c i r c u m s c r i p t i o n w i t h fixed terms. However, we can relax the r e s t r i c t i o n on denotations by using M1N theories, w i t h o u t the sentences f i x i n g equality.
References [Etherington and Reiter, 1984] R. E. Mercer D. W. Etherington and R. Reiter. On the adequacy of predicate circumscription for closed-world reason ing. In AAA! Workshop on Non-Mono!ante Reasoning. American Association for A r t i f i c i a l Intelligence, 1984. de Kleer and Konolige, 1989] Johan de Kleer and K u r t Konolige. E l i m i n a t i n g the fixed predicates from a cir cumscription. Artificial Intelligence, page to appear, 1989. [Etherington, 1986] David W i l l i a m E t h e r i n g t o n . Reasoning with Incomplete Information: Investigations of Non-Monotonia Reasoning. P h D thesis, University of British C o l u m b i a , Vancouver, British C o l u m b i a , 1986. [Imielinski, 1987] T. Imielinski. Results on translating defaults to circumscription. Artificial Intelligence, 32(1):131-146, 1987. [Konolige, 1984] K u r t Konolige. A Deduction Model of Belief and its Logics. P h D thesis, Stanford University, 1984. [Konolige, 1987] K u r t Konolige. On the relation be tween default logic and autoepistemic theories Artificial Intelligence, 35(3):343-382, 1987. [Levesque, 1982] Hector J. Levesque. A formal treat ment of incomplete knowledge bases. Technical Re port 614, Fairchild A r t i f i c i a l Intelligence Laboratory, Palo A l t o , C a l i f o r n i a , 1982. [Levesque, 1987] Hector J. Levesque. A l l 1 know; an abridged report. In Proceedings of the American Association of Artificial Intelligence. Seattle, W a s h i n g t o n , 1987.
M I N theories w i t h o u t e q u a l i t y f i x a t i o n are thus inter mediate between a fixed and v a r y i n g i n t e r p r e t a t i o n of equality, and seem to be the r i g h t level of v a r i a t i o n for commonsense reasoning in a b n o r m a l i t y theories.
5
Conclusion
We have extended the language and semantics of autoepistemic logic in a n a t u r a l way to the case of q u a n t i f i e d - i n variables. By l o o k i n g at a class of A10 the ories, the M I N = theories, we showed t h a t all of the firstorder consequences of predicate c i r c u m s c r i p t i o n could be expressed in a simple way in autoepistemic logic. T h i s is the first result on the r e l a t i o n s h i p of these two logics for the case of n o n f m i t e d o m a i n s . T h e results have been used to shed some l i g h t on the t r e a t m e n t of equality in commonsense reasoning.
[Lifschitz, 1984] V l a d i m i r Lifschitz. Some results on cir cumscription. In AAAl Workshop on Non-Monotonic Reasoning, 1984. [Lifschitz, 1985] V l a d i m i r Lifschitz. C o m p u t i n g circum scription. In Proceedings of the International Joint Conference on Artificial Intelligence, pages 121 127, Los Angeles, California, 1985. [McCarthy, 1986] John M c C a r t h y . A p p l i c a t i o n s of cir cumscription to formalizing commonsense knowledge. Artificial Intelligence, 28, 1986. [Moore, 1985] Robert C. Moore. Semantical considera tions on nonmonotonic logic. Artificial Intelligence, 25(1), 1985. [Perlis, 1986] Donald Perlis. On the consistency of com monsense reasoning. Computational Intelligence, 2, 1986. [Reiter, 1980] R a y m o n d Reiter. A logic for default rea soning. Artificial Intelligence, 13(1-2), 1980. [Shoham, 1987] Yoav Shoh a m . Reasoning about Change: Time and Causation from the Standpoint of Artificial Intelligence. M I T Press, Cambridge, Massachusetss, 1987.
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Knowledge Representation
