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Reasoning with incomplete information : investigations of non-monotonic reasoning Etherington, David William 1986

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Reasoning With Incomplete Information Investigationsof Non-Monotonic  Reasoning  By  DAVID WILLIAM B.Sc,  ETHERINGTON  The University of Lethbridge, 1977  M.Sc,  T h e University British Columbia, 1982  A T H E S I S IS S U B M I T T E D I N P A R T I A L F U L L F I L M E N T O F THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE F A C U L T Y O F G R A D U A T E STUDIES ( D e p a r t m e n t o f C o m p u t e r Science)  We accept this thesis as c o n f o r m i n g to the required s t a n d a r d .  THE UNIVERSITY OF BRITISH C O L U M B I A A p r i l 1986 0  D a v i d W i l l i a m E t h e r i n g t o n , 1986.  In  presenting  requirements  this thesis  f o r an a d v a n c e d  of  British  it  freely available  agree for  that  Columbia,  I agree  that  the L i b r a r y  shall  and s t u d y .  I  f o r extensive  p u r p o s e s may  f u l f i l m e n t of the  degree a t the U n i v e r s i t y  f o r reference  permission  scholarly  in partial  for  that  copying  f i n a n c i a l gain  Department o f  Date  of this  It is thesis  n o t be a l l o w e d w i t h o u t my  permission.  The U n i v e r s i t y o f B r i t i s h 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  thesis  be g r a n t e d by t h e h e a d o f my  or publication  shall  further  copying of t h i s  d e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . understood  make  Columbia  written  J a n u a r y 2 9 , 1987  ERRATA Reasoning with Incomplete Information Investigations of Nonmonotonic Reasoning David W. Etherington The following is a (partial) list of errata.  Page 44:  In the definition of the result of a sequence of defaults, the three occurrences of <8,> should be <8 >. ;  Page 50:  In (2.ii), delete "and 7, € Oi....,&}"  Page 77:  In point 3, x = ux should be .r = u.  Page 96:  The last two occurrence of Qa in Example 8.2 should be -Qa.  Page 116:  In (2.ii), delete "and 7, £ {0i,...,&}" line -2:  Page 131/:  LITERALS  (a) should be LITERALS  (a A 7 ) .  Every occurrence of (J should be [J. r =0  r=1  -Px  • _/>  Page 148:  The three occurrences of  Page 149:  In the proof of Lemma 8.2.2, after Pcxj i CONSEQUENTS (GE(E.\))., insert "(The remaining terms can be put into GD(E, A) in like manner — again, the existence of M and the domain-closure axiom guarantee that this is possible.)"  i  should be  r  Abstract I n t e l l i g e n t b e h a v i o u r relies h e a v i l y o n the a b i l i t y to reason i n the absence of c o m p l e t e information.  U n t i l r e c e n t l y , there has been l i t t l e w o r k done o n d e v e l o p i n g a f o r m a l u n d e r s t a n d i n g of  h o w s u c h r e a s o n i n g c a n be p e r f o r m e d .  W e focus o n t w o aspects of this p r o b l e m : d e f a u l t or p r o t o -  t y p i c a l r e a s o n i n g , a n d c l o s e d - w o r l d or c i r c u m s c r i p t i v e r e a s o n i n g . A f t e r s u r v e y i n g the w o r k i n the field, we c o n c e n t r a t e o n R e i t e r ' s d e f a u l t l o g i c a n d the v a r i ous c i r c u m s c r i p t i v e f o r m a l i s m s d e v e l o p e d b y M c C a r t h y a n d others.  T a k i n g a largely semantic  a p p r o a c h , we d e v e l o p a n d / o r e x t e n d m o d e l - t h e o r e t i c s e m a n t i c s f o r the f o r m a l i s m s i n q u e s t i o n . T h e s e a n d o t h e r tools are t h e n used to c h a r t the c a p a b i l i t i e s , l i m i t a t i o n s , a n d i n t e r r e l a t i o n s h i p s of the v a r i o u s a p p r o a c h e s . It is a r g u e d t h a t the f o r m a l s y s t e m s c o n s i d e r e d , w h i l e i n t e r e s t i n g i n t h e i r o w n r i g h t s , h a v e a n i m p o r t a n t role as s p e c i f i c a t i o n / e v a l u a t i o n tools vis-a-vis e x p l i c i t l y c o m p u t a t i o n a l a p p r o a c h e s . A n a p p l i c a t i o n of these p r i n c i p l e s is g i v e n i n the f o r m a l i z a t i o n of i n h e r i t a n c e n e t w o r k s i n the presence of e x c e p t i o n s , u s i n g d e f a u l t l o g i c .  ii  Table of Contents Abstract  ii  T a b l e of C o n t e n t s  iii  L i s t of F i g u r e s  viii  L i s t of T a b l e s  ix  Acknowledgements 1  Chapter 1 - Incomplete Information  1.1 2  x 1  O v e r v i e w of the thesis  3  C h a p t e r 2 - Approaches to Incomplete Knowledge  2.1  Closed-World Reasoning  5 6  2.1.1  Naive Closure  7  2.1.2  Negation A s Failure T o Derive  9  2.1.3  Database Completion  2.1.4  G e n e r a l i z e d R e a l i z a t i o n s of the C W A  13  2.1.5  Circumscription  15  2.1.5.1  9  Predicate Circumscription  15  2.1.5.2  Formula Circumscription  18  2.1.5.3  Domain Circumscription  20  2.1.6  R e s t r i c t i n g C l o s e d - W o r l d Inferences  22  2.1.7  Semantic Interconnections  23  2.2  D e f a u l t or P r o t o t y p i c a l R e a s o n i n g  2.2.1  24  Default Logic  24  2.2.1.1  Default Theories  2.2.1.2  Closed Default Theories and Their Extensions  26  2.2.1.3  General Default Theories  27  Interacting Defaults  28  2.2.1.4  24  2.2.2  Minimizing Abnormality  30  2.2.3  Non-Monotonic Logic  32  2.2.4  Autoepistemic Logic  34  2.2.5  KFOPC  35  2.2.6  O b j e c t i o n s to N o n - M o n o t o n i c F o r m a l i s m s  3 3.1  Chapter 3 - Default Logic  40 42  T h e S e m a n t i c s of D e f a u l t T h e o r i e s  42  Definition:  Satisfiability, admissibility, and applicability  44  Definition:  R e s u l t of a default  44  Definition:  R e s u l t of a sequence of defaults  44  Definition:  Stability  44  T h e o r e m 3.1 - S o u n d n e s s  45  T h e o r e m 3.2 - C o m p l e t e n e s s  45  E x a m p l e 3.1  45  E x a m p l e 3.2  46  3.2  C o h e r e n c e of D e f a u l t T h e o r i e s  47  3.3  Ordered Default Theories  49  Definition:  50  «:  and  «:  iii  Definition:  Orderedness  51  T h e o r e m 3.3 - C o h e r e n c e 3.4  4  Constructing Extensions  51  Definition:  53  Network Default Theory  T h e o r e m 3.4 - C o n v e r v e n c e  54  T h e o r e m 3.5 - S t r o n g C o n v e r g e n c e  54  C h a p t e r 4 - Inheritance Networks w i t h Exceptions  55  T h e o r e m 4.1  61  C o r o l l a r y 4.2  61  C o r o l l a r y 4.3  61  C o r o l l a r y 4.4  61  4.1 4.2 5  51  P a r a l l e l N e t w o r k Inference A l g o r i t h m s  61  T h e o r y Preference  65  T h e o r e m 4.5  66  Chapter 5 - Predicate Circumscription  5.1  69  Formal Preliminaries  5.2  5.3  5.4 5.4.1  69  O n t h e C o n s i s t e n c y of P r e d i c a t e C i r c u m s c r i p t i o n  70  E x a m p l e 5.1 - A n i n c o n s i s t e n t c i r c u m s c r i p t i o n  70  T h e o r e m 5.1  70  T h e o r e m 5.2  71  C o r o l l a r y 5.3  71  W e i l - F o u n d e d Theories and Predicate Circumscription  71  T h e o r e m 5.4  71  T h e o r e m 5.5  72  Equality  73  The Unique-Names Assumption  73  T h e o r e m 5.6  74  (Reiter)  T h e o r e m 5.7 5.4.2  5.4.3  74  C o r o l l a r y 5.8  74  The D o m a i n Closure Assumption  75  T h e o r e m 5.9  76  T h e o r e m 5.10  76  Some Misconceptions  76  T h e o r e m 5.11  76  C o r o l l a r y 5.12 5.5 6 6.1  6.2  76  W h a t to C i r c u m s c r i b e ?  78  C h a p t e r 6 - Generalizations of C i r c u m s c r i p t i o n  79  Formula Circumscription Definition:  M<  Definition:  £(P,i)-Minimal Model  79  ^p^Af*  80 80  T h e o r e m 6.1 - S o u n d n e s s  80  T h e o r e m 6.2 - F i n i t a r y C o m p l e t e n e s s ( P e r l i s a n d M i n k e r )  80  T h e o r e m 6.3  81  Generalized Circumscription Definition:  M<  81  (x,fl)Af'  :  iv  82  Definition:  (X,.R)-Minimal Model  82  T h e o r e m 6.4 - S o u n d n e s s  •  P r o p o s i t i o n 6.5  6.3  P r o p o s i t i o n 6.6  82  E x a m p l e 6.1  83  W e l l - F o u n d e d Theories  83  Definition - Well-Foundedness  83  P r o p o s i t i o n 6.7  83  E x a m p l e 6.2  (Lifschitz)  (Lifschitz)  84  P r o p o s i t i o n 6.8  7 7.1 7.2  84  E x a m p l e 6.3  84  T h e o r e m 6.9  85  C o r o l l a r y 6.10  85  T h e o r e m 6.11  85  T h e o r e m 6.12  85  T h e o r e m 6.13  86  T h e o r e m 6.14  86  Chapter 7 - Domain Circumscription  87  A Revised Domain Circumscription A x i o m Schema  87  T h e o r e m 7.1 - S o u n d n e s s  88  S o m e P r o p e r t i e s of D o m a i n C i r c u m s c r i p t i o n  88  E x a m p l e 7.1  88  P r o p o s i t i o n 7.2  (Jios-Tarski Theorem)  89  T h e o r e m 7.3  7.3 8  90  T h e o r e m 7.4 - F i n i t a r y C o m p l e t e n e s s  90  C o r o l l a r y 7.5  90  Related Formalisms  91  Chapter 8 - Connections Between Default Logic and Circumscription  92  P r o p o s i t i o n 8.1  92  E x a m p l e 8.1 8.1 8.2  92  " T r a n s l a t i o n " from Default Logic to Circumscription T r a n s l a t i o n s f r o m C i r c u m s c r i p t i o n to D e f a u l t L o g i c  94 94  C o r o l l a r y 8.3  95  C o r o l l a r y 8.4  95  C o r o l l a r y 8.5  95  P r o p o s i t i o n 8.6  95  T h e o r e m 8.7  95  P r o p o s i t i o n 8.8  96 (Gelfond, Przymusinska, and Przymusinski)  P r o p o s i t i o n 8.9 9.1  P r i n c i p l e s of N o n - M o n o t o n i c R e a s o n i n g Update  10.1  96 97  Chapter 9 - Open Problems  9.2 10  93  T h e o r e m 8.2  E x a m p l e 8.2  9  82 82  98 98 100  C h a p t e r 10 - C o n c l u s i o n s  102  Default Logic and Inheritance  102  v  10.2  Predicate Circumscription  10.3  G e n e r a l i z a t i o n s of C i r c u m s c r i p t i o n  104  10.4  Domain Circumscription  105  Relations Between Circumscription and Default Logic  105  10.5 A  103  A p p e n d i x A - P r o o f s of T h e o r e m s  Ill  Background Information  Ill  Definition:  Satisfiability, admissibility, and applicability  112  Definition:  R e s u l t of a d e f a u l t  112  Definition:  R e s u l t o f a sequence of defaults  112  Definition:  Stability  112  T h e o r e m 3.1 - S o u n d n e s s  113  T h e o r e m 3.2 - C o m p l e t e n e s s  113  L e m m a 3.3.1  115  D e f i n i t i o n 3.3.2:  <K  and  <K  116  D e f i n i t i o n 3.3.3:  Orderedness  116  D e f i n i t i o n 3.3.4:  U n i v e r s e of A  116  D e f i n i t i o n 3.3.5:  I : U(A) j-* N  117  D e f i n i t i o n 3.3.6:  JMAX> ' M I N  1^  L e m m a 3.3.7  117  C o r o l l a r y 3.3.8  118  C o r o l l a r y 3.3.9  118  L e m m a 3.3.10  118  T h e o r e m 3.3 - C o h e r e n c e  120  L e m m a 3.4.1  124  L e m m a 3.4.2  124  D e f i n i t i o n 3.4.3:  Network Default Theory  125  L e m m a 3.4.4  125  L e m m a 3.4.5  127  L e m m a 3.4.6  128  L e m m a 3.4.7  129  L e m m a 3.4.8  129  T h e o r e m 3.4 - C o n v e r g e n c e  130  T h e o r e m 3.5 - S t r o n g C o n v e r g e n c e  131  T h e o r e m 4.1  133  T h e o r e m 4.5  133  T h e o r e m 5.2  136  T h e o r e m 5.4  136  T h e o r e m 5.5  136  T h e o r e m 5.6  (Reiter)  137  T h e o r e m 5.7  138  C o r o l l a r y 5.8  139  T h e o r e m 5.9  139  T h e o r e m 5.10  140  T h e o r e m 5.11  •  Definition:  Formula Circumscription  Definition:  M<  140 141  j ^ M *  141  vi  Definition:  £(P,5)-Minimal Model  141  T h e o r e m 6.1 - S o u n d n e s s  141  T h e o r e m 6.3  142  Definition:  Generalized Circumscription  Definition:  M<  Definition:  (X,.R)-Minimal Model  142  (x,/j)Af'  142 143  T h e o r e m 6.4 - S o u n d n e s s  143  Definition:  143  Well-Foundedness  T h e o r e m 6.9  143  T h e o r e m 6.11  144  T h e o r e m 6.12  144  T h e o r e m 6.13  144  T h e o r e m 6.14  144  T h e o r e m 7.1 - S o u n d n e s s  146  T h e o r e m 7.3  146  T h e o r e m 7.4 - F i n i t a r y C o m p l e t e n e s s  146  C o r o l l a r y 7.5  147  T h e o r e m 8.2  148  L e m m a 8.2.1  148  L e m m a 8.2.2  148  P r o p o s i t i o n 8.6  -  149  T h e o r e m 8.7  149  P r o p o s i t i o n 8.9  149  B  A p p e n d i x B - D i c t i o n a r y of S y m b o l s  150  C  A p p e n d i x C - Useful Logical Definitions  151  vii  List of Figures F i g u r e 4.1 — F r a g m e n t of a t a x o n o m y  56  F i g u r e 4.2 —  L i n k s w i t h exceptions  58  F i g u r e 4.3 —  N e t w o r k r e p r e s e n t a t i o n of our k n o w l e d g e a b o u t M o l l u s c s  59  F i g u r e 4.4 —  N E T L - l i k e r e p r e s e n t a t i o n s of o u r k n o w l e d g e a b o u t M o l l u s c s  59  F i g u r e 4.5 —  P r o b l e m s for local inheritance algorithms  62  F i g u r e 4.6 —  A multi-level inheritance graph  63  F i g u r e 4.7 —  A genuinely ambiguous inheritance graph  67  viii  List of Tables T a b l e 4.1 — L e v e l s of literals  64  T a b l e 4.2 — P o s s i b l e o u t c o m e s u s i n g different p r o p a g a t i o n schemes  64  be  Acknowledgements M y s u p e r v i s o r , R a y R e i t e r , s p a r k e d m y interest i n A l , a n d p a r t i c u l a r l y i n f o r m a l approaches thereto, t h e n fostered i t w i t h b o t h h i s insight a n d his resistance to i n a d e q u a t e l y reasoned ideas (as w e l l as w i t h  financial  a n d m o r a l s u p p o r t ) . H e , together w i t h R o b e r t M e r c e r , e n g a g e d m e i n h u n -  dreds of f r u i t f u l ( a n d n o t - s o - f r u i t f u l ) discussions w h i c h h a v e shaped b o t h t h e f o r m a n d c o n t e n t of m y research career. P a r t s o f c h a p t e r s 4, 5, a n d 7 a p p e a r i n [ E t h e r i n g t o n a n d R e i t e r 1983], [ E t h e r i n g t o n , M e r c e r , a n d R e i t e r 1985], a n d [ E t h e r i n g t o n  a n d M e r c e r 1986].  I gratefully  acknowledge Mercer's a n d  R e i t e r ' s s i g n i f i c a n t c o n t r i b u t i o n s t o t h i s thesis. M a n y t h a n k s are d u e t o A n d r e w A d l e r , A l e x B o r g i d a , P a u l G i l m o r e , A k i r a K a n d a , a n d D a v i d T o u r e t z k y f o r t h e i r generosity w i t h t h e i r t i m e , t h e i r ideas, a n d t h e i r e n c o u r a g e m e n t , a n d t o R o n B r a c h m a n for a l l that and more. F i n a n c i a l l y , I h a v e benefitted f r o m p r e d o c t o r a l scholarships f r o m the N a t u r a l Sciences a n d E n g i n e e r i n g R e s e a r c h C o u n c i l o f C a n a d a a n d t h e Isaac W a l t o n K i l l a m t r u s t , as w e l l as a research assistantship s p o n s o r e d b y N S E R C g r a n t A 7 6 4 2 t o R a y m o n d R e i t e r .  I a m also t h a n k f u l f o r  A T & T B e l l L a b o r a t o r i e s ' s u p p o r t , w h i c h I r e c e i v e d as a v i s i t i n g scientist i n M u r r a y H i l l , d u r i n g the f a l l of 1985. L a s t a n d m o s t , t h a n k s t o J a n i n e for s h a r i n g m y victories, c o m m i s e r a t i n g o v e r m y defeats, a n d t a k i n g o n t h e t a s k o f e x p l a i n i n g n o n - m o n o t o n i c i t y t o o u r friends; of tea a n d l o v e .  x  a n d t o those friends f o r lots  CHAPTER 1  Incomplete Information  T h e perverse m a x i m t h a t w h a t e v e r y o u c a n get a w a y w i t h is r i g h t has i t s c o u n t e r p a r t i n t h e c l a i m t h a t w h a t e v e r w o r k s is clear. I m i g h t n o t understand the devices I employ i n making useful computations or predictions a n y more t h a n [one] u n d e r s t a n d s t h e c a r [one] d r i v e s t o b r i n g h o m e the groceries. T h e u t i l i t y of a n o t i o n testifies n o t to i t s c l a r i t y , b u t r a t h e r to t h e p h i l o s o p h i c a l i m p o r t a n c e of clarifying it. — N e l s o n G o o d m a n [1955]. H u m a n c o m m o n - s e n s e r e a s o n i n g appears t o r e l y h e a v i l y u p o n t h e a b i l i t y t o use g e n e r a l rules subject t o e x c e p t i o n s ; w h a t h a s been c a l l e d p r o t o t y p i c o r default i n f o r m a t i o n . V i r t u a l l y none of the decisions one m a k e s e v e r y d a y are m a d e w i t h c o m p l e t e c e r t a i n t y . W i t h l i t t l e effort, a n endless s u p p l y of m o r e o r less p r o b a b l e scenarios c a n be c o n s t r u c t e d w h i c h c o n t r a i n d i c a t e a n y chosen course.  Y e t p e o p l e are n o t p a r a l y z e d b y i n d e c i s i o n ; t h e y c o n t i n u e t o a c t a n d to decide i n spite of  a l l this u n c e r t a i n t y . Science  fiction  fans w i l l recognize " I n s u f f i c i e n t  D a t a " as a f a v o u r i t e c l i c h e : c o m p u t e r s are  f r e q u e n t l y c h a r a c t e r i z e d as p a r a l y z e d b y n o t h a v i n g e n o u g h i n f o r m a t i o n t o a r r i v e a t a l o g i c a l l y s o u n d c o n c l u s i o n . If c o m p u t e r s are r e s t r i c t e d t o s o u n d m o d e s o f r e a s o n i n g b a s e d o n c o m p l e t e i n f o r m a t i o n t h e n A r t i f i c i a l Intelligence is a f u t i l e goal. F o r a v a r i e t y o f reasons, " I n t e l l i g e n c e " ( w h a t e v e r i t m a y be) m u s t i n v o l v e t h e a b i l i t y t o f u n c t i o n w i t h o u t c o m p l e t e i n f o r m a t i o n a b o u t the world. I n t h e first p l a c e , c o m p l e t e i n f o r m a t i o n is h a r d t o come b y , e v e n i n t h e most  contrived  s i t u a t i o n s . C o n s i d e r , f o r e x a m p l e , a s i m p l e " b l o c k s - w o r l d " s i t u a t i o n i n w h i c h there are three b l o c k s of k n o w n d i m e n s i o n s , masses, a n d l o c a t i o n s , a n d a robot m a n i p u l a t o r a r m w i t h a k n o w n l i f t i n g c a p a c i t y , effective r a d i u s , a n d p o s i t i o n .  If a l l of the b l o c k s are o f a size a n d mass w i t h i n  the tolerances o f t h e a r m , c a n the a r m be used t o s t a c k the b l o c k s ? A t first g l a n c e , the answer seems a n o b v i o u s " y e s " . s i t u a t i o n is i n c o m p l e t e .  R e f l e c t i o n s h o w s t h a t this m i g h t be h a s t y .  O u r i n f o r m a t i o n a b o u t the  T h e r e m a y be t h i n g s w e k n o w n o t h i n g a b o u t w h i c h m a y interfere. F o r  e x a m p l e , t h e a r m m a y be b r o k e n . ( T h i s a r g u m e n t m a y n o t c o n v i n c e those w h o s a y , " I f so, the a c t u a l l i f t i n g c a p a c i t y o f t h e a r m (Og) w a s n o t r e a l l y k n o w n ! " . ) G r a n t i n g t h i s , there m a y s t i l l b e a w a l l b e t w e e n t h e a r m a n d t h e b l o c k s — w e d o n o t k n o w . W e c a n improve o u r specification of the problem to avoid such incomplete information b y saying t h a t there is n o t h i n g b e t w e e n t h e a r m a n d t h e b l o c k s ( n o t e v e n a i r ? ) , b u t w e s t i l l c a n n o t uneq u i v o c a l l y a n s w e r t h e q u e s t i o n . A m o n k e y m a y be h o l d i n g b a c k t h e a r m - t h e perverse m i n d c a n  - 1-  -2generate possible reasons for f a i l u r e i n d e f i n i t e l y .  W i t h o u t m o r e i n f o r m a t i o n , these c a n n o t be r u l e d  out. T h e n e x t s t e p is t o a d d the i n f o r m a t i o n t h a t n o t h i n g prevents the a r m f r o m g e t t i n g to a n d l i f t i n g the b l o c k s . N o w we c a n safely decide t h a t the a r m c a n lift the b l o c k s . O f course, if nothing p r e v e n t s the g o a l , we d o not n e e d a n y k n o w l e d g e of b l o c k s a n d a r m s t o a n s w e r the q u e s t i o n : we h a v e g i v e n t o o m u c h a w a y . P u t t i n g a finer p o i n t o n o u r k n o w l e d g e , we m i g h t say " n o t h i n g p r e v e n t s the a r m f r o m funct i o n i n g a c c o r d i n g t o s p e c i f i c a t i o n " . W e w i l l be c h a r i t a b l e , for the m o m e n t , a n d assume t h a t this p r e c l u d e s m o n k e y s . C a n the a r m l i f t the b l o c k s n o w ? W e l l , the b l o c k s m a y be t o o s l i p p e r y , m a y e x p l o d e w h e n t o u c h e d , o r a n y of a n u m b e r of t h i n g s " t o o r i d i c u l o u s to c o n s i d e r " m a y h a p p e n . It seems t h a t , s h o r t of b e i n g e x p l i c i t l y t o l d - or a c t u a l l y t r y i n g - we c a n never k n o w e n o u g h to decide w h e t h e r a n a t t e m p t e d lift w i l l succeed. E v e n i n s i t u a t i o n s where one i n t u i t i v e l y  s h o u l d h a v e c o m p l e t e k n o w l e d g e , incompleteness  m a y result from t h e i m p r a c t i c a l i t y of r e p r e s e n t i n g a l l of the r e l e v a n t i n f o r m a t i o n . F o r e x a m p l e , a n a i r l i n e d a t a b a s e w h i c h records flights a n d the cities w h i c h t h e y connect w o u l d be o v e r w h e l m e d i f it h a d to keep t r a c k of a l l of the p a i r s of cities not c o n n e c t e d b y e a c h flight. If this " n e g a t i v e " information  is n o t e x p l i c i t l y s t o r e d , h o w e v e r , h o w c a n we decide w h e t h e r P W 8 1 9 , w h i c h connects  V a n c o u v e r a n d G u y a m a s , connects T o k y o a n d H o n g K o n g ?  T h e t r a d i t i o n a l a p p r o a c h to  this  p r o b l e m has been to i n v o k e the Closed-World Assumption. If we assume t h a t we h a v e c o m p l e t e k n o w l e d g e a b o u t a l l of the p o s i t i v e facts true of the w o r l d , we c a n infer t h a t a n y t h i n g we d o not k n o w t o be t r u e - s u c h as CONNECTS(PW819,Tokyo,HongKong)  - is false.  If o u r k n o w l e d g e a b o u t a n y aspect of the w o r l d m a y be i n c o m p l e t e , however, t h i s a s s u m p t i o n is o b v i o u s l y suspect. S u p p o s e , for e x a m p l e , t h a t we w a n t to start a flight from V a n c o u v e r to the N e w Y o r k C i t y a r e a , b u t d o n o t y e t k n o w w h e t h e r i t w i l l a c t u a l l y g o t o N e w Y o r k o r N e w a r k . P e r h a p s the d a t a b a s e also stores i n f o r m a t i o n a b o u t  flights'  " h o m e p o r t " for m a i n t e n a n c e  pur-  poses. W e m a y w a n t to enter F i c t i c t i o u s A i r l i n e s 001, w i t h h o m e p o r t V a n c o u v e r , so t h a t the m a i n t e n a n c e d e p a r t m e n t c a n gear u p for the e x t r a aircraft. T h e c l o s e d - w o r l d a s s u m p t i o n w o u l d t h e n a l l o w us t o i n f e r t h a t F A L 0 0 1 connects V a n c o u v e r to n e i t h e r N e w Y o r k n o r N e w a r k  (nor  a n y w h e r e else, for t h a t m a t t e r ) . T o prevent  such unwarranted  inferences, we must  retract  o u r a s s u m p t i o n of  k n o w l e d g e . T h u s w e c a n n o longer use t h e c l o s e d - w o r l d a s s u m p t i o n .  complete  A s a side-effect, o u r uncer-  t a i n t y a b o u t F A L 0 0 1 r e i n t r o d u c e s u n c e r t a i n t y a b o u t w h e t h e r P W 8 1 9 connects T o k y o a n d H o n g K o n g . I n t h i s case, w e m i g h t d e c i d e t o m a n a g e the u n c e r t a i n t y b y e x p l i c i t l y s t a t i n g for w h i c h flights  we h a v e c o m p l e t e k n o w l e d g e . T h e c l o s e d - w o r l d a s s u m p t i o n c a n t h e n be u s e d w h e r e i t is  a p p r o p r i a t e , a n d a v o i d e d elsewhere. The guaranteed.  closed-world  assumption  is  often  made  even  when  its  applicability  cannot  O n e c a n i m a g i n e s i t u a t i o n s - i n d o m a i n s less s t r u c t u r e d t h a n a i r l i n e databases -  be in  w h i c h it m a y not be k n o w n w h e t h e r t h e i n f o r m a t i o n at h a n d is c o m p l e t e . P h y s i c i s t s , f o r e x a m p l e , p e r i o d i c a l l y believe t h a t t h e y h a v e t r a c k e d d o w n the f u l l suite of s u b a t o m i c p a r t i c l e s , a n d w o r k u s i n g this a s s u m p t i o n .  S o far, no-one h a s b e e n able t o s a y h o w we w i l l know w h e n a l l s u c h p a r t i -  cles h a v e b e e n d i s c o v e r e d . I n s u c h s i t u a t i o n s , the best course o f a c t i o n is often t o a c t as t h o u g h  -3one has c o m p l e t e i n f o r m a t i o n u n t i l one has reason to suspect otherwise.  T h e q u e s t i o n of w h e n to  suspect o t h e r w i s e t h e n b e c o m e s q u i t e i m p o r t a n t . T h e p r i n c i p l e s w h i c h guide this t y p e of r e a s o n i n g a p p e a r d i f f i c u l t t o e l u c i d a t e . C e r t a i n l y , k n o w i n g n o t h i n g is reason t o d o u b t t h a t one k n o w s e v e r y t h i n g , b u t w h e r e does one d r a w the l i n e ? C l o s e d - w o r l d r e a s o n i n g t a k e s p o s i t i v e facts as g i v e n , a n d sanctions n e g a t i v e c o n c l u s i o n s . C o m m o n s e n s e r e a s o n i n g often requires a different sort of a s s u m p t i o n t o be m a d e .  B e c a u s e of the  need t o a c t , a n d the pervasiveness o f i n c o m p l e t e i n f o r m a t i o n , h u m a n s are u s u a l l y w i l l i n g assume -  often q u i t e u n c o n s c i o u s l y -  justification.  v a s t n u m b e r s of " n o r m a l c y " c o n d i t i o n s w i t h o u t  to  explicit  In p l a n n i n g to get to the a i r p o r t b y going out the f r o n t d o o r , g e t t i n g i n t o one's c a r ,  a n d d r i v i n g , one assumes t h a t the d o o r w i l l o p e n , t h e c a r w i l l s t a r t , the a i r p o r t h a s n ' t m o v e d , a n d t h a t one's u s u a l r o u t e is s t i l l passable. T h e s e a s s u m p t i o n s r a r e l y r e a c h the c o n s c i o u s l e v e l , unless c i r c u m s t a n c e s m a k e it l i k e l y t h a t t h e y w i l l be v i o l a t e d . F o r e x a m p l e , at - 4 0 ' C , one m i g h t m a k e c o n t i n g e n c y p l a n s for the c a r ' s f a i l i n g to start. S h o u l d subsequent i n f o r m a t i o n o r r e f l e c t i o n v i o l a t e a n y of these " i m p l i c i t " a d j u s t m e n t s are m a d e ; b u t the absence of v i o l a t i o n need not be proven  assumptions,  before a s s u m p t i o n s are  m a d e . T h e k i n d s o f a s s u m p t i o n s w h i c h are m a d e t o d e a l w i t h the v a r i o u s f o r m s of i n c o m p l e t e i n f o r m a t i o n c a n n o t be s o u n d , i n the u s u a l sense of n e v e r l e a d i n g f r o m true premises t o false c o n clusions. T h i s is d i s a p p o i n t i n g t o the p u r i s t . U n f o r t u n a t e l y , if one w a n t s to get a n y t h i n g done, c e r t a i n a s s u m p t i o n s m u s t be m a d e . If we are w i l l i n g t o forsake soundness, h o w d o we a v o i d e m b r a c i n g i r r a t i o n a l i t y ?  T h e best  one c a n hope f o r is some f o r m of " j u s t i f i c a t i o n " for one's a s s u m p t i o n s ; p r i n c i p l e s w h i c h a l l o w gaps i n one's k n o w l e d g e to be filled a n d w h i c h guarantee t h a t — most o f the t i m e - these a s s u m p t i o n s w i l l not l e a d t o o w i l d l y a s t r a y .  D e c i d i n g w h a t c o n s t i t u t e s the " n o r m a l " state-of-affairs a n d w h e n  t o assume t h a t t h i n g s are i n d e e d " n o r m a l " are i m p o r t a n t p r o b l e m s . C l e a r l y , one m u s t be v e r y good at d e t e c t i n g a b n o r m a l c o n d i t i o n s before a s s u m i n g t h a t e v e r y t h i n g is n o r m a l .  Furthermore,  once s u c h a s s u m p t i o n s h a v e been m a d e , one m u s t be p r e p a r e d t o detect a n d d e a l w i t h a n y c o n f l i c t i n g (or a p p a r e n t l y conflicting) i n f o r m a t i o n w h i c h turns u p .  1.1. Overview of the thesis T h e thesis a t t e m p t s to p u l l together a n u m b e r of t h r e a d s — aspects of v a r i o u s approaches to reasoning w i t h i n c o m p l e t e i n f o r m a t i o n .  T h e results p r e s e n t e d f a l l i n t o t w o m a i n categories: those  w h i c h e x t e n d o u r u n d e r s t a n d i n g of the c a p a b i l i t i e s a n d l i m i t a t i o n s o f p a r t i c u l a r a p p r o a c h e s , a n d those w h i c h e x p l o r e t h e i n t e r c o n n e c t i o n s , s i m i l a r i t i e s , a n d differences b e t w e e n a p p r o a c h e s . ( H o p e f u l l y the l a t t e r c a t e g o r y is s u b s u m e d b y t h e former.) C h a p t e r 2 presents a d e t a i l e d s u r v e y of a n u m b e r of i m p o r t a n t systems f o r n o n - m o n o t o n i c reasoning. tions.  W e d r a w together a n u m b e r o f results f r o m the l i t e r a t u r e a n d some o r i g i n a l o b s e r v a -  T h e e m p h a s i s is o n p r e s e n t i n g a cohesive p i c t u r e of the  field.  T h e presentation. thus  a t t e m p t s t o stress t h e c o m m o n a l i t i e s a n d essential differences o f the v a r i o u s a p p r o a c h e s .  The  r e a d e r s h o u l d b e a b l e t o c o m e a w a y w i t h a n u n d e r s t a n d i n g of b o t h t h e p r o b l e m s a n d state of the  -4art of the  field.  C h a p t e r 3 consists of i n v e s t i g a t i o n s of the properties of a p a r t i c u l a r f o r m a l s y s t e m , R e i t e r ' s l o g i c for d e f a u l t r e a s o n i n g .  W e present a g e n e r a l s e m a n t i c s for default theories, a n d show h o w  1  this s e m a n t i c s h i g h l i g h t s the essential s i m i l a r i t i e s a n d d i s s i m i l a r i t i e s between d e f a u l t logic a n d other non-monotonic systems.  W e t h e n c h a r a c t e r i z e a b r o a d class of default theories w h i c h are  w e l l - b e h a v e d , i n the sense of preserving the coherence of the u n d e r l y i n g w o r l d - d e s c r i p t i o n . W e t u r n , i n c h a p t e r 4, t o a n i n v e s t i g a t i o n of i n h e r i t a n c e n e t w o r k s w i t h e x c e p t i o n s . W e develop  a correspondence  between  such networks  a n d default  theories.  We  then  use  this  correspondence to p r o v e a n u m b e r of i n t e r e s t i n g results, i n c l u d i n g sufficient c o n d i t i o n s for the correctness o f a n e t w o r k representation  of a body  inference a l g o r i t h m a n d for the coherence of a n i n h e r i t a n c e of k n o w l e d g e .  We  conclude b y  showing  that Touretzky's  network [1984a]  " i n f e r e n t i a l d i s t a n c e " a l g o r i t h m satisfies these c r i t e r i a . C h a p t e r s 5 t h r o u g h 7 t u r n f r o m default logic to discuss a q u i t e different a p p r o a c h t o i n c o m plete i n f o r m a t i o n , the v a r i o u s forms of m i n i m a l e n t a i l m e n t o r c i r c u m s c r i p t i o n . I n c h a p t e r 5, we discuss a n u m b e r  of s e m a n t i c a l l y - m o t i v a t e d  predicate circumscription. formula circumscription.  p e s s i m i s t i c results c o n c e r n i n g t h e c a p a b i l i t i e s  of  C h a p t e r 6 l o o k s at a g e n e r a l i z a t i o n of p r e d i c a t e c i r c u m s c r i p t i o n , c a l l e d M o d e l - t h e o r i e s are presented for some v a r i a n t s of t h i s a p p r o a c h , a n d a  n u m b e r of results ( b o t h p o s i t i v e a n d negative) are p r o v e d c o n c e r n i n g t h e i r power. T h e " l o n g - d e a d " d o m a i n c i r c u m s c r i p t i o n f o r m a l i s m is " r e s u r r e c t e d " in c h a p t e r 7. W e argue t h a t this a p p r o a c h p r o v i d e s a n i m p o r t a n t c a p a b i l i t y for common-sense a n d d a t a b a s e reasoning systems.  W e u n c o v e r a n d correct a n error i n the o r i g i n a l p r e s e n t a t i o n , a n d we show t h a t a niche  r e m a i n s for d o m a i n c i r c u m s c r i p t i o n b y r e f u t i n g s u b s u m p t i o n c l a i m s m a d e i n f a v o u r of predicate (and formula)  circumscription.  W e c o n c l u d e the c h a p t e r w i t h some results c o n c e r n i n g d o m a i n  circumscription's capabilities and limitations. In c h a p t e r 8, we r e t u r n t o d e f a u l t l o g i c , this t i m e i n the c o n t e x t of o u r d i s c u s s i o n of circumscription.  W e present a n u m b e r o f results d e t a i l i n g the r e l a t i o n s h i p b e t w e e n these r a t h e r  d i s p a r a t e f o r m a l i s m s , s h o w i n g t h e i r p o i n t s of correspondence a n d t h e i r  (unfortunately)  more-  frequent p o i n t s o f d i v e r g e n c e . The  thesis c o n c l u d e s w i t h a l e n g t h y  d i s c u s s i o n of some i m p o r t a n t o p e n p r o b l e m s  and  i n t e r e s t i n g r e s e a r c h d i r e c t i o n s , i n c h a p t e r 9, a n d a s u m m a r y a n d e v a l u a t i o n o f the significance of t h e w o r k p r e s e n t e d , i n c h a p t e r 10. E v e r y a t t e m p t has b e e n m a d e to m a k e t h e thesis as s e l f - c o n t a i n e d as possible. A (at t i m e s , i n t i m a t e ) w i t h for a n i n t r o d u c t i o n . )  first-order  l o g i c is a s s u m e d t h r o u g h o u t , h o w e v e r .  T o preserve c o n t i n u i t y ,  familiarity  (See [ M e n d e l s o n 1964]  the proofs of t h e theorems h a v e been r e l e g a t e d to  A p p e n d i x A , w h i l e A p p e n d i c e s B a n d C c o n t a i n n o t a t i o n a l c o n v e n t i o n s a n d d e f i n i t i o n s of l o g i c a l t e r m s a s s u m e d elsewhere i n the thesis. T h e i n t e n t i o n has been t o k e e p t h e degree of l o g i c a l sophist i c a t i o n r e q u i r e d t o r e a d t h e b u l k of the thesis to a m i n i m u m . Some have objected to the use of the term " l o g i c " (and even "formal") for the systems we discuss here. Bather than debate this issue, we encourage those who find the terminology objectionable to substitute whatever term(s) they find appropriate. 1  CHAPTER 2  Approaches to Incomplete Knowledge  T r a d i t i o n a l logics suffer f r o m the ' M o n o t o n i c i t y P r o b l e m ' — Drew McDermott I n t r a d i t i o n a l l o g i c a l s y s t e m s , e x t e n d i n g a set o f a x i o m s c a n n e v e r p r e v e n t t h e d e r i v a t i o n of c o n c l u s i o n s d e r i v a b l e f r o m t h e o r i g i n a l set. M o r e f o r m a l l y , i f 5 a n d S' are a r b i t r a r y sets of f o r m u lae t h e n : S C S' — The  {w | 5 | - w} C { « ; | S' f - w}.  1  a d d i t i o n o f f o r m u l a e t o a set monotonically  hence s u c h logics are s o m e t i m e s c a l l e d  increases w h a t c a n b e p r o v e d f r o m t h a t set;  monotonic.  R e c e n t l y , i t has b e e n n o t e d [ M c C a r t h y 1977, M i n s k y 1975] t h a t m o n o t o n i c logics seem i n a d e q u a t e to c a p t u r e t h e t e n t a t i v e n a t u r e of h u m a n reasoning. S i n c e people's k n o w l e d g e about the w o r l d is n e c e s s a r i l y i n c o m p l e t e , there w i l l a l w a y s be t i m e s w h e n t h e y w i l l b e f o r c e d t o d r a w c o n c l u s i o n s b a s e d o n a n i n c o m p l e t e s p e c i f i c a t i o n o f p e r t i n e n t details of the s i t u a t i o n . U n d e r such c i r c u m s t a n c e s , assumptions  are m a d e ( i m p l i c i t l y o r e x p l i c i t l y ) a b o u t t h e state o f the u n k n o w n fac-  tors. B e c a u s e these a s s u m p t i o n s are n o t i r r e f u t a b l e , t h e y m a y h a v e t o b e w i t h d r a w n a t some later t i m e s h o u l d n e w e v i d e n c e p r o v e t h e m i n v a l i d . If t h i s h a p p e n s , the n e w e v i d e n c e w i l l p r e v e n t some a s s u m p t i o n s f r o m b e i n g used; hence a l l c o n c l u s i o n s w h i c h c a n be a r r i v e d at o n l y i n c o n j u n c t i o n w i t h those a s s u m p t i o n s w i l l n o longer be d e r i v a b l e . T h i s causes a n y s y s t e m w h i c h a t t e m p t s t o reason c o n s i s t e n t l y u s i n g a s s u m p t i o n s t o e x h i b i t n o n - m o n o t o n i c b e h a v i o u r . C o m m o n - s e n s e c o n c l u s i o n s a r e often based o n b o t h s u p p o r t i n g e v i d e n c e a n d t h e absence of contradictory evidence.  T r a d i t i o n a l logics c a n n o t e m u l a t e this f o r m of r e a s o n i n g because they  l a c k a n y m e a n s f o r c o n s i d e r i n g t h e absence of k n o w l e d g e .  A n u m b e r o f systems h a v e been  d e v e l o p e d t o address t h i s s h o r t c o m i n g , b y a u g m e n t i n g a t r a d i t i o n a l  first-order  logic w i t h some  m e c h a n i s m f o r p r e d i c a t i n g c o n c l u s i o n s o n t h e absence o f specific k n o w l e d g e . I n A l , l o g i c - b a s e d a t t e m p t s t o solve the p r o b l e m s presented b y i n c o m p l e t e i n f o r m a t i o n have f a l l e n i n t o t w o categories. ( F o r t h e purposes o f t h i s thesis, we ignore " p r o b a b i l i s t i c " approaches.) T h e first c a t e g o r y i n c l u d e s those w h i c h assume t h a t a l l o f the r e l e v a n t positive  i n f o r m a t i o n (e.g.,  w h i c h i n d i v i d u a l s e x i s t , w h i c h p r e d i c a t e s a r e satisfied b y w h i c h i n d i v i d u a l s ) is k n o w n . F r o m this 1  S |— w means w is provable from premises S.  -5 -  -6a s s u m p t i o n , it f o l l o w s t h a t a n y t h i n g w h i c h is n o t " k n o w n * to be t r u e m u s t be false. N e g a t i v e f a c t s c a n t h u s be o m i t t e d , since t h e y c a n be inferred f r o m the absence o f t h e i r p o s i t i v e c o u n t e r 2  parts.  Such  a s s u m p t i o n s of  complete  positive  knowledge  underlie  PLANNER'S  "THNOT"  [ H e w i t t 1972] a n d r e l a t e d n e g a t i o n o p e r a t o r s i n A l p r o g r a m m i n g languages, s e m a n t i c n e t w o r k s , a n d d a t a b a s e s [ R e i t e r 1978a, b], as w e l l as m o r e f o r m a l r e a s o n i n g t e c h n i q u e s s u c h as p r e d i c a t e c o m p l e t i o n [ C l a r k 1978], a n d c i r c u m s c r i p t i o n [ M c C a r t h y 1980, 1986]. I n c o n t r a s t , m a n y h a v e w a n t e d to represent a n d use w h a t w o u l d g e n e r a l l y d e s c r i b e d as " d e f a u l t " o r " p r o t o t y p i c " i n f o r m a t i o n . D e f a u l t s are used to fill gaps i n k n o w l e d g e . In the absence of specific evidence, t h e y a l l o w a s y s t e m to m a k e (hopefully) e n l i g h t e n e d " g u e s s e s " , i n s t e a d of reserving judgement  or  assuming that  whatever  is u n k n o w n  is false.  Non-monotonic  logic  [ M c D e r m o t t & D o y l e 1980], default logic [Reiter 1980a], t r u t h m a i n t e n a n c e s y s t e m s [Doyle 1979, M c A l l e s t e r 1978, 1980], a n d v a r i o u s n e t w o r k - a n d f r a m e - b a s e d p r o c e d u r a l k n o w l e d g e r e p r e s e n t a t i o n schemes [ Q u i l l i a n 1968, M i n s k y 1975] a l l e m b o d y this i d e a . T h e t w o a p p r o a c h e s are n o t m u t u a l l y e x c l u s i v e - e a c h o f these r e a s o n i n g t e c h n i q u e s has been u s e d to a c h i e v e t h e o t h e r .  C o m p a r i s o n s of the power of the t w o p a r a d i g m s are m o s t n o t a b l e  for t h e i r absence f r o m the l i t e r a t u r e , h o w e v e r .  T h e d i s c u s s i o n i n the r e m a i n d e r of t h i s c h a p t e r  does n o t p r o v i d e s u c h a c o m p a r i s o n , a l t h o u g h some p o i n t s of correspondence are i n d i c a t e d .  2.1.  Closed-WorldReasoning  N e g a t i v e facts - those w h i c h state w h a t is not true a b o u t the w o r l d - v a s t l y  outnumber  p o s i t i v e facts. F o r e x a m p l e , i n a d i s c u s s i o n at a sufficiently h i g h l e v e l , e v e r y t h i n g w h i c h is at some place is not at every o t h e r p l a c e . S i m i l a r l y , if T u m n u s is a c a t , he is n o t a d o g , a fish, o r a tree (among other things).  T h e a m o u n t of n e g a t i v e i n f o r m a t i o n a b o u t a w o r l d increases g e o m e t r i c a l l y  w i t h the size of the H e r b r a n d U n i v e r s e . O n e w o u l d l i k e t o a v o i d h a v i n g to e x p l i c i t l y represent a l l s u c h i n f o r m a t i o n . T h e i n f o r m a t i o n m u s t s o m e h o w be a v a i l a b l e , h o w e v e r - at some p o i n t it m a y become u s e f u l t o k n o w t h a t T u m n u s is n o t a d o g . In c e r t a i n s i t u a t i o n s , it is reasonable to assume t h a t one k n o w s a l l of the r e l e v a n t  truths.  F o r e x a m p l e , i t is r e a s o n a b l e to assume t h a t a c o m p a n y ' s i n v e n t o r y d a t a b a s e lists a l l p a r t s s u p p l i e d b y t h a t c o m p a n y , t h a t one's T 4 s l i p s list a l l d e d u c t i o n s f r o m one's i n c o m e , a n d t h a t one's e l e c t r i c i t y w i l l n o t be c u t off t o m o r r o w .  S u c h a s s u m p t i o n s are j u s t i f i e d e i t h e r b y the d e s i g n a n d  i n t e n d e d f u n c t i o n of the i n s t r u m e n t i n q u e s t i o n or, as i n the l a t t e r e x a m p l e , b y the i m p l i c i t belief t h a t if a f a c t w e r e i m p o r t a n t e n o u g h - s u c h as the i m p e n d i n g cessation of one's electric services one w o u l d p r e s u m a b l y h a v e h e a r d a b o u t it. If one assumes " t o t a l k n o w l e d g e a b o u t the d o m a i n b e i n g r e p r e s e n t e d " , it is n o l o n g e r necess a r y t o e x p l i c i t l y represent n e g a t i v e i n f o m a t i o n . absence of t h e i r p o s i t i v e c o u n t e r p a r t s . Assumption 3  N e g a t i v e facts m a y s i m p l y be i n f e r r e d f r o m the  R e i t e r [1978a] c a l l s t h i s a s s u m p t i o n the  ( C W A ) , since i t i m p l i e s a c l o s e d d o m a i n i n w h i c h a l l t r u t h s are k n o w n .  A fact is negative iff a l l of the literals in its clausal form are negative.  Closed- World T h e closed-  - 7 -  w o r l d a s s u m p t i o n o n a k n o w l e d g e - b a s e , KB, corresponds r o u g h l y to a n inference r u l e o f the f o r m : If KB  \f- Pthen infer -<P  a p p l i c a b l e t o p o s i t i v e facts, P. T h i s r u l e c a n be p a r a p h r a s e d as " I f P is not p r o v a b l e f r o m the k n o w l e d g e - b a s e , assume ->P."  2.1.1.  Naive Closure  R e i t e r p r o v i d e s t h e f o l l o w i n g s y n t a c t i c r e a l i z a t i o n of the C W A , w h i c h we w i l l c a l l naive closure (NC)?  D e f i n e EKB,  the negative e x t e n s i o n of K B , as follows:  = { ->Pa\P is a n n-ary p r e d i c a t e letter,  EKB  a is a n nrtuple o f g r o u n d t e r m s , a n d KB T h e n the n a i v e closure of KB KB  \—  N  C  \f- Pet. }.  is defined as those f o r m u l a e p r o v a b l e f r o m KB  U EKB.  W e write  .  It is i m p o r t a n t to notice t h a t n a i v e closure extends the knowledge-base b y a d d i n g a set of g r o u n d literals.  U n i v e r s a l s t a t e m e n t s c a p t u r i n g the C W A for p a r t i c u l a r p r e d i c a t e s d o n o t gen-  erally  from  KB  follow  = { Penguin(Opus) EKB  the  naive  closure  of  the  knowledge-base.  4  For  example,  if  }, t h e n  = { ->Penguin(Tweety), ->Penguin(Fred), ... }  but KB  \/r NC V x  x  fr Opus D -'Penguin(x).  T o see this, n o t i c e t h a t we c a n c o n s t r u c t a m o d e l for KB w h i c h does n o t c o r r e s p o n d to a n y n a m e d b y KB  o r EKB  U EKB  w i t h a d o m a i n e l e m e n t , say a,  a n d set Penguin{ct) true.  A s e m a n t i c c h a r a c t e r i z a t i o n of this t y p e of c l o s e d - w o r l d r e a s o n i n g c a n be g i v e n i n t e r m s of m i n i m a l H e r b r a n d m o d e l s , as o u t l i n e d below.  W e i n t r o d u c e the n o t i o n of m i n i m a l m o d e l i n  greater g e n e r a l i t y t h a n is i m m e d i a t e l y r e q u i r e d . T h i s w i l l help s i m p l i f y subsequent d i s c u s s i o n a n d c l a r i f y the r e l a t i o n s h i p s b e t w e e n the v a r i o u s f o r m a l i s m s we w i l l be d i s c u s s i n g . In g e n e r a l , if we are g i v e n a n o r d e r i n g r e l a t i o n ,  <  o n some class of i n t e r p r e t a t i o n s , I, we  s a y t h a t IG I is minimal i n I iff V / ' € I. -<(/' < I) o r (/' = J). F o r o u r present purposes, let P , Q , a n d Z be d i s j o i n t sets of p r e d i c a t e - l e t t e r s w h i c h j o i n t l y e x h a u s t the s u p p l y of predicate-letters of the language. W e c a n define a n o r d e r i n g h < h =  VP e  P . \P\ C l i ^ , a n d h  <  on sets of H e r b r a n d i n t e r p r e t a t i o n s as f o l l o w s :  V<? £  Q . \Q\  h  5  '  6  = IQ^ .  I n o t h e r w o r d s , t h e extensions of p r e d i c a t e s i n Q are i d e n t i c a l , a n d those i n P are (not necessarily 8  A confusion prevalent in the literature conflates 'the C W A ' w i t h what we are calling 'naive closure'.  4  T o simplify the discussion, we assume here that there is at least one ground term.  6  Note that this is not the standard mathematical notion of substructure or submodel.  We will use \I\ to represent the domain of the interpretation, / , and \P\p \t\jto represent the interpretation in / of the predicate, P, and term, t, respectively. 6  -8proper) subsets. O b s e r v e t h a t n o t h i n g is s a i d a b o u t the i n t e r p r e t a t i o n s of t h e p r e d i c a t e s i n Z . R e t u r n i n g t o the s e m a n t i c s of the C W A , it c a n be s h o w n [van E m d e n a n d K o w a l s k i 1976] t h a t the n a i v e closure of KB the f o l l o w i n g sense. KB  corresponds t o m i n i m a l i t y i n the set of H e r b r a n d m o d e l s o f KB,  in  L e t P be the set of a l l p r e d i c a t e s y m b o l s of L (hence Q = Z = { }). T h e n , if  is H o r n a n d c o n s i s t e n t , there is a u n i q u e m i n i m a l element, M, i n the class of H e r b r a n d m o d e l s  of KB  ( i n fact, M=  L, KB  U EKB  D { M | M i s a H e r b r a n d m o d e l of  \- L iff M\=  KB  }).  F u r t h e r m o r e , for g r o u n d clause,  L.  T h e class of H e r b r a n d m o d e l s of a t h e o r y is i n t e r e s t i n g for c o m m o n - s e n s e r e a s o n i n g because each H e r b r a n d m o d e l c o n t a i n s precisely the i n d i v i d u a l s for w h i c h the t h e o r y p r o v i d e s names. I n t u i t i v e l y , t h i s is a t t r a c t i v e for c l o s e d - w o r l d r e a s o n i n g , since one w o u l d i m a g i n e t h a t a closed w o r l d w o u l d c o n t a i n no spurious i n d i v i d u a l s . a n d KB  U EKB  KB  m a y h a v e m o d e l s w i t h i n d i v i d u a l s not c o r r e s p o n d i n g t o a n y n a m e . T h i s a c c o u n t s  for the fact t h a t , w h i l e KB clauses, there  U n f o r t u n a t e l y , as we h a v e seen, i n g e n e r a l b o t h  may  U EKB  agrees w i t h i f B ' s m i n i m a l H e r b r a n d m o d e l , M, for  be facts true i n M  w h i c h d o not f o l l o w f r o m the  (specifically, those w h i c h e n t a i l there being e x a c t l y the n a m e d i n d i v i d u a l s ) . e n t a i l s t h a t there are o n l y  finitely-many  t o i s o m o r p h i s m ) for KB  EKB.  U  ground  n a i v e closure of  KB  If the knowledge-base  i n d i v i d u a l s , it c a n be s h o w n t h a t M i s the o n l y m o d e l (up  D e s p i t e its a t t r a c t i v e n e s s as a m e a n s of i m p l i c i t l y representing n e g a t i v e k n o w l e d g e , closedw o r l d r e a s o n i n g is not w i t h o u t s h o r t c o m i n g s a n d p i t f a l l s .  T h e most o b v i o u s of these is t h a t there  is n o r o o m f o r g e n u i n e l y i n c o m p l e t e knowledge u n d e r the C W A - a n y t h i n g w h i c h is not  known  w i l l be a s s u m e d false. T o see the p r o b l e m s presented b y i n c o m p l e t e i n f o r m a t i o n , c o n s i d e r a d a t a base  consisting  BLOCK(A) ->BLOCK(B).  nor  of  only  BLOCK(A)  BLOCK(B),  naive  V BLOCK(B). closure  allows  Since the  it  is  possible  derivation  of  to  derive  ^BLOCK(A)  neither and  It is easy t o see t h a t s u c h s i t u a t i o n s l e a d t o i n c o n s i s t e n t c o n c l u s i o n s .  T h e fact t h a t some c l a s s i c a l l y consistent databases are n o t consistent w i t h n a i v e closure leads to the q u e s t i o n , " U n d e r w h a t c i r c u m s t a n c e s c a n n a i v e closure be c o n s i s t e n t l y e m p l o y e d ? " T h e r e is n o c o m p l e t e c h a r a c t e r i z a t i o n o f suitable databases, a n d t h e o n l y k n o w n sufficient c o n d i t i o n is t h a t the d a t a b a s e be H o r n a n d consistent.  P u r e l y negative i n f o r m a t i o n  (clauses w i t h o u t  p o s i t i v e literals) p l a y s n o p a r t i n c l o s e d - w o r l d q u e r y e v a l u a t i o n for s u c h databases. Since negat i v e i n f o r m a t i o n c a n be r e c o n s t r u c t e d u s i n g the C W A , i t c a n be i g n o r e d w i t h o u t loss of d e d u c t i v e p o w e r [Reiter 1978a]. A m o r e s u b t l e d r a w b a c k is t h a t t h e " \f- " r e l a t i o n is not effectively c o m p u t a b l e , since firstorder p r o v a b i l i t y is o n l y s e m i - d e c i d a b l e . T h u s , e v e n where n a i v e closure preserves c o n s i s t e n c y , it m a y be i m p o s s i b l e t o e v e n e n u m e r a t e a l l of its consequences. W h i l e  first-order  d e c i d a b l e , a n d i t s t h e o r e m s r e c u r s i v e l y e n u m e r a b l e , n e i t h e r of these h o l d s for n a i v e closure.  l o g i c is s e m i -  first-order  logic  +  -9 2.1.2. Negation As Failure To Derive A l p r o g r a m m i n g languages (e.g., P R O L O G [Roussel 1975], P L A N N E R [ H e w i t t 1972]) h a v e often addressed t h e p r o b l e m of n e g a t i v e k n o w l e d g e b y a d o p t i n g a w e a k e n e d f o r m of t h e C W A . T h e y represent o n l y p o s i t i v e i n f o r m a t i o n , a s s u m i n g t h a t w h a t e v e r c a n n o t be s h o w n t o be true m u s t be false. S u c h systems e m b o d y a w e a k e n e d f o r m of t h e C W A because t h e y d o n o t f u l l y i m p l e m e n t t h e " )f- " r e l a t i o n .  A d e r i v a t i o n o f ->P t y p i c a l l y consists o f a n unsuccessful e x h a u s t i v e  search f o r a d e r i v a t i o n of P. T h i s t e c h n i q u e i s c a l l e d negation as failure ( N A F ) .  Because the  search-space m a y n o t be finite, t h e search f o r a d e r i v a t i o n o f P m a y never f a i l , even w h e n P t r u l y does not f o l l o w f r o m t h e k n o w l e d g e - b a s e . T h u s , N A F m a y n o t be able t o find a l l o f t h e negative facts i m p l i e d b y t h e C W A . In P R O L O G , a n a t t e m p t t o p r o v e t h e l i t e r a l , ->P, consists of (recursively) a t t e m p t i n g t o p r o v e P. If this fails, h a v i n g e x h a u s t e d t h e p o t e n t i a l proofs f o r P, t h e n t h e p r o o f of ->P succeeds. T h i s i s t h e o n l y inference r u l e f o r n e g a t i o n , a n d i t i s a p p l i c a b l e o n l y w h e n P is a p o s i t i v e g r o u n d literaL  C l a r k [1978] justifies this a p p r o a c h t o n e g a t i o n b y s h o w i n g t h a t t h e inference o f ->P f r o m  a d a t a b a s e , D B , b y N A F corresponds t o a p r o o f o f ->P f r o m a n e x t e n d e d d a t a b a s e w h i c h i s i m p l i c i t l y g i v e n b y D B . ( T h i s e x t e n d e d database i s discussed i n d e t a i l i n t h e n e x t section.) C l a r k shows t h a t N A F c a n be v i e w e d as a d e r i v e d inference rule, a heuristic f o r d e r i v i n g n e g a t i v e facts w h i c h are ( u n d e r t h e C W A ) i m p l i c i t i n t h e database. B e c a u s e o f t h e r e q u i r e m e n t of finite f a i l u r e , t h e s y n t a c t i c f o r m of t h e d a t a b a s e , as w e l l as i t s l o g i c a l c o n t e n t , c a n p l a y a role i n w h a t c a n be d e r i v e d b y N A F . F o r e x a m p l e , S h e p e r d s o n [1984] p o i n t s o u t t h a t w h i l e t h e databases: DB  1  =  {Pa}  DB  2  = { --Pa Z> Pa }  and:  are l o g i c a l l y e q u i v a l e n t , P R O L O G c a n p r o v e Pa o n l y f r o m DB DB  2  leads t o a n i n f i n i t e p r o o f tree:  V  A n a t t e m p t t o p r o v e Pa f r o m  T h e s u b g o a l ->Pa i s set u p , l e a d i n g t o a f u r t h e r s u b g o a l of  (failure t o p r o v e ) Pa, ad infinitum. A l t h o u g h t h e a t t e m p t t o p r o v e Pa o b v i o u s l y fails, i t does n o t finitely f a i l , so t h e f a i l u r e p r o o f n e v e r r e t u r n s .  O f course, b o t h databases l o g i c a l l y e n t a i l Pa, a n d  the C W A f u n c t i o n s c o r r e c t l y i n either.  2.1.3. Database Completion T h e C W A a l l o w s a s y s t e m t o a c t o n t h e a s s u m p t i o n t h a t " t h e objects t h a t c a n be s h o w n t o h a v e a c e r t a i n p r o p e r t y b y r e a s o n i n g f r o m c e r t a i n facts are a l l t h e o b j e c t s t h a t h a v e t h a t p r o p e r t y " [ M c C a r t h y 1980]. It does n o t , h o w e v e r , a l l o w t h e reasoner t o d e r i v e this a s s u m p t i o n . S u c h systems c a n n e v e r b e " c o n s c i o u s * o f t h e u n d e r l y i n g p r i n c i p l e s w h i c h t h e y are i m p l i c i t l y a s s u m i n g . C l a r k [1978] r e m e d i e s t h i s s h o r t c o m i n g b y m a k i n g t h e completeness a s s u m p t i o n s e x p l i c i t i n t h e database.  A l l of the information about a particular relation i n the database, D B , is gathered  - 10 -  together a n d a completion axiom is a d d e d w h i c h states t h a t a p a r t i c u l a r t u p l e satisfies t h e r e l a t i o n o n l y i n those cases where D B says it m u s t . A p p l y i n g this process t o a l l of the r e l a t i o n s i n D B yields the  completed database ( C ( D B ) ) . T h i s c o m p l e t i o n of the d a t a b a s e m a k e s e x p l i c i t  the  a s s u m p t i o n s of t o t a l w o r l d k n o w l e d g e . The  d a t a b a s e is v i e w e d as a set o f clauses, e a c h w i t h at m o s t one d i s t i n g u i s h e d p o s i t i v e  l i t e r a l . A c l a u s e is s a i d t o be about the p r e d i c a t e o c c u r r i n g i n i t s d i s t i n g u i s h e d p o s i t i v e l i t e r a l .  All  of the clauses i n D B a b o u t e a c h n-ary p r e d i c a t e , P, are g a t h e r e d together a n d c o n v e r t e d to e q u i v a l e n t i m p l i c a t i o n s w i t h i ' ( z . . . , z ) as t h e i r consequents. T h i s i m p l i c a t i v e f o r m m a k e s c l e a r 1 /  n  a l l of the c o n d i t i o n s w h i c h D B gives as sufficient for P. P r e d i c a t e c o m p l e t i o n asserts t h a t these c o n d i t i o n s are also necessary, thus y i e l d i n g a definition for P. If Ei(x),...,E (x) are t h e left-hand k  sides of a l l of the i m p l i c a t i o n s for P{x\ i n D B , t h e n the completion axiom for P i n D B i s : Vz*. P(t) D [^(5)  V...V  E (tj]. k  If there are n o a x i o m s a b o u t a p r e d i c a t e , the c o m p l e t i o n a x i o m says t h a t t h a t p r e d i c a t e is u n i v e r s a l l y false. T h e completed database, C ( D B ) , is the o r i g i n a l d a t a b a s e , together w i t h the c o m p l e t i o n a x i o m s for e a c h p r e d i c a t e .  F o r e x a m p l e , the t h e o r y :  Bird[ Tweety)  (l)  V s . Penguin(x) D Bird(x)  (2)  V z . Bird(x) A ->Penguin(x) D Flies(x)  (3)  gives rise t o the f o l l o w i n g i m p l i c a t i o n s a b o u t Bird: V z . x = Tweety D Bird(x)  (l')  V z . Penguin(x) Z> Bird(x)  (2*)  (Bird does not o c c u r p o s i t i v e l y i n (3)).  T h u s , the c o m p l e t i o n a x i o m for Bird, g i v e n these a x i o m s  is: V z . Bird(x) D x =  Tweety V Penguin(x) .  (4)  S i m i l a r l y , the c o m p l e t i o n a x i o m for Flies is: V z . Flies(x) Z> Bird(x) A ->Penguin(x) ,  (5)  a n d the c o m p l e t i o n a x i o m f o r Penguin is: V z . -<Penguin(x) . ( W e h a v e a s s u m e d t h a t (3) is a b o u t Flies.) H e n c e , C ( { ( 1 ) ,  (6) (2), (3)})  =  {(l'), (2'),  (3)-(6)},  w h i c h s a y s t h e o n l y b i r d s are Tweety a n d the p e n g u i n s , a n d a l l n o n - p e n g u i n b i r d s fly. F u r t h e r m o r e , there are n o p e n g u i n s , so a l l ( a n d o n l y ) b i r d s fly. B e s i d e s the o r i g i n a l t h e o r y a n d the c o m p l e t i o n a x i o m s , C l a r k a d d s " U n i q u e N a m e s A x i o m s " [Reiter 1978a]. T h e s e are i n e q u a l i t y a x i o m s s t a t i n g t h a t different n a m e s denote different objects. T h u s , for e x a m p l e , if we a d d : Penguin(Opus) t o the D B ( l ) - ( 3 ) , w e get t h e n e w c o m p l e t e d d a t a b a s e :  -11 C(DB') = { ( l ' ) , ( 2 ' ) , ( 3 ) - ( 5 ) , Vx. Penguin(x) = x = Opus, Opus f  Tweety },  w h i c h e n t a i l s Flies(Tweety) a n d ->Flies(Opus). W i t h o u t the u n i q u e n a m e s a x i o m , C(DB') w o u l d e n t a i l n e i t h e r Flies(Tweety) n o r ->Flies(Opus). W h e n r e s t r i c t e d to H o r n databases, w h i c h h a v e at m o s t one p o s i t i v e l i t e r a l , d a t a b a s e c o m p l e t i o n preserves c o n s i s t e n c y .  H o w e v e r , if clauses are a l l o w e d to h a v e more t h a n one p o s i t i v e  l i t e r a l p r o b l e m s m a y result. F o r e x a m p l e , the c l a u s a l f o r m of (3), a b o v e , is: ->Bird(x) V Penguin(x) V Flies(x) . W e a r b i t r a r i l y d e c i d e d t h a t (3) w a s a b o u t Flies (because it i l l u s t r a t e d o u r p o i n t ) , b u t we c o u l d as easily h a v e c h o s e n Penguin. It is easy t o see t h a t o u r choice makes the c o m p l e t i o n of: { ( l ) - ( 3 ) , ->Flies(Tweety) }  DB=  inconsistent.  B e c a u s e (3) is t a k e n t o be a b o u t Flies, it is not t a k e n i n t o a c c o u n t w h e n c a l c u l a t i n g  the c o m p l e t i o n o f Penguin, e v e n t h o u g h it c a n be used t o infer Penguin( Tweety). H e n c e , the c o m p l e t i o n a x i o m s t a t i n g t h a t there are n o p e n g u i n s c a n s t i l l be d e r i v e d , even t h o u g h it is n o w i n c o n sistent w i t h  DB.  D a t a b a s e c o m p l e t i o n c a n s o m e t i m e s be c o n s i s t e n t l y e x t e n d e d t o n o n - H o r n theories b y treating a clause w i t h p o s i t i v e literals, L ...,L , v  k  as k clauses, e a c h about a different  .  This may  a l l o w d a t a b a s e c o m p l e t i o n to be a p p l i e d to databases c o n t a i n i n g i n c o m p l e t e i n f o r m a t i o n  without  i n t r o d u c i n g i n c o n s i s t e n c i e s . F o r e x a m p l e , the d a t a b a s e : BLOCK(A)  V  BLOCK(B),  w h i c h is n o t H o r n a n d is i n c o n s i s t e n t w i t h n a i v e closure, c a n be r e w r i t t e n as: V i . \-^BLOCK\A) A i =  Bo  BLOCK{x)},\  \fx. [-^BLOCK(B) A x = A D BLOCK(x)}  j  T h e s e result i n the consistent c o m p l e t e d d a t a b a s e : {Vx.  [BLOCK(x)  = (-^BLOCK(A) A x = B) V (-iBLOCK(B)  /\x=  A)}, A  B },  or equivalently, {[Vx.  BLOCK(x)  = x = A] V [Vx.  BLOCK(x)  = x = B], Aj= B)  .  N o t i c e t h a t t h e c o m p l e t e d d a t a b a s e states t h a t there is e x a c t l y one b l o c k , a n d it m u s t be e i t h e r A or B.  T h e d i s j u n c t i o n i n the o r i g i n a l d a t a b a s e , w h i c h d i d not e x c l u d e the p o s s i b i l i t y o f  two  b l o c k s , has become " e x c l u s i v e " i n the c o m p l e t e d d a t a b a s e . T h i s a p p r o a c h has t w o d r a w b a c k s . F i r s t , the price p a i d f o r p r e s e r v i n g c o n s i s t e n c y is w e a k ened c o n j e c t u r e s . F o r e x a m p l e , i f a x i o m (3) is t r e a t e d as also being a b o u t p e n g u i n s , t h e c o m p l e t i o n a x i o m for Penguin i n ( l ) - ( 3 ) b e c o m e s : Vx. Penguin(x) 3 Bird(x) A ->Flies(x) , a n d the c o m p l e t e d d a t a b a s e n o longer a l l o w s us to c o n c l u d e t h a t Tweety does not fly. I n fact, i t is a s i m p l e c o r o l l a r y o f results b y R e i t e r [1982] a n d those i n c h a p t e r 5 t h a t p r e d i c a t e c o m p l e t i o n c a n n o t be u s e d t o c o n j e c t u r e p o s i t i v e facts (such as Flies(Tweety)) w i t h o u t risk of i n c o n s i s t e n c y .  - 12 -  A  m o r e serious d r a w b a c k , h o w e v e r , is t h a t this w e a k e n e d f o r m s t i l l does not guarantee c o n -  sistency. S h e p h e r d s o n [1984] shows t h a t the database: P[a)yP(a)  (7)  has a n i n c o n s i s t e n t c o m p l e t i o n , n a m e l y : P{x)  Vs.  =  x =  a A ~>P(a) .  T h i s is e s p e c i a l l y d i s t u r b i n g , since (7) is e q u i v a l e n t to the t r i v i a l d a t a b a s e , P(a).  Perhaps con-  sistency c a n be g u a r a n t e e d b y r e s t r i c t i n g databases t o some n o r m a l f o r m w h i c h p r e c l u d e s (7), b u t e x c l u d i n g a l l p r o b l e m a t i c cases w o u l d p r e s u m a b l y require a s o p h i s t i c a t e d a l g o r i t h m , c a p a b l e of d e t e r m i n i n g w h e n one set of clauses subsumes a n o t h e r . S u c h a n a l g o r i t h m w o u l d lose some of the a d v a n t a g e s o f s i m p l i c i t y a n d directness w h i c h p r e d i c a t e c o m p l e t i o n enjoys.  N o r m a l forms aside,  the precise l i m i t s of t h e consistent a p p l i c a b i l i t y of p r e d i c a t e c o m p l e t i o n are as y e t u n k n o w n . T h i s i l l u s t r a t e s w h a t is s i m u l t a n e o u s l y a s t r e n g t h a n d a weakness o f d a t a b a s e c o m p l e t i o n . T h e m a n i p u l a t i o n s i n v o l v e d i n c o m p l e t i n g the d a t a b a s e are d e t e r m i n i s t i c s y n t a c t i c t r a n s f o r m a tions.  A n y d a t a b a s e c a n t h u s be effectively c o m p l e t e d w i t h r e l a t i v e l y l i t t l e effort.  T h i s same  fact, h o w e v e r , m e a n s t h a t l o g i c a l l y e q u i v a l e n t databases m a y h a v e different c o m p l e t i o n s . T h u s , the s y n t a c t i c f o r m s of f o r m u l a e t a k e o n s e m a n t i c significance, w h i c h is foreign to most l o g i c a l systems.  B e s i d e s s o m e t i m e s l e a d i n g t o i n c o n s i s t e n c y , this seems to argue against C l a r k ' s v i e w t h a t  the c o m p l e t i o n a x i o m s are s o m e h o w i m p l i c i t i n the d a t a b a s e . R e i t e r [1984] e x p l o r e s the effects of a d d i n g c o m p l e t i o n a x i o m s to n o r m a l r e l a t i o n a l d a t a bases. H e d e m o n s t r a t e s a p p l i c a t i o n s of these techniques t o p r o b l e m s i n v o l v i n g some t y p e s of i n c o m p l e t e i n f o r m a t i o n c o m m o n l y e n c o u n t e r e d i n the d a t a b a s e field, s u c h as n u l l v a l u e s a n d disjunctive information. D a t a b a s e c o m p l e t i o n is more p o w e r f u l t h a n a  first-order  system augmented by N A F . Clark  shows t h a t , for P R O L O G p r o g r a m s , the s t r u c t u r e of a f a i l u r e p r o o f is i s o m o r p h i c t o t h a t of a first-order DB  p r o o f f r o m the c o m p l e t e d d a t a b a s e . C o n v e r s e l y , the c o m p l e t i o n of the d a t a b a s e : =  {Penguin(Opus)}  is: C(DB)  =  {Vs.  [Penguin(x)  =  x =  Opus}}  f r o m w h i c h Vs. [s J= Opus D ->Penguin[x)]  (8)  follows b y  s j= Opus, N A F a p p l i e d to D B c a n s h o w -<Penguin(x),  first-order  reasoning.  F o r any particular  b u t the u n i v e r s a l s u m m a r y (8) is b e y o n d  its c a p a b i l i t i e s . ( T h i s f o l l o w s f r o m t h e fact t h a t N A F is w e a k e r t h a n n a i v e closure a n d n a i v e c l o sure c a n n o t d e r i v e the u n i v e r s a l s u m m a r y . ) D a t a b a s e c o m p l e t i o n does n o t a v o i d a l l o f the p r o b l e m s of N A F s i m p l y because a l l of the d e d u c t i o n s are  first-order.  T h e r e w i l l s t i l l be p r o p o s i t i o n s w h i c h are n o t d e c i d e d b y t h e c o m p l e t e d  database - f o r e x a m p l e , p r o p o s i t i o n s c o r r e s p o n d i n g to those for w h i c h the e x h a u s t i v e s e a r c h f o r a failure proof never terminates. DB  =  { Penguin(Opus),  Vs.  C o n s i d e r the d a t a b a s e : Penguin(father(x))  D Penguin(x)  }  - IS -  w h i c h says t h a t the p r o p e r t y o f b e i n g a p e n g u i n is h a n d e d d o w n f r o m f a t h e r t o son. N A F c a n n o t p r o v e ->Penguin(Bruce) because the search for a d e r i v a t i o n o f Penguin(Bruce) w i l l s e a r c h forever for a p e n g u i n a m o n g Bruce'8 p a t e r n a l ancestors. T h e c o m p l e t e d d a t a b a s e , = { V x . Penguin(x) = x = Opus V Penguin(father(x)), Bruce ^= Opus }  C(DB)  also fails t o e n t a i l -iPenguin(Bruce).  B e c a u s e of the c i r c u l a r i t y i n the d e f i n i t i o n for Penguin, it  c a n n o t p r o v e the n o n p e n g u i n i t y of his father.  2.1.4. G e n e r a l i z e d R e a l i z a t i o n s o f the C W A T h e C W A is t h e a s s u m p t i o n o f c o m p l e t e k n o w l e d g e a b o u t w h i c h p o s i t i v e facts are true in the w o r l d . A s we h a v e seen, t h i s a s s u m p t i o n is not a l w a y s a p p r o p r i a t e , a n d c a n l e a d t o i n c o n sistency if m a d e i n s i t u a t i o n s where k n o w l e d g e is genuinely i n c o m p l e t e . T h i s has l e d a n u m b e r of researchers to d e v e l o p m o r e s o p h i s t i c a t e d k n o w l e d g e - c l o s i n g operators w h i c h are able to h a n d l e i n c o m p l e t e n e s s i n c e r t a i n aspects of the K B w i t h o u t c o m p l e t e l y r e t r e a t i n g t o the " O p e n - W o r l d A s s u m p t i o n " t h a t w h a t is k n o w n is p r e c i s e l y w h a t follows f r o m w h a t is e x p l i c i t l y s t a t e d . To  specify  the  generalized closed-world assumption  Minker  (GCWA),  [1982] also uses  m i n i m a l m o d e l s to c h a r a c t e r i z e w h a t f o l l o w s f r o m t h e closure of the d a t a b a s e .  R e s t r i c t i n g his  a t t e n t i o n t o c l a u s a l databases (hence to u n i v e r s a l theories) w i t h a finite set o f t e r m s , M i n k e r c o n siders the set of m i n i m a l H e r b r a n d m o d e l s of the d a t a b a s e . ( F o r n o n - H o r n theories, there m a y not be a u n i q u e m i n i m a l H e r b r a n d model.) T h e G C W A a u g m e n t s the d a t a b a s e w i t h the negations of a l l the literals w h i c h are false i n a l l o f its m i n i m a l H e r b r a n d models. It c a n be s h o w n t h a t the r e s u l t i n g e x t e n d e d d a t a b a s e is c o n sistent iff the o r i g i n a l database is, a n d t h a t n o new p o s i t i v e clauses are d e r i v a b l e f r o m the a u g mented database. To  illustrate  the  idea,  V ->BLOCK(D)}.  BLOCK(C)  consider  the  theory  Mi = {BLOCK(A),  BLOCK(B),  BLOCK(C),  BLOCK(D)}  M  2  = {BLOCK(A),  BLOCK(B),  BLOCK(C),  -^BLOCK(D)}  M  3  = {BLOCK(A),  BLOCK(B),  ^BLOCK(C),  A * 4 = {BLOCK(A), M  s  = {BLOCK(A),  BLOCK(D)}  --BLOCK(B),  -iBLOCK(D)}  {BLOCK(A),  M=  {-^BLOCK(A), BLOCK(B),  M  8  BLOCK(C),  ->BLOCK(B), --BLOCK(C),  = {^BLOCK(A),  BLOCK(B),  Mg = {->BLOCK{A), BLOCK(B), of w h i c h  M$  and  M  9  are  V  BLOCK(B),  ->BLOCK(D)}  -<BLOCK(B), BLOCK(C),  A^= 7  {BLOCK(A)  T h i s database has n i n e H e r b r a n d m o d e l s :  ->BLOCK(D)}  BLOCK(C),  BLOCK(D)}  BLOCK(C),  -^BLOCK(D)}  ^BLOCK(C),  minimal.  ~>BLOCK(D)}  Accordingly,  the  GCWA  s a n c t i o n s -•BLOCK(C)  and  ->BLOCK(D), s i n c e t h e y are b o t h false i n a l l m i n i m a l H e r b r a n d m o d e l s , b u t y i e l d s n o c o n c l u s i o n s a b o u t w h i c h o f A a n d B are b l o c k s . T h u s , where the d a t a b a s e c o u l d c o n s i s t e n t l y be c o n s t r u e d as closed,  the  BLOCK(A)  GCWA V BLOCK(B)),  closes  it,  but  where  n o c o n c l u s i o n is d r a w n .  it  is  known  to  be  incomplete  (i.e.,  - 14 -  B e c a u s e H o r n theories h a v e u n i q u e m i n i m a l H e r b r a n d models, it is easily seen t h a t this d e f i n i t i o n of the G C W A  corresponds to n a i v e closure f o r H o r n theories. T h e G C W A  has the  a d v a n t a g e t h a t it does not o v e r c o m m i t itself t o the p r i n c i p l e t h a t a l l p o s i t i v e i n f o r m a t i o n  is  k n o w n . F a c e d w i t h a s i t u a t i o n where some p o s i t i v e i n f o r m a t i o n is c l e a r l y not k n o w n , j u d g e m e n t is reserved, r a t h e r t h a n b l u n d e r i n g i n t o i n c o n s i s t e n c y . M i n k e r also p r o v i d e s a s y n t a c t i c d e f i n i t i o n of the G C W A , w h i c h he proves c o r r e s p o n d s to the s e m a n t i c c h a r a c t e r i z a t i o n g i v e n a b o v e .  T h e d a t a b a s e , DB, is e x t e n d e d b y a d d i n g EDB,  the  set of n e g a t i o n s of g r o u n d a t o m i c f o r m u l a e o c c u r r i n g i n m i n i m a l p o s i t i v e clauses d e r i v a b l e f r o m Specifically:  DB.  EDB  - { ->Pc | MK.  DB  \/- (PcV  K), where K is a disjunction of  0 or more positive literals such that DB  \f- K }  It is easily seen t h a t , for H o r n theories, this reduces to: EDB=  EDB  — { ->Pc*  \DB\f-Pc}  - the closure set g e n e r a t e d b y n a i v e closure - since for H o r n DB (PcV  K) iff DB  ( - P c * o r DB  \- K.  M i n k e r p r o v e s the G C W A preserves c o n s i s t e n c y - DB introduced no new positive information DB  |— K.  a n d a p o s i t i v e clause, K, DB |—  if K  U EDB  is consistent iff DB  is a p o s i t i v e clause, t h e n DB  U EDB  is - a n d |— i f  iff  T h e s e f a c t s , t o g e t h e r w i t h the fact t h a t the G C W A subsumes n a i v e closure i n d i c a t e  t h a t the G C W A  is a n i n t e r e s t i n g e x t e n s i o n .  O f course, the G C W A is e v e n less t r a c t a b l e  than  n a i v e closure (to the e x t e n t t h a t either c a n be s a i d to be t r a c t a b l e ) , since it i n v o l v e s m u l t i p l e \ftests for e a c h l i t e r a l .  T h i s suggests t h a t n a i v e closure m i g h t be preferred i n those cases ( H o r n  theories) w h e r e it is a p p l i c a b l e .  G e l f o n d a n d P r z y m u s i n s k a [1985] h a v e d e v e l o p e d a n e x t e n s i o n of the G C W A a n d n a i v e closure.  T h e i r " c a r e f u l closure p r o c e d u r e " differs f r o m the G C W A ( a n d n a i v e closure) i n t h a t the  effects of c l o s i n g the w o r l d c a n be c o n s t r a i n e d b y i n d i c a t i n g precisely w h i c h p r e d i c a t e s m a y be affected. T h e p r e d i c a t e s of t h e t h e o r y are d i v i d e d i n t o three sets, P , Q , a n d Z .  P consists of those  aspects of the w o r l d w h i c h are to be closed; Q c o n t a i n s the predicates w h i c h are not to be affected b y the closure; a n d the  predicates i n Z  m a y be affected in a n y w a y  (consistent w i t h  the  knowledge-base) necessary t o a c h i e v e m a x i m u m " c l o s e d - m i n d e d n e s s " a b o u t P . T h i s a r r a n g e m e n t a l l o w s greater  flexibility  i n c l o s i n g the w o r l d .  Firstly, b y requiring that  c e r t a i n p r e d i c a t e s n o t be affected b y the closure (those i n Q ) , one c a n a v o i d i n a d v e r t e n t l y m a k i n g c o n c l u s i o n s a b o u t , for e x a m p l e , t h e p r i c e of tea i n C h i n a w h i l e one's i n t e n t i o n w a s to c o n c l u d e t h a t the a v a i l a b i l i t y of tea a t the l o c a l s u p e r m a r k e t has not c h a n g e d .  S e c o n d l y , a l l o w i n g the  p r e d i c a t e s i n Z t o v a r y w e a k e n s the G C W A / n a i v e closure r e s t r i c t i o n t h a t n o n e w p o s i t i v e facts be d e r i v a b l e f r o m the closure of t h e d a t a b a s e . T h i s means t h a t i f one is confident t h a t one has a l l the p o s i t i v e i n f o r m a t i o n a b o u t P , b u t k n o w s o n l y c e r t a i n c o n s t r a i n t s o n the r e l a t i o n s h i p between P a n d Z , t h e n Z c a n v a r y as necessary t o e s t a b l i s h t h e m i n i m a l extensions for P .  - 15 -  T h e " c a r e f u l c l o s u r e " o f DB w i t h respect t o (P, Q , Z ) i s defined as DB* — DB U EDB, where EDB=  { - . P c * | V{L ...,L } C (P+ U Q + U Q " ) . DB \f- L V...V L h  n  x  or3k<n.  m  DB\-L  X  P?£ {L,}, V...V L }  7  k  I n t u i t i v e l y , one c a n assume ->Pc' unless this w o u l d a l l o w the d e r i v a t i o n o f n e w facts a b o u t Q a n d / o r p o s i t i v e P. T h e s e m a n t i c d e f i n i t i o n o f c a r e f u l closure a g a i n i n v o l v e s a v a r i a n t o f the n o t i o n o f m i n i m a l H e r b r a n d m o d e l o u t l i n e d earlier, this t i m e i n i t s f u l l generality ( P , Q , a n d Z m a y a l l be n o n empty).  G e l f o n d a n d P r z y m u s i n s k a show  that,  f o r a u n i v e r s a l k n o w l e d g e - b a s e , KB, every  m i n i m a l H e r b r a n d m o d e l o f KB satisfies KB , a n d t h a t KB  is consistent iff KB is.  It is easy t o see t h a t i f Q = Z = { } t h e n the a b o v e s e m a n t i c c h a r a c t e r i z a t i o n is the same as t h a t f o r the G C W A . n a i v e closure.  F u r t h e r m o r e , i f the knowledge-base is also H o r n , the same is true f o r  S i n c e G e l f o n d a n d P r z y m u s i n s k a d o not require t h a t the knowledge-base be func-  t i o n free n o r h a v e a finite set o f c o n s t a n t s , this o b s e r v a t i o n shows t h a t these r e s t r i c t i o n s g i v e n i n the d e v e l o p m e n t o f the G C W A were unnecessary.  2.1.5.  Circumscription  M c C a r t h y [1977, 1980, 1986] has presented a n u m b e r of rules o f conjecture for c l o s e d - w o r l d reasoning. T h e s e rules are based on s y n t a c t i c m a n i p u l a t i o n s , rather t h a n consistency. Instead of the u n d e c i d a b i l i t y o f appeals t o n o n - p r o v a b i l i t y o n w h i c h some approaches t o n o n - m o n o t o n i c r e a s o n i n g are b a s e d , these " c i r c u m s c r i p t i v e " f o r m a l i s m s s i m p l y a d d new a x i o m s (conjectures). T h e s e conjectures force m i n i m a l , " c l o s e d - w o r l d " , interpretations o n p a r t i c u l a r aspects of the u n d e r l y i n g incomplete theory.  2.1.5.1. P r e d i c a t e C i r c u m s c r i p t i o n  The  most  [ M c C a r t h y 1980].  widely  studied  of these  rules  of conjecture  is " p r e d i c a t e  circumscription"  P r e d i c a t e c i r c u m s c r i p t i o n a l l o w s e x p l i c i t completeness a s s u m p t i o n s , s i m i l a r t o  C l a r k ' s c o m p l e t i o n a x i o m s , t o be c o n j e c t u r e d as they are r e q u i r e d .  T h i s p r o v i d e s a means f o r  c l o s i n g off the w o r l d w i t h respect t o a p a r t i c u l a r predicate at a p a r t i c u l a r t i m e . set o f first-order sentences is generated.  A s c h e m a for a  T h i s s c h e m a is then i n s t a n t i a t e d b y s u b s t i t u t i n g suitable  predicates for the predicate v a r i a b l e s i t c o n t a i n s . T h e p a r t i c u l a r substitution(s)  chosen d e t e r m i n e  w h i c h i n d i v i d u a l s are c o n j e c t u r e d t o c o m p r i s e the entire e x t e n s i o n o f the c i r c u m s c r i b e d p r e d i c a t e . T h e s e m a n t i c i n t u i t i o n u n d e r l y i n g predicate c i r c u m s c r i p t i o n is the n o w - f a m i l i a r n o t i o n t h a t c l o s e d - w o r l d reasoning a b o u t one o r m o r e predicates of a theory corresponds t o t r u t h i n a l l models If R is a get of predicates, we use R ground literals over prediatea in R . 7  +  and R~, respectively, to indicate the positive and negative  - 16 -  of the t h e o r y w h i c h are m i n i m a l in those predicates. S p e c i f i c a l l y , let T ( P , . „ , P J be a  first-order  theory, some (but not necessarily all) of whose predicates are those i n P = { P . . . , P } .  A model  1  1 )  M o f T is a P-submodel of a m o d e l M ' of T ( w r i t t e n M <  n  p M ' ) iff the e x t e n s i o n of e a c h P,- i n M i s  a subset of its e x t e n s i o n i n M ' , a n d M a n d M ' are otherwise i d e n t i c a l . M i s a P-minimal T iff e v e r y P - s u b m o d e l of M i s i d e n t i c a l to For  finitely  model of  M  a x i o m a t i z a b l e theories, ! T ( P , . . . , P ) , M c C a r t h y [1980] proposes r e a l i z i n g p r e d i 1  n  cate c i r c u m s c r i p t i o n s y n t a c t i c a l l y b y a d d i n g the f o l l o w i n g a x i o m s c h e m a to T:  21(*i,...,*J A A [Vx*. ( S ^ D Pg)\ Here  are p r e d i c a t e v a r i a b l e s , w i t h the same arities as P ...,P r e s p e c t i v e l y . T($ ...,$„) lt  m  lr  is the sentence o b t a i n e d b y c o n j o i n i n g the sentences of T, t h e n r e p l a c i n g e v e r y occurrence of Pi,...,P i n T b y $ ! , . . . , $ „ , r e s p e c t i v e l y . n  T h e above s c h e m a is c a l l e d the (joint)  schema of P ,...,P in T. L e t CLOSUREp(T) 1  n  circumscription  - the closure of T with respect to P = { P . . . , P J D  denote the t h e o r y c o n s i s t i n g of T together w i t h the above a x i o m s c h e m a .  M c C a r t h y formally  identifies reasoning a b o u t T u n d e r the c l o s e d - w o r l d a s s u m p t i o n w i t h respect to the predicates P with  first-order  d e d u c t i o n s f r o m the theory  CLOSURE?(T).  M c C a r t h y [1980] shows t h a t a n y i n s t a n c e of the s c h e m a r e s u l t i n g f r o m c i r c u m s c r i b i n g a s i n gle p r e d i c a t e P i n a sentence T(P) is true i n a l l { P } - m i n i m a l m o d e l s of T. T h i s generalizes d i r e c t l y to the j o i n t c i r c u m s c r i p t i o n o f m u l t i p l e predicates.  A n a r g u m e n t due to D a v i s [1980] c a n  be used t o s h o w t h a t no g e n e r a l " c o m p l e t e n e s s " result can be o b t a i n e d i d e n t i f y i n g  the  c u m s c r i p t i v e t h e o r e m s " w i t h precisely those f o r m u l a e true in a l l m i n i m a l m o d e l s of the M i n k e r a n d P e r l i s [1983, 1984a] prove a " f i n i t a r y "  "cir-  theory.  completeness result, however. S p e c i f i c a l l y , if  the o r i g i n a l t h e o r y (or the c i r c u m s c r i b e d version) e n t a i l s that the m i n i m i z e d predicates h a v e  finite  extensions, the m i n i m a l m o d e l s of t h e o r i g i n a l t h e o r y are a l l ( a n d o n l y ) the models of the circumscribed theory. M c C a r t h y considers the b l o c k s - w o r l d e x a m p l e , discussed p r e v i o u s l y , i n w h i c h a l l t h a t is k n o w n is: BLOCK(A)  V BLOCK(B)  (9)  8  If the p r e d i c a t e v a r i a b l e , 0 , i n the c i r c u m s c r i p t i o n of (9): [ 0 ( A ) V © ( B ) ] A Vx. \e(x) D BLOCK(x)\  D Vx. \BLOCK{x) D  O(x)]  is r e p l a c e d s u c c e s s i v e l y b y the predicates x = A a n d x = B, the conjecture: Vx. [BLOCK(x)  D x = A] V Vx. \BLOCK{x) D x = B\  (10)  c a n be d e r i v e d . A s d i d the c o m p l e t e d database, (10) says t h a t there is o n l y one b l o c k : A o r B . A g a i n , t h e c o n j e c t u r e closes the w o r l d a n d puts the " e x c l u s i v e " i n t e r p r e t a t i o n o n the o r i g i n a l d i s junction.  Recall that this theory is N O T consistent with its naive closure.  -  - 17 T h e choice of s u b s t i t u e n d s is c r u c i a l in d e t e r m i n i n g w h a t c a n be o b t a i n e d b y c i r c u m s c r i p t i o n . It is not c l e a r , i n general, h o w these substituends are to be chosen. M c C a r t h y suggests t h a t the desired g o a l d i r e c t s the choice o f a p p r o p r i a t e s u b s t i t u t i o n s . It r e m a i n s to be seen w h e t h e r this can be t r a n s l a t e d i n t o g e n e r a l rules. T h e r e l a t i o n s h i p s between p r e d i c a t e c i r c u m s c r i p t i o n a n d the v a r i o u s forms of c l o s e d - w o r l d reasoning are o n l y p a r t i a l l y u n d e r s t o o d .  R e i t e r [1982] shows t h a t p r e d i c a t e c i r c u m s c r i p t i o n c a n  s o m e t i m e s be used t o d e r i v e t h e database c o m p l e t i o n a x i o m s . M c C a r t h y c i r c u m s c r i p t i v e l y derives the i n d u c t i o n a x i o m for a r i t h m e t i c , w h i c h shows t h a t predicate c i r c u m s c r i p t i o n is m o r e p o w e r f u l than database completion. D o y l e [1984] has o b s e r v e d t h a t c i r c u m s c r i p t i o n is r e l a t e d to the i d e a of i m p l i c i t as it o c c u r s i n M a t h e m a t i c a l L o g i c .  definability  A set of a x i o m s , A, implicitly defines a p r e d i c a t e , P, if A  forces a " u n i q u e " i n t e r p r e t a t i o n for P, or, more f o r m a l l y , if A{$) Z>  [VF. PF = *2]  is v a l i d for e a c h e x p r e s s i o n ,  of the same a r i t y as P. It is easy to see t h a t t h i s s c h e m a i m p l i e s  the c i r c u m s c r i p t i o n s c h e m a . B e t h ' s D e f i n a b i l i t y T h e o r e m [ B e t h 1953] guarantees t h a t if A i m p l i c i t l y defines P t h e n A explicitly defines P. T h a t is, A  |f- VF.  Px = $x  where <j> is some e x p r e s s i o n u s i n g o n l y s y m b o l s of A (exclusive of P). T h i s result is m u c h - s t u d i e d i n logic, a n d the k n o w n consequences i n c l u d e m e t h o d s for  finding  a n a p p r o p r i a t e <f>.  In those  cases where the c i r c u m s c r i p t i o n s c h e m a a c t u a l l y i m p l i c i t l y defines P, these techniques c a n be used to r e d u c e the s c h e m a to a n e x p l i c i t d e f i n i t i o n a x i o m . C i r c u m s c r i p t i o n does not a l w a y s result in a n i m p l i c i t definition for P. I n general, it is not even d e c i d a b l e w h e t h e r P is i m p l i c i t l y defined. for e x a m p l e , a l l t h a t is o b t a i n a b l e is a disjunctive [Vz.  Block(x) = x = A] V [Vz.  T h e r e are t e c h n i q u e s for  finding  In the Block(A) V Block(B) e x a m p l e c i t e d a b o v e , definition,  Block(x) = x = B] . d i s j u n c t i v e definitions w i t h k disjuncts, where s u c h definitions  exist, b u t it is u n d e c i d a b l e i n general w h e t h e r a d i s j u n c t i v e d e f i n i t i o n (or a d i s j u n c t i v e  definition  of size k) exists. D o y l e suggests t h a t there m a y be p r o f i t i n searching the M a t h e m a t i c a l L o g i c l i t e r a t u r e ( a n d e n q u i r i n g of m a t h e m a t i c a l logicians) for results w h i c h m a y shed l i g h t o n s u c h questions as: 1) W h e n does c i r c u m s c r i p t i o n i m p l i c i t l y define PI D i s j u n c t i v e l y ? i n t e r e s t i n g cases w h i c h c a n be c h a r a c t e r i z e d ? R e c o g n i z e d ?  W h e n does it f a i l ? A r e there  2) W h a t does c i r c u m s c r i p t i o n d o w h e n it fails to define PI 3) W h e n are n e w a x i o m s i r r e l e v a n t to p r i o r c i r c u m s c r i p t i o n s ? T h a t is, w h e n is the a d d i t i o n of n e w i n f o r m a t i o n g u a r a n t e e d not t o i n v a l i d a t e c i r c u m s c r i p t i v e l y d e r i v e d e x p l i c i t definitions? 4) H o w c a n the r e v i s i o n of c i r c u m s c r i p t i v e conclusions i n the face of new i n f o r m a t i o n be m e c h a n ized?  - 18 -  W e h a v e d i s c o v e r e d a n u m b e r of s u r p r i s i n g l i m i t a t i o n s on the a p p l i c a b i l i t y a n d efficacy of predicate circumscription.  T h e s e are d e t a i l e d i n c h a p t e r 5.  2.1.5.2. Formula Circumscription M a n y of the l i m i t a t i o n s  of p r e d i c a t e c i r c u m s c r i p t i o n s t e m f r o m the f a c t t h a t o n l y those  p r e d i c a t e s b e i n g m i n i m i z e d are a l l o w e d to v a r y . M c C a r t h y [1986] has d e v e l o p e d a g e n e r a l i z e d f o r m of c i r c u m s c r i p t i o n w h i c h addresses t h i s p r o b l e m .  T h i s new formalism, formula circumscrip-  t i o n , r e t a i n s m a n y of the a t t r a c t i v e features of its predecessor, w i t h o u t some of its  limitations.  T h e f o r m u l a circumscription axiom looks like: T{$)  V$. where  A [V?.  E(P,x)  is  E($,x) D E[P,x)] Z> [Vz*.  any  well-formed  expression  E(?,X) D whose  E@,X)} free  individual  variables  are  among  ~x = x ,...,x , a n d i n w h i c h some of the p r e d i c a t e v a r i a b l e s P = P ,...,P o c c u r free; E(3>,x) is the x  k  1  n  result of r e p l a c i n g e a c h free o c c u r r e n c e of the p r e d i c a t e v a r i a b l e s , F , - , i n E(P,x) w i t h p r e d i c a t e variables,  , o f the s a m e a r i t y .  T h e r e are three m a i n differences b e t w e e n the p r e d i c a t e c i r c u m s c r i p t i o n s c h e m a a n d the form u l a c i r c u m s c r i p t i o n a x i o m . F i r s t , the f o r m e r is a first-order a x i o m s c h e m a , w h i l e the l a t t e r is a second-order a x i o m .  M c C a r t h y suggests t h a t t h i s is a d v a n t a g e o u s because it a l l o w s the results of  one c i r c u m s c r i p t i o n to p a r t i c i p a t e i n subsequent c i r c u m s c r i p t i o n s . H o w e v e r , this feature is not essential; the s e c o n d - o r d e r a x i o m c a n be r e p l a c e d w i t h a  first-order  first-order-schema  applications  variant  appears a d e q u a t e for  many  s c h e m a . A l t h o u g h w e a k e r , the [Perlis a n d M i n k e r  I n v e s t i g a t i o n s i n t o the r e l a t i v e a d v a n t a g e s a n d d i s a d v a n t a g e s of second-order a x i o m s vs  1986].  first-order  s c h e m a s for c i r c u m s c r i p t i o n are s t i l l c o n t i n u i n g , a n d the q u e s t i o n of the v a l u e of a d o p t i n g  a  s e c o n d - o r d e r logic r e m a i n s u n d e c i d e d . T h e s e c o n d n e w feature of f o r m u l a c i r c u m s c r i p t i o n is t h a t a r b i t r a r y p r e d i c a t e expressions, r a t h e r t h a n s i m p l e p r e d i c a t e s , m a y be m i n i m i z e d . M c C a r t h y [1984, p e r s o n a l c o m m u n i c a t i o n ]  sug-  gests t h a t t h i s is a n i n e s s e n t i a l c h a n g e , since the same effect c o u l d be i n d i r e c t l y o b t a i n e d b y i n t r o d u c i n g n e w p r e d i c a t e s , w i t h a x i o m s defining these predicates as e q u i v a l e n t to the r e q u i r e d expression. W h i l e t h i s is true for f o r m u l a c i r c u m s c r i p t i o n , we show i n c h a p t e r 5 t h a t p r e d i c a t e  cir-  c u m s c r i p t i o n c a n n o t d e a l w i t h s u c h d e f i n i t i o n s . W e also discuss a d d i t i o n a l m e c h a n i s m s w h i c h are s o m e t i m e s u s e d to a u g m e n t predicate c i r c u m s c r i p t i o n w h i c h a l l o w d e f i n i t i o n s t o be c i r c u m s c r i b e d . T h e s e m e c h a n i s m s d o n o t a l w a y s preserve c o n s i s t e n c y , however. T h e t h i r d , a n d m o s t s i g n i f i c a n t , i n n o v a t i o n is t h a t the p r e d i c a t e s a l l o w e d to v a r y are no l o n g e r i d e n t i f i e d w i t h those b e i n g m i n i m i z e d . T h i s is reflected i n the fact t h a t P [ a l t e r n a t e l y , m a y c o n t a i n p r e d i c a t e v a r i a b l e s n o t o c c u r r i n g i n E(P,x) [respectively, i ? ( $ , z ) ]  $]  ( a n d vice versa).  T h i s s e p a r a t i o n a l l o w s c i r c u m s c r i p t i o n to operate i n r i c h l y c o n n e c t e d w o r l d s . P r o v i d e d p r e d i c a t e s which  would  be a l t e r e d b y the m i n i m i z a t i o n of t h e expression i n q u e s t i o n are a m o n g  those  i d e n t i f i e d as " v a r i a b l e " , c i r c u m s c r i p t i o n c a n have the desired effect. C h a p t e r 6 describes a m o d e l t h e o r y we h a v e d e v e l o p e d for f o r m u l a c i r c u m s c r i p t i o n , a l o n g the  lines  of  McCarthy's  [1980]  semantics  for  predicate  circumscription.  For  formula  - 19 -  c i r c u m s c r i p t i o n , the a p p r o p r i a t e n o t i o n of s u b m o d e l is one i n w h i c h the extensions of the v a r i a b l e predicates are a l l o w e d to e x p a n d or c o n t r a c t , p r o v i d e d t h a t the e x t e n s i o n o f E(P,x)  contracts.  T h e extensions of the p r e d i c a t e p a r a m e t e r s (those predicates w h i c h are not a m o n g t h e predicates designated as v a r i a b l e ) m u s t be i d e n t i c a l i n a m o d e l a n d its s u b m o d e l s . A m o d e l is m i n i m a l if it has no p r o p e r s u b m o d e l s . It is s h o w n t h a t f o r m u l a c i r c u m s c r i p t i o n is s o u n d w i t h respect to this m o d e l t h e o r y ; a n y t h i n g d e r i v a b l e f r o m the c i r c u m s c r i b e d t h e o r y is true i n a l l m i n i m a l models of the o r i g i n a l t h e o r y . P e r l i s a n d M i n k e r 1986] c o n s i d e r the completeness of the f i r s t - o r d e r - s c h e m a v a r i a n t of form u l a c i r c u m s c r i p t i o n w i t h respect to t h i s m o d e l t h e o r y . finitary  T h e y present results analogous to their  completeness results for predicate c i r c u m s c r i p t i o n [ M i n k e r a n d P e r l i s 1983, 1984a].  results p a r t i a l l y  These  a n s w e r some of D o y l e ' s [1984] questions a b o u t the r e l a t i o n s h i p b e t w e e n  cir-  c u m s c r i p t i o n a n d e x p l i c i t / d i s j u n c t i v e d e f i n a b i l i t y , a t least i n a s m u c h as t h e y establish e x p l i c i t a n d disjunctive definability  as sufficient c o n d i t i o n s for the completeness o f f o r m u l a  circumscription.  T h e s e results h a v e yet to be e x t e n d e d t o the case of second-order f o r m u l a c i r c u m s c r i p t i o n . L i f s c h i t z [1984] has s t u d i e d second-order f o r m u l a c i r c u m s c r i p t i o n a n d d e r i v e d c e r t a i n c o n d i tions u n d e r w h i c h the second-order c i r c u m s c r i p t i o n a x i o m c a n be r e d u c e d to a n e q u i v a l e n t  first-  o r d e r a x i o m . S u c h e q u i v a l e n c e s i m p r o v e the usefulness of f o r m u l a c i r c u m s c r i p t i o n , i n some cases, b y e l i m i n a t i n g b o t h the need for a second-order logic a n d the p r o b l e m of finding the " r i g h t " s u b stitutions. L i f s c h i t z defines a f o r m u l a t o be separable in P iff it c a n be w r i t t e n i n the f o r m : V  A [Vz.  Ei{x) z> P f S ) ] A [V?.  P(i)  where C,-, E , a n d F,- are P-free f o r m u l a e . t  D  Fffl)  E s s e n t i a l l y , a formula is separable if it is not recursive  i n P. L i f s c h i t z p r o v e s t h a t the second-order f o r m u l a r e s u l t i n g f r o m c i r c u m s c r i b i n g P i n a separable f o r m u l a , A, a l l o w i n g o n l y P to v a r y is e q u i v a l e n t to a  first-order  formula w i t h about  the  same l o g i c a l c o m p l e x i t y as A. In itself, this result is n o t v e r y e x c i t i n g , since second-order c i r c u m s c r i p t i o n of P w i t h o n l y P v a r i a b l e is subject to the same l i m i t a t i o n s c h a p t e r 5 outlines for p r e d i c a t e c i r c u m s c r i p t i o n .  Lifs-  c h i t z also shows, h o w e v e r , t h a t the c i r c u m s c r i p t i o n of P i n A w i t h P a n d Y v a r i a b l e is e q u i v a l e n t to the c i r c u m s c r i p t i o n of P i n [3 Y. A] w i t h o n l y P v a r i a b l e . t h e n [3 Y. A] is e q u i v a l e n t to a  first-order  F u r t h e r m o r e , if A is separable i n Y,  f o r m u l a w i t h c o m p l e x i t y l o w e r t h a n A.  W h i l e these  t r a n s f o r m a t i o n s d o not a l w a y s preserve s e p a r a b i l i t y [Reiter, p e r s o n a l c o m m u n i c a t i o n ] , it appears t h a t these t e c h n i q u e s m a y be useful for e l i m i n a t i n g the second-order quantifiers i n t r o d u c e d b y form u l a c i r c u m s c r i p t i o n - w i t h o u t r e - i n t r o d u c i n g the a w k w a r d n e s s of a x i o m s c h e m a t a a n d " r i g h t " substitutions. Another  innovation  due  to  Lifschitz  is to  minimize  according to  (reflexive, t r a n s i t i v e b i n a r y r e l a t i o n s ) , r a t h e r t h a n s i m p l e subset r e l a t i o n s .  arbitrary  pre-orders  S p e c i f i c a l l y , if X is a n  n-tuple of p r e d i c a t e , f u n c t i o n , a n d / o r c o n s t a n t letters of T, a n d X ' is a n n-tuple of p r e d i c a t e , f u n c t i o n , a n d / o r i n d i v i d u a l v a r i a b l e s of c o r r e s p o n d i n g types a n d arities, t h e n the g e n e r a l i z e d circ u m s c r i p t i o n a x i o m has the f o r m :  - 20 -  ?pq where  A V x ' . Tpc') A ( x ' < x ) R  D  (x <  K  x')  < ^ is a n a p p r o p r i a t e pre-order. T h e use of pre-orders a l l o w s a n u m b e r of interesting a n d p o t e n t i a l l y useful extensions to cir-  cumscription.  F o r e x a m p l e , the pre-order X < RY defined b y  (Vz. X  Z>  lX  Y ) lX  A ((Vz. Y  lX  D  X ) lX  Z>  (Vz. X  D  2X  a l l o w s the j o i n t m i n i m i z a t i o n of the u n a r y p r e d i c a t e s X  x  K z)) 2  a n d X% w i t h the m i n i m i z a t i o n of X± h a v -  ing a " h i g h e r p r i o r i t y " . T h e effect of a l l o w i n g X to i n c l u d e c o n s t a n t a n d f u n c t i o n letters is to a l l o w c o n s t a n t s a n d f u n c t i o n s t o v a r y d u r i n g the m i n i m i z a t i o n process. It appears t h a t - for languages w i t h finite sets of c o n s t a n t s - it is possible to c i r c u m s c r i p t i v e l y conjecture new facts a b o u t e q u a l i t y , i n c l u d i n g u n i q u e n a m e s a x i o m s , b y a l l o w i n g c o n s t a n t s to v a r y . U n f o r t u n a t e l y , L i f s c h i t z n e i t h e r m o t i v a t e s n o r discusses t h e v a r i a b i l i t y of t e r m s i n d e t a i l . has  yet  to  appear.  In  c h a p t e r 6, we  A s e m a n t i c e x p l a n a t i o n of the process i n v o l v e d  show that  allowing circumscriptively  variable  terms  corresponds to w e a k e n i n g the d e f i n i t i o n of s u b m o d e l i n the s e m a n t i c c h a r a c t e r i z a t i o n of f o r m u l a c i r c u m s c r i p t i o n b y d r o p p i n g the r e q u i r e m e n t  t h a t a m o d e l a n d its s u b m o d e l s share i d e n t i c a l  interpretations of constant and function symbols.  W e also s h o w t h a t this a p p r o a c h c a n lead to  some u n e x p e c t e d consequences.  2.1.5.3.  Domain Circumscription  In d a t a b a s e a n d c o m m o n s e n s e r e a s o n i n g , it is often necessary t o assume t h a t the o n l y i n d i v i d u a l s whose existence is r e l e v a n t t o some task are those r e q u i r e d to exist b y w h a t is k n o w n a b o u t the t a s k .  In s u c h s i t u a t i o n s , the domain-closure assumption is m a d e [Reiter 1980a]. T h i s is  the a s s u m p t i o n t h a t the " w o r l d " c o n t a i n s o n l y i n d i v i d u a l s whose existence is r e q u i r e d b y the available information.  R e i t e r observes t h a t this a s s u m p t i o n is i m p l i c i t  i n r e l a t i o n a l database  t h e o r y , where it is e n t a i l e d b y the m a n n e r i n w h i c h u n i v e r s a l queries are t r e a t e d . T h u s , for e x a m ple, i n the e d u c a t i o n database: Teacher(Smith)  Student(Brown)<  Teacher(Jones)  Student(Black)  Teacher(Plato)  Student(Aristotle)  w i t h a n i n t e g r i t y c o n s t r a i n t s p e c i f y i n g t h a t the sets of teachers a n d s t u d e n t s are d i s j o i n t , even the s i m p l e q u e r y , " W h o are a l l o f the t e a c h e r s ? " c a n n o t be answered w i t h o u t i m p l i c i t l y a s s u m i n g t h a t the d o m a i n consists of o n l y the listed i n d i v i d u a l s . I n cases where there are o n l y  finitely  m a n y i n d i v i d u a l s , this a s s u m p t i o n c a n be s t a t e d using  domain-closure aiioms. T h e s e are a x i o m s of the f o r m :  Vz. z = r V . . . V z = t 1  (11)  n  w h e r e the t,- are g r o u n d t e r m s .  A n y m o d e l s a t i s f y i n g (11) w i l l h a v e at m o s t n d i s t i n c t i n d i v i d u a l s  i n its d o m a i n , those c o r r e s p o n d i n g t o the  R e i t e r [1980a, 1984] shows t h a t d o m a i n - c l o s u r e  - 21 a x i o m s have a n i m p o r t a n t role i n l o g i c a l l y f o r m a l i z i n g the theory of r e l a t i o n a l databases. E v e n w h e n the d o m a i n cannot be e n u m e r a t e d t o f o r m a d o m a i n closure a x i o m , useful rest r i c t i o n s c a n s o m e t i m e s be p u t o n the size a n d c o m p o s i t i o n of the d o m a i n b y c o n j e c t u r i n g t h a t i t coincides w i t h the e x t e n s i o n of some p r e d i c a t e or f u n c t i o n whose e x t e n s i o n is ( p a r t l y ) k n o w n . F o r e x a m p l e , i n the e d u c a t i o n database discussed above, i f i t is k n o w n i s t h a t teachers are employees a n d s t u d e n t s are not, a s s u m i n g d o m a i n closure a l l o w s one t o conjecture t h a t teachers are the o n l y employees. Vx.  B y conjecturing  that  the domain  consists o n l y  o f teachers a n d students (i.e.,  Teacher(x) V Student(x)), i t becomes possible t o deduce t h a t  there  are n o non-teacher  employees (regardless of w h e t h e r a l l of the teachers a n d students are k n o w n ) . D o m a i n - c l o s u r e a x i o m s are also i m p o r t a n t w i t h respect t o a v a r i e t y o f c l o s e d - w o r l d reasoni n g f o r m a l i s m s . P e r l i s a n d M i n k e r [1986], for e x a m p l e , show t h a t the effects o f p r e d i c a t e a n d form u l a c i r c u m s c r i p t i o n [ M c C a r t h y 1980, 1986] c a n be more precisely c h a r a c t e r i z e d i n c o n j u n c t i o n with  c l o s e d - d o m a i n theories.  S i m i l a r l y , C l a r k [1978] requires d o m a i n - c l o s u r e a x i o m s i n t h e  d e v e l o p m e n t of his predicate c o m p l e t i o n a p p r o a c h . G i v e n the i m p o r t a n c e o f d o m a i n - c l o s u r e a x i o m s , the question arises: W h y not e x p l i c i t l y a d d t h e m t o theories?  P r o b a b l y t h e m o s t i m p o r t a n t reason is t h a t the a p p r o p r i a t e d o m a i n - c l o s u r e  a x i o m m a y n o t be o b v i o u s .  T h e repercussions o f choosing t o o strong o r t o o w e a k a n a x i o m  (inconsistency o r loss o f useful conjectures, respectively) argues i n f a v o u r o f a m o r e a u t o m a t i c approach.  F u r t h e r m o r e , as the state o f the w o r l d (or the system's knowledge) changes t o b r i n g  more entities i n t o c o n s i d e r a t i o n , the same m e c h a n i s m c o u l d be used t o generate new d o m a i n closure a x i o m s . I n c e r t a i n cases, d o m a i n c i r c u m s c r i p t i o n provides s u c h a n a u t o m a t i c m e c h a n i s m . A c t u a l l y t h e first o f t h e c i r c u m s c r i p t i v e f o r m a l i s m s , d o m a i n c i r c u m s c r i p t i o n  [McCarthy  1977, 1980; D a v i s 1980] is i n t e n d e d t o be a s y n t a c t i c r e a l i z a t i o n o f the m o d e l - t h e o r e t i c d o m a i n closure a s s u m p t i o n .  It p r o v i d e s a m e c h a n i s m for c o n j e c t u r i n g d o m a i n closure a x i o m s , e l i m i n a t i n g  the need t o e x p l i c i t l y state t h e m . T o c i r c u m s c r i b e the d o m a i n of a sentence, A, M c C a r t h y proposes a d d i n g the s c h e m a : Axiom($) A A * D to  A.  V i . $(x)  Axiom(<&) i s  (12)  the conjunction  of  $ a  for  each  Vzi-.-in. [ $ Z i A---A 3>x„] 3> $/xi...i for each »-ary f u n c t i o n s y m b o l /. n  A, r e p l a c i n g e a c h u n i v e r s a l o r e x i s t e n t i a l quantifier,  constant  symbol  a and  A * is the result of r e w r i t i n g  'Vx.' o r '3z.', i n A w i t h 'Vz.$x  Z)  ' or  '3x.$x A ', respectively. T h i s a x i o m s c h e m a represents the conjecture t h a t the d o m a i n of discourse i s no larger t h a n it m u s t be g i v e n the sentence A. existence  is g i v e n  F o r any predicate,  b y the constant  terms,  i f $ is true f o r a l l i n d i v i d u a l s whose  through function  application,  or b y existential  q u a n t i f i c a t i o n , a n d i f a l l i n d i v i d u a l s i n $ ' s e x t e n s i o n satisfy a l l of the u n i v e r s a l l y q u a n t i f i e d form u l a e , t h e n $ i s a s s u m e d t o c o n t a i n the entire d o m a i n . If the extension of some p r e d i c a t e meeti n g these r e q u i r e m e n t s is k n o w n , t h e n the d o m a i n is (assumed t o be) c o m p l e t e l y k n o w n . T h e s e m a n t i c i n t u i t i o n u n d e r l y i n g d o m a i n c i r c u m s c r i p t i o n is minimal entailment: o n l y those models w i t h m i n i m a l d o m a i n s s h o u l d be considered i n d e t e r m i n i n g the consequences o f the g i v e n information.  I n t h i s c o n n e c t i o n , a m o d e l , M, o f a sentence is s a i d t o be a submodel o f a n o t h e r  - 22 m o d e l , N, if M is the r e s t r i c t i o n of N to a subset of ffs d o m a i n . A m o d e l is s a i d t o be minimal if it has n o p r o p e r s u b m o d e l s . D a v i s [1980] shows t h a t every instance of (12) is true i n a l l m i n i m a l m o d e l s of the o r i g i n a l sentence A.  T h i s result is correct for those theories w i t h at least one c o n -  stant s y m b o l . In c h a p t e r 7, h o w e v e r , we s h o w t h a t i n c o n s i s t e n c y results w h e n c i r c u m s c r i b i n g theories whose p r e n e x n o r m a l forms c o n t a i n no l e a d i n g e x i s t e n t i a l quantifiers a n d n o c o n s t a n t symbols.  W e also present a s i m p l e , easily m o t i v a t e d s o l u t i o n . T h i s leads t o a r e v i s e d v e r s i o n of  d o m a i n c i r c u m s c r i p t i o n w h i c h is s h o w n t o preserve consistency.  2.1.6. Restricting Closed-World Inferences O n e m a y w a n t t o d o c l o s e d - w o r l d reasoning t o f o r m conjectures a b o u t the u n d e r l y i n g p r i n c i ples g o v e r n i n g a s i t u a t i o n . I n this case, one is m a k i n g u n i v e r s a l ( i n d u c t i v e ) c o n j e c t u r e s a b o u t the state of the w o r l d .  T h i s is the type of reasoning w h i c h is i n v o l v e d i n d e d u c i n g l a w s , s u c h as " a n  u n s u p p o r t e d object d r o p s w h e n r e l e a s e d " . In m a n y cases, however, c l o s e d - w o r l d r e a s o n i n g y i e l d s stronger conjectures t h a n m a y be d e s i r a b l e . F o r e x a m p l e , it is often sufficient t o c o n c l u d e t h a t the s i t u a t i o n i m m e d i a t e l y at h a n d does not h a v e c e r t a i n properties.  In d a y - t o - d a y r e a s o n i n g , one is  u s u a l l y i n t e r e s t e d in f o r m i n g p a r t i c u l a r conjectures i n a i d of c o m p l e t i n g a p a r t i c u l a r d e d u c t i o n . T h e s e conjectures s h o u l d be of as l i m i t e d scope as possible w h i l e s t i l l s t r o n g e n o u g h to a l l o w the desired g o a l t o be a c h i e v e d . T h u s , for e x a m p l e , if we k n e w t h a t T w e e t y is a b i r d a n d t h a t a l l b i r d s except p e n g u i n s fly, we m i g h t w a n t to conjecture t h a t T w e e t y c o u l d fly ( a n d hence t h a t T w e e t y is not a p e n g u i n ) . It  is u n l i k e l y t h a t we w o u l d w a n t to conjecture t h a t there are no  penguins at all, h o w e v e r . " P r o t e c t e d C i r c u m s c r i p t i o n " [ M i n k e r & P e r l i s 1984b] p r o v i d e s one means for d e l i m i t i n g the effects o f c l o s e d - w o r l d r e a s o n i n g . T o p r e v e n t the c i r c u m s c r i p t i o n of P i n a t h e o r y , A, f r o m conject u r i n g t h a t iS*s are not  P°s, the p r e d i c a t e , S, is p r o t e c t e d b y w e a k e n i n g the  circumscription  s c h e m a to:  A($) A [Vz*. ($z*A -Si) 3 Pxl D [Vi*. (Px A ~>Sx) D *x] . T h e c o n c l u s i o n s of p r o t e c t e d c i r c u m s c r i p t i o n a p p l y o n l y to those i n d i v i d u a l s t h a t d o not satisfy the p r o t e c t e d p r e d i c a t e .  T h u s , for e x a m p l e , M c C a r t h y [1984, p e r s o n a l c o m m u n i c a t i o n ] has sug-  gested t h a t one m a y w i s h to c o n c l u d e o n l y t h a t there are no p e n g u i n s present. A s s u m i n g t h a t there is a p r e d i c a t e , Present(x), w h i c h says t h a t a n i n d i v i d u a l is i n the i m m e d i a t e v i c i n i t y , p r o t e c t i n g -<Present w h i l e c i r c u m s c r i b i n g Penguin w i l l result i n conjectures w h i c h say n o t h i n g about those p e n g u i n s w h i c h are n o t present. U s i n g f o r m u l a c i r c u m s c r i p t i o n , the scope of conjectures c a n be l i m i t e d b y c o n j o i n i n g a p r o t e c t i n g p r e d i c a t e w i t h the expression t o be m i n i m i z e d , a n d not a l l o w i n g the p r o t e c t i n g p r e d i c a t e to vary.  F o r e x a m p l e , to m i n i m i z e present penguins w i t h respect to a theory, A, w h i l e p r o t e c t i n g  possible " a b s e n t " p e n g u i n s , the f o l l o w i n g c i r c u m s c r i p t i o n a x i o m suffices: V$.  vl($)  A [Vz.  <£z A Present(x) D Penguin(x) A Present(x)\  D [Vz.  Penguin(x) A Present(x) D $ i A Present(x)\ ,  w h i c h says n o t h i n g n e w a b o u t absent penguins.  - 23 2.1.7. Semantic Interconnections G e l f o n d , P r z y m u s i n s k a , a n d P r z y m u s i n s k i [1985] have e x t e n d e d the " c a r e f u l c l o s u r e " n o t i o n of G e l f o n d a n d P r z y m u s i n s k a [1985], b y a l l o w i n g t h e theory to be a u g m e n t e d w i t h the negations of a r b i t r a r y f o r m u l a e m e e t i n g a d m i s s i b i l i t y c r i t e r i a . t i o n s of g r o u n d a t o m i c f o r m u l a e .  T h i s is more p o w e r f u l t h a n a d d i n g o n l y nega-  G e l f o n d , P r z y m u s i n s k a , a n d P r z y m u s i n s k i restrict t h e i r a t t e n -  t i o n t o fixed-domain theories, those w i t h a x i o m s s t a t i n g t h a t there are f i n i t e l y m a n y i n d i v i d u a l s , a n d t h a t e a c h t e r m of the language denotes a u n i q u e i n d i v i d u a l . 2.1.  L e t P , Q , a n d Z be as i n section  T h e n a f o r m u l a , K, n o t i n v o l v i n g literals f r o m Z , is free for negation iff there is n o g r o u n d  clause, B, m a d e u p o f l i t e r a l s i n P + U Q + U extended CWA ECWA(T)  Q ~ s u c h t h a t T \- K V B a n d T \f- B.  T h e n the  for T is d e n n e d as: = T U { ^K\K is free for n e g a t i o n i n T } .  U s i n g t h e same p a r t i a l - o r d e r r e l a t i o n on m o d e l s as G e l f o n d a n d P r z y m u s i n s k a [1985] (see section 2.1.4), G e l f o n d , P r z y m u s i n s k a , a n d P r z y m u s i n s k i c l a i m t h a t the set of f o r m u l a e free f o r n e g a t i o n i n T are p r e c i s e l y those whose negations are true i n every m i n i m a l m o d e l of T. (Here w e refer to m i n i m a l i t y o v e r a l l , n o t j u s t H e r b r a n d , models.) T h u s , for consistent, function-free, theories,  T, ECWA[T)  is c o n s i s t e n t ,  9  a n d corresponds precisely to the f o r m u l a e  fixed-domain true i n a l l  m i n i m a l m o d e l s o f T. It f o l l o w s t h a t t h e free-for-negation f o r m u l a e c h a r a c t e r i z e t h e results o f form u l a c i r c u m s c r i p t i o n for s u c h theories. I n fact, because o f the junctions of literals from P  fixed-domain +  U Q  +  p r o p e r t y , one need o n l y consider those K w h i c h are c o n -  U Q ~ It c a n be s h o w n t h a t ECWA(DB)  corresponds, s y n t a c -  t i c a l l y , to the c a r e f u l closure o f DB if DB is a  fixed-domain  t h e o r y . T h e s e m a n t i c correspondence  follows f r o m the fact t h a t every m o d e l of a  fixed-domain  theory is i s o m o r p h i c t o a H e r b r a n d  m o d e l , a n d hence e v e r y m i n i m a l m o d e l t o a m i n i m a l H e r b r a n d m o d e l . B y s u i t a b l y m a t c h i n g the m o d e l - t h e o r y to the proof-theory, it is possible to show t h a t , for fixed-domain  theories,  predicate  circumscription  corresponds  t o the G C W A  a n d , for  Horn  theories, t o n a i v e closure. T h e s e o b s e r v a t i o n s show h o w c e n t r a l t h e n o t i o n of m i n i m a l m o d e l is to the v a r i o u s f o r m a l i s m s for c l o s e d - w o r l d r e a s o n i n g . T h e t w o forms of m i n i m i z a t i o n - of extensions of p r e d i c a t e s a n d of t h e d o m a i n of the m o d e l (hence p r o d u c i n g a  fixed-domain  connect t h e m a l l .  0  Every consistent, finite-domain theory has at least one minimal model.  model) - suffice to  - 24 -  2.2. D e f a u l t o r P r o t o t y p i c a l R e a s o n i n g  N e v e r u t t e r these w o r d s : 'I d o not k n o w this, therefore it is false.' O n e m u s t s t u d y t o k n o w , k n o w to u n d e r s t a n d , u n d e r s t a n d to judge. — A p o t h e g m of N e r u d a  A l l o f t h e a p p r o a c h e s discussed so f a r p r o v i d e w a y s o f b e c o m i n g more " c l o s e d - m i n d e d " . E a c h f u n c t i o n s b y r e s t r i c t i n g the set o f m o d e l s for the g i v e n a x i o m s . T h e g o a l has b e e n t o a l l o w o n l y m i n i m a l m o d e l s , i n w h i c h o n l y a m i n i m a l set o f p r e d i c a t e instances o r d o m a i n elements necessary t o satisfy the a x i o m s is a l l o w e d . T h e c o m p l e m e n t a r y a p p r o a c h also i n v o l v e s r e s t r i c t i n g the set o f m o d e l s c o n s i d e r e d . R a t h e r t h a n focussing o n m i n i m a l i t y , the systems discussed i n the sequel p r o v i d e more  flexibility  i n deter-  m i n i n g w h i c h m o d e l s are considered " i n t e r e s t i n g " .  2.2.1. D e f a u l t L o g i c  R e i t e r [1978a, 1980a] addresses t h e p r o b l e m of i n c o m p l e t e i n f o r m a t i o n  b y allowing  new  inference rules t o be a d d e d t o a s t a n d a r d first-order logic. T h e s e rules s a n c t i o n t h e i r conclusions p r o v i d e d t h a t the set o f beliefs satisfies the c o n d i t i o n s o u t l i n e d i n t h e i r premises. U n l i k e s t a n d a r d logic, these premises are a l l o w e d t o refer b o t h t o w h a t is k n o w n a n d t o w h a t i s not k n o w n . T h e l a t t e r p r o p e r t y a l l o w s rules t o be a d d e d t h a t specify inferences t h a t w i l l be m a d e o n l y  when  specific i n f o r m a t i o n is m i s s i n g . T h e s e inferences c a n be used to t a i l o r the c o m p l e t i o n o f p a r t i a l knowledge, unlike closed-world reasoning, w h i c h involves a uniform completion strategy.  2.2.1.1. D e f a u l t T h e o r i e s  A default is a n y e x p r e s s i o n o f the f o r m : A(x):  frffl,...,  1 0  B {x) m  VJ{X)  where A(x), J3,(i*), a n d UJ(X) are a l l f o r m u l a e whose free v a r i a b l e s are a m o n g those i n 5 * = xi,...,x . n  A, Bfy a n d w are c a l l e d the prerequisite, justifications, a n d consequent o f the default, r e s p e c t i v e l y . If none of A, Bj, a n d w c o n t a i n free v a r i a b l e s , the default is said t o be closed. If the prerequisite is empty,  i t m a y be t a k e n t o be a n y t a u t o l o g y .  T w o classes o f defaults h a v i n g o n l y a single  j u s t i f i c a t i o n , B(x), are d i s t i n g u i s h e d . T h o s e w i t h B(x) = u>(~x), are s a i d t o be normal, w h i l e those w i t h B[x) = w(x) A G{x), f o r some C(x), are c a l l e d semi-normal.  V i r t u a l l y a l l o f the defaults  This notation differs from Reiter'a i n the omission of the " M " preceeding each of the B/s. Since they are implicit in the positional notation, they have been omitted as a notational convenience. 1 0  - 25 -  o c c u r r i n g i n the l i t e r a t u r e f a l l i n t o one o f these t w o categories, (tyukaszewicz [1985] argues t h a t the r e m a i n i n g class of s i n g l e - j u s t i f i c a t i o n defaults, where B(x) k n o w of n o a p p l i c a t i o n for m u l t i - j u s t i f i c a t i o n  \f= w{x) are i l l - m o t i v a t e d , a n d we  defaults.)  D e f a u l t s serve as rules of inference or conjecture, a u g m e n t i n g those n o r m a l l y p r o v i d e d b y first-order  l o g i c . U n d e r c e r t a i n c o n d i t i o n s , t h e y s a n c t i o n inferences w h i c h c o u l d n o t be m a d e  within a strictly  first-order  framework.  If t h e i r prerequisites are k n o w n a n d t h e i r j u s t i f i c a t i o n s are  " c o n s i s t e n t " (i.e., t h e i r negations are n o t p r o v a b l e ) , t h e n t h e i r consequents c a n be inferred. T h u s the t e r m " j u s t i f i c a t i o n " is seen to be s o m e w h a t m i s l e a d i n g , since j u s t i f i c a t i o n s need not be k n o w n , merely consistent.  T h e c o n s e q u e n t ' s s t a t u s is a k i n t o t h a t of a belief, subject to r e v i s i o n s h o u l d  11  the j u s t i f i c a t i o n s be d e n i e d at some f u t u r e t i m e . It is t h i s c h a r a c t e r i s t i c w h i c h i n d u c e s the n o n m o n o t o n i c b e h a v i o r of defaults. D e f a u l t rules c a n be seen t o h a v e a great d e a l i n c o m m o n w i t h m a n y p r e v i o u s l y m e n t i o n e d approaches. F o r e x a m p l e , the C l o s e d - W o r l d A s s u m p t i o n states: VJ infer  If \f-  ->w  w h i c h c a n be represented i n default logic b y :  In fact, (13) w i l l l a t e r be referred to as the " C l o s e d - W o r l d " default. T h e D E F A U L T a s s i g n m e n t s w h i c h c a n be a t t a c h e d to f r a m e slots i n K R L [ B o b r o w & W i n o g r a d 1977] also a p p e a r to be r e l a t e d . K R L p r o v i d e s a m e c h a n i s m for o b t a i n i n g a v a l u e for a slot i n the absence of a " b e t t e r " value.  A K R L d e f a u l t v a l u e , d, for a slot, s, i n a frame instance, /, c a n be v i e w e d as:  ¥- (f) J=  I f  s  d  i n f e  r «W =  d  or, i n default l o g i c , as: :s(f)  =  d  s(f)  =  d  S i m i l a r m e c h a n i s m s are a v a i l a b l e in m a n y other f r a m e - b a s e d k n o w l e d g e r e p r e s e n t a t i o n schemes [ M i n s k y 1975]. A closely r e l a t e d a p p r o a c h is S a n d e w a l l ' s [1972] "Unless" p r e t e d as " \f- P", a n d "Unless" A A  Unless(B)  D  operator.  "Unless[P)"  is inter-  terms are a l l o w e d i n the c o n s t r u c t i o n of wffs, w i t h results l i k e :  C  w h i c h c o r r e s p o n d s r o u g h l y to: A  :^B C  "Unless"  '  w a s o r i g i n a l l y p r o p o s e d as a s o l u t i o n to the f r a m e p r o b l e m [Hayes 1973].  Rather  t h a n h a v i n g t o h a v e e x p l i c i t a x i o m s s t a t i n g t h a t the properties of objects r e m a i n e d i n v a r i a n t f r o m s i t u a t i o n t o s i t u a t i o n unless e x p l i c i t l y c h a n g e d , S a n d e w a l l suggested t h a t these " f r a m e a x i o m s " be 1 1  In a modal logic with the operator K (know) the justifications 5,- might appear as ->K->Bi.  - 26 -  r e p l a c e d b y a frame inference rule l i k e : IS[obje c t,prope rty, situation) Unless(ENDS(obiectpropertv.Successor(  situation.act)))  IS(object,property,Successor(situation,act)) w h i c h c a n be i n t e r p r e t e d : If a n object has a p r o p e r t y i n a s i t u a t i o n , it c a n be c o n c l u d e d t o r e t a i n t h a t p r o p e r t y i n the successor s i t u a t i o n r e s u l t i n g f r o m p e r f o r m i n g ' a c t ' , unless it c a n be s h o w n otherwise. N o f o r m a t i o n r u l e s were p r o v i d e d for "Unless", however, so q u e s t i o n a b l e f o r m u l a e such as: A 3 Unless(B) c a n be c o n s t r u c t e d . T h e s e m a n t i c s of s u c h f o r m u l a e are, at best, difficult to d e t e r m i n e . S a n d e w a l l also fails to p r o v i d e a n y f o r m a l u n d e r s t a n d i n g o f t h e i m p a c t of the " Unless" r u l e o n the u n d e r l y i n g logic. D e f a u l t logic has, to some e x t e n t , r e m e d i e d these s h o r t c o m i n g s .  2.2.1.2. C l o s e d D e f a u l t Theories a n d T h e i r E x t e n s i o n s  A default theory, A , is a n ordered p a i r , (D, W). order f o r m u l a e .  D is a set of defaults;  W is a set of  first-  R e i t e r [1980a] describes the extensions of a default t h e o r y as " a c c e p t a b l e sets of  beliefs t h a t one m a y h o l d a b o u t a n i n c o m p l e t e l y specified w o r l d , W°. D is v i e w e d as e x t e n d i n g the  first-order  k n o w l e d g e of W i n order to p r o v i d e i n f o r m a t i o n not d e r i v a b l e f r o m  W.  S i n c e defaults a l l o w reference to w h a t is n o t p r o v a b l e i n the d e t e r m i n a t i o n of w h a t is p r o v able, the " t h e o r e m s " of a default t h e o r y are not so easy to generate as are those of a first-order t h e o r y . W h a t is p r o v a b l e b o t h determines a n d is d e t e r m i n e d by w h a t is not p r o v a b l e . T o a v o i d this a p p a r e n t c i r c u l a r i t y , the theorems of a default theory are defined b y a f i x e d - p o i n t c o n s t r u c tion.  A n e x t e n s i o n , E, f o r A is r e q u i r e d t o have the f o l l o w i n g properties: W C E Th {F) = L  E  „ , , .. , F o r each default,  A: Bi,...,B w  G D , if A G E, a n d - i 5 , , . . . , - . B & E m  t h e n w e E. T h e s e p r o p e r t i e s state t h a t E m u s t c o n t a i n a l l the k n o w n facts, t h a t E must be closed u n d e r the |— r e l a t i o n , a n d t h a t the consequent of a n y default whose prerequisite is satisfied b y E, a n d whose j u s t i f i c a t i o n s are consistent w i t h E, m u s t also be in E. R e i t e r defines a n e x t e n s i o n for a closed d e f a u l t t h e o r y t o be a m i n i m a l  fixed-point  of a n o p e r a t o r h a v i n g t h e a b o v e c h a r a c t e r i s t i c s .  T h e e x t e n s i o n s of a default t h e o r y select r e s t r i c t e d subsets of the m o d e l s of the u n d e r l y i n g first-order  theory,  W.  A n y m o d e l for a n e x t e n s i o n of A w i l l also be a m o d e l for W, b u t the c o n -  verse is g e n e r a l l y n o t t r u e .  D e f a u l t theories need not a l w a y s h a v e extensions, e v e n w h e n W is  consistent. T h e r e are, h o w e v e r , c e r t a i n classes of theories for w h i c h the existence o f at least one  - 27 -  e x t e n s i o n is g u a r a n t e e d .  T h e o r i e s w i t h o n l y n o r m a l defaults have been s h o w n a l w a y s to have  extensions [Reiter 1980a].  In c h a p t e r 3, we prove the same result for c e r t a i n classes of theories  w i t h s e m i - n o r m a l defaults. R e i t e r [1980a] presents a n i t e r a t i v e m e c h a n i s m for d e c i d i n g w h e t h e r a set of f o r m u l a e f o r m s a n e x t e n s i o n f o r a t h e o r y , A . T h e m e t h o d is, u n f o r t u n a t e l y , not s u i t a b l e for c o n s t r u c t i n g e x t e n sions. T h i s is because a n oracle is r e q u i r e d w h i c h c a n decide w h e t h e r a p a r t i c u l a r f o r m u l a ' s negat i o n w i l l be i n the set. R e i t e r [1980a] a n d E t h e r i n g t o n [1982] also present c o n s t r u c t i v e m e c h a n i s m s a p p l i c a b l e to n o r m a l theories a n d t o a r b i t r a r y finite theories, r e s p e c t i v e l y . S o m e e x a m p l e s of defaults were presented i n the preceeding s e c t i o n . T h e f o l l o w i n g e x a m p l e illustrates the e x t e n s i o n s i n d u c e d b y the c l o s e d - w o r l d default o n the t h e o r y : W=  {BLOCK(A)  V  BLOCK(B)}.  T h e c l o s e d - w o r l d d e f a u l t is r e a l l y a default s c h e m a w h i c h is a p p l i c a b l e t o a n y p o s i t i v e g r o u n d l i t e r a l . I n t h i s case, it results i n the f o l l o w i n g set of n o r m a l defaults: ( \  -.-iBLOCK(A) ->BLOCK(A)  :^BLOCK(B)  j  T h e t h e o r y , (D, W), has t w o e x t e n s i o n s , E E  t  = Th({^BLOCK(A),  E  2  = Th({BLOCK(A),  N o t e t h a t E=  )  ' ^BLOCK(B)  a n d E^.  1  BLOCK(B)}) -^BLOCK(B)})  Tk({BLOCK(A),  BLOCK(B)})  is not a n e x t e n s i o n . L i k e d a t a b a s e c o m p l e -  t i o n a n d c i r c u m s c r i p t i o n , the c l o s e d - w o r l d default sanctions the e x c l u s i v e i n t e r p r e t a t i o n of d i s j u n c t i o n s t o w h i c h it is a p p l i e d . I n t u i t i v e l y , this is because the defaults force as m a n y things to be false as possible, r e s u l t i n g i n extensions whose models m a y be m i n i m a l m o d e l s for  W.  More  p r e c i s e l y , E is n o t a n e x t e n s i o n because it v i o l a t e s the m i n i m a l i t y c o n d i t i o n o f the d e f i n i t i o n of extensions. ( W e r e  W  also to c o n t a i n b o t h BLOCK(A)  a n d BLOCK(B),  E w o u l d be the o n l y  extension.) N o t i c e h o w the extensions E  x  sistent a s s i g n m e n t s f o r BLOCK(A)  and E  2  manifest W ' s i n c o n s i s t e n c y w i t h the C W A . T h e i n c o n -  a n d BLOCK(B)  are s t i l l o b t a i n a b l e , b u t t h e y are separated  i n t o o r t h o g o n a l , self-consistent extensions. In fact, R e i t e r has s h o w n t h a t the e x t e n s i o n s of a n y default t h e o r y w i l l a l w a y s be self-consistent p r o v i d e d t h a t the  first-order  theory  W is consistent,  a n d t h a t a l l the e x t e n s i o n s of a n o r m a l default t h e o r y w i l l be (pairwise) m u t u a l l y i n c o n s i s t e n t .  2.2.1.3. General Default Theories I n c o n t r a s t t o c l o s e d defaults, a n open d e f a u l t is one i n w h i c h at least one of A ( z ) , B^x), o r w(x) c o n t a i n free v a r i a b l e s i n z*. A n o p e n default is i n t e r p r e t e d as s t a n d i n g f o r t h e set of closed defaults o b t a i n a b l e b y r e p l a c i n g its free v a r i a b l e s b y g r o u n d terms. i n f i n i t e t h i s r e s u l t s i n a d e f a u l t t h e o r y w i t h a n infinite set of defaults.  If the set of g r o u n d terms is  - 28 -  M o s t i n t e r e s t i n g default theories are n o t closed.  C o n s i d e r w h a t , b y n o w , m u s t b e the arche-  t y p a l default t h e o r y : Vz. Penguin(x) ~D Bird(x), Vz. Penguin(x) D ->Can-Fly(x), Vz. Dead-Bird(x) D Bird(x), 4 Vz. Dead-Bird(x) Z> -> Can-Fly (x),  W=  Vz. Ostrich(x) Z> Bird(x), Vz. Ostrich(x) D ->Can-Fly(x), Bird( Tweety)  D  ( Bird(x) : Can-Fly{x) \  =  \  Can-Fly(x)  )  T h e default, w h i c h is n o t c l o s e d , m i g h t be i n t e r p r e t e d as " I f z is a b i r d , a n d it is consistent t h a t x c a n fly, c o n c l u d e t h a t it c a n " .  T h i s theory a l l o w s one to c o n c l u d e , f o r a n a r b i t r a r y b i r d (e.g.,  T w e e t y ) , t h a t it c a n fly - unless one i s t o l d t h a t i t c a n n o t , or t h a t it is a p e n g u i n , a n o s t r i c h , o r dead.  T h e c o n c l u s i o n m a y l a t e r h a v e t o be r e v o k e d s h o u l d T w e e t y t u r n o u t t o be a p e n g u i n , b u t  c o m m o n sense seems t o s a n c t i o n the same c o n c l u s i o n .  T h i s is p a r t l y because people t e n d t o  assume t h a t t h e y h a v e the r e l e v a n t i n f o r m a t i o n i n most s i t u a t i o n s (cf. l i n g u i s t s ' use o f G r i c e ' s C o n v e r s a t i o n a l I m p l i c a t u r e s [Griee 1975]: one o f these is t h a t a l l i n f o r m a t i o n necessary to interpret a n u t t e r a n c e i s e x p e c t e d t o be c o n t a i n e d i n the utterance.)  2.2.1.4. Interacting Defaults T h e i r b r o a d a p p l i c a b i l i t y a n d t h e guarantee of coherence m a k e s n o r m a l d e f a u l t s a t t r a c t i v e for k n o w l e d g e r e p r e s e n t a t i o n a n d r e a s o n i n g . T h e r e are, h o w e v e r , some t y p e s o f k n o w l e d g e w h i c h n o r m a l d e f a u l t s c a n n o t c o m p l e t e l y c h a r a c t e r i z e . F o r e x a m p l e , R e i t e r a n d C r i s c u o l o [1983] have n o t i c e d t h a t defaults s o m e t i m e s i n t e r a c t w i t h one another, a n d t h a t n o r m a l defaults c a n n o t adeq u a t e l y c o n s t r a i n these i n t e r a c t i o n s . O n e m a n i f e s t a t i o n of this occurs w h e n t w o defaults w i t h distinct  but not mutually  e x c l u s i v e prerequisites have c o n t r a d i c t o r y  c u m s t a n c e s i t is n o t a l w a y s c l e a r w h i c h default s h o u l d be a p p l i e d .  consequents. I n s u c h cir-  C o m m o n s e n s e reasoning u s u -  a l l y prefers o n e of t h e c o m p e t i n g defaults b y v i r t u e of i t s prerequisite b e i n g more specific, m a k i n g the default a p p l i c a b l e f o r o n l y a subset o f those i n d i v i d u a l s for w h i c h the c o m p e t i n g d e f a u l t is applicable.  T h i s preference c a n n o t be enforced using o n l y n o r m a l defaults.  we are g i v e n : T y p i c a l a d u l t s are e m p l o y e d . T y p i c a l h i g h - s c h o o l d r o p o u t s are a d u l t s . T y p i c a l h i g h - s c h o o l d r o p o u t s are n o t e m p l o y e d . T h i s m a y be expressed b y t h e f o l l o w i n g n o r m a l defaults:  F o r e x a m p l e , assume  - 29 -  Adult(x) : Employed(x)  { For  Employed(x)  Dropout(x) : Adult(x) '  Adult(x)  Dropout(x) : ->Employed(x) '  -<Employed(x)  a g i v e n a d r o p o u t , this theory c a n be seen to h a v e t w o extensions w h i c h differ o n h i s / h e r  state o f e m p l o y m e n t .  I n t u i t i o n d i c t a t e s t h a t we assume s / h e is u n e m p l o y e d . C a r e f u l c o n s i d e r a -  t i o n shows t h a t the c o n f l i c t arises because t y p i c a l d r o p o u t s are not typical a d u l t s ; this a t y p i c a l i t y s h o u l d b l o c k the t r a n s i t i v i t y f r o m Dropout t h r o u g h Adult to Employed. T h e first d e f a u l t i n c o r porates no e x p l i c i t reference to these e x c e p t i o n a l c i r c u m s t a n c e s w h i c h s h o u l d b l o c k its a p p l i c a t i o n . O n e w a y t o address t h i s p r o b l e m is t o require t h a t the case u n d e r c o n s i d e r a t i o n not be a k n o w n e x c e p t i o n a l case. T h i s r e q u i r e m e n t is t h e n a d d e d to the j u s t i f i c a t i o n . T h u s the first default a b o v e becomes: Adult(x) : Employed(x) A -'Dropout(x) Employed(x) w h i c h is not a p p l i c a b l e to k n o w n d r o p o u t s . S e m i - n o r m a l defaults c a n be used to resolve t h e a m b i g u i t i e s r e s u l t i n g f r o m the i n t e r a c t i o n s between defaults. of defaults.  T h i s is done b y m a k i n g i n t e r a c t i o n s e x p l i c i t , as e x c e p t i o n s to the a p p l i c a b i l i t y  T h e r e are three m a j o r o b j e c t i o n s t o t h i s a p p r o a c h , h o w e v e r .  F i r s t , the c o m p l e x i t y of theories w i t h s e m i - n o r m a l defaults is s u b s t a n t i a l l y greater t h a n of theories w i t h n o r m a l defaults.  A p p l i c a t i o n of a default m a y force c o n c l u s i o n s o b t a i n e d f r o m pre-  v i o u s l y a p p l i e d d e f a u l t s to be r e t r a c t e d .  This phenomenon, which cannot occur w i t h  normal  default theories, p r e c l u d e s the t y p e o f s t r a i g h t f o r w a r d proof theory d e v e l o p e d b y R e i t e r [1980a] for n o r m a l theories. S e c o n d l y , it is possible to so o v e r c o n s t r a i n the i n t e r a c t i o n s b e t w e e n defaults t h a t the resulting theory has no e x t e n s i o n . C h a p t e r 3 explores w a y s of g u a r a n t e e i n g t h a t this does not h a p p e n , b u t , for c o m p l i c a t e d theories w i t h m a n y i n t e r a c t i o n s , it m a y be difficult to detect s u c h o v e r c o n st r a i n i n g . F i n a l l y , i n t e r a c t i o n s m u s t be n o t i c e d a n d e x p l i c i t l y dealt w i t h at the t i m e new k n o w l e d g e is g i v e n to the s y s t e m . I n a large, c o m p l i c a t e d , s y s t e m this is l i k e l y t o be a n e n o r m o u s task. T h e c o n t r i b u t o r s of new k n o w l e d g e m a y n o t be aware of a l l possible i n t e r a t i o n s b e t w e e n t h e i r c o n t r i b u t i o n s a n d the r e m a i n d e r of the k n o w l e d g e base.  In s e c u r i t y - c o n s c i o u s e n v i r o n m e n t s , c o n t r i b u -  tors m a y not e v e n be a l l o w e d access to some of the i n f o r m a t i o n w h i c h i n t e r a c t s w i t h t h e i r c o n t r i bution.  T o u r e t z k y [l984a,b] argues t h a t e x p l i c i t c o n t r o l of i n t e r a c t i o n s i n d e f a u l t theories is i n a p -  p r o p r i a t e , f o r the reasons o u t l i n e d a b o v e a n d because m a n y of the a m b i g u i t i e s i n t r o d u c e d b y s u c h i n t e r a c t i o n s c a n be r e s o l v e d u s i n g more g e n e r a l p r i n c i p l e s .  In semantic network systems, w h i c h  c a n be v i e w e d as c o r r e s p o n d i n g t o default theories (see c h a p t e r 4), the s t a n d a r d s u c h p r i n c i p l e is the " s h o r t e s t - p a t h h e u r i s t i c " , w h i c h resolves a m b i g u i t i e s b y preferring w h i c h e v e r c o n c l u s i o n c a n be r e a c h e d b y t r a v e r s i n g the smallest n u m b e r of n e t w o r k arcs. E t h e r i n g t o n [1982] shows h o w to c o n s t r u c t n e t w o r k s w h i c h defeat the s h o r t e s t - p a t h h e u r i s t i c a n d o t h e r s i m p l e - m i n d e d a m b i g u i t y r e s o l u t i o n techniques. T o u r e t z k y [1984a] presents a more s o p h i s t i c a t e d a m b i g u i t y r e s o l u t i o n d e v i c e , the inferential distance topology, w h i c h appears to c a p t u r e the i n t e n t i o n of the s h o r t e s t - p a t h h e u r i s t i c w i t h o u t its  - 30 -  naive realization.  H e e x p l o i t s the s u b c l a s s / s u p e r c l a s s relations, w h i c h are one of the raisons  d'etre for s e m a n t i c n e t w o r k s , to a r b i t r a t e between r i v a l c o n c l u s i o n s . In the " D r o p o u t " e x a m p l e , above, since Dropout is a (default) subclass of Adult, the i n f e r e n t i a l d i s t a n c e o r d e r i n g perfers c o n clusions a s s o c i a t e d w i t h  Dropout  (i.e., u n e m p l o y e d )  employed), in accord w i t h our intuitions.  o v e r those associated w i t h  Adult (i.e.,  I n c h a p t e r 4, we discuss T o u r e t z k y ' s a p p r o a c h in more  d e t a i l , a n d s h o w its r e l a t i o n s h i p to d e f a u l t logic. I n spite of the f a c t t h a t i t is a p p l i c a b l e o n l y i n s u b c l a s s / s u p e r c l a s s h i e r a r c h i e s , the success of T o u r e t z k y ' s a p p r o a c h i n agreeing w i t h the i n t u i t i v e l y a c c e p t a b l e c o n c l u s i o n s ( a v a g u e c r i t e r i o n , t o be sure) suggests t h a t it m a y be possible to e l u c i d a t e some set of general p r i n c i p l e s w h i c h a v o i d the necessity o f ad hoc m a n i p u l a t i o n s of the k n o w l e d g e base. F i n d i n g a n d e v a l u a t i n g s u c h p r i n c i ples r e m a i n s a n i m p o r t a n t o p e n p r o b l e m .  2.2.2. Minimizing Abnormality D e f a u l t r e a s o n i n g c a n i n v o l v e c o n j e c t u r i n g b o t h positive a n d negative instances of p r e d i cates.  T h i s w o u l d s e e m to p r e c l u d e the use of a n y o f the c l o s e d - w o r l d o r c i r c u m s c r i p t i v e f o r m a l -  isms, discussed earlier, i n s i t u a t i o n s where general default reasoning is r e q u i r e d . (In c h a p t e r 5, t h i s is  s h o w n c o n c l u s i v e l y i n the  presented (to  our knowledge)  case of p r e d i c a t e c i r c u m s c r i p t i o n . ) by  E x p a n d i n g on an idea  first  L e v e s q u e [1982], M c C a r t h y [1986] a n d G r o s o f [1984] h a v e  e x p l o r e d the p o s s i b i l i t y of using f o r m u l a c i r c u m s c r i p t i o n for default r e a s o n i n g . E s s e n t i a l l y , the i d e a i n v o l v e d is t h a t if defaults represent the properties of " n o r m a l " i n d i v i d u a l s , t h e n there is " s o m e t h i n g a b n o r m a l " a b o u t a n i n d i v i d u a l w h o does not fit the default p a t e r n . B y  appropriately  a x i o m a t i z i n g a b n o r m a l i t y , it is possible t o do default reasoning b y c i r c u m s c r i b i n g a b n o r m a l i t y . A n i n d i v i d u a l m a y be n o r m a l i n some respects a n d a b n o r m a l i n others; few, if a n y , are ever totally " t y p i c a l " .  T h u s , some a l l o w a n c e m u s t be m a d e for these differing aspects of a b n o r m a l i t y .  M c C a r t h y e x p l i c i t l y i n t r o d u c e s these aspects i n t o his o n t o l o g y , s p e a k i n g of the (ab) n o r m a l i t y p a r t i c u l a r aspects of a n i n d i v i d u a l .  of  G r o s o f , preferring not to proliferate objects u n d u l y , i n s t e a d  has a v a r i e t y of a b n o r m a l i t y predicates, each c o r r e s p o n d i n g t o a b n o r m a l i t y of a p a r t i c u l a r aspect in M c C a r t h y ' s notation. A n e x a m p l e helps t o c l a r i f y the m e t h o d . W e follow M c C a r t h y ' s n o t a t i o n :  - 31 -  Vz. Thing(x) A ->ab(aspect (x)) D -*Fly(x) l  Vz. Bird(x) D ab(aspect (x)) l  Vz. Bt'rd(z) A ->ab(aspect (x)) D F/y(z) 2  Vz. Penguin(x) D a6(a5pect2(z)) Vz. i>enou»*n(z) 3 B»'rd(z) Vz. Penjtttn(z) A -<ab(aspect (x)) D ->F/y(z) 3  Vz. Penguin—in-his-dreams(x) D ab^aspect^x)) Vz. Penguin-in-his-dreams(x) 3 Peri(/uin(z) Vz. Penguin-in—his-dreams(x) A -"atfaspeefc^a;)) Z> Fly(x)  Vz. OsfricA(z) D a6(aapec<2(z)) Vz. 0s<ri'cA(z) D Birrf(z)  Vz. Ostrich(x) A ->a&(aspect(z)) 5  aspecti(x)-aspect^x)  D  ->.Fty(z)  are t h e aspects, a n d a t is the a b n o r m a l i t y p r e d i c a t e o n aspects o f i n d i v i d u -  als. G i v e n Thing(Theodore), c i r c u m s c r i b i n g ab(x) v a r y i n g ab a n d Fly allows us t o c o n c l u d e ->ab(aspecti(Theodore)) a n d hence ~>Fly{Theodore). G i v e n Bird(Tweety), y i e l d ab(aspect (Tweety)), 1  circumscription will  ->ab[aspect2XTweety)), a n d Fly[Tweety). If Opus i s a p e n g u i n , t h e c o n -  jectures w i l l be ab(aspect (Opus)), ab(aspect {Opus)), ->ab(aspect (Opus)), a n d -<Fly(Opus). 1  2  3  T h i s r e f o r m u l a t i o n o f default reasoning as c l o s e d - w o r l d r e a s o n i n g a b o u t a b n o r m a l i t y c a n d e a l w i t h m a n y of the p r o b l e m s o f i n t e r a c t i n g defaults t h a t forced t h e c o n s i d e r a t i o n o f s e m i n o r m a l default theories. T h e d i r e c t default r e p r e s e n t a t i o n of the above e x a m p l e looks l i k e : Thing(x) : ->Fly[x) ->Fly{x)  '  Ostrich(x) : -<Fly(x) ^Fly(x)  Bird(x) : Fly(x) A -iOstrich{x) A —>(Penguin(x) A ->Penguin-in-his-dreams(x)) Fiyjx) Penguin(x) : Fly(x) A -iPenguin-in-his-dreams(x)) ^Fly(x)  '  I n order t o preserve a u n i q u e e x t e n s i o n , t h e c o m p l i c a t e d i n t e r a c t i o n s between t h e d e f a u l t statem e n t s m u s t be e x p l i c i t l y reflected i n the rules. I n t r o d u c i n g a b n o r m a l i t y a l l o w s a normal default r e p r e s e n t a t i o n c o n s i s t i n g o f t h e first-order a x i o m s (14), together w i t h t h e single closed-world default: : ->ab(x) -iab(x) G r o s o f [1984] h a s d e v e l o p e d a t r a n s l a t i o n scheme, using a b n o r m a l i t y p r e d i c a t e s , w h i c h he c l a i m s p r o d u c e s representations of n o r m a l default theories i n a f o r m s u i t a b l e f o r c i r c u m s c r i p t i v e default r e a s o n i n g .  1 2  H e is c u r r e n t l y seeking a w a y o f e x t e n d i n g t h i s a p p r o a c h t o a r b i t r a r y s e m i -  n o r m a l d e f a u l t theories. T h e r e l a t e d p r o b l e m - w h e t h e r 06 c a n be used t o reduce s e m i - n o r m a l default theories t o n o r m a l default theories - also r e m a i n s o p e n .  In fact, the representation correctly translates only prerequisite-free normal defaults.  - 32 -  M c C a r t h y [1984, p e r s o n a l c o m m u n i c a t i o n ] has d i s c o v e r e d t h a t i t is possible to r u n i n t o i n t e r a c t i o n p r o b l e m s w i t h 06 predicates. A u g m e n t i n g (14) w i t h : Vz. Canary(x) A ->ab(aspect (x)) D Bird(x) G  Vz. Gangster-Canary[x)  Z> Canary(x)  Vz. Gangster-Canary(x)  D ab(aspect (x)) , 6  to a l l o w the p o s s i b i l i t y t h a t " c a n a r y " m a y be used i n the sense o f o l d gangster-movies, m a y result i n a m b i g u i t y . Dinsdale t h e Canary m u s t be a b n o r m a l w i t h respect t o e i t h e r aspect^ o r aspect^ C i r c u m s c r i b i n g ab c a n o n l y conjecture e i t h e r t h a t Dinsdale flies (because he is a b i r d a n d hence a b n o r m a l i n aspect^, o r t h a t he is a n a b n o r m a l c a n a r y ( i n aspect^. M c C a r t h y [1986] has proposed a v a r i a n t  of formula circumscription, prioritized circumscription, w h i c h allows several  expressions t o be s i m u l t a n e o u s l y m i n i m i z e d a c c o r d i n g t o some p a r t i c u l a r precedence. T h i s c a n e l i m i n a t e u n d e s i r a b l e i n t e r a c t i o n s , b u t at the cost t h a t the precedence m u s t be e x p l i c i t l y w o r k e d out before c i r c u m s c r i b i n g . T h e c r i t i c i s m s a p p l i e d t o s e m i - n o r m a l default representations,  that  i n t e r a c t i o n s m u s t be k n o w n a n d a c c o m m o d a t e d w h e n k n o w l e d g e is represented, a p p l y e q u a l l y to the p r i o r i t i z e d c i r c u m s c r i p t i o n o f a b n o r m a l i t y representations. W h e t h e r s u c h i n t e r a c t i o n s c a n be dealt w i t h w i t h o u t d e s t r o y i n g the c o n c e p t u a l c l a r i t y a n d naturalness  o f the ab r e p r e s e n t a t i o n  scheme is u n k n o w n .  2.2.3. Non-Monotonic  Logic  M c D e r m o t t and Doyle  [1980, M c D e r m o t t 1982] propose a f o r m a l i s m c o m p l e m e n t a r y  default logic, w h i c h t h e y c a l l n o n - m o n o t o n i c logic ( N M L ) .  to  U n l i k e default logic, w h i c h uses the  n o t i o n of consistency o n l y at the " m e t a " l e v e l ( i n the inference rules), N M L centres a r o u n d the i n t r o d u c t i o n o f consistency i n t o the object language. T h e first i n c a r n a t i o n o f N M L [ M c D e r m o t t & D o y l e 1980] consists o f a s t a n d a r d  first-order  logic, a u g m e n t e d w i t h a n " A f operator, r o u g h l y  e q u i v a l e n t t o t h e f a m i l i a r " \f-—1 " . T h e set o f theorems is defined as the i n t e r s e c t i o n o f a l l o f the fixed-points  of a n operator,  NM. E s s e n t i a l l y , NM  produces the l o g i c a l closure o f the o r i g i n a l  t h e o r y together w i t h as m a n y assertions o f the f o r m Mq as possible. T h e set o f theorems c a n be c o n t r a s t e d w i t h the extensions o f a default theory, each o f w h i c h is a  fixed-point.  T h i s indicates  t h a t n o n - m o n o t o n i c t h e o r e m h o o d is, i n some sense, a more c o n s e r v a t i v e or r e s t r i c t i v e concept than extension membership. viewing  fixed-points  intersection of the  M o o r e [1983a] suggests t h a t this difference c a n be u n d e r s t o o d b y  as sets of beliefs a n agent m i g h t come to h o l d g i v e n his premises, w h i l e the fixed-points  d e t e r m i n e s w h a t a n outside observer c o u l d infer a b o u t the agent's  beliefs k n o w i n g o n l y h i s premises.  I n fact, the extensions o f default theories a n d the  of n o n - m o n o t o n i c theories are i n c o m p a r a b l e i n general. i n t u i t i v e l y be e x p e c t e d , g i v e n t h a t a n y default: A : B ...,B v  m  w c a n be a p p r o x i m a t e d i n N M L b y :  fixed-points  T h e t w o f o r m a l i s m s often agree, as w o u l d  - 33 -  A MB  A  X  A - A MB  m  D  w.  T h e r e are, h o w e v e r , default theories w h i c h h a v e extensions even t h o u g h the c o r r e s p o n d i n g n o n m o n o t o n i c theories h a v e no  fixed-points,  a n d vice versa (see [Reiter 1980a] for e x a m p l e s ) .  D a v i s [1980] suggests t h a t it m i g h t be impossible to assign a reasonable s e m a n t i c s to the o p e r a t o r were it i n c l u d e d i n the object language. M c D e r m o t t a n d D o y l e p o i n t out t h a t MP,  M  intui-  t i v e l y r e a d as "P is c o n s i s t e n t " , is not necessarily inconsistent w i t h -<P\ M o o r e [1983a] observes t h a t this is c a u s e d b y the l a c k of a n y p r o h i b i t i o n , i n the MP  being c o n t a i n e d i n a single  fixed-point.  fixed-point  c o n s t r u c t i o n , a g a i n s t ->P a n d  T h u s , i n M o o r e ' s terms, -<P m a y be b e l i e v e d w i t h o u t  the s t a t e m e n t "->P is b e l i e v e d " (->MP or L->P) b e i n g b e l i e v e d . T h i s a l l o w s w e a k e r i n t e r p r e t a t i o n s t o be p l a c e d o n M t h a n the i n t e n d e d " i s c o n s i s t e n t " .  T h e s e a n d o t h e r p r o b l e m s led t o the recast-  i n g of the t h e o r y i n t e r m s of a more c l a s s i c a l m o d a l logic [ M c D e r m o t t 1 9 8 2 ] .  13  T h e resulting non-  m o n o t o n i c S5 is u n f o r t u n a t e l y r e d u n d a n t , since it is no more p o w e r f u l t h a n S 5 . B e c a u s e of this, M c D e r m o t t suggests f a l l i n g b a c k to n o n - m o n o t o n i c S 4 or n o n - m o n o t o n i c T . T h i s suggestion is p e c u l i a r , since M c D e r m o t t a c k n o w l e d g e s t h a t the c h a r a c t e r i s t i c a x i o m of S5 (->LP D L->LP) - if P is not b e l i e v e d , it is b e l i e v e d n o t t o be b e l i e v e d - seems a p p r o p r i a t e for a n y belief s y s t e m .  How-  ever, t h e c o l l a p s e of n o n - m o n o t o n i c S5 was seen to force this retreat. M o o r e [1983a, b] argues t h a t t h i s retreat is i l l - m o t i v a t e d .  H e goes o n to s h o w t h a t the c o l -  lapse of n o n - m o n o t o n i c S5 is a c t u a l l y due t o the a x i o m LP Z> P, w h i c h says t h a t w h a t e v e r is b e l i e v e d is true.  W h i l e this a x i o m is a p p r o p r i a t e for k n o w l e d g e , M o o r e c l a i m s t h a t a n o n -  m o n o t o n i c s y s t e m is a c t u a l l y d e a l i n g w i t h belief, a n d t h a t an a x i o m s t a t i n g the i n f a l l i b i l i t y of an agent s h o u l d be expected to lead to p e c u l i a r consequences. A s i d e f r o m the q u e s t i o n of t h e i r a p p r o p r i a t e n e s s , M c D e r m o t t presents no proofs of the c o n sistency of n o n - m o n o t o n i c T a n d S 4 . S u c h proofs are a necessary step i n the d e v e l o p m e n t of n o n monotonic T and S4. In the second p a p e r o n N M L , M c D e r m o t t [1982] acknowledges the r e s t r i c t i v e n e s s of believi n g o n l y those f o r m u l a e i n the i n t e r s e c t i o n of a l l the "brave robot"  fixed-points  of a theory. H e proposes a  w h i c h w o u l d believe a l l of the f o r m u l a e of some p a r t i c u l a r  fixed-point.  Such an  a p p r o a c h is r e q u i r e d i n order to p r o v i d e a n i n t u i t i v e l y satisfactory s e m a n t i c s for Mp: "p is c o n sistent w i t h w h a t is b e l i e v e d " . T h e a v a i l a b i l i t y of the " A f  t e r m s i n the language has a d v a n t a g e s a n d d i s a d v a n t a g e s . F o r  e x a m p l e , it c a n be s h o w n t h a t sentences of the f o r m : p  3  Mq  where p a n d q are a r b i t r a r y f o r m u l a e , are either r e d u n d a n t or inconsistent [ E t h e r i n g t o n & M e r c e r , 1982, u n p u b l i s h e d notes]. a l l f o r m u l a e Mp  ( T h i s follows because the " t h e o r e m s " of a n y N M L theory m u s t i n c l u d e  w h i c h are not inconsistent.)  S u c h sentences c a n n o t be f o r m e d i n d e f a u l t logic,  b u t are r e a d i l y a v a i l a b l e i n N M L (as they are i n S a n d e w a l l ' s f o r m a l i s m ) .  A discussion of modal logics is beyond the scope of this proposal. See [Hughes and Cresswel 1972] for an introduction. 1 3  - 34 -  O n the p o s i t i v e side, the default rules c a n be m a n i p u l a t e d b y the t h e o r y . F o r e x a m p l e , i n the n o r m a l default t h e o r y w i t h no a x i o m s a n d the defaults:  n o t h i n g c a n be i n f e r r e d a b o u t B. T h e c o r r e s p o n d i n g n o n - m o n o t o n i c theory: {A  A MB  D  B, ->A  i m p l i e s M B a n d MB  A MB  D  B}  Z> B, f r o m w h i c h B c a n be inferred. T h i s a p p e a r s to be m o r e i n a c c o r d w i t h  n o r m a l commonsense reasoning. F i n a l l y , Lp = p is a thesis of N M L . W h i l e most m o d a l l o g i c i a n s w o u l d agree t h a t "p is prova b l e " i m p l i e s " p is t r u e " , the converse is u s u a l l y not a c c e p t e d . H u g h e s a n d C r e s s w e l l [1972, p28] c o n c l u d e t h a t " n o i n t u i t i v e l y p l a u s i b l e m o d a l s y s t e m " w o u l d h a v e s u c h a thesis. T h i s i n d i c a t e s t h a t there m a y be f u n d a m e n t a l p r o b l e m s w i t h N M L .  2.2.4.  Autoepistemic Logic  M o o r e [1983a, b] p r o v i d e s a d e t a i l e d c r i t i c i s m a n d r e c o n s t r u c t i o n of N M L . H e begins b y dist i n g u i s h i n g b e t w e e n d e f a u l t r e a s o n i n g a n d " a u t o e p i s t e m i c " reasoning. T h e l a t t e r is defined t o be w h a t goes on i n a n i d e a l l y r a t i o n a l agent reasoning a b o u t h e r o w n beliefs. It is t h i s t y p e of r e a s o n ing - not default r e a s o n i n g - t h a t N M L a t t e m p t s t o m o d e l , a c c o r d i n g to M o o r e . M o o r e sees a N M L a x i o m of the f o r m : Vx.Bird(x) A M[Can-Fly(x)) D Can-Fly(x)  (15)  as s a y i n g n o t " T y p i c a l b i r d s c a n fly", as M c D e r m o t t a n d D o y l e interpret it, b u t r a t h e r " T h e o n l y b i r d s w h i c h do not fly are those known not to  fly".  R e a d i n this w a y , a x i o m s s u c h as (15) become  s t a t e m e n t s a b o u t the state of a n a g e n t ' s k n o w l e d g e , not a b o u t t y p i c a l i n d i v i d u a l s .  1 4  H a v i n g m a d e t h i s d i s t i n c t i o n , M o o r e p o i n t s o u t t h a t default a n d a u t o e p i s t e m i c reasoning are n o n m o n o t o n i c for different reasons. D e f a u l t reasoning is t e n t a t i v e , a n d thus defeasible.  It p r o -  vides p l a u s i b l e g r o u n d s for h o l d i n g c e r t a i n beliefs, b u t these beliefs m a y have to be r e t r a c t e d s h o u l d those g r o u n d s p r o v e to have been m e r e l y p l a u s i b l e , r a t h e r t h a n true. soning m a k e s o n l y v a l i d inferences.  A u t o e p i s t e m i c rea-  P r o v i d e d t h a t the premises are true, the c o n c l u s i o n s f o l l o w  w i t h a l l of the force of logic b e h i n d t h e m . N o n - m o n o t o n i c i t y enters because a u t o e p i s t e m i c statem e n t s are context-sensitive or indexical. T h e y e x p l i c i l y refer to the entire k n o w l e d g e c o n t e x t t h a t c o n t a i n s t h e m . T h u s , t h e i r m e a n i n g changes d e p e n d i n g o n w h a t is k n o w n . O b v i o u s l y , w h a t f o l lows f r o m not k n o w i n g a w i l l h o l d w h e n a is not k n o w n , b u t m a y not h o l d if a is l e a r n e d . M o o r e argues t h a t the possible sets o f beliefs a n i d e a l l y r a t i o n a l agent c a n h o l d based on a consistent set of p r e m i s e s , A, are those sets, T, s u c h t h a t " Similar arguments can be applied to default logic and other consistency-based non-monotonic formalisms.  - 35 -  T = Th(A U {LP\Pe  T) U {^LP\P<£  T}) ,  where LP means "P is b e l i e v e d " . T h e s e sets he calls the stable expansions of A.  A stable e x p a n -  sion i n c l u d e s the premises, a c c u r a t e l y c h a r a c t e r i z e s w h a t is a n d w h a t is not b e l i e v e d , a n d i n c l u d e s no beliefs n o t s u p p o r t e d b y the premises. M o o r e shows t h a t stable e x p a n s i o n s c o n t a i n a l l a n d o n l y those f o r m u l a e w h i c h are true i n e v e r y i n t e r p r e t a t i o n w h i c h satisfies a l l o f the premises a n d m a k e s LP true for e v e r y f o r m u l a , P , i n the e x p a n s i o n . T h e i m p o r t a n t t h i n g t o n o t i c e here is t h a t if the premises ( w h i c h m a y c o n t a i n i m p l i c a t i o n s f r o m w h a t i s / i s not believed) are true, a n d t h e set of beliefs corresponds to the beliefs c o n t a i n e d i n a p a r t i c u l a r e x p a n s i o n , t h e n a l l o f ( a n d o n l y ) the f o r m u l a e i n t h a t e x p a n s i o n c a n be t r u e . T h i s i n t u i t i v e l y corresponds to the i d e a t h a t the different c o n c l u s i o n s one c a n d r a w f r o m i n c o m p l e t e l y specified knowledge w i l l be c o m p l e t e l y d e t e r m i n e d b y w h a t one chooses t o believe. U n l i k e N M L , a u t o e p i s t e m i c logic ( A E L ) is a p r o p o s i t i o n a l m o d a l l o g i c . N o p r o v i s i o n is m a d e for i n d i v i d u a l v a r i a b l e s o r q u a n t i f i e r s . M o o r e [1984, p e r s o n a l c o m m u n i c a t i o n ] s h o u l d be a r e l a t i v e l y easy m a t t e r to e x t e n d A E L to its  first-order  suggests t h a t  it  c o u n t e r p a r t , p r o v i d e d t h a t the  M o p e r a t o r is a p p l i e d o n l y to closed f o r m u l a e . T h i s m e a n s t h a t no M occurs w i t h i n the scope of a quantifier, so the p r o b l e m s of " q u a n t i f y i n g i n " to the scope of a m o d a l i t y are a v o i d e d . U n f o r t u n a t e l y , m a n y of the s t a t e m e n t s one w o u l d l i k e t o m a k e using a  first-order  v e r s i o n of A E L  i n v o l v e q u a n t i f y i n g i n . F o r e x a m p l e , to say " A l l of the a's are k n o w n " seems to r e q u i r e a n a x i o m of the f o r m :  Vx. a(x)  La(x)  D  ( i n N M L , t h i s w o u l d be w r i t t e n as  Vx. Af->a(x)  Z>  ->a(x)), but the free 'x' i n 'La(x)' is c a p t u r e d b y  the u n i v e r s a l q u a n t i f i e r outside the m o d a l c o n t e x t .  T h e p r o b l e m s of q u a n t i f y i n g i n are a n i m p o r -  t a n t t o p i c i n the s t u d y of m o d a l logics (c./. [ L i n s k y 1971]), b u t the p r o b l e m has y e t to be s t u d i e d i n d e p t h f r o m the perspective of n o n - m o n o t o n i c i t y . L i k e m a n y o t h e r n o n - m o n o t o n i c r e a s o n i n g systems, A E L was presented n o n - c o n s t r u c t i v e l y [Moore 1983a, b]. N e i t h e r the s e m a n t i c basis n o r the s y n t a c t i c r e a l i z a t i o n of t h a t s e m a n t i c s p r o v i d e d a m e c h a n i s m for e n u m e r a t i n g the t h e o r e m s of a g i v e n theory. M o o r e [1984] has r e c e n t l y d e v e l o p e d a n a l t e r n a t i v e s e m a n t i c c h a r a c t e r i z a t i o n based on the f a m i l i a r K r i p k e - s t y l e possiblew o r l d s structures. In t h i s s e m a n t i c s , it is possible to enumerate a l l of the i n t e r p r e t a t i o n s m o d e l is  finitely  specifiable a n d , if the language is  finite,  there are o n l y  finitely  (each  m a n y ) , decide  w h i c h o f these are m o d e l s , a n d d e t e r m i n e w h a t is true i n those. M o o r e [1984, p e r s o n a l c o m m u n i cation] p o i n t s o u t t h a t , f o r theories w i t h few p r o p o s i t i o n a l c o n s t a n t s , this is an easy task. O n e m i g h t hope for a m o r e d i r e c t m e a n s o f a r r i v i n g at the theorems of a n a u t o e p i s t e m i c t h e o r y , b u t this r e m a i n s a n o p e n p r o b l e m .  2.2.5.  K F O P C  Levesque  [1982,  1984]  a p p r o a c h e s the  problem  of  incomplete  information  differently.  Instead of i m m e d i a t e l y a d d r e s s i n g the task of c o m p l e t i n g the i n c o m p l e t e n e s s , he considers w h a t an i n c o m p l e t e k n o w l e d g e - b a s e m i g h t be e x p e c t e d t o k n o w about its o w n k n o w l e d g e , a n d w h a t one  - 36 -  m i g h t r e a s o n a b l y ask o r t e l l s u c h a k n o w l e d g e base.  T h i s entails q u e s t i o n s of w h a t constitutes a  reasonable a n s w e r to a q u e r y u n d e r i n c o m p l e t e n e s s , a n d w h a t a n i n c o m p l e t e k n o w l e d g e base c a n be e x p e c t e d to k n o w after b e i n g t o l d a p a r t i c u l a r fact. A f t e r d e v e l o p i n g a l o g i c a l f r a m e w o r k w h i c h to t a l k a b o u t (incomplete) k n o w l e d g e , Levesque discusses w a y s i n w h i c h t h i s  in  framework  c a n be a p p l i e d t o c o m p l e t i n g k n o w l e d g e bases. I n f o r m a l i z i n g k n o w l e d g e (or " r a t i o n a l b e l i e f ) , L e v e s q u e [1982] m a k e s three basic a s s u m p tions a b o u t t h e n a t u r e o f k n o w l e d g e bases.  These are:  1) C o n s i s t e n c y : T h e k n o w l e d g e i n a k n o w l e d g e base is self-consistent. state-of-affairs t h a t m a k e s e v e r y t h i n g t h a t is k n o w n true.  T h e r e is some possible  2) C o m p e t e n c e : E v e r y u n k n o w n sentence is false i n some w o r l d c o m p a t i b l e w i t h e v e r y t h i n g t h a t is k n o w n . I.e., a l l l o g i c a l consequences o f w h a t is k n o w n are k n o w n . 3) C l o s u r e : T h e k n o w l e d g e base has c o m p l e t e a n d accurate self-knowledge. A n y sentence w h i c h deals o n l y w i t h the state of the k n o w l e d g e base w i l l be k n o w n to be t r u e o r false a c c o r d i n g as it is t r u e o r false, r e s p e c t i v e l y , for the k n o w l e d g e base. These assumptions lead to a  first-order  m o d a l logic of k n o w l e d g e ( K F O P C ) , r o u g h l y s i m i l a r  t o " w e a k " S 5 [Stalnaker 1980]. T h i s logic characterizes the beliefs of a n i d e a l l y r a t i o n a l agent c a p a b l e of reflecting o n h e r o w n beliefs. A n y sentence w h i c h is n o t k n o w n is k n o w n to be u n k n o w n ( a n d vice versa), k n o w l e d g e is l o g i c a l l y closed, a n d the agent believes i n the v e r a c i t y of her beliefs. T h i s last s t a t e m e n t does not m e a n t h a t e v e r y belief is true, m e r e l y that the agent believes a l l of h e r beliefs t o be true. O n e of the m o s t s u r p r i s i n g aspects of this s y s t e m is t h a t , a l t h o u g h K F O P C a l l o w s one to t e l l / a s k the k n o w l e d g e base t h i n g s t h a t c a n n o t generally be phrased i n a n d u p d a t e m e c h a n i s m s , a n d the k n o w l e d g e base itself, are a l l  first-order  first-order  logic, the q u e r y  representable. L e v e s q u e  presents a f u n c t i o n t h a t translates a n y u p d a t e or query i n t o a n e q u i v a l e n t  first-order  sentence  w i t h respect t o a p a r t i c u l a r k n o w l e d g e base.  U n f o r t u n a t e l y , the m a p p i n g is not a p a r t i a l recur-  sive f u n c t i o n : I n the g e n e r a l case, c h o o s i n g  first-order  must  r e p r e s e n t a b i l i t y means t h a t effectiveness  be t r a d e d for some h e u r i s t i c c o m p o n e n t . L e v e s q u e does not explore w h e t h e r  there  are  effective subcases. N o n - m o n o t o n i c i t y enters K F O P C a s s u m p t i o n of closure. C l e a r l y , t h o u g h , KB  T h a t is, if KB  in two ways.  T h e first, m o s t o b v i o u s , comes f r o m the  does not k n o w P, it k n o w s this (i.e., KB  U {P} \- KP, a n d hence KB  U {P} \j- K^KP.  \—K--KP).  A s L e v e s q u e p o i n t s out,  m e a n s o n l y t h a t P is not c u r r e n t l y k n o w n , not t h a t it w i l l never be k n o w n .  L e v e s q u e ' s s o l u t i o n to  this p r o b l e m is to h a v e self-knowledge c o m e o n l y f r o m i n t r o s p e c t i o n : f r o m i m p l i c i t , r a t h e r e x p l i c i t , s t a t e m e n t s . S u c h a KB  ^KP than  w i l l n e v e r c o n t a i n statements of the f o r m ->KP:  " . . . a n y assertion w i l l be a s t a t e m e n t a b o u t the w o r l d a n d not the KB. If the assertion t a l k s a b o u t w h a t is k n o w n ... it is o n l y d o i n g so to h e l p m a k e a s t a t e m e n t a b o u t the w o r l d . T h u s , there is no w a y to t e l l a KB a b o u t itself... . T h i s is to be e x p e c t e d , h o w e v er, since a KB has b e e n assumed t o h a v e c o m p l e t e a n d a c c u r a t e k n o w l e d g e of itself at any time." T h e n o n - m o n o t o n i c i t y of s t a t e m e n t s a b o u t l a c k of k n o w l e d g e is not p a r t i c u l a r l y p r o b l e m a t i c w h e n treated in this way.  - 37 A s e c o n d f o r m of n o n - m o n o t o n i c i t y arises w h e n l a c k of k n o w l e d g e is used as a premise i n deductions.  F o r e x a m p l e , the " F l y i n g B i r d s " default c a n be expressed in K F O P C as:  V*. Bird(x) A ^K-^Fly(x) D Fly(x)  (16)  - a n y b i r d n o t k n o w n n o t t o fly c a n fly. S u c h s t a t e m e n t s a l l o w c o n c l u s i o n s a b o u t w h a t is t r u e i n the w o r l d t o be based o n w h a t is not k n o w n b y the c u r r e n t knowledge base. S i n c e the state of the k n o w l e d g e base c a n c h a n g e , n o n - m o n o t o n i c i t y c a n e x t e n d to encompass more t h a n p u r e l y introspective statements. T h i s r e p r e s e n t a t i o n of defaults i n K F O P C is subject to the p r o b l e m s of i n t e r a c t i o n t h a t plague n o r m a l d e f a u l t theories [Reiter & C r i s c u o l o 1983].  F o r e x a m p l e , the fact t h a t A u s t r a l i a n  b i r d s are non-fliers b y default c a n be w r i t t e n : Vz. Australian-bird(x)  D Bird(x)  Vz. Australian-bird(x)  A ->KFly(x) ^ ->Fly(x)  b u t A u s t r a l i a n b i r d s c a n also b e c o n j e c t u r e d t o fly b y v i r t u e of being b i r d s , t h r o u g h (16).  Again,  the logic p r o v i d e s no m e a n s of d e c i d i n g b e t w e e n these a l t e r n a t i v e s . A m o r e serious p r o b l e m arises w h e n the k n o w l e d g e base k n o w s t h a t some u n k n o w n i n d i v i d u a l is a t y p i c a l . F o r e x a m p l e , the k n o w l e d g e base: Bird(Bl), Bird(B2), ^Fly{Bl)  V ^Fly{B2)  is inconsistent w i t h the default (16) because there is a b i r d ( 5 1 o r B2), not k n o w n not to w h i c h nonetheless does not  fly.  fly,  I n fact, this type of reasoning is not r e a l l y d e f a u l t r e a s o n i n g at  a l l , b u t w h a t M o o r e [1983a] calls " A u t o e p i s t e m i c R e a s o n i n g " . W h a t is r e a l l y i n v o l v e d is a v a l i d f o r m o f inference: f r o m the premise t h a t a l l e x c e p t i o n a l cases are k n o w n to the c o n c l u s i o n t h a t a n i n d i v i d u a l , not k n o w n t o be e x c e p t i o n a l , is t y p i c a l .  If a l l of the e x c e p t i o n a l cases are not k n o w n ,  as i n the a b o v e e x a m p l e , t h e n the " d e f a u l t " is s i m p l y false. T h i s means t h a t d e f a u l t s c a n n o t s i m p l y be s t a t e d as a x i o m s , since the logic is too r i g i d to a l l o w for o c c a s i o n a l v i o l a t i o n s b y p a r t i c u l a r individuals. L e v e s q u e addresses these p r o b l e m s b y " p r e p r o c e s s i n g " the defaults. T h i s is done b y means of a m a p p i n g o n d e f a u l t s t h a t rejects a n y w h i c h are c o n t r a d i c t e d b y the k n o w l e d g e base or w h i c h , given the current  state of the k n o w l e d g e base, c o n f l i c t w i t h a n o t h e r default.  All c o n f l i c t i n g  defaults are r e j e c t e d . T h i s leads t o a s t r o n g l y c o n s e r v a t i v e i n t e r p r e t a t i o n of defaults.  If t w o d e f a u l t s c a n n o t be  a p p l i e d t o g e t h e r because t h e y are m u t u a l l y i n c o n s i s t e n t , n e i t h e r w i l l be a p p l i e d . N o t h a v i n g a n y grounds for deciding between them, K F O P C m o n o t o n i c logic [ M c D e r m o t t p o i n t o f its theories.  chooses t o reject b o t h . T h i s is s i m i l a r to n o n -  & D o y l e 1980], w h i c h sanctions o n l y those beliefs i n e v e r y f i x e d -  In c o n t r a s t , d e f a u l t logic [Reiter 1980] a n d a u t o - e p i s t e m i c logic  [Moore  1983a] s a n c t i o n m u l t i p l e sets of beliefs, one s u p p o r t i n g e a c h of the a l t e r n a t i v e s . C i r c u m s c r i p t i v e default r e a s o n i n g c o n j e c t u r e s the d i s j u n c t i o n of the t w o a l t e r n a t i v e s .  T h e o n l y defaults a s s u m e d  i n K F O P C are those w h i c h are a s s u m a b l e i n d e p e n d e n t l y of a l l other defaults.  - 38 -  L e v e s q u e discusses some i n t e r e s t i n g techniques f o r c i r c u m v e n t i n g these difficulties i n c e r t a i n cases.  T o a v o i d t h e p r o b l e m o f i n t e r a c t i n g defaults w i t h o u t r e q u i r i n g defaults t o e x p l i c i t l y a l l o w  for e x c e p t i o n a l cases, L e v e s q u e suggests a r e p r e s e n t a t i o n scheme i n v o l v i n g a " t y p i c a l " - p r e d i c a t e f o r m i n g o p e r a t o r , \7:k. w i t h respect t o t h e k  tl>  order axioms.  If P is a p r e d i c a t e letter, S?:kP(x) is i n t e r p r e t e d as s a y i n g t h a t x*is t y p i c a l  aspect of P n e s s .  T h e properties of t y p i c a l i n d i v i d u a l s a r e s t a t e d as first-  F o r e x a m p l e , t h e KB:  V z . V : l 5 i ' r d ( z ) 3 Fly(x) Vz. \7:lBird(x) 3 -<Australian-bird(x) Vz. Australian-bird(x)  3 Bird(x)  V z . V:1 Australian-bird(x)  3 -iFly(x)  says t h a t a l l b i r d s t y p i c a l i n aspect 1 of " B i r d n e s s " fly a n d are n o t A u s t r a l i a n - b i r d s , S7:lAustralian-birds  while  a r e b i r d s w h i c h d o n o t fly. S u c h a x i o m s d o n o t state d e f a u l t properties of  classes of i n d i v i d u a l s . R a t h e r , they state properties t h a t a l l typical i n d i v i d u a l s of those classes must ( o r m u s t n o t ) h a v e . D e f a u l t r e a s o n i n g is p e r f o r m e d b y c o n j e c t u r i n g t h a t i n d i v i d u a l s are t y p i c a l . F o r e x a m p l e , the s t a t e m e n t s : V z . Bird(x) A -.if->V:l5i'r<i(z) 3 S7:lBird(x) Vz. Australian—bird(x)  (17)  A ->K->S7:lAustralian-bird(x) Z> \/:\Australian-bird(x)  (18)  say t h a t - unless k n o w n otherwise - b i r d s a n d A u s t r a l i a n birds are t y p i c a l i n t h e specified aspects. If Tweety is a b i r d n o t k n o w n n o t t o fly ( n o r be otherwise a t y p i c a l i n aspect 1), (17) says t h a t she is t y p i c a l i n a s p e c t 1, a n d hence flies. A n A u s t r a l i a n b i r d , Oscar, o n the o t h e r h a n d , i s k n o w n to be a n a t y p i c a l b i r d i n aspect 1, so (17) i s i n a p p l i c a b l e . T h e default (18) is a p p l i c a b l e , h o w e v e r , so ^:lAustralian—bird(Oscar)  c a n be c o n j e c t u r e d , a n d hence -<Fly(Oscar).  T o p r e v e n t d e f a u l t s w h i c h are c o n t r a d i c t e d f o r p a r t i c u l a r i n d i v i d u a l s (or classes) f r o m b e i n g rejected o u t r i g h t , a x i o m s c a n be a d d e d w h i c h e x p l i c i t l y state t h a t those i n d i v i d u a l s ( o r m e m b e r s of those classes) are a t y p i c a l i n t h e r e l e v a n t aspects. F o r e x a m p l e , g i v e n t h e k n o w l e d g e base: Quaker(john),  V z . S7Quaker(x) 3 Pacifist(x),  Republican(george),  V z . VRepublican(x) 3  -iPacifist(x),  Quaker(nixon) A Republican(nixon), V z . Quaker(x) A -<K-> 7Quaker(x) 3 VQuaker(x) J  V z . Republican(x) A -<K-i\7Republican(x) 3 VRepublican(x)  (19) (20)  the defaults, (19) a n d ( 2 0 ) , s t a t i n g t h a t R e p u b l i c a n s a n d Q u a k e r s are t y p i c a l R e p u b l i c a n s a n d Q u a k e r s , r e s p e c t i v e l y , are n o t a p p l i c a b l e f o r a n y i n d i v i d u a l because t h e y are m u t u a l l y c o n t r a d i c t o r y f o r nixon. nixon v i o l a t e s t h e defaults:  he is n o t k n o w n t o be a n a t y p i c a l Q u a k e r , so (19)  s a n c t i o n s VQuaker(nixon); s i m i l a r l y , (20) s a n c t i o n s S7Republican(nixon); b u t these c o n c l u s i o n s are m u t u a l l y i n c o n s i s t e n t .  U n d e r t h e c o n s e r v a t i v e i n t e r p r e t a t i o n o f defaults, K F O P C rejects b o t h  (19) a n d (20) f o r this k n o w l e d g e - b a s e . H e n c e , John a n d george c a n n o t be c o n c l u d e d t o be t y p i c a l , a n d so Pacifist(john) a n d ->Pacifist(george) c a n n o t be c o n c l u d e d . T o r e m e d y t h i s , a x i o m s s t a t i n g t h a t t h e t y p i c a l R e p u b l i c a n is n o t a Q u a k e r ( a n d vice versa) c a n be a d d e d :  - 39 -  Vx. VQuaker(x) D -iRepublican(x) Vx. VRepublican(x) O -<Q,uaker(x) . W i t h these a d d i t i o n a l facts, nixon n o longer c o n s t i t u t e s a v i o l a t i o n of the defaults, since K->S7Republican(nixon) A i f - ' V Quaker(nixon) follows f r o m t h e k n o w l e d g e base. T y p i c a l - p r e d i c a t e s c a n also be used to specify a precedence-hierarchy a m o n g p o t e n t i a l l y - c o n f l i c t i n g , defaults.  multiple,  F o r e x a m p l e , the a x i o m s :  Vx. V-lstudent(x) D undergrad(x) Vx. S7:1student(x) D undergrad(x) V MSc(x) Vx. V:3student(x) D undergrad(x) V MSc(x) V PhD(x) together w i t h t h e defaults: Vx. student(x) A -<K-<S7:kstudent(x) D S7:kstudent(x)  for k = 1,2,3.  (21)  w i l l result i n a t h e o r y t h a t w i l l assume students are undergrades if possible, o t h e r w i s e MSc's i f possible, a n d o t h e r w i s e PhD's if possible. L e v e s q u e gives n u m e r o u s e x a m p l e s s h o w i n g t h a t these strategies c a n be c o m b i n e d t o o b t a i n r e m a r k a b l y subtle c o n t r o l of t h e i n t e r a c t i o n s b e t w e e n defaults, w i t h o u t m o d i f y i n g t h e s t r u c t u r e of the defaults t h e m s e l v e s . A l l t h a t is r e q u i r e d is the a d d i t i o n of n e w a x i o m s . H e cites three a d v a n tages of t h i s f o r m a l i z a t i o n of default r e a s o n i n g : 1) A default n e e d n o t be d i s c a r d e d a n d r e p l a c e d w h e n a subclass t h a t t y p i c a l l y fails t o satisfy the default is d i s c o v e r e d . A d d i t i o n a l a x i o m s c a n be a d d e d , s t a t i n g the i n a p p l i c a b i l i t y of the default for m e m b e r s o f t h a t subclass. 2) T h e k n o w l e d g e g i v e n t o the k n o w l e d g e base is m o r e s t r u c t u r e d . Instead of a r b i t r a r y defaults, properties of t y p i c a l i n d i v i d u a l s are l i s t e d . 3) O n l y a single t y p e of default (e.g., (21)) need be considered, a n d o n l y one of these for each t y p i c a l - p r e d i c a t e , S7:kP. L e v e s q u e ' s use of t y p i c a l - p r e d i c a t e s as a representation scheme for defaults corresponds d i r e c t l y to M c C a r t h y ' s subsequent use of a b n o r m a l i t y - p r e d i c a t e s .  T h e r e is a s t r a i g h t f o r w a r d m a p -  p i n g b e t w e e n L e v e s q u e ' s a x i o m a t i z a t i o n s using V - p r e d i c a t e s a n d M c C a r t h y ' s u s i n g  ab-predicates.  T h e defaults, Vx. Px A ~<K->S7'-kPx D *7:kPx, t h e n c o r r e s p o n d to m i n i m i z a t i o n s of the c o r r e s p o n d ing abnormality-predicates.  It is n o t y e t k n o w n w h e t h e r this m a p p i n g c o n s t i t u t e s a t r a n s l a t i o n ,  w h e t h e r t h e t w o a p p r o a c h e s l e a d t o the same conjectures for c o r r e s p o n d i n g default T h e r e are s t r i k i n g s i m i l a r i t i e s , however.  theories.  F o r e x a m p l e , t a n g l e d hierarchies, w h e r e i n m e m b e r s of  one class m a y t y p i c a l l y — b u t not a l w a y s - be m e m b e r s of another, are p r o b l e m a t i c i n b o t h p a r a digms.  McCarthy's "Gangster and Canaries"  e x a m p l e , discussed earlier, requires  K F O P C rules, b e y o n d those r e q u i r e d b y the s t r a i g h f o r w a r d a b n o r m a l i t y / t y p i c a l i t y to express p r i o r i t i e s o r preferences for p a r t i c u l a r k i n d s of a t y p i c a l i t y .  additional  representation,  C a p t u r i n g these p r i o r i t i e s i n  K F O P C a p p e a r s t o i n v o l v e a loss o f c l a r i t y a n d naturalness of r e p r e s e n t a t i o n s i m i l a r  to that  i n c u r r e d b y M c C a r t h y ' s i n t r o d u c t i o n of priorities i n t o c i r c u m s c r i p t i v e a b n o r m a l i t y theories.  r - 40 -  In spite o f L e v e s q u e ' s insights i n t o representing default k n o w l e d g e , default r e a s o n i n g i n K F O P C remains largely unexplored. (abnormal-)  S i m i l a r l y , the a p p l i c a t i o n of L e v e s q u e ' s i d e a s o n t y p i c a l -  p r e d i c a t e s to default reasoning based o n other f o r m a l i s m s has o n l y b e g u n . B o t h of  these areas p r o m i s e to p r o v i d e i m p o r t a n t  insights i n t o reasoning a b o u t i n c o m p l e t e l y specified  w o r l d s , a n d deserve f u r t h e r e x p l o r a t i o n .  2.2.6.  Objections to N o n - M o n o t o n i c Formalisms  K r a m o s i l [1975] c l a i m s to have s h o w n t h a t a n y f o r m a l i z e d t h e o r y w h i c h a l l o w s u n p r o v a b i l i t y as a  premise  corresponding  in  deductions must  first-order  either  be " m e a n i n g l e s s ' ' , or  no  more  powerful  than  the  t h e o r y w i t h o u t rules i n v o l v i n g s u c h premises. H e presents t w o " p r o o f s "  to s u p p o r t h i s c l a i m . C a r e f u l e x a m i n a t i o n shows t h a t the first result follows f r o m a d e f i n i t i o n of "formalized theory"  w h i c h expressly excludes a n y t h e o r y w h i c h e x h i b i t s the t y p e s o f b e h a v i o r  common to non-monotonic  theories. T h e second result is based o n a n i n c o r r e c t d e f i n i t i o n  of  " p r o o f " a n d hence of " t h e o r e m h o o d " a n d is itself meaningless. A s the p a p e r stands, it shows o n l y t h a t n o n - m o n o t o n i c theories m u s t behave differently t h a n m o n o t o n i c theories i n those cases where the f o r m e r c a n d e r i v e results u n o b t a i n a b l e u s i n g the l a t t e r . K r a m o s i l w a s not the o n l y one to be u n c o m f o r t a b l e w i t h o p e n i n g the " P a n d o r a ' s B o x " of n o n - m o n o t o n i c i t y . S a n d e w a l l [1972] notes that the " U n l e s s " o p e r a t o r has " s o m e d i r t y l o g i c a l p r o p e r t i e s " . C o n s i d e r i n g the e x a m p l e : A A A Urdess(B) ^  C  A A Utdess(C) D B he observes t h a t e i t h e r B a n d C c a n be theorems, b u t , i n general, not b o t h s i m u l t a n e o u s l y .  Reiter  [1978b] m a k e s a s i m i l a r o b s e r v a t i o n i n a n e a r l y paper, s t a t i n g t h a t : S u c h b e h a v i o r , [is] c l e a r l y u n a c c e p t a b l e ; A t the v e r y least, we m u s t d e m a n d of a default t h e o r y t h a t it satisfy a k i n d of ' C h u r c h - R o s s e r ' p r o p e r t y : N o m a t t e r w h a t the order i n w h i c h the t h e o r e m s of a t h e o r y are d e r i v e d , the r e s u l t i n g set of theorems w i l l be u n i q u e .  It  appears t h a t  the  Church-Rosser  property  is a necessary c a s u a l t y if  non-monotonicity  is  accepted'. A f u r t h e r p r o b l e m w h i c h m u s t be faced b y those e m b r a c i n g c o n s i s t e n c y - or u n p r o v a b i l i t y based a p p r o a c h e s to n o n - m o n o t o n i c i t y is t h a t the n o n - t h e o r e m s of a  first-order  t h e o r y are not  r e c u r s i v e l y e n u m e r a b l e . T h i s means t h a t the rules of inference in theories i n v o l v i n g the  •)/- o p e r a -  tor c a n n o t be effective i n generaL It follows that the t h e o r e m s are not r e c u r s i v e l y e n u m e r a b l e . B y c o n t r a s t , i n m o n o t o n i c logics,, the r u l e s of inference M U S T be effective a n d the t h e o r e m s M U S T be r e c u r s i v e l y e n u m e r a b l e . Finally,  the  very  non-monotonicity  which  m a k e s s u c h theories interesting  means  that  " t h e o r e m s " m a y h a v e t o be r e t r a c t e d i f the a s s u m p t i o n s o n w h i c h t h e y are b a s e d are refuted (either b y new k n o w l e d g e o r changes i n the state of the w o r l d ) .  T o be useful, a n o n - m o n o t o n i c  - 41 -  reasoning s y s t e m m u s t be able to r e m e m b e r w h i c h a s s u m p t i o n s u n d e r l y e a c h t h e o r e m a n d be able to u n w i n d the p o t e n t i a l l y c o m p l e x c h a i n of d e d u c t i o n s f o u n d e d on r e t r a c t e d j u s t i f i c a t i o n s .  CHAPTER  3  Default Logic  If the w h e e l is fixed, I w o u l d still take a chance. If w e ' r e t r e a d i n g o n t h i n ice, T h e n we m i g h t as w e l l d a n c e . — Jesse W i n c h e s t e r I n t h i s c h a p t e r , we explore default  logic i n some d e t a i l .  W e present a m o d e l - t h e o r e t i c  s e m a n t i c s for a r b i t r a r y default theories, thus r e c t i f y i n g a m a j o r deficiency. t i o n s i n v e s t i g a t e t h e causes of incoherence i n c e r t a i n default theories.  T h e r e m a i n i n g sec-  T h i s leads to a strong  sufficient ( a l t h o u g h n o t necessary) s y n t a c t i c c o n d i t i o n for the existence of extensions for p a r t i c u l a r theories.  3.1. T h e S e m a n t i c s of D e f a u l t T h e o r i e s  In his d e v e l o p m e n t of default logic, R e i t e r p r o v i d e d a  fixed-point  c h a r a c t e r i z a t i o n of the  extensions of a default t h e o r y , b u t no m o d e l - t h e o r e t i c s e m a n t i c s for the logic.  E t h e r i n g t o n [1982,  1983] observes t h a t the s e m a n t i c s c a n be v i e w e d i n terms of restrictions of the set of m o d e l s of the underlying  theory,  tyukaszewicz  [1985] formalizes this i n t u i t i o n for  n o r m a l default  theories.  B e c a u s e of the w e l l - b e h a v e d nature of these theories, this is r e l a t i v e l y s t r a i g h t f o r w a r d .  The  r e s u l t i n g s e m a n t i c c h a r a c t e r i z a t i o n a m o u n t s to c o n s i d e r i n g the T a r s k i a n s e m a n t i c s of e a c h of the p a r t i a l extensions c o n s t r u c t e d b y p r o c e d i n g m o n o t o n i c a l l y t o w a r d a n e x t e n s i o n b y s a t i s f y i n g , at e a c h step, the  next  a p p l i c a b l e n o r m a l default  defaults) b y m a k i n g its consequent true.  ( a c c o r d i n g to some a r b i t r a r y  o r d e r i n g of  s i d e r e d , n o m o r e defaults f r o m D are a p p l i c a b l e , the r e s u l t i n g set, together w i t h the theory,  the  If, after e a c h default i n the sequence has been c o n first-order  W, y i e l d s a n e x t e n s i o n . S i n c e e a c h step affirms a f o r m u l a consistent w i t h those affirmed  p r e v i o u s l y , the set of m o d e l s c o n t r a c t s m o n o t o n i c a l l y .  T h e i n t e r s e c t i o n of the sets o f m o d e l s f r o m  each stage is p r e c i s e l y the set of m o d e l s o f the e x t e n s i o n . T h i s s e m a n t i c s c a n p e r h a p s best be envisaged as a t r a n s i t i o n n e t w o r k , whose nodes are s u b sets o f M , the set of a l l m o d e l s of W, w i t h arcs l a b e l l e d b y defaults, as follows: F r o m the node c o r r e s p o n d i n g t o a set o f m o d e l s N , f o r e v e r y 6 — if n o m o d e l i n N  a  * ^ € D, a n arc l a b e l l e d 6 leads (i) b a c k t o P  N  satisfies f$ o r some satisfy - i a , o r (ii) to the node c o r r e s p o n d i n g to the set:  - 42 -  - 43 -  { N | N €E N a n d N |= p }, otherwise. reachable from M  E a c h leaf - a node a l l of whose o u t b o u n d l i n k s l o o p b a c k -  c o r r e s p o n d s t o the set o f m o d e l s of some e x t e n s i o n of A . F u r t h e r m o r e , the set  of m o d e l s of e a c h e x t e n s i o n of A corresponds to s u c h a leaf node. T h e set of a r c - l a b e l s for every p a t h f r o m r o o t t o leaf gives the g e n e r a t i n g defaults for the e x t e n s i o n c o r r e s p o n d i n g to the node. T h i s a p p r o a c h does n o t a p p l y d i r e c t l y to n o n - n o r m a l defaults, since the p r o p e r t y of s e m i m o n o t o n i c i t y w h i c h guarantees its success holds o n l y f o r n o r m a l defaults [Reiter 1980a, t h e o r e m 3.2].  J i u k a s z e w i c z p a r t i a l l y addresses this p r o b l e m b y p r e s e n t i n g a t r a n s l a t i o n scheme f r o m n o n -  n o r m a l defaults to n o r m a l defaults.  H e argues t h a t , of single-justification d e f a u l t s , o n l y n o r m a l  a n d s e m i - n o r m a l d e f a u l t s h a v e reasonable i n t e r p r e t a t i o n s . N o n - s e m i - n o r m a l defaults are therefore t r a n s l a t e d to s e m i - n o r m a l defaults b y c o n j o i n i n g the consequent to the j u s t i f i c a t i o n :  a •• P _ 7 The  « : M 7 7  t r a n s l a t i o n f r o m s e m i - n o r m a l t o n o r m a l , w h i c h is s o m e w h a t more c o n t r o v e r s i a l , involves  r e p l a c i n g the c o n s e q u e n t w i t h the j u s t i f i c a t i o n :  a : PA 7 _ 7  ~*  T h i s m a k e s sense,  PA 7 P/\l  <* =  fyukaszewicz  argues, so l o n g as a ' s w h i c h are also •y's are t y p i c a l l y /?'s. T h a t is,  so l o n g as one c o u l d r e a s o n a b l y a u g m e n t the t h e o r y w i t h  a A 7 :P P O n e c a n i m a g i n e s i t u a t i o n s where this is not a p p r o p r i a t e .  F o r e x a m p l e , a s y s t e m for l e g a l  reasoning m i g h t w a n t to h a v e a r u l e suggesting t h a t those w i t h m o t i v e s w h o might be g u i l t y s h o u l d be suspects: has-motive(x) : guilty(x) suspect{x) It is c l e a r l y reasonable to t r a n s l a t e this to: has-motive(x) : suspect(x) A guilty(x) suspect(x)  '  a l l o w i n g t h a t there m a y be reasons not to i n c l u d e someone o n the list of suspects even w i t h o u t knowing their innocence.  It  is not reasonable to f o l l o w t h r o u g h  b y asserting the g u i l t of a l l  suspects: has-motive(x) : suspect(x) A guilty(x) ^ suspect(x) A guilty(x) T h u s , w h i l e the s e m a n t i c s  tyukaszewicz  o u t l i n e s covers m a n y cases, there is reason to w a n t  a  s e m a n t i c s w h i c h c o v e r s m o r e t h a n n o r m a l defaults. T o this we now t u r n . B e c a u s e o f the f a i l u r e of s e m i - m o n o t o n i c i t y for n o n - n o r m a l theories, s i m p l y a p p l y i n g one default after a n o t h e r w i l l not, i n general, lead to extensions.  It is necessary to ensure t h a t the  a p p l i c a t i o n of e a c h d e f a u l t does not v i o l a t e the j u s t i f i c a t i o n s of a l r e a d y - a p p l i e d defaults.  If we  a u g m e n t ^ u k a s z e w i c z ' s s e m a n t i c s b y e n c o d i n g some i n f o r m a t i o n i n e a c h state a b o u t the set of  - 44 -  defaults w h i c h l e d to a p a r t i c u l a r state, we c a n d e t e r m i n e w h e t h e r a node is o n a v i a b l e p a t h t o w a r d a n e x t e n s i o n . T h e precise details are these:  Definition:  Satisfiability, admissibility, a n d applicability  L e t X be a set of m o d e l s ; T a set o f f o r m u l a e ; a, B, a n d w f o r m u l a e , a n d 8 =  a : B — w  a  default. T h e n i)  a is X-satisfiable (X-valid) iff ^x € X. x j= a  ii)  r is X-admissible (X permits T) iff V 7 e" I\ 3 z € X . x {= 7  ( V i G X. x \= a)  iii) 8 is X-applicable iff a is X v a l i d a n d B is X - s a t i s f i a b l e .  Definition:  |  R e s u l t of a default  L e t X, T , a n d 8 be as above. T h e n the result o f 8 i n (X, T) i s :  {  ( X , T) i f 8 is n o t X - a p p l i c a b l e a n d T is X - a d m i s s i b l e , \{X-{N\  Definition:  I  N\= n w } ) , ( r U {6})) i f 8 is X - a p p l i c a b l e a n d T is X - a d m i s s i b l e , a n d  otherwise.  |  R e s u l t o f a sequence o f defaults  L e t X a n d T be as a b o v e , a n d let <£,-> be a sequence of defaults.  T h e n the result o f <£,->  is: < 5 , > ( X , T) = ( D X,-, U r ) where J X = X ; t  \(X , i+1  Definition:  r  0  T ) i+1  0  = T;  =S,{X„I\),  and i > 0 .  I  Stability  L e t Y be a n o n - e m p t y set o f models, T a set of f o r m u l a e , a n d A = (D, W) a d e f a u l t theory. T h e n (Y, T) is stable for A iff (1)  (Y, T) = <8i>(X, { }) for X = {M\ Mf=  (2)  W e D . <5( K, T) = ( 7 , T) , a n d  (3)  r is F - a d m i s s i b l e .  W), a n d some { £ , } C £ > ,  |  In o t h e r w o r d s , a set o f m o d e l s a n d a set of c o n s t r a i n t s is stable f o r a default t h e o r y , (D,W), if t h e y are t h e result o f some sequence o f defaults i n D a p p l i e d to the set of m o d e l s o f W a n d n o c o n s t r a i n t s , i f n o default i n D produces a n y change i n t h i s result, a n d the c o n s t r a i n t s are satisfied  - 45 -  b y the set of models.  N o t e t h a t c o n d i t i o n ( 2 ) , together w i t h the d e f i n i t i o n of " r e s u l t " means t h a t  c o n d i t i o n (3) i s r e d u n d a n t .  W e i n c l u d e it f o r c o n c e p t u a l c l a r i t y .  T h e soundness a n d c o m p l e t e -  ness results f o r t h i s s e m a n t i c s are g i v e n b y T h e o r e m s 3.1 a n d 3.2, r e s p e c t i v e l y .  Theorem 3.1 — Soundness If E is a n e x t e n s i o n f o r A , t h e n there is some set T s u c h t h a t  ({M\M\= E), Y) is stable f o r A .  Theorem 3.2 — Completeness If (X, T ) is stable f o r A t h e n X is t h e set of m o d e l s for some e x t e n s i o n of A . (I.e., Th(X) is a n e x t e n s i o n f o r A . )  |  R e t u r n i n g t o t h e t r a n s i t i o n n e t w o r k a n a l o g y , the nodes are n o w pairs c o n s i s t i n g of a subset of M a n d a subset of t h e j u s t i f i c a t i o n s o f the defaults i n D. N o w A ' s extensions c o r r e s p o n d t o those leaf nodes, (X, T), w h e r e X p e r m i t s I\ are  t h e t h e o r y h a s n o extensions.  W e s a y t h a t s u c h nodes are viable. If a l l leaf nodes  A g a i n , t h e generating d e f a u l t s f o r t h e e x t e n s i o n Th(X) are  those defaults l a b e l l i n g arcs o n a n y p a t h f r o m ( M , { }) to (X, T ), f o r a n y 1 .  Example 3.1 C o n s i d e r t h e default t h e o r y :  T h i s p r o d u c e s the f o l l o w i n g t r a n s i t i o n  network.  ({Af | Af j=  ({M| M ( = A,B), {BA-iC})  A}, { })  ({M\ M(= A,C\, {Cf\^B})  B o t h leaves are v i a b l e , so t h e t h e o r y h a s t w o extensions, Th({A, B}) a n d Th({A, C}). |  - 46 -  Example 3.2 T h e incoherent theory:  { } gives rise t o :  ({M\M\=py^p},{  })  ( { M | M\= A}, {^A})  i n w h i c h t h e leaf is not v i a b l e . H e n c e t h i s t h e o r y h a s no e x t e n s i o n .  I  It i s i n s t r u c t i v e t o c o m p a r e this model-set r e s t r i c t i o n semantics w i t h t h e m i n i m a l - m o d e l s e m a n t i c s of c l o s e d - w o r l d r e a s o n i n g presented i n c h a p t e r 2. T h e r e , the s e m a n t i c s o f the closure of a theory w a s defined i n t e r m s of a r e s t r i c t i o n o f the set of models of t h e u n d e r l y i n g a c c o r d i n g t o t h e p r i n c i p l e of m i n i m i z a t i o n .  theory,  T h e model-set r e s t r i c t i o n s e m a n t i c s f o r default logic  s i m i l a r l y p r o v i d e s a p r i n c i p l e f o r d e t e r m i n i n g subsets o f the models o f a  first-order  theory w h i c h  c h a r a c t e r i z e a c c e p t a b l e belief-sets, o n the basis of m a x i m a l s a t i s f a c t i o n of t h e set o f defaults. T h e r e are s e v e r a l significant differences, however.  Firstly, rather than an ordering o n individual  m o d e l s , this s e m a n t i c s imposes a n o r d e r i n g o n sets o f m o d e l s . S e c o n d l y , t h e o r d e r i n g is defined i n t e r m s of a c c e s s i b i l i t y v i a a sequence o f defaults, r a t h e r t h a n s t r i c t l y i n t e r m s o f i n t r i n s i c features of the m o d e l s themselves. F i n a l l y , e a c h e x t e n s i o n i s d e t e r m i n e d b y a single e x t r e m u m o f the o r d e r i n g , r a t h e r t h a n b y the set of a l l e x t r e m a . T h e first o f these differences results because t h e extensions o f a default t h e o r y — u n l i k e t h e m o d e l s of a first-order t h e o r y - are n o t c o m p l e t e . T h e y d o not decide every f o r m u l a . B e c a u s e t h e y i n c o m p l e t e l y specify t h e w o r l d , sets o f m o d e l s - r a t h e r t h a n single m o d e l s - are r e q u i r e d to a l l o w for  undecided  formulae.  Using  situations  (Barwise  and Perry  1983] -  incomplete  model-  d e s c r i p t i o n s - i n s t e a d of sets of m o d e l s m i g h t l e a d t o a closer correspondence. I n t u i t i v e l y , cert a i n l y , one c a n s i m p l y v i e w the model-sets as p a r t i a l m o d e l - d e s c r i p t i o n s w i t h o u t i l l effect. T h e s e c o n d d e v i a t i o n results f r o m t h e fact t h a t defaults are general inference rules. quently,  t h e submodel(-set)  relation is potentially  more c o m p l e x f o r default l o g i c .  Conse-  Lifschitz'  [1984] r e c e n t w o r k a l l o w i n g a r b i t r a r y pre-orders as w e l l as simple subset orderings m a y v o i d t h i s difference, b u t the q u e s t i o n r e m a i n s o p e n . T h e f a c t t h a t i n d i v i d u a l e x t r e m a d e t e r m i n e extensions is the result o f t h e " b r a v e " ( i n M c D e r m o t t ' s [1982] t e r m i n o l o g y ) c h a r a c t e r o f default logic. R e i t e r ' s p r e s e n t a t i o n of default logic defined e a c h e x t e n s i o n as a n a c c e p t a b l e set o f beliefs, w i t h t h e i n t e n t i o n t h a t a reasoner w o u l d s o m e h o w " c h o o s e " a single e x t e n s i o n w i t h i n w h i c h t o reason a b o u t t h e w o r l d .  Other non-  m o n o t o n i c f o r m a l i s m s (see c h a p t e r 2) are based o n " c a u t i o u s " approaches w h i c h a c c e p t a default  - 47 c o n c l u s i o n o n l y if it o c c u r s i n all a c c e p t a b l e sets of beliefs.  O n e c a n easily c o n s t r u c t a v a r i a n t of  default logic w h i c h pursues a " c a u t i o u s " course. ( T h e converse is not o b v i o u s l y true for a l l " c a u t i o u s " s y s t e m s , as we see i n c h a p t e r 8.)  S u c h a s y s t e m w o u l d define the t h e o r e m s o f a default  t h e o r y to be those f o r m u l a e true i n a l l extensions, w i t h the o b v i o u s change to the s e m a n t i c s : the t h e o r e m s w o u l d t h e n be defined as those f o r m u l a e true i n a l l m o d e l s of a l l v i a b l e leaves.  3.2. Coherence of Default Theories E x t e n s i o n s p l a y a f u n d a m e n t a l role i n default logic. A n e x t e n s i o n is a set of beliefs w h i c h are i n some sense " j u s t i f i e d " o r " r e a s o n a b l e " i n l i g h t of w h a t is k n o w n a b o u t a w o r l d . F o r m a l l y , extensions are a t t r a c t i v e  because t h e y are b o t h g r o u n d e d a n d c o m p l e t e : A f o r m u l a enters a n  e x t e n s i o n , E, o n l y if it is i n W, if it is p r o v a b l e f r o m o t h e r f o r m u l a e i n E, or if it is the consequent of a default whose prerequisites are i n E a n d whose j u s t i f i c a t i o n s are not d e n i e d b y E; f u r t h e r m o r e , e v e r y f o r m u l a w h i c h meets these r e q u i r e m e n t s is i n E. prevents justified  extensions f r o m beliefs are not  theorems of a  first-order  containing ignored.  spurious, u n s u p p o r t e d  The  first  of these r e s t r i c t i o n s  beliefs. T h e second ensures  T h e s e r e s t r i c t i o n s are analogous to those w h i c h define  that the  theory.  S i n c e the i n d i v i d u a l extensions of a default theory are b o t h g r o u n d e d a n d c o m p l e t e , it is q u i t e n a t u r a l t o require a n y default inference s y s t e m to restrict its c o n c l u s i o n s to a single c o m m o n e x t e n s i o n . If n o e x t e n s i o n of a t h e o r y c o n t a i n s a f o r m u l a , t h e n it is n o t in a n y a c c e p t a b l e set of beliefs a s s o c i a t e d w i t h t h a t theory. If c o n c l u s i o n s are d r a w n f r o m different extensions, t h e y m a y be i n c o m p a t i b l e .  C o n s i d e r the b l o c k s - w o r l d e x a m p l e f r o m the previous c h a p t e r .  In t h a t e x a m p l e ,  b o t h ->Block(A) a n d ->Block(B) are reasonable a s s u m p t i o n s . T h e y are d r a w n f r o m different e x t e n sions, h o w e v e r , a n d c o n c l u d i n g b o t h leads to i n c o n s i s t e n c y . S i n c e reasonable c o n c l u s i o n s m u s t reside i n a n e x t e n s i o n of the default theory u n d e r c o n s i d e r a t i o n , it is c l e a r l y i m p o r t a n t t o k n o w w h e t h e r e v e r y theory has extensions. S i m p l y p u t , the answer is " N o " . F o r e x a m p l e , the t h e o r y : W={}  has no e x t e n s i o n . S u c h theories are i n c o h e r e n t ; t h e y s u p p o r t n o reasonable set of beliefs a b o u t the world.  B e y o n d p o i n t i n g o u t the existence o f i n c o h e r e n t theories, the m o s t useful a n s w e r w o u l d  i n c l u d e a s y n t a c t i c c h a r a c t e r i z a t i o n of w h i c h theories h a v e o r d o not h a v e extensions. W h i l e n o s u c h c h a r a c t e r i z a t i o n is k n o w n , there are sufficient c o n d i t i o n s w h i c h guarantee extensions. W e present three s u c h c o n d i t i o n s below, i n o r d e r of i n c r e a s i n g u t i l i t y . A t h e o r y , ({ the u n d e r l y i n g  },W), w i t h no d e f a u l t s has a u n i q u e e x t e n s i o n , Th( W), t h e l o g i c a l closure of  first-order  theory.  O f course, this is a t r i v i a l default t h e o r y . W e m e n t i o n it o n l y to  e m p h a s i z e t h a t , since d e f a u l t l o g i c is a superset of the a r e a of o v e r l a p .  first-order  l o g i c , the r e q u i r e d results o b t a i n for  - 48 -  T h e d i s t i n c t i o n s b e t w e e n c o m m o n l y e n c o u n t e r e d types of defaults lead to m o r e e n l i g h t e n i n g results.  A n y default of the f o r m :  a:p P is s a i d to be normal. N o r m a l defaults are sufficient for knowledge r e p r e s e n t a t i o n a n d reasoning i n m a n y n a t u r a l l y o c c u r r i n g c o n t e x t s . In fact, they c a n express a n y rule whose a p p l i c a t i o n is subject o n l y to  first-order  prerequisites a n d the consistency of its c o n c l u s i o n w i t h t h e rest of w h a t is  believed. Rules like: " A s s u m e a b i r d c a n fly unless y o u k n o w o t h e r w i s e . " , or " A s s u m e a t h i n g is not a b l o c k unless it is r e q u i r e d t o b e . " t r a n s l a t e e a s i l y i n t o n o r m a l defaults: Bird(x) : Can-fly(x)  ^  : -iBlock(x)  Can-fly(x)  -iBlock(x)  T h e consequent of a n o r m a l default is e q u i v a l e n t to its j u s t i f i c a t i o n . I n t u i t i v e l y , this m a k e s the default i n a p p l i c a b l e where the consequent has been d e n i e d . S u c h defaults c a n n o t i n t r o d u c e inconsistencies, t h e y c a n n o t refute the j u s t i f i c a t i o n s of other, a l r e a d y a p p l i e d , n o r m a l defaults, nor c a n they refute t h e i r o w n j u s t i f i c a t i o n s .  T h i s gives rise to w e l l - b e h a v e d theories.  Any  theory  i n v o l v i n g o n l y n o r m a l d e f a u l t s (a normal theory) m u s t have at least one e x t e n s i o n [Reiter 1980a]; A n y default of the f o r m :  <* = P A P is  said  to  be  7 semi-normal.  S e m i - n o r m a l defaults  differ  from  normal  j u s t i f i c a t i o n s w h i c h e n t a i l b u t are not e n t a i l e d b y t h e i r consequents.  defaults  by  having  T h e assurances of w e l l -  behavedness a s s o c i a t e d w i t h n o r m a l theories do not c a r r y over to theories w i t h s e m i - n o r m a l defaults. F o r e x a m p l e , the t h e o r y : W={} /  :A A - B  \  A  :B A : C / \ - > A '  B  '  C  \  j  (1)  has n o e x t e n s i o n . T h i s a p p e a r s t o be a s o m e w h a t a r t i f i c i a l e x a m p l e , i n a s m u c h as we h a v e been u n a b l e t o find a n a t u r a l s i t u a t i o n w h i c h fits this p a t t e r n . W h i c h s e m i - n o r m a l theories, t h e n , are assured of extensions? D o till " n a t u r a l " theories h a v e extensions? P e r h a p s p a t h o l o g i c a l e x a m p l e s are m e r e l y f o r m a l curiosities? W e do not p u r p o r t to a n s w e r these questions — p a r t l y because of the d i f f i c u l t y of d e l i m i t i n g the class of " n a t u r a l " theories. T h e r e is, however, a large class of s e m i - n o r m a l theories w h i c h are coherent. W e c h a r a c t e r i z e this class, w h i c h appears t o be sufficient for m a n y c o m m o n a p p l i c a t i o n s , i n the n e x t section.  - 49 -  3.S.  Ordered Default Theories  T h e r e a p p e a r s to be a u n i f y i n g c h a r a c t e r i s t i c a m o n g default theories w i t h o u t extensions. C o n s i d e r a g a i n the theory: W={}  w h i c h has n o e x t e n s i o n . T h e o n l y reasonable c a n d i d a t e s are Ey = is c o n s i s t e n t w i t h E , so t o be a n e x t e n s i o n E x  x  i n c o n s i s t e n t w i t h E , so E 2  2  Th({ }) or E  2  =  Th({->A}). A  m u s t c o n t a i n -<A, w h i c h it does not. S i m i l a r l y , A is  c a n n o t c o n t a i n ->A. T h e p r o b l e m is t h a t the d e f a u l t ' s j u s t i f i c a t i o n is  d e n i e d b y its c o n s e q u e n t ; not a p p l y i n g the default forces its a p p l i c a t i o n , a n d v i c e v e r s a . R e t u r n i n g to the s e m i - n o r m a l t h e o r y ( l ) , we see t h a t a p p l y i n g a n y one default leaves one o t h e r a p p l i c a b l e . A p p l y i n g a n y t w o , h o w e v e r , results i n the d e n i a l of the n o n - n o r m a l p a r t of the j u s t i f i c a t i o n s of at least one of t h e m . A n y set s m a l l e n o u g h to be a n e x t e n s i o n is too s m a l l ; a n y set large e n o u g h is t o o large.  T h i s b e h a v i o u r is c h a r a c t e r i s t i c of theories w i t h no extension; the r e q u i r e m e n t  that  extensions be c l o s e d u n d e r the default rules forces the a p p l i c a t i o n of defaults whose consequents l e a d to the d e n i a l of j u s t i f i c a t i o n s of o t h e r a p p l i e d defaults. T h e e x a c t source of the p r o b l e m c a n be f u r t h e r i s o l a t e d b y recalling t h a t a l l n o r m a l theories have e x t e n s i o n s . S i n c e the j u s t i f i c a t i o n a n d consequent of n o r m a l defaults are i d e n t i c a l , no a p p l i cable d e f a u l t c a n refute the j u s t i f i c a t i o n s of a n a l r e a d y a p p l i e d default: a p p l i e d n o r m a l defaults have a l r e a d y asserted t h e i r j u s t i f i c a t i o n s .  T h i s means t h a t a n y n o r m a l default c a p a b l e of r e f u t i n g  those j u s t i f i c a t i o n s is i n a p p l i c a b l e , since its j u s t i f i c a t i o n s have a l r e a d y been r e f u t e d .  It  follows  t h a t t h a t p a r t o f the j u s t i f i c a t i o n w h i c h distinguishes n o n - n o r m a l defaults f r o m n o r m a l defaults is integrally default  i n v o l v e d in making a theory  theories, we  see t h a t  incoherent.  once a default  R e s t r i c t i n g o u r a t t e n t i o n to  has been a p p l i e d , o n l y  semi-normal  those c o n j u n c t s  j u s t i f i c a t i o n n o t e n t a i l e d b y its consequent are susceptible to r e f u t a t i o n b y o t h e r defaults.  of  its  These  c o n j u n c t s p l a y a k e y role i n the d i s c u s s i o n below. T h e conflict b e t w e e n closure u n d e r defaults a n d consistency of j u s t i f i c a t i o n s c a n o c c u r o n l y if some f o r m u l a d e p e n d s o n the absence of a n o t h e r a n d at the same t i m e m a y serve to s u p p o r t the inference o f t h a t f o r m u l a . In the t h e o r y ( l ) a b o v e , for e x a m p l e , A depends o n the absence of B, B o n t h a t of C, a n d C o n t h a t of A. H e n c e i n f e r r i n g A w o u l d b l o c k the inference of C, a l l o w i n g the inference B, w h i c h w o u l d i n v a l i d a t e the inference of A, a n d s i m i l a r l y for B a n d C. T h e e x a m p l e s presented so f a r h a v e i n v o l v e d defaults i n t h e i r simplest f o r m :  a : p\ A ... A  0  n  where a, u> a n d ft are a l l l i t e r a l s (i.e., a t o m i c f o r m u l a e or negations o f a t o m i c f o r m u l a e ) . T h e p r o b l e m of d e t e r m i n i n g dependencies is m o r e c o m p l i c a t e d w h e n a, w a n d ft are a l l o w e d to be arbitrary  first-order  f o r m u l a e . F o r e x a m p l e , the consequent o f a default m a y be a n i m p l i c a t i o n ;  a p p l y i n g t h a t d e f a u l t w o u l d i n t r o d u c e n e w dependencies. T h e essential i d e a r e m a i n s the same,  - 50 -  h o w e v e r : d e t e r m i n e w h e t h e r the dependencies i n v o l v e p o t e n t i a l l y u n r e s o l v a b l e c i r c u l a r i t i e s . T h e f o l l o w i n g d e f i n i t i o n s o u t l i n e a s y n t a c t i c m e t h o d for d e t e r m i n i n g w h e t h e r s u c h c i r c u l a r i t i e s exist w i t h i n a semi-normal theory.  Definition:  Let  <c  and  A = (D, W)  assume  all  be a c l o s e d ,  formulae  are  s e m i - n o r m a l default theory.  1  in  clausal  form.  The  partial  Without  loss of  relations,  generality,  and  <§; ,  on  Literals X Literals, are defined as follows: If a € W t h e n a = (a V ... V a j  (1)  , for some n > 1 .  y  For all a  it  aj € {a ,...,a }, if 1  CK :  let -la;  Jt  a . t  3 A 7  If 8 e D t h e n 8 = —'——  (2)  ^= a  n  . L e t o^, ... a , p\, ... B , a n d 7 ^ ... 7 be the l i t e r a l s of the r  s  t  P c l a u s a l f o r m s of a,fi,a n d 7, r e s p e c t i v e l y . T h e n  A€ { A , A } let o « A . { 7 i , - , 7 t } . A € { & , . . . , & } a n d 7, £ { & , . . . , & } let - 7 i « A .  (i)  If a G { « ! , . . . , a } a n d  (ii)  If 7i S  {  r  (iii) A l s o , B = At  An 1 f °  A ••• A  F o r each i < m, A T h u s if (3)  t  (Ai  =  r  some m > 1. , where m j > 1 .  V ... V A,nJ  flj, A,k e { A ^ . v A ^ m J  and  T h e e x p e c t e d t r a n s i t i v i t y r e l a t i o n s h i p s h o l d for (i)  If a ^  /? a n d /9 «  7 then a ^  (ii)  If a <K 8 a n d /3 <SC 7 t h e n a <§; 7.  A,j £ A, let -Aj ^ A,kk  <K a n d <C . I.e.,  7.  (iii) li a <K B a n d /? <C 7 or a «C /9 a n d /3 <K 7 t h e n a <C 7. The is  any  definition  way  intuition  x  V...V  a  is  parts  as  an  aj  complex,  could  behind  interpreted (a  that  = [(-"a*! A---A  figure  (l)  and  but in  (2.iii)  implication ""Xj-i  the an  intention  is  inference  of  is  any  of  A "'C'j+i A---A  that any ")  -,  n  B in  that the  | a <£. B or a theory  disjunction  one ^> ct\ ]•  of  of those  as i t n  if  /3  there  stands.  literals  The  can  literals.  be  E.g.,  T h e s p e c i a l p r o m i n e n c e we have  a l l u d e d t o f o r the c o n j u n c t s i n a j u s t i f i c a t i o n not e n t a i l e d b y the consequent is reflected i n p a r t (2.ii) b y the use o f the d i s t i n g u i s h e d " <§: " r e l a t i o n . T h e n e g a t i o n , ->7; , occurs i n p a r t (2.ii) since it is not k n o w i n g -17; w h i c h m a k e s 7; consistent.  1  T h e definition is readily extensible to open theories using a technique given in [Reiter 1980a].  - 51 Definition:  Orderedness  A s e m i - n o r m a l default theory is s a i d to be ordered if a n d o n l y if there is n o l i t e r a l , a, s u c h that a «  a.  |  A n ordered theory has no potentially  unresolvable c i r c u l a r dependencies.  e x a m p l e (1) is n o t o r d e r e d , since B <s: A, C <K B, a n d A <£. C; hence A « W = {  T h e theory i n  i . T h e theory:  }  : B/\^D  /j_A_/WB \  A  *  B  : (C D D) A ->A \ '  ( C ^ D )  (2)  j  is also n o t o r d e r e d . T h e defaults give rise to t h e f o l l o w i n g r e l a t i o n s h i p s : {B<s:A},  {D<KB},  respectively. Hence A «  and  O «  { C «  D,  -i£>«->C,  A <sz ->C,  A<^D},  B « A.  T h e significance of orderedness f o r s e m i - n o r m a l default theories is s h o w n b y T h e o r e m 3.3.  Theorem S.S — Coherence If a s e m i - n o r m a l default theory is o r d e r e d , then i t h a s at least one e x t e n s i o n .  |  N o r m a l theories are c l e a r l y o r d e r e d , since o n l y n o n - n o r m a l defaults give rise t o " <SC " r e l a tionships.  T h u s t h e coherence of a l l n o r m a l theories is a c o r o l l a r y  of T h e o r e m 3 . 3 . T h i s is  e n c o u r a g i n g i n a s m u c h a s it suggests t h a t orderedness is n o t m e r e l y a s p e c i a l purpose g i m m i c k b u t , r a t h e r , it subsumes a n e x i s t i n g , w i d e l y a p p l i c a b l e c h a r a c t e r i z a t i o n . It is i m p o r t a n t t o notice t h a t orderedness is o n l y a sufficient  c o n d i t i o n f o r existence o f  extensions. N o n - o r d e r e d theories have p o t e n t i a l l y unresolvable c i r c u l a r i t i e s b u t , f o r o n e reason o r another, these c i r c u l a r i t i e s d o not a l w a y s interfere.  T h e theory (2) is n o t o r d e r e d , b u t i t does  have a n e x t e n s i o n : Th({B, (C D D)}). T h e c i r c u l a r i t y w o u l d cause p r o b l e m s , h o w e v e r , i f C were a d d e d t o W: t h e r e s u l t i n g theory has no extensions. I n other cases, t w o o r m o r e p o t e n t i a l c i r c u l a r ities m a y c a n c e l e a c h o t h e r o u t . A t present, we d o n o t k n o w w h e t h e r t h e g i v e n c o n d i t i o n c a n be strengthened  to one w h i c h is b o t h necessary a n d sufficient  for t h e coherence of s e m i - n o r m a l  theories a n d y e t is s t i l l d e c i d a b l e .  3.4. Constructing Extensions  H a v i n g d e l i n e a t e d a large class o f theories w h i c h h a v e extensions, w e t u r n to t h e p r o b l e m of g e n e r a t i n g e x t e n s i o n s . R e i t e r [1980a] shows t h a t extensions need not b e r e c u r s i v e l y  enumerable,  a n d t h a t i t is n o t g e n e r a l l y s e m i - d e c i d a b l e w h e t h e r a f o r m u l a is i n a n y e x t e n s i o n o f a theory. F a c e d w i t h s u c h p e s s i m i s m , f u r t h e r e x p l o r a t i o n m i g h t seem pointless. S t i l l , there are t r a c t a b l e  - 52 -  subcases. E t h e r i n g t o n [1982] presents a procedure w h i c h c a n generate a l l the extensions of a n a r b i t r a r y finite default t h e o r y .  2  T h e procedure centres o n a r e l a x a t i o n style c o n s t r a i n t  propagation  t e c h n i q u e . E x t e n s i o n s are c o n s t r u c t e d b y a series o f successive a p p r o x i m a t i o n s . E a c h a p p r o x i m a t i o n , H , is b u i l t u p f r o m the  first-order  }  c o m p o n e n t s i n W b y a p p l y i n g defaults, one at a t i m e . A t  each step, the default t o be a p p l i e d is chosen f r o m those, not yet a p p l i e d , whose prerequisites are " k n o w n " a n d whose j u s t i f i c a t i o n s are consistent w i t h b o t h the p r e v i o u s a p p r o x i m a t i o n a n d the c u r r e n t s t a t e of the c u r r e n t a p p r o x i m a t i o n . W h e n n o m o r e defaults are a p p l i c a b l e , the procedure c o n t i n u e s w i t h the n e x t a p p r o x i m a t i o n . If t w o successive a p p r o x i m a t i o n s are the same, the procedure is s a i d to converge. T h e choice o f w h i c h default to a p p l y at each step o f the i n n e r l o o p m a y i n t r o d u c e a degree of n o n - d e t e r m i n i s m . G e n e r a l i t y requires t h i s n o n - d e t e r m i n i s m , h o w e v e r , since theories d o  not  necessarily h a v e u n i q u e extensions. D e t e r m i n i s t i c procedures c a n be c o n s t r u c t e d for theories w h i c h have u n i q u e extensions, o r i f f u l l g e n e r a l i t y is not r e q u i r e d . a • 8 I n the p r e s e n t a t i o n o f the procedure, below, CONSEQUENT^——) is defined to be 7. 7  repeat * o -  W\  GD *-{};  G D  I  0  i«-0;  repeat  I  A «-  ^  h  a), (^ \f-  ^0),  [H  hl  \h  ^  if ->null(Di - GDi) t h e n c h o o s e S f r o m (£>; GD  i+1  <- GDi U {6};  hi i«- hi U +  *'«- t +  GDj;  {CONSEQUENT^)};  endif;  1;  u n t i l nu/i(P _ i  1  GD^);  H\ = h-i until H  }  =  T o see h o w this p r o c e d u r e w o r k s , c o n s i d e r the theory: W=  {A}  A finite theory is one with only finitely many variables, constant symbols, predicate letters, and defaults. N o function symbols are allowed, except the 0-ary function symbols, the constants. These restrictions make the universe of discourse (or Herbrand Universe) finite, ensuring only a finite number of closed instances of open defaults. 2  - 5 3 -  / A : B \  B  A :C '  C  B : D '  D  B : ~>D  A ~>C \  -Z?  '  /'  w h i c h h a s t h e u n i q u e e x t e n s i o n , Th({A,B, C,D}). T h e p r o c e d u r e c a n generate a n y o f t h e f o l l o w i n g sequences o f a p p r o x i m a t i o n s : Ho={A}  HM^B^C)  H^{A}  H ={A}  H^{A,B,Cf)  H^i^C&D}  H^{A,B,C,D}  Hs={A,B,D,C)  E^H  H*=H  0  H^Ht  X  3  ( T h e f o r m u l a e i n e a c h a p p r o x i m a t i o n are l i s t e d i n t h e order i n w h i c h t h e y are derived.) I n t h e first  sequence o f a p p r o x i m a t i o n s , ->D o c c u r s i n H  x  because it c a n be i n f e r r e d i n  before C is  inferred i n k . 3  E t h e r i n g t o n [1982] p r o v e s : There is a converging computation such that H  n  =  and Th(H^) = E if and only if E is  an extension for the default theory (D,W). In o t h e r w o r d s , t h e p r o c e d u r e c a n r e t u r n e v e r y e x t e n s i o n , a n d o n l y extensions are r e t u r n e d . T h i s result falls short i n t w o respects: F i r s t , w h i l e the procedure c a n converge o n e v e r y e x t e n s i o n , there are a p p e a l s t o non-provability. I n g e n e r a l , s u c h tests are not c o m p u t a b l e , since a r b i t r a r y order f o r m u l a e are i n v o l v e d . T h e r e a r e c o m p u t a b l e subcases, however.  first-  If t h e set:  W  belongs t o a d e c i s i o n class f o r  first-order  provability,  extensions are c o m p u t a b l e .  Propositional  theories a n d f u n c t i o n - f r e e , m o n a d i c theories f a l l i n t o this class, as d o finite theories, p r o v i d e d W is also  finite. T h e s e c o n d s h o r t c o m i n g is t h a t some  finite  theories a d m i t n o n - c o n v e r g i n g  computations.  T h e p r o c e d u r e m a y n e v e r t e r m i n a t e even t h o u g h t h e theory has a n e x t e n s i o n a n d each step is c o m p u t a b l e . I n s u c h cases, t h e p r o c e d u r e cycles forever between t w o o r m o r e d i s t i n c t #j's. F o r t u n a t e l y this c y c l i c b e h a v i o u r seems t o be c a u s e d b y features s i m i l a r to those w h i c h m a k e theories i n c o h e r e n t . W e h a v e c h a r a c t e r i z e d c e r t a i n classes o f o r d e r e d theories f o r w h i c h t h e p r o c e d u r e is more well-behaved.  T h e o r e m 3.4 shows t h a t one s u c h class is t h e class o f o r d e r e d ,  network  theories.  Definition: Network Default Theory A d e f a u l t t h e o r y , A = (D, W), is a network theory iff i t satisfies t h e f o l l o w i n g c o n d i t i o n s : (1)  (2)  W contains only: a)  L i t e r a l s (i.e., A t o m i c f o r m u l a e or t h e i r negations), a n d  b)  D i s j u n c t s o f t h e f o r m (a V 8) where o; a n d B are l i t e r a l s .  D c o n t a i n s o n l y n o r m a l a n d s e m i - n o r m a l defaults o f the f o r m :  a: BA  a : B —TT—  P  or  li f\  P  ...  f\ la  - 54 -  where a, B, a n d -y; are literals.  I  Theorem 3.4 — Convergence F o r finite, o r d e r e d , n e t w o r k theories, t h e p r o c e d u r e g i v e n above a l w a y s converges o n a n e x t e n s i o n .  |  W e w i l l h a v e m o r e t o s a y a b o u t n e t w o r k theories i n the next chapter. W e conjecture t h a t T h e o r e m 3.4 c a n be generalized to a p p l y t o a r b i t r a r y o r d e r e d s e m i n o r m a l theories, b u t we h a v e no proof. T h e proof m a y require a more r e s t r i c t i v e d e f i n i t i o n of  in  the p r o c e d u r e , viz:  i n s t e a d of:  b u t it c a n be s h o w n t h a t a l l t h e results of [ E t h e r i n g t o n 1982] a n d those of t h i s c h a p t e r s t i l l h o l d for t h e s t r o n g e r v e r s i o n , so t h i s s h o u l d present no p r o b l e m . F o r n o r m a l theories, a n e v e n stronger result c a n be p r o v e d :  Theorem 3.5 — Strong Convergence F o r finite n o r m a l theories, t h e p r o c e d u r e g i v e n above a l w a y s converges o n a n e x t e n s i o n i m m e d i a t e l y — i.e., Th(Hi)  is a l w a y s a n e x t e n s i o n .  |  CHAPTER 4  Inheritance Networks with Exceptions  A centipede was h a p p y , q u i t e , U n t i l a frog, i n f u n , S a i d , " P r a y , w h i c h leg comes after w h i c h ? " T h i s raised his m i n d to such a pitch He l a y distracted i n a ditch, Considering how to run.  O n e o f the p r o b l e m s w i t h the n o n - m o n o t o n i c f o r m a l i s m s we h a v e discussed t o this p o i n t is their intractability.  D e f a u l t logic, i n the g e n e r a l case, is not even s e m i - d e c i d a b l e . B e c a u s e o f the  need t o b u i l d s y s t e m s w h i c h h a v e good c o m p u t a t i o n a l properties, m a n y researchers h a v e sacrificed formal precision.  W h i l e this has s o m e t i m e s l e d t o v e r y fast " i n f e r e n c e " m e c h a n i s m s , there has  often been l i t t l e m o r e t h a n v a g u e i n t u i t i o n s a b o u t e x a c t l y what these m e c h a n i s m s infer. A s the field m a t u r e s a n d systems c a p a b l e o f a s s u m i n g r e s p o n s i b i l i t y f o r s u c h t h i n g s as n u c l e a r reactors a n d m e d i c a l diagnosis are t o u t e d as " o n the h o r i z o n " , it becomes i n c r e a s i n g l y i m p o r t a n t t h a t it be u n d e r s t o o d w h a t s u c h systems " c o n s i d e r " justifiable inferences. T h e a r g u m e n t h a s long been m a d e t h a t , because o f the general i n t r a c t a b i l i t y of f o r m a l systems, it is u n r e a s o n a b l e to consider t h e m for p r a c t i c a l a p p l i c a t i o n s .  T h i s is t a k e n as s u p p o r t for  the use o f s y s t e m s s u c h as s e m a n t i c n e t w o r k s w h i c h , a l t h o u g h not c o m p l e t e l y u n d e r s t o o d , c a n compute quickly.  T h i s a r g u m e n t falls d o w n o n t w o points. T h e first is t h a t most o f these fast  inference a l g o r i t h m s are a p p l i c a b l e to a l i m i t e d class o f p r o b l e m s . It c o u l d w e l l be t h a t - for these p r o b l e m s - f o r m a l systems s u c h as default logic are just as t r a c t a b l e , a n d fast  implementations  m a y be possible. S e c o n d l y , even if f o r m a l systems are n o t i m p l e m e n t e d d i r e c t l y i n a n inference s y s t e m , t h e y m a y be u s e f u l as s p e c i f i c a t i o n tools.  I n t h i s w a y , a n i m p l e m e n t a t i o n c o u l d either be  s h o w n a l w a y s t o r e a c h j u s t i f i e d c o n c l u s i o n s or, a t t h e v e r y least, t o d e v i a t e i n w e l l - u n d e r s t o o d w a y s from j u s t i f i e d c o n c l u s i o n s . I n the f o r m e r case, t h e fast a l g o r i t h m c o u l d a c t u a l l y be v i e w e d as a n implementation o f a n a p p r o p r i a t e l y - r e s t r i c t e d v e r s i o n of the general f o r m a l s y s t e m ; i n t h e l a t t e r case, at least w o u l d - b e purchasers o f s u c h systems c o u l d m a k e e n l i g h t e n e d decisions a b o u t the risks i n v o l v e d . I n t h i s c h a p t e r , w e e m p l o y d e f a u l t logic t o o u t l i n e a s p e c i f i c a t i o n for " i n h e r i t a n c e " reasoning i n the presence o f e x c e p t i o n s .  S e m a n t i c n e t w o r k s h a v e been w i d e l y a d o p t e d as a r e p r e s e n t a t i o n a l  m e c h a n i s m f o r A l . I n s u c h n e t w o r k s , " i n f e r e n c e " is e q u a t e d w i t h i n h e r i t a n c e of p r o p e r t i e s b y nodes f r o m t h e i r superiors. R e c e n t w o r k has c o n s i d e r e d the effects of a l l o w i n g e x c e p t i o n s t o i n h e r i t a n c e w i t h i n n e t w o r k s [ B r a c h m a n 1982; E t h e r i n g t o n a n d R e i t e r 1983; F a h l m a n 1979; F a h l m a n et al 1981; T o u r e t z k y  1982, 1984a; W i n o g r a d 1980]. S u c h exceptions represent e i t h e r e x p l i c i t o r  - 55 -  - 56-  i m p l i c i t c a n c e l l a t i o n of the n o r m a l p r o p e r t y i n h e r i t a n c e w h i c h n e t w o r k s enjoy. I n the absence of e x c e p t i o n s , a n i n h e r i t a n c e n e t w o r k is a t a x o n o m y o r g a n i z e d b y the u s u a l I S - A r e l a t i o n , as i n F i g u r e 4 . 1 . S c h u b e r t [1976] a n d H a y e s [1977] h a v e a r g u e d t h a t s u c h n e t w o r k s c o r r e s p o n d q u i t e n a t u r a l l y t o c e r t a i n theories of first-order logic. E.g., NAUTILUS(Fred)  V  Vx. NAUTILUS(x)  D  Vx. CEPHALOPOD(x)  z  M  0  L  L  U  S  C  t  x  \  D  WVERTEBRATE(x)  CEPHALOPOD(x) D  MOLLUSC(x)  S u c h a c o r r e s p o n d e n c e c a n be v i e w e d as p r o v i d i n g the semantics w h i c h " s e m a n t i c " n e t w o r k s h a d p r e v i o u s l y l a c k e d [ W o o d s 1975].  INVERTEBRATE  INSECT  MOLLUSC  CEPHALOPOD  NAUTILUS  ARACHNID  BIVALVE  CUTTLEFISH  T  Fred  Figure 4-1 — F r a g m e n t o f a t a x o n o m y . T h e s i g n i f i c a n t features of t h i s s e m a n t i c s are these: (1)  I n h e r i t a n c e is a l o g i c a l p r o p e r t y of the r e p r e s e n t a t i o n . G i v e n t h a t NAUTILUS(Fred), MOLLUSC(Fred) is p r o v a b l e f r o m the g i v e n f o r m u l a e . Inheritance is the r e p e a t e d a p p l i c a t i o n of modus ponens.  (2)  T h e node labels of s u c h a n e t w o r k TEBRATE^).  (3)  N o e x c e p t i o n s to i n h e r i t a n c e are possible. If F r e d is a n a u t i l u s , he m u s t be a n i n v e r t e b r a t e , regardless of a n y o t h e r properties he enjoys. Unfortunately,  are u n a r y predicates:  e.g., NAUTILUS(*),  INVER-  this correspondence n o longer applies w h e n e x c e p t i o n s t o i n h e r i t a n c e are  a l l o w e d . T h e l o g i c a l p r o p e r t i e s of n e t w o r k s change d r a s t i c a l l y w h e n e x c e p t i o n s are p e r m i t t e d .  For  e x a m p l e , c o n s i d e r the f o l l o w i n g facts a b o u t elephants: (1) (2)  E l e p h a n t s are g r a y , e x c e p t for a l b i n o elephants. A l l a l b i n o e l e p h a n t s are elephants.  C o m m o n - s e n s e r e a s o n i n g a b o u t " e l e p h a n t s " a l l o w s one, g i v e n a n i n d i v i d u a l e l e p h a n t not k n o w n to be a n a l b i n o , to infer t h a t she is g r a y . S u b s e q u e n t d i s c o v e r y — perhaps b y o b s e r v a t i o n — t h a t she is a n a l b i n o e l e p h a n t forces the r e t r a c t i o n of the c o n c l u s i o n a b o u t h e r grayness. T h u s , c o m m o n - s e n s e r e a s o n i n g a b o u t e x c e p t i o n s is n o n - m o n o t o n i c ; new i n f o r m a t i o n c a n i n v a l i d a t e prev i o u s l y d e r i v e d facts. T h i s n o n - m o n o t o n i c i t y precludes the use of  first-order  representations, l i k e  - 57 those used for t a x o n o m i e s , for f o r m a l i z i n g n e t w o r k s w i t h exceptions. We  e s t a b l i s h a correspondence between n e t w o r k s  w i t h exceptions and network  default  theories. T h i s correspondence p r o v i d e s a f o r m a l s e m a n t i c s a n d a n o t i o n of correct inference for such networks.  A s w a s the case for t a x o n o m i e s , i n h e r i t a n c e emerges as a l o g i c a l feature of the  r e p r e s e n t a t i o n . T h o s e properties Pi,...,P w h i c h a n i n d i v i d u a l , 6, i n h e r i t s prove to be precisely n  those for w h i c h P ( 6 ) , . . . , P ( 6 ) a l l belong to a c o m m o n e x t e n s i o n of the default t h e o r y . S h o u l d the 1  n  t h e o r y h a v e m u l t i p l e extensions — a n u n d e s i r a b l e feature, as we s h a l l see — t h e n b m a y i n h e r i t different  sets of properties d e p e n d i n g o n w h i c h e x t e n s i o n is chosen.  W e consider two radically  different r e m e d i e s for this p r o b l e m . T o see h o w d e f a u l t s m i g h t  be used to represent n e t w o r k s w i t h e x c e p t i o n s , c o n s i d e r the  elephant e x a m p l e , w h i c h c a n be represented b y the default theory:  W —  jy _  |vz.  f Elephant(x) : Gray(x) A ->Albino—Elepkant(x) \  It  is  D Elephant(x)  Albino-Elephant(x)  we  know,  Gray(Fred) A ->Albino-Elephant(Fred) is consistent; hence Gray(Fred) m a y be i n f e r r e d .  Given  only  easy  Gray(x) to  see  that  Albino—Elephant(Sue)  if  we  told  only  then,  Elephant(Fred)  one c a n c o n c l u d e Elephant(Sue) using  "blocks"  Albino-Elephant(Sue)  are  the  application  of  the  default,  so  far  first-order  as  knowledge,  preventing  the  but  derivation  of  Gray(Sue), as required . 1  W e a d o p t a n e t w o r k r e p r e s e n t a t i o n w i t h seven l i n k types. O t h e r approaches to i n h e r i t a n c e m a y o m i t one o r m o r e o f these, b u t o u r f o r m a l i s m subsumes these.  T h e seven l i n k t y p e s ,  1  with  t h e i r t r a n s l a t i o n s to default logic, are:  (1)  S t r i c t I S - A : A. • . 5 : A ' s are a l w a y s B's. S i n c e this is u n i v e r s a l l y t r u e , we i d e n t i f y w i t h the first-order f o r m u l a : Vx. A(xj Z> B(x).  (2)  M e m b e r s h i p : ao w i t h the first-order  (3)  S t r i c t I S N ' T - A : A. | | \ >.B: A's are never B's. A g a i n , t h i s is a u n i v e r s a l s t a t e m e n t , i d e n t i f i e d w i t h :  • . A : T h e i n d i v i d u a l a belongs to the class A. fact A(a).  W e represent  N o n - m e m b e r s h i p : ao | j | >• .A: T h e i n d i v i d u a l represent t h i s w i t h the first-order fact -iA(a).  (5)  D e f a u l t I S - A : A. >.B: N o r m a l l y A's are B's, b u t there m a y be e x c e p t i o n s . T o p r o v i d e f o r e x c e p t i o n s , we i d e n t i f y this w i t h a default:  tively.  this  Vx. A ( z ) Z> -<B(x).  (4)  1  it  a does not  belong to the class A.  We  Note that strict and default links are distinguished graphically by solid and open arrowheads, respec-  - 58 -  A(x) : B(x) B(x) (6)  D e f a u l t ISN'T-A: Identified with:  A. \ \ \ >.B:  N o r m a l l y A's are not B's, b u t  e x c e p t i o n s are a l l o w e d .  A(x) : -iB(x) ^B(x) E x c e p t i o n : A. > T h e e x c e p t i o n l i n k has no i n d e p e n d e n t s e m a n t i c s ; i t serves o n l y t o m a k e e x p l i c i t the exceptions, i f a n y , to the a b o v e default l i n k s . T h e r e m u s t a l w a y s be a default l i n k at the head of a n e x c e p t i o n l i n k ; the e x c e p t i o n t h e n alters the s e m a n t i c s of t h a t default l i n k . T h e r e are t w o types of default l i n k s w i t h exceptions; their g r a p h i c a l structures a n d t r a n s l a t i o n s are s h o w n i n F i g u r e 4.2.  (7)  B  A{x) : B{x) A -iC^x) A - A -^C {x) n  B(x)  B  A(x) : ^B{x) A - . C i ( i ) A ... A - . C ( x ) n  ^B(x) 1\  4^ Cx  ...  C7„  Figure 4.2 — L i n k s w i t h exceptions. W e i l l u s t r a t e w i t h a n e x a m p l e f r o m [ F a h l m a n et al 1981]. M o l l u s c s are n o r m a l l y shell-bearers. C e p h a l o p o d s m u s t be M o l l u s c s b u t n o r m a l l y are not shell-bearers. N a u t i l i m u s t be C e p h a l o p o d s a n d m u s t be shell-bearers. O u r n e t w o r k r e p r e s e n t a t i o n of these facts is g i v e n i n F i g u r e 4.3.  - 59 -  Shell-bearer  Mollusc  Cephalopod  Nautilus Figure 4-$ — N e t w o r k r e p r e s e n t a t i o n of o u r k n o w l e d g e a b o u t M o l l u s c s .  T h e c o r r e s p o n d i n g default logic r e p r e s e n t a t i o n is:  M(x) : Sb{x) A ~>C{x)  D =  W  Sb{x)  =  (x). C{x)  D  M[x),  C(x) : -ngfc(z) A -^(x)  '  ^Sb[x)  (x). N(x) z> C{x), (x). N{x)  \  J ' D  Sb{x)  )  G i v e n a p a r t i c u l a r N a u t i l u s , this t h e o r y has a u n i q u e e x t e n s i o n i n w h i c h i t is also a C e p h a l o p o d , a M o l l u s c , a n d a S h e l l - b e a r e r . A C e p h a l o p o d not k n o w n t o be a N a u t i l u s w i l l t u r n out t o be a M o l lusc w i t h n o s h e l l . It is i n s t r u c t i v e t o c o m p a r e our n e t w o r k representations w i t h those of N E T L [ F a h l m a n et al 1981]. A b a s i c difference is t h a t i n N E T L there are no strict l i n k s ; a l l I S - A a n d I S N ' T - A l i n k s are p o t e n t i a l l y c a n c e l l a b l e a n d hence are defaults. M o r e o v e r , F a h l m a n et al a l l o w e x p l i c i t e x c e p t i o n ( • U N C A N C E L ) l i n k s o n l y for I S N ' T - A ( * C A N C E L ) l i n k s . If we restrict the g r a p h of F i g u r e 4.3 to N E T L - l i k e l i n k s , we get F i g u r e 4.4(a), w h i c h is essentially the g r a p h g i v e n b y F a h l m a n .  a)  Shell-bearer  .  b)  Shell-bearer  Mollusc  Mollusc  Cephalopod  Cephalopod  Nautilus  Nautilus  .  Figure 4-4 — N E T L - l i k e n e t w o r k r e p r e s e n t a t i o n s of o u r knowledge about Molluscs.  T h e n e t w o r k i n F i g u r e 4.4(a) corresponds to the defaults:  - 60 M[x)  : Sb(x)  Sb{x)  C(x) : M(x) '  M{x)  C{x) : -,Sb{x) A ^N{x)  -.56 (z)  N{x) : C(x) ' '  C{x)  '  N{x) : Sb{x)  '  Sb(x)  A s before, a g i v e n N a u t i l u s w i l l also be a C e p h a l o p o d , a M o l l u s c , a n d a S h e l l - b e a r e r . A C e p h a l o p o d not  k n o w n to be a N a u t i l u s , h o w e v e r , gives rise t o two extensions, c o r r e s p o n d i n g to a n  a m b i v a l e n c e a b o u t w h e t h e r or not it has a shell. W h i l e c o u n t e r - i n t u i t i v e , this m e r e l y i n d i c a t e s that  an  exception  to  shell-bearing, namely  represented i n the n e t w o r k .  being  a  C e p h a l o p o d , has  not  been  explicitly  T h e a m b i g u i t y c a n be resolved b y m a k i n g the e x c e p t i o n e x p l i c i t , as  i n F i g u r e 4.3. O n the o t h e r h a n d , representations w h i c h do not p e r m i t e x c e p t i o n l i n k s to p o i n t to I S - A l i n k s c a n n o t m a k e this e x c e p t i o n e x p l i c i t i n the g r a p h i c a l r e p r e s e n t a t i o n . O t h e r v e r s i o n s o f N E T L ( a n d m a n y o t h e r inheritance reasoners) do not a l l o w e x p l i c i t except i o n l i n k s at a l l .  If o n l y default I S - A a n d I S N ' T - A l i n k s are a l l o w e d , the r e p r e s e n t a t i o n of the  N a u t i l u s e x a m p l e becomes t h a t of F i g u r e 4.4(b), w h i c h corresponds to the defaults: M(x) : Sb(x) Sb(x)  C{x) : M(x)  '  M(x)  C{x) : -,Sb{x)  !  ^Sb{x)  N(x) : C(x) '  '  C(x)  '  (  N(x) : Sb{x)  '  Sb{x)  In s u c h theories, there is a f u r t h e r a m b i g u i t y a b o u t w h e t h e r a N a u t i l u s is a S h e l l - b e a r e r . H o w t h e n do s u c h systems c o n j e c t u r e t h a t a C e p h a l o p o d is not a S h e l l - b e a r e r , w i t h o u t also c o n j e c t u r i n g t h a t it is a S h e l l - b e a r e r ? S u c h a m b i g u i t i e s are t y p i c a l l y resolved b y m e a n s of a n inference p r o c e d u r e w h i c h prefers shortest p a t h s . I n t e r p r e t e d in terms of default l o g i c , this " s h o r test p a t h h e u r i s t i c " is i n t e n d e d to f a v o u r one e x t e n s i o n of the default theory. T h u s , i n the netw o r k s of F i g u r e 4.4, the p a t h s f r o m C e p h a l o p o d t o - • S h e l l - b e a r e r are shorter t h a n those to S h e l l bearer so t h a t the f o r m e r w i n .  Unfortunately,  e x c l u d e d e x c e p t i o n t y p e i n a l l cases.  this heuristic is n o t sufficient  to replace the  R e i t e r a n d C r i s e u o l b [1983]; a n d E t h e r i n g t o n [1982] s h o w  t h a t it c a n l e a d to c o n c l u s i o n s w h i c h are u n i n t u i t i v e or even i n v a l i d — i.e., not i n a n y e x t e n s i o n . F a h l m a n et al [1981] a n d T o u r e t z k y [1981, p e r s o n a l c o m m u n i c a t i o n ; 1982] have also o b s e r v e d t h a t shortest p a t h a l g o r i t h m s c a n lead to a n o m a l o u s c o n c l u s i o n s . T h e y describe a t t e m p t s to restrict the f o r m of n e t w o r k s to e x c l u d e structures w h i c h a d m i t such p r o b l e m s .  O n e effect of these res-  t r i c t i o n s a p p e a r s t o be to p e r m i t o n l y n e t w o r k s whose corresponding default theories have u n i q u e extensions. A n inference a l g o r i t h m for n e t w o r k structures is correct o n l y i f it c a n be s h o w n to derive c o n c l u s i o n s a l l of w h i c h lie w i t h i n a single e x t e n s i o n of the u n d e r l y i n g default t h e o r y . T h i s c r i t e r i o n rules out shortest p a t h inference for u n r e s t r i c t e d n e t w o r k s . T h i s is u n f o r t u n a t e , since shortest p a t h inference has been p o p u l a r for its r e l a t i v e efficiency a n d ease o f i m p l e m e n t a t i o n . O n a m o r e p o s i t i v e note, a n y n e t w o r k c o n s t r u c t e d using the seven l i n k - t y p e s g i v e n a b o v e corresponds t o  a network  default  theory.  By  insisting t h a t  any network  constructed  must  c o r r e s p o n d t o a n o r d e r e d t h e o r y , the coherence of a n e t w o r k k n o w l e d g e r e p r e s e n t a t i o n s y s t e m c a n be assured.  F o r s u c h systems, the p r o c e d u r e g i v e n i n c h a p t e r 3 is a correct a n d a l w a y s c o n v e r g i n g  inference a l g o r i t h m .  - 61 It t u r n s o u t t h a t orderedness c a n be assured w i t h o u t reference to the f u l l c o m p l e x i t y of the m e c h a n i s m d e s c r i b e d i n c h a p t e r 3.  It is easy t o see t h a t a n y a c y c l i c n e t w o r k gives rise to a n  o r d e r e d t h e o r y . T h e same is true i f o n l y the s u b g r a p h consisting of a l l I S - A l i n k s a n d e x p l i c i t e x c e p t i o n s t h e r e t o has no cycles i n v o l v i n g at least one e x c e p t i o n l i n k , o r if there are n o e x p l i c i t e x c e p t i o n s to I S - A l i n k s .  Theorem 4.1 A n y n e t w o r k i n w h i c h the s u b g r a p h of I S - A l i n k s a n d e x p l i c i t e x c e p t i o n s t h e r e t o is a c y c l i c corresponds to a n ordered theory. |  Corollary 4.2 A n y a c y c l i c n e t w o r k corresponds to a n ordered theory.  I  Corollary 4.3 A n y n e t w o r k w i t h no e x p l i c i t e x c e p t i o n s t o I S - A l i n k s corresponds t o a n o r d e r e d t h e o r y . |  Corollary 4.4 T h e n e t w o r k s m e n t i o n e d i n t h e o r e m 4.1 a n d corollaries 4.2 a n d 4.3 are coherent. | In a d d i t i o n t o p o i n t i n g out the i n a d e q u a c i e s of shortest p a t h inferencing a n d to p r o v i d i n g sufficient c o n d i t i o n s for coherence a n d a correct inference m e c h a n i s m , the f o r m a ! r e c o n s t r u c t i o n of i n h e r i t a n c e we h a v e presented clarifies some of the o u t s t a n d i n g p r o b l e m s i n n e t w o r k  inference.  O n e of these, h o w t o p e r f o r m inferences i n p a r a l l e l , is considered in the n e x t s e c t i o n .  4.1. Parallel Network Inference Algorithms T h e c o m p u t a t i o n a l c o m p l e x i t y of i n h e r i t a n c e problems, c o m b i n e d w i t h some e n c o u r a g i n g e x a m p l e s , has s p a r k e d interest i n the p o s s i b i l i t y of d e t e r m i n i n g i n h e r i t a n c e i n p a r a l l e l . F a h l m a n [1979] has p r o p o s e d a m a s s i v e l y p a r a l l e l m a c h i n e a r c h i t e c t u r e , N E T L . T h i s a r c h i t e c t u r e assigns one processor to e a c h p r e d i c a t e i n the k n o w l e d g e base. " I n f e r e n c i n g " is p e r f o r m e d b y nodes passi n g " m a r k e r s " to a d j a c e n t nodes i n response to t h e i r o w n state a n d t h a t of t h e i r i m m e d i a t e n e i g h bours.  F a h l m a n suggests t h a t s u c h a r c h i t e c t u r e s c o u l d achieve l o g a r i t h m i c speed i m p r o v e m e n t s  over traditional serial machines. T h e f o r m a l i z a t i o n of i n h e r i t a n c e n e t w o r k s as default theories suggests, h o w e v e r , t h a t there m i g h t be severe l i m i t a t i o n s to this a p p r o a c h . F o r e x a m p l e , correct inference requires t h a t a l l c o n clusions share a c o m m o n e x t e n s i o n . F o r n e t w o r k s w i t h more t h a n one e x t e n s i o n , i n t e r - e x t e n s i o n interference effects m u s t be p r e v e n t e d . T h i s seems i m p o s s i b l e for a one pass p a r a l l e l a l g o r i t h m w i t h p u r e l y l o c a l c o m m u n i c a t i o n , e s p e c i a l l y in v i e w of the i n a d e q u a c i e s of t h e shortest heuristic.  path  - 62 -  E v e n i n k n o w l e d g e bases w i t h u n i q u e extensions, structures r e q u i r i n g a n a r b i t r a r i l y r a d i u s o f c o m m u n i c a t i o n c a n be c r e a t e d . F o r e x a m p l e [ E t h e r i n g t o n  1982], the d e f a u l t  c o r r e s p o n d i n g t o t h e n e t w o r k s i n F i g u r e 4.5 e a c h have u n i q u e extensions. A n e t w o r k  large  theories inference  a l g o r i t h m m u s t r e a c h F before p r o p a g a t i n g t h r o u g h B i n the first n e t w o r k a n d c o n v e r s e l y i n the second.  T h e s a l i e n t d i s t i n c t i o n s between the t w o n e t w o r k s are n o t l o c a l ; hence t h e y c a n n o t be  u t i l i z e d t o g u i d e a p u r e l y l o c a l inference m e c h a n i s m t o the correct choices.  Similar networks can  be c o n s t r u c t e d w h i c h defeat m a r k e r - p a s s i n g a l g o r i t h m s w i t h a n y fixed r a d i u s .  Figure 4-5 — Problems for local inheritance algorithms.  This has prompted Touretzky [1981, personal communication; 1984a] to characterize a restricted class of network structures which admit parallel inferencing algorithms.  In part, his res-  trictions appear to exclude networks whose corresponding default theory has more than one extension.  Unfortunately, it is unclear how these restrictions affect the expressive power of the result-  ing networks.  Moreover, Touretzky [1982, personal communication; 1983] has shown that it is not  possible to determine on a parallel marker-passing machine whether a network satisfies these restrictions. Provided the network in question corresponds to an ordered theory, a form of limited parallelism can be achieved without sacrificing correctness.  T h e key to this result lies in partitioning  the network into subnetworks which are suitable for parallel processing.  Essentially, each node in  the network is numbered according to the number of exception links apon which it depends. This assigns each node to the lowest "level" possible while preserving the ordering amongst the nodes induced by the " <§: " and " <Z " relations.  Since the network is ordered, this can be done in  parallel, in finite time proportional to the longest chain in the network.  Processing then proceeds  in k parallel steps, where k is the number of the highest level to which nodes were assigned.  At  step n, all links having exceptions which were asserted at step n-1 are disabled. The resulting sub-network, consisting of all remaining links impinging on nodes at levels less than or equal to n, is processed in parallel, ignoring exception links, with markers propagating from nodes asserted at step n-1.  T h e "nodes asserted at level 0" are those in Th( W). These correspond to the nodes for  - 63 -  w h i c h the n e t w o r k is " a c t i v a t e d " . T h e result after step k is a n e x t e n s i o n .  2  T h e r e are t w o c a v e a t s associated w i t h this procedure: If b o t h p o s i t i v e a n d n e g a t i v e m a r k e r s r e a c h a node i n the same step, one m u s t be chosen. E i t h e r choice w i l l l e a d to a n e x t e n s i o n ; we do not c o n s i d e r o t h e r r a m i f i c a t i o n s of s u c h choices here. Second, the a l g o r i t h m assumes t h a t a l l strict links propagate instantaneously.  If t h i s is not the case, each step i n the a l g o r i t h m m u s t be f o l -  l o w e d b y p r o p a g a t i o n a l o n g strict l i n k s , resolving conflicts as a b o v e . N o t e t h a t conflicts are a l w a y s r e s o l v e d b y c h a n g i n g assignments at the c u r r e n t l e v e l . P r o v i d e d t h a t the i n v i o l a b i l i t y of strict l i n k s is m a i n t a i n e d , t h a t default l i n k s are a c t i v e o n l y if t h e i r p r e r e q u i s i t e s are asserted a n d t h e i r j u s t i f i c a t i o n s h a v e not been d e n i e d , a n d t h a t no node a n d its n e g a t i o n are asserted t o g e t h e r (conflict resolution), any reasonable p r o p a g a t i o n a l g o r i t h m ( p a r a l l e l o r otherwise) m a y be used at e a c h s t e p .  3  T o i l l u s t r a t e t h e c o n s t r u c t i o n , we a p p l y it to the m o d e r a t e l y c o m p l e x n e t w o r k o f F i g u r e 4 . 6 . R a t h e r t h a n r e s t r i c t ourselves t o a p a r t i c u l a r p a r a l l e l p r o p a g a t i o n a l g o r i t h m at e a c h step, we present a t a b l e s h o w i n g a l l p o s s i b i l i t i e s .  A  Figure 4-6— A m u l t i - l e v e l i n h e r i t a n c e g r a p h .  T h e c o r r e s p o n d i n g default t h e o r y , s i m p l i f i e d to the p r o p o s i t i o n a l case a n d " a c t i v a t e d " for A , is:  W =  {A, ( A o f l ) , ( A O  ( A.^D \  2  -.£> '  A  C))  : -iF -,F  '  B: D D  C: F '  F  B : E '  E  E :-^H  '  -itf  This construction is that used in the proof of Theorem 3.3, where it is shown to yield an extension.  T o see this, it is necessary only to note that each step is, effectively, dealing with a normal theory. Arguments similar to those used in the proof of Theorem 3.5 can be used to show that the order of propagation is immaterial. 3  - 64 -  E: G f\ -<D G  G: H  '  H  E: If\ ->F '  I  A ~-H  I:  -<J T h e d e f a u l t s a b o v e h a v e been g r o u p e d a c c o r d i n g t o the l e v e l t o w h i c h their consequents are assigned Level  (see T a b l e  4.1).  Table  4.2  shows the  possibilities  at  e a c h step; a l t e r n a t i v e s  are  Literals  1  A , B , C , D , n D , E , F , - . F , ->H  2  G,  H,I  3 T a b l e 4.1 — L e v e l s of literals. s h o w n i n separate c o l u m n s , w i t h m a j o r rows corresponding to steps i n the a l g o r i t h m . Step 1  A , B, C , E, -.H D, F  Step 2  D, - F  -.D, F  -.D, -,F  I  G  I, G  Step 3 T a b l e 4.2 — P o s s i b l e outcomes using different p r o p a g a t i o n schemes.  T h u s the a l g o r i t h m c a n , d e p e n d i n g o n the nature of the p a r a l l e l m a r k e r p r o p a g a t i o n p r o c e d u r e , find: E Ex E E 0  2  3  = Th( W = Th( W = Th\ W = Th( W  U U U U  {A, {A, {A, {A,  B, B, B, B,  C, C, C, C,  E, D, F}) E, ->H, D, -,F, I, ->J}) E, ->H, ->D, F, G}) E, ->H, -<D, /, G, ->J})  a l l of w h i c h are extensions. S i g n i f i c a n t l y , no choice of p a r a l l e l m a r k e r - p a s s i n g p r o c e d u r e  will  enable the a l g o r i t h m t o find the t h e o r y ' s o t h e r t w o extensions: E E  A  5  = =  Th( W Th( W  U U  {A, B, C, E, H, -.£>, F, G}) {A, B, C, E, H, ->F, G, /})  because ->H is at l e v e l 1 a n d so c a n ( a n d must) be inferred at step 1. H, being at l e v e l 2, is t h u s p r e c l u d e d before it c a n be inferred. W e have not yet c h a r a c t e r i z e d the biases w h i c h t h i s i n a b i l i t y to find a l l e x t e n s i o n s w o u l d induce i n a reasoner. A n o t h e r p o t e n t i a l p r o b l e m w i t h t h i s a p p r o a c h stems f r o m the fact t h a t m a n y n e t w o r k  infer-  ence s y s t e m s " p r e f e r " one l i n k - t y p e o v e r a n o t h e r (e.g., n e g a t i o n m a y o v e r r i d e assertion).  By  b r e a k i n g the n e t w o r k i n t o s u b - n e t w o r k s w h i c h are processed i n t u r n , the a b i l i t y t o g l o b a l l y assert these preferences m a y be lost.  W e have three responses t o this.  F i r s t l y , if n e t w o r k s t r u c t u r e is  r e s t r i c t e d , i n the m a n n e r suggested b y T o u r e t z k y [1981, p e r s o n a l c o m m u n i c a t i o n ] , so t h a t resulti n g theories h a v e u n i q u e extensions, the a b o v e a l g o r i t h m produces the same results as a n y correct  - 65 -  procedure.  Secondly, many of these preferences are not well-defined, and break down when  pressed (c.f. race conditions in [Fahlman et al 1981]). The inability to exhibit incorrect behaviour can hardly be called a liability. Finally, given a well-defined preference scheme, it must preserve correctness: all inferences must lie in a single extension. If such a scheme exists which cannot be implemented within the confines outlined above, some other inference procedure will be required. Given the problems already observed with parallelism, we doubt that a parallel or quasi-parallel, single-pass, marker-passing algorithm can be found which takes global considerations into account (at least in unrestricted networks).  4  Touretzky [1984a] has recently developed a well-defined notion of preference, which we discuss in the next section. The above algorithm does not necessarily produce the conclusions this scheme sanctions, but Touretzky observes that there appears to be no parallel marker-passing algorithms which respect this preference-order for all networks.  4.2. Theory Preference The  formalisation of inheritance, above, uses semi-normal links to represent default links  with explicit exceptions.  We argue that such explicit exceptions are generally necessary to ensure  that the resulting theory has a unique extension.  This is important for systems whose inference  mechanism is incapable of guaranteeing that all the conclusions it draws from the network representation lie within a single extension of the corresponding default theory.  Otherwise, the  correctness of the system's "beliefs" must be questionable. Touretzky [1984a, 1984b] argues that our reformulation of inheritance in terms of seminormal defaults knowledge-base  is inappropriate for two  reasons:  Firstly,  adding new  requires modification of the defaults already in the  become increasingly complex as the knowledge-base grows. depends on other links in the network.  information to  knowledge-base.  the  These  Secondly, the translation of a link  The translation, Touretzky claims, ignores the essentially  "hierarchical" nature of inheritance networks, which he views as their chief asset - both in terms of representational conciseness and computational efficiency. These criticisms suggest that a (common) misapprehension about default logic has occurred. It is commonly believed that a default logic based reasoning system must be able to find any of the extensions of a default theory, and must view them all as equally acceptable sets of beliefs.  In  fact, while extensions are all acceptable, the logic says nothing about preference of one to another. It has always been assumed that an agent would "choose" one extension or another, but nothing has been said about how such choices should be made.  There is no reason not to (and, perhaps,  good reason to) exploit extra-logical properties of the knowledge-base (e.g., hierarchical structure) to establish preferred extensions.  C o t t r e l l [1985] has experimented with a multi-pass, "connectionist", parallel architecture which shows some promise here, although no correctness proofs have been forthcoming. Connectionist architectures are beyond the scope of this thesis, however. 4  - 66 -  T o o u r k n o w l e d g e , the first a l g o r i t h m c a p a b l e of c o r r e c t l y reasoning w i t h a n i n h e r i t a n c e network  in  parallel was that  presented i n the preceding section (see also [ E t h e r i n g t o n  1983]).  B e c a u s e of the p a r t i t i o n i n g of the n e t w o r k , the a l g o r i t h m is i n c a p a b l e of finding some extensions of some d e f a u l t theories;  it is not c o m p l e t e .  T h i s a l g o r i t h m is correct;  a l l of its c o n c l u s i o n s lie  w i t h i n a single e x t e n s i o n . H o w e v e r , it does not necessarily p r o d u c e the preferred e x t e n s i o n , based o n the i n t u i t i v e s e m a n t i c s for i n h e r i t a n c e n e t w o r k s . T o u r e t z k y [1984a] has d e v e l o p e d a m o r e s o p h i s t i c a t e d a l g o r i t h m , based o n the distance" topology.  "inferential  T h i s i n f e r e n t i a l d i s t a n c e a l g o r i t h m is a p p l i c a b l e to n e t w o r k s w i t h o u t e x p l i c i t  e x c e p t i o n l i n k s , a n d is correct, i n the sense t h a t a l l of its conclusions lie w i t h i n a single e x t e n s i o n . F u r t h e r m o r e , the i n f e r e n t i a l distance c o n c e p t is b a s e d o n the p r i n c i p l e t h a t a m b i g u o u s i n h e r i t a n c e s h o u l d be, w h e n possible, r e s o l v e d b y a p p e a l i n g to the s u b c l a s s / s u p e r c l a s s r e l a t i o n w h i c h forms the basis o f i n h e r i t a n c e . I n f e r e n t i a l d i s t a n c e is somewhere b e t w e e n the " b r a v e " a n d " c a u t i o u s " ends of the s p e c t r u m of n o n - m o n o t o n i c r e a s o n i n g systems. E s s e n t i a l l y , if a n i n d i v i d u a l c o u l d inherit p r o p e r t y P b y v i r tue of the fact t h a t she I S - A B, a n d p r o p e r t y ->P because she I S - A C, t h e n the a m b i g u i t y is r e s o l v e d as f o l l o w s :  If C I S - A B a n d not vice versa, inherit ~<P,  vice versa, i n h e r i t P; otherwise, i n h e r i t neither.  otherwise, if B I S - A C a n d not  C o n c e p t u a l l y , the i n f e r e n t i a l d i s t a n c e a l g o r i t h m  e l i m i n a t e s those e x t e n s i o n s w h i c h do not satisfy the h i e r a r c h i c a l n a t u r e of the r e p r e s e n t a t i o n , then d r a w s those c o n c l u s i o n s w h i c h h o l d i n a l l of the r e m a i n i n g extensions. T h i s a p p r o a c h c a p t u r e s the s e m a n t i c i n t u i t i o n (properties associated w i t h subclasses s h o u l d o v e r r i d e those a s s o c i a t e d w i t h superclasses) w h i c h is the f u n d a m e n t a l raison d'etre for i n h e r i t a n c e representations.  It also a v o i d s the p i t f a l l s of i n c o r r e c t b e h a v i o u r w h i c h curse s h o r t e s t - p a t h infer-  ence a l g o r i t h m s , as e v i d e n c e d b y T h e o r e m 4.5.  T h e o r e m 4.5  In the absence of " n o - c o n c l u s i o n " l i n k s , a l l of the g r o u n d facts r e t u r n e d b y T o u r e t z k y ' s i n f e r e n t i a l d i s t a n c e a l g o r i t h m lie w i t h i n a single extension of the default t h e o r y w h i c h c o r r e s p o n d s to the i n h e r i t a n c e g r a p h i n q u e s t i o n . |  T o i l l u s t r a t e the i n f e r e n t i a l d i s t a n c e a l g o r i t h m , consider the n e t w o r k f r o m F i g u r e 4.4(b). Because Nautilus is a subclass of Cephalopod, w h i c h is a subclass of Mollusc, i n f e r e n t i a l distance gives the d e s i r e d results: Nautili are Shell-Bearers, w h i l e Cephalopods not k n o w n to be Nautili are not.  I n t h e n e t w o r k of F i g u r e 4.7, n e i t h e r Republican nor Quaker is a subclass of the other.  i n f e r e n t i a l d i s t a n c e s a n c t i o n s n o c o n c l u s i o n s a b o u t w h e t h e r Nixon is a Pacifist.  Thus  - 67 .  Quaker  Pacifist  '. R e p u b l i c a n  .  Figure J^.l— A g e n u i n e l y a m b i g u o u s i n h e r i t a n c e g r a p h . T h e o r e m 4.5 o n l y begins to e x p l o r e the connections between T o u r e t z k y ' s w o r k a n d t h a t r e p o r t e d i n c h a p t e r s 3 a n d 4 o f t h i s thesis ( a n d i n [Reiter 1980a]). facts r e t u r n e d b y i n f e r e n t i a l d i s t a n c e -  W e have shown that ground  e.g., " C l y d e is a n e l e p h a n t " , o r " C l y d e is not g r e y "  belong to a c o m m o n e x t e n s i o n of the c o r r e s p o n d i n g default theory.  I n f e r e n t i a l d i s t a n c e also sanc-  tions n o r m a t i v e c o n c l u s i o n s , s u c h as " A l b i n o - e l e p h a n t s are [typically] h e r b i v o r e s " .  W e h a v e not  e x p l o r e d the r e l a t i o n s h i p s u c h s t a t e m e n t s inferred u n d e r i n f e r e n t i a l distance b e a r to the u n d e r l y ing default t h e o r y . T o u r e t z k y also a l l o w s w h a t he calls " n o - c o n c l u s i o n " l i n k s .  T h e s e l i n k s a l l o w i n h e r i t a n c e to  be b l o c k e d w i t h o u t e x p l i c i t c a n c e l l a t i o n . D e f a u l t logic has n o analogue for the n o - c o n c l u s i o n l i n k , a n d we h a v e e x c l u d e d t h e m f r o m c o n s i d e r a t i o n here.  It appears that it w o u l d be  straightforward  to a d d a s i m i l a r c a p a c i t y to default logic, a s s u m i n g t h a t such l i n k s a c t u a l l y p r o v e useful.  The  proof of t h e o r e m 4.5 suggests t h a t its g e n e r a l i z a t i o n t o n e t w o r k s w i t h n o - c o n c l u s i o n l i n k s vis-a-vis s u c h a n e x t e n d e d d e f a u l t logic w o u l d present n o p r o b l e m s . T o u r e t z k y [1984a] p r o v i d e s a d e t a i l e d e x p l o r a t i o n of the properties of i n f e r e n t i a l d i s t a n c e i n h e r i t a n c e r e a s o n i n g , i n c l u d i n g a c o n s t r u c t i v e m e c h a n i s m for d e t e r m i n i n g the ' g r o u n d e d e x p a n sions' (analogous to extensions) of a n e t w o r k .  M a n y of his results bear a s u p e r f i c i a l s i m i l a r i t y i n  f o r m a n d p r o o f to those in [Reiter 1980a] a n d i n c h a p t e r 3 of this thesis. H i s proofs r e l y o n p a r t i a l a c y c l i c i t y c o n d i t i o n s w h i c h seem s i m i l a r to the orderedness c o n d i t i o n s we describe.  W e have  s p e c u l a t e d (as has T o u r e t z k y ) t h a t the results i n [ T o u r e t z k y 1984a] a n d those c o n t a i n e d herein m a y p r o v e to be closely r e l a t e d . F i n a l l y , T o u r e t z k y [1984a, 1985] explores the a p p l i c a t i o n s of i n f e r e n t i a l distance to " i n h e r i t able r e l a t i o n s " , c i t i n g e x a m p l e s s u c h as Citizens dislike crooks. E l e c t e d c r o o k s are crooks. G u l l i b l e c i t i z e n s d o n ' t d i s l i k e elected crooks. In t h i s e x a m p l e , c i t i z e n s generally d i s l i k e elected crooks, b u t F r e d , the g u l l i b l e c i t i z e n , doesn't disl i k e D i c k , t h e e l e c t e d crook. default  logic should try  A c o m p l e t e t r e a t m e n t of the r e l a t i o n between T o u r e t z k y ' s w o r k a n d  t o e x t e n d the correspondence presented here t o i n c l u d e  Touretzky's  i n f e r e n t i a l d i s t a n c e t r e a t m e n t of i n h e r i t a b l e r e l a t i o n s . T o u r e t z k y shows t h a t , i n general, p a r a l l e l m a r k e r - p a s s i n g a l g o r i t h m s c a n n o t d e r i v e the c o n c l u s i o n s s a n c t i o n e d b y the i n f e r e n t i a l d i s t a n c e a l g o r i t h m .  H e also shows t h a t a n a r b i t r a r y  net-  w o r k c a n be " c o n d i t i o n e d " , b y a d d i n g l o g i c a l l y - r e d u n d a n t l i n k s , i n s u c h a w a y t h a t a p a r a l l e l m a r k e r - p a s s i n g a l g o r i t h m can r e t u r n correct results. U n f o r t u n a t e l y , this c o n d i t i o n i n g , w h i c h m u s t  -  - 68 -  be p e r f o r m e d  each time  the  network  is m o d i f i e d , is expensive ( T o u r e t z k y  p o l y n o m i a l - t i m e a l g o r i t h m w h i c h a d d s O^N ) 2  [1984a] gives a  l i n k s i n the worst case) a n d is a p p a r e n t l y not a m e n -  able to p a r a l l e l m a r k e r - p a s s i n g i m p l e m e n t a t i o n [ T o u r e t z k y 1982, p e r s o n a l c o m m u n i c a t i o n ; 1983]. We  c o n c l u d e t h a t , for c o n d i t i o n e d n e t w o r k s ,  described) exceptions.  there  are correct  p a r a l l e l , m a r k e r - p a s s i n g a l g o r i t h m s for d e t e r m i n i n g  (in the sense we  i n h e r i t a n c e i n the  have  presence of  S u c h a l g o r i t h m s c a n be v i e w e d as fast inference a l g o r i t h m s f o r r e a s o n i n g w i t h the  t r a c t a b l e class of d e f a u l t theories w h i c h c o r r e s p o n d to c o n d i t i o n e d n e t w o r k s .  CHAPTER 5  Predicate Circumscription  I n t h i s c h a p t e r we focus o n predicate c i r c u m s c r i p t i o n , as presented i n [ M c C a r t h y  1980].  O u r o b j e c t i v e is t o e s t a b l i s h v a r i o u s results c o n c e r n i n g the consistency of this f o r m a l i s m , a n d to describe some l i m i t a t i o n s of its a b i l i t y to conjecture new i n f o r m a t i o n .  O n e s u c h l i m i t a t i o n is t h a t  p r e d i c a t e c i r c u m s c r i p t i o n c a n n o t a c c o u n t for the s t a n d a r d k i n d s of default r e a s o n i n g . l i m i t a t i o n relates t o e q u a l i t y ;  Another  p r e d i c a t e c i r c u m s c r i p t i o n yields no new g r o u n d facts a b o u t  e q u a l i t y p r e d i c a t e for a large class o f  first-order  theories.  the  T h i s has i m p o r t a n t consequences for the  so-called " u n i q u e n a m e s " a n d " d o m a i n closure" assumptions.  5.1. F o r m a l P r e l i m i n a r i e s  P r e d i c a t e c i r c u m s c r i p t i o n w a s discussed i n d e t a i l i n c h a p t e r 2. t e c h n i c a l d e t a i l s here for c o n v e n i e n c e .  W e repeat some of  the  T h e s e m a n t i c i n t u i t i o n u n d e r l y i n g predicate c i r c u m s c r i p -  t i o n is t h a t c l o s e d - w o r l d reasoning a b o u t one or more predicates of a theory corresponds to t r u t h i n a l l m o d e l s of the t h e o r y w h i c h are m i n i m a l in those predicates. S p e c i f i c a l l y , let T[Pi,...,P^ be a  first-order  t h e o r y , some (but not necessarily all) of whose predicates are P  = {P,...P }. 1  /  re  A  m o d e l M of T is a "P-submodel of a m o d e l M ' of T iff the e x t e n s i o n of e a c h P,- i n M is a subset of its e x t e n s i o n i n M * , a n d M a n d M * are otherwise i d e n t i c a l .  M i s a P-minimal model of T iff every  P - s u b m o d e l of M i s i d e n t i c a l to M . For  finite  theories,  T(Pi,...,P^ M c C a r t h y [1980] proposes r e a l i z i n g predicate c i r c u m s c r i p t  t i o n s y n t a c t i c a l l y b y a d d i n g the f o l l o w i n g a x i o m s c h e m a to T:  Here  are p r e d i c a t e v a r i a b l e s w i t h the same arities as P ^ - . ^ P n , r e s p e c t i v e l y .  !!($!,...,3> ) n  is t h e sentence o b t a i n e d b y c o n j o i n i n g t h e sentences of T, t h e n r e p l a c i n g e v e r y occurrence of P i , . . P „ i n T b y $i,...,$„ r e s p e c t i v e l y . v  T h e above s c h e m a is c a l l e d the (joint)  schema of P ...,P in T. L e t CLOSURE (T) u  n  P  circumscription  - the closure of T with respect to P = { P . . . , P J l 7  denote the t h e o r y c o n s i s t i n g of T together w i t h the above a x i o m s c h e m a .  M c C a r t h y formally  identifies r e a s o n i n g a b o u t T u n d e r the c l o s e d - w o r l d a s s u m p t i o n w i t h respect to the predicates P with  first-order  d e d u c t i o n s f r o m the t h e o r y  CLOSURE-p{T).  M c C a r t h y [1980] shows t h a t a n y i n s t a n c e of t h e s c h e m a r e s u l t i n g f r o m c i r c u m s c r i b i n g a s i n gle p r e d i c a t e P i n a sentence 1\P) is true i n a l l { P } - m i n i m a l m o d e l s of T. T h i s generalizes  - 69 -  -  - 70 d i r e c t l y t o the j o i n t c i r c u m s c r i p t i o n of m u l t i p l e predicates;  we o m i t the proof of t h i s .  this g e n e r a l i z a t i o n e x t e n s i v e l y i n the proofs of the results of this c h a p t e r . c u m s c r i p t i o n is a p p l i c a b l e o n l y to  finitely  W e use  B e c a u s e p r e d i c a t e cir-  a x i o m a t i z a b l e theories, we w i l l restrict o u r a t t e n t i o n to  s u c h theories.  5.2.  O n the Consistency of Predicate Circumscription  T h e m i n i m a l m o d e l s e m a n t i c s of predicate c i r c u m s c r i p t i o n guarantees t h a t CLOSUREp[ is consistent w h e n e v e r T has P - m i n i m a l models.  T h i s suggests t h a t c e r t a i n consistent  T)  first-order  theories l a c k i n g m i n i m a l m o d e l s m a y h a v e inconsistent closures. Indeed, this c a n h a p p e n , as we now show.  E x a m p l e 5.1 — A n i n c o n s i s t e n t  circumscription  C o n s i d e r the f o l l o w i n g consistent theory: 3z. N i A Vy. [Ny D x £ succ(y)]  Vz. N z Z> Nsucc(z) Vxy. succ(x); = succ(y)  Z>  z=  y  In a n y m o d e l of T, the e x t e n s i o n of N J c o n t a i n s a sequence of elements i s o m o r p h i c t o the n a t u r a l n u m b e r s . A n { N } - s u b m o d e l c a n a l w a y s be c o n s t r u c t e d b y d e l e t i n g a finite i n i t i a l segment of this sequence. H e n c e every m o d e l of T has a p r o p e r { N } - s u b m o d e l , so T has no { N } - m i n i m a l m o d e l s . C i r c u m s c r i b i n g N i n this theory, a n d l e t t i n g $z be [Nz A 3y- z = succ(y^\ Ny] V z . N i D 3 y- [Ny l\ x = succ(y)\ w h i c h c o n t r a d i c t s the first a x i o m . fl  yields  :  I n v i e w of t h i s e x a m p l e , it is n a t u r a l t o seek classes of  first-order  theories for w h i c h p r e d i -  cate c i r c u m s c r i p t i o n does not i n t r o d u c e inconsistencies. T h e " w e l l - f o u n d e d ' ' theories f o r m s u c h a class.  W e say t h a t a  first-order  t h e o r y is well-founded iff e a c h of its m o d e l s has a P - m i n i m a l s u b -  m o d e l f o r e v e r y finite set o f predicates P . least one P - m i n i m a l m o d e l .  A n y consistent w e l l - f o u n d e d theory o b v i o u s l y has at  S i n c e e v e r y instance of the c i r c u m s c r i p t i o n s c h e m a of P i n a t h e o r y  T is t r u e i n a l l P - m i n i m a l m o d e l s of T, we have:  Theorem  5.1  If T is a c o n s i s t e n t w e l l - f o u n d e d theory, t h e n CLOSURE-^ T) is consistent for a n y set P of p r e d i c a t e s of theories.  I.e., p r e d i c a t e c i r c u m s c r i p t i o n  T.  preserves c o n s i s t e n c y for  well-founded  |  W h i c h theories are w e l l - f o u n d e d ?  W e k n o w of no c o m p l e t e s y n t a c t i c c h a r a c t e r i z a t i o n , b u t a  p a r t i a l a n s w e r comes f r o m a g e n e r a l i z a t i o n of a result on u n i v e r s a l theories d u e t o B o s s u a n d S i e g e l [1985].  A  first-order  t h e o r y is universal iff t h e prenex n o r m a l f o r m of e a c h of its f o r m u l a e  - 71 c o n t a i n s n o e x i s t e n t i a l quantifiers.  Theorem  5.2  U n i v e r s a l theories are w e l l - f o u n d e d .  |  In v i e w of T h e o r e m 5.1, we k n o w t h a t predicate c i r c u m s c r i p t i o n preserves c o n s i s t e n c y for u n i v e r s a l theories: Corollary  If  5.3  T is a consistent u n i v e r s a l theory, t h e n CLOSUREp(T)  p r e d i c a t e s o f T.  is consistent for a n y set P  of  |  N o t i c e t h a t the class of u n i v e r s a l theories includes the H o r n theories, w h i c h h a v e a t t r a c t e d c o n s i d e r a b l e a t t e n t i o n f r o m the P R O L O G , A l , a n d D a t a b a s e c o m m u n i t i e s . L i f s c h i t z [1985b] has g e n e r a l i z e d t h e o r e m 5.2 t o i n c l u d e the class of " a l m o s t theories.  universal"  A t h e o r y is almost universal in P iff it has the f o r m V x t A, where A does not c o n t a i n  p o s i t i v e occurrences of P € P w i t h i n the scope of quantifiers.  A l m o s t u n i v e r s a l theories i n c l u d e  u n i v e r s a l theories as w e l l as the " s e p a r a b l e " theories of L i f s c h i t z [1985a] (see § 2.1.5.2).  5.3.  Weil-Founded Theories and Predicate Circumscription  T h e property  of well-foundedness, t a k e n together w i t h the " s o u n d n e s s " of p r e d i c a t e cir-  c u m s c r i p t i o n w i t h respect to the set of m i n i m a l m o d e l s allows us to characterize the p o w e r of predicate circumscription.  This  leads t o some r a t h e r  surprising results.  In  this  s e c t i o n we  describe some l i m i t a t i o n s of p r e d i c a t e c i r c u m s c r i p t i o n w i t h respect to w e l l - f o u n d e d theories.  The  first s u c h result is t h a t p r e d i c a t e c i r c u m s c r i p t i o n y i e l d s no new p o s i t i v e g r o u n d instances of a n y of the p r e d i c a t e s b e i n g c i r c u m s c r i b e d .  Theorem  5.4  S u p p o s e t h a t T is a w e l l - f o u n d e d theory, P € P is a n n-ary p r e d i c a t e , a n d ct ,... '3 are nl  t u p l e s of g r o u n d terms. CLOSUREDT)|- i ra  t  k  Then v  -  v  P^k  *=»  T \— P&i V . - V Pct . k  |  O n r e f l e c t i o n , t h i s is not too s u r p r i s i n g , since c i r c u m s c r i p t i o n is i n t e n d e d to m i n i m i z e the e x t e n sions of those p r e d i c a t e s being c i r c u m s c r i b e d . n o t arise f r o m t h i s m i n i m i z a t i o n .  N e w positive instances o f s u c h predicates s h o u l d  - 72 -  A  more i n t e r e s t i n g - e v e n s t a r t l i n g - result is t h a t no n e w g r o u n d i n s t a n c e s , p o s i t i v e o r  n e g a t i v e , of u n c i r c u m s c r i b e d predicates c a n be d e r i v e d b y predicate c i r c u m s c r i p t i o n .  T h e o r e m 5.5  S u p p o s e t h a t T i s a w e l l - f o u n d e d theory, P £ P is a n n-ary predicate, a n d <£ ...,a are nv  tuples of ground terms. (i) (ii)  CLOSUREp(T) CLOSURE^  k  Then \-P3 y...y i  Pet  T |— P a j V . . . V Pa ,  k  k  T) \- -PoTi V . . . V - . P a  fc  and  T \- - P o ^ V . . . V ^Pa  k  I  .  In s u m m a r y , T h e o r e m s 5.4 a n d 5.5 t e l l us t h a t the o n l y n e w g r o u n d l i t e r a l s t h a t c a n be c o n j e c t u r e d b y p r e d i c a t e c i r c u m s c r i p t i o n of w e l l - f o u n d e d theories are negative instances of one of t h e p r e d i c a t e s b e i n g c i r c u m s c r i b e d ; A n u n f o r t u n a t e consequence of this result i s t h a t t h e u s u a l k i n d s of default r e a s o n i n g c a n n o t be r e a l i z e d b y predicate c i r c u m s c r i p t i o n . standard A l example concerning whether  T o see w h y , c o n s i d e r t h e  b i r d s fly, g i v e n t h a t " b y d e f a u l t "  b i r d s fly. T h e  r e l e v a n t f a c t s m a y be represented i n v a r i o u s ways, t w o o f w h i c h f o l l o w : 1) I n this r e p r e s e n t a t i o n , a l l o f the exceptions t o flight are listed e x p l i c i t l y i n t h e a x i o m s a n c t i o n i n g t h e c o n c l u s i o n t h a t b i r d s c a n fly. Vz. Bird{x) A -*Penguin(x) A -<Ostrich(x) A -*Dead(x) A ••• 3 Can-Fly(x) In a d d i t i o n , there are v a r i o u s I S - A a x i o m s , as w e l l as m u t u a l e x c l u s i o n a x i o m s : Vz. Vz. Vz. Vz.  Canary(x) Z5 Birdix) Penguin(x) p 5tra(z) -if Canary(x) A Penguin(x\\ -i(Pen</tttn(z) A Ostrich(x))  2) I n t h i s r e p r e s e n t a t i o n , due t o M c C a r t h y [19861, a new p r e d i c a t e , ab, s t a n d i n g f o r " a b n o r m a l " , is i n t r o d u c e d . O n e then states t h a t " n o r m a l " b i r d s c a n fly: Vz. Bird(x) A --a6(z) D Can-Fly(x) T h e a b n o r m a l b i r d s are l i s t e d : Vz. Penguin(x) D ab(x) Vz. Ostrich(x) D ab[x) F i n a l l y , one i n c l u d e s t h e I S - A a n d m u t u a l e x c l u s i o n a x i o m s as i n (1) above. B o t h r e p r e s e n t a t i o n s (1) a n d (2) are u n i v e r s a l , a n d hence w e l l - f o u n d e d , theories. if  Bird(Tweety)  is given,  Theorems  5.4 a n d 5.5(i)  tell  us t h a t  t h e default  Therefore, assumption  Can-Fly( Tweety) c a n n o t be c o n j e c t u r e d b y predicate c i r c u m s c r i p t i o n . C a r e f u l readers o f [ M c C a r t h y 1980] m i g h t find T h e o r e m s 5.4 a n d 5.5 inconsistent w i t h t h e results i n S e c t i o n 7 of t h a t paper.  I n t h e b l o c k s - w o r l d e x a m p l e presented there t o i l l u s t r a t e p r e d i -  cate c i r c u m s c r i p t i o n , t h e g r o u n d instance on(A,C,result{move(A,C),SQJ) c a n be d e r i v e d b y c i r c u m s c r i b i n g a different p r e d i c a t e , Xz.prevents(z,move(A,C),s ). T h i s appears t o v i o l a t e T h e o r e m 0  5.5(i).  T h i s d i s c r e p a n c y stems f r o m t h e fact t h a t i n f o r m u l a t i n g t h e c i r c u m s c r i p t i o n s c h e m a f o r  - 73 -  this example, M c C a r t h y uses specializations of some of the original axioms (i.e., the axioms which specify what can prevent a move from succeeding), and omits one of the axioms (i.e., the axiom which states that if nothing prevents a move from succeeding, the move will be successful).  Thus,  only part of the theory enters into the circumscription for his example, whereas Theorems 5.4 and 5.5 suppose that the entire theory is used in proposing a circumscription schema.  5.4.  Equality  We  now consider some limitations of predicate circumscription with respect to the treat-  ment of equality.  These limitations will be seen to have consequences for two special cases of  closed-world reasoning, namely deriving the "unique names assumption" and the "domain closure assumption".  5.4.1.  TheUnique-Names Assumption  When told that T o m , Dick and Harry are friends, one naturally assumes that ' T o m ' , 'Dick' and 'Harry' denote distinct individuals: T o m j= Dick, T o m j= Harry, Dick £ Harry.  For a more  general example, consider a setting in which one is told that Tom's telephone number is the same as Sue's, and that Bill's number is 555-1234, which is different from Mary's number.  Thus, we  have: tel-no(Tom) = tel-no(Sue)i tel-no(Bill) = 555-1234 tel-no(Mary) £ 555-1234 One  would naturally assume from this information that tel-no(Tom) j= 555-1234, and that tel-  no(Tom) j= tel-no(Mary). In general, the unique-names assumption is invoked whenever one can assume that all of the relevant information about the equality of individuals has been specified. not specified as identical are assumed to be different.  A l l pairs of individuals  This assumption arises in a number of set-  tings, for example in the theory of databases [Reiter 1980b], and in connection with the semantics of negation in P R O L O G [Clark 1978]. Virtually every A l reasoning system, with the exception of those based on theorem-provers, implicitly makes this assumption.  Because of Clark's results, we  know that this is also the case for P R O L O G based A l systems. Unique-names axioms are also important for closed-world reasoning using predicate circumscription.  F o r example, if all we know is that Opus is a Penguin, we can circumscriptively  conjecture Vx. Penguin(x) = x = Opus. W e cannot use this to deduce -<Penguin(Tweety), however, unless we know Opus How  Tweety.  then can we formalize reasoning under the unique-names assumption?  T h e natural  first attempt is to circumscribe the equality predicate in the theory under consideration. T o that end,  we shall assume that the theory T contains the following axioms which define the equality  predicate, = , for the theory:  - 74 -  Vz. x = x Vzy. x = y  y =  D  Vxyz. x —  x  y/\y=z'Dx=z  Vx ...,x yi,...,y„. z i = yi A ... A i „ = y„ A P f o , . . . , * , , ) v  m  D P(yy.--,y^, for e a c h n-ary p r e d i c a t e s y m b o l P of T . 1  Vzi,...,z yi,... y„. zi = yi A ... A z„ = y n)  D  /  /(zi,...,in)  A i.>"->yn)> v  =  f°  r  n  e a c  h w-ary f u n c t i o n s y m b o l / o f  T.  W h e n T is finite, it is therefore possible to c i r c u m s c r i b e the e q u a l i t y p r e d i c a t e , since the r e s u l t i n g s c h e m a is  finite.  T h e o r e m 5.6  Let  T be a  T h e next result informs us t h a t d o i n g so yields n o t h i n g new.  (Reiter)  first-order  t h e o r y c o n t a i n i n g a x i o m s w h i c h define the e q u a l i t y p r e d i c a t e ,  =.  I  T h e n T \- CLOSURE {T). [=)  In v i e w of t h i s r e s u l t , one m i g h t a t t e m p t to c a p t u r e the u n i q u e names a s s u m p t i o n b y j o i n t l y c i r c u m s c r i b i n g s e v e r a l predicates of the theory, n o t j u s t the e q u a l i t y p r e d i c a t e .  W e do not k n o w  w h e t h e r there are a n y theories for w h i c h this m i g h t w o r k , b u t it c a n n o t succeed for w e l l - f o u n d e d theories.  N o new g r o u n d e q u a l i t i e s or i n e q u a l i t i e s c a n be d e r i v e d b y c i r c u m s c r i b i n g a w e l l -  f o u n d e d t h e o r y , regardless of the p r e d i c a t e s c i r c u m s c r i b e d . T h e o r e m 5.7  Suppose t h a t T is a w e l l - f o u n d e d t h e o r y c o n t a i n i n g a x i o m s w h i c h define the e q u a l i t y p r e d i cate;  ai,...,a ; k  0 ,...,0 1  k  are g r o u n d terms, a n d P is a set of some of the predicates of T.  Then it  (i)  CLOSURE (T) P  Jfc  | - ( V a,- = 0,) < = •  T \- ( V a,- = 0.) ,  (ii)  CLOSURE^  and  i=l  £=1  T) \- ( Vat £ 0,) <=>  I  £ 0,).  T — (  C o r o l l a r y 5.8 S u p p o s e t h a t T is a w e l l - f o u n d e d t h e o r y c o n t a i n i n g a x i o m s w h i c h define t h e e q u a l i t y p r e d i c a t e ; P is a n n-ary p r e d i c a t e ; a n d a , . . a f o 01,...,0ka.re g r o u n d terms. T h e n 1  CLOSURED  T) \- ^P{a ...,a ) => u  k  v  T \- ( V a, £ :  0.)  or T  \f-P(0u-,0k) •  I  1=1  R e t u r n i n g t o the " P e n g u i n " e x a m p l e a b o v e , we see t h a t predicate c i r c u m s c r i p t i o n c a n n o t c o n j e c t u r e -<Penguin( Tweety) unless it is k n o w n t h a t Opus j= Tweety, otherwise w e c o u l d derive Opus j= Tweety f r o m CLOSURE({Penguin(Opus)}),  c o n t r a d i c t i n g T h e o r e m 5.7.  - 75 -  T h i s last r e s t r i c t i o n  is s o m e w h a t p u z z l i n g .  T h e model-theory  fixes  the d o m a i n a n d the  i n t e r p r e t a t i o n s of c o n s t a n t s a n d f u n c t i o n s y m b o l s w h e n d e t e r m i n i n g m i n i m a l models.  G i v e n the  soundness of p r e d i c a t e c i r c u m s c r i p t i o n w i t h respect t o t h i s m o d e l - t h e o r y , it is easy to see w h y i d e n t i t y is n o t i n f l u e n c e d b y the set of m i n i m a l m o d e l s . If the e q u a l i t y predicate is i n t e r p r e t e d as a c o n g r u e n c e r e l a t i o n , r a t h e r t h a n as i d e n t i t y (i.e., if n o n - n o r m a l m o d e l s are a l l o w e d , where p a i r s of d i s t i n c t d o m a i n elements are p e r m i t t e d to be i n the e x t e n s i o n of ' = ' ) , the s i t u a t i o n is less clear. E s s e n t i a l l y , it c a n be s h o w n t h a t , for a n y p a i r of terms for w h i c h one m i g h t hope t o c i r c u m s c r i p t i v e l y c o n j e c t u r e ( i n ) e q u a l i t y , there are m i n i m a l m o d e l s w h i c h s u p p o r t e i t h e r side of the issue. S o m u c h for  the s e m a n t i c e x p l a n a t i o n . T h e r e r e m a i n t w o  questions.  What  feature  of the  cir-  c u m s c r i p t i o n s c h e m a g i v e s rise to this a n o m a l y ? W h a t does this t e l l us a b o u t c i r c u m s c r i p t i o n ?  A  p a r t i a l a n s w e r t o the first q u e s t i o n is t h a t L e i b n i z ' p r i n c i p l e of s u b s t i t u t i v i t y - equals are e v e r y where i n t e r s u b s t i t u t i b l e  preserving t r u t h - m a k e s a stronger s t a t e m e n t a b o u t e q u a l i t y t h a n the  c i r c u m s c r i p t i o n s c h e m a . T h e second q u e s t i o n r e m a i n s unanswered. In a recent p a p e r , M c C a r t h y [1986] proposes a c i r c u m s c r i p t i v e a p p r o a c h to the n a m e s a s s u m p t i o n b y i n t r o d u c i n g t w o e q u a l i t y predicates.  unique-  O n e of these is the s t a n d a r d e q u a l i t y  p r e d i c a t e , b u t r e s t r i c t e d to a r g u m e n t s w h i c h are names of objects. e(x,y), means t h a t the names x a n d y denote the same object,  T h e other e q u a l i t y p r e d i c a t e ,  e is a x i o m a t i z e d as a n e q u i v a l e n c e  r e l a t i o n w h i c h does not, however, satisfy the f u l l p r i n c i p l e of s u b s t i t u t i o n , i n c o n t r a s t t o " n o r m a l " equality.  T h i s f a i l u r e of f u l l s u b s t i t u t i v i t y for the predicate e prevents o u r T h e o r e m s 5.6 a n d 5.7  f r o m a p p l y i n g to e.  B e n j a m i n G r o s o f (personal c o m m u n i c a t i o n )  s i m i l a r a p p r o a c h to the u n i q u e - n a m e s a s s u m p t i o n .  has i n d e p e n d e n t l y p r o p o s e d a  H e has also o b s e r v e d t h a t o u r T h e o r e m 5.6  applies to M c C a r t h y ' s [1986] more g e n e r a l n o t i o n of c i r c u m s c r i p t i o n .  5.4.2.  The Domain Closure Assumption  T h e d o m a i n - c l o s u r e a s s u m p t i o n is the a s s u m p t i o n t h a t , i n a g i v e n  first-order  t h e o r y T, the  universe of discourse is r e s t r i c t e d to the s m a l l e s t set w h i c h c o n t a i n s those i n d i v i d u a l s m e n t i o n e d i n T, a n d w h i c h is closed u n d e r the a p p l i c a t i o n of those functions m e n t i o n e d i n T. D o m a i n circ u m s c r i p t i o n [ M c C a r t h y 1977, 1980] is a proposed f o r m a l i z a t i o n of this a s s u m p t i o n .  McCarthy  [1980] suggests t h a t d o m a i n c i r c u m s c r i p t i o n m i g h t be reduced to predicate c i r c u m s c r i p t i o n .  This  is i n fact false, as s h o w n b y T h e o r e m s 5.9 a n d 5.10. T h e s i m p l e s t s e t t i n g i n w h i c h the d o m a i n - c l o s u r e a s s u m p t i o n c a n arise is for a t h e o r y w i t h a finite  Herbrand Universe {c ,...,c }. 1  n  I n t h i s case we m i g h t w a n t to conjecture the domain-closure  axiom for t h i s t h e o r y : V z . z = c V . . . V z = c . S u c h a n a x i o m is i m p o r t a n t for the t h e o r y of x  o r d e r databases [ R e i t e r 1980b]. f o u n d e d theories.  n  first-  N o s u c h a x i o m c a n arise f r o m predicate c i r c u m s c r i p t i o n for w e l l -  - 76 -  Theorem 5.9 S u p p o s e t h a t T is a w e l l - f o u n d e d theory;  t ...,t are g r o u n d terms; v  n  a n d P is a set of some  of t h e p r e d i c a t e s y m b o l s o f T. T h e n CLOSURE (T) P  |— Vx. x = t V . . . V x = t <=> x  T  n  (— V z .  x=  t V...V x  x - <„ .  |  Theorem 5.10 If  T is a w e l l - f o u n d e d t h e o r y a n d T has a m o d e l w i t h some d o m a i n , D, t h e n so does  CLOSURE^!).  I  5.4.3. Some Misconceptions T h e r e are a n u m b e r o f c o m m o n m i s c o n c e p t i o n s a b o u t the use o f p r e d i c a t e c i r c u m s c r i p t i o n , w h i c h w e discuss b r i e f l y , b e l o w . It has been p r o p o s e d t h a t a r b i t r a r y f o r m u l a e c o u l d be c i r c u m s c r i b e d u s i n g p r e d i c a t e c i r c u m s c r i p t i o n b y i n c l u d i n g a n e w p r e d i c a t e letter a n d a d e f i n i t i o n d e c l a r i n g it t o be e q u i v a l e n t t o the e x p r e s s i o n t o be c i r c u m s c r i b e d . T h i s w i l l not w o r k , i n general.  Theorem 5.11 If T (— Vxt Px= $ x * f o r some expression <5z* not i n v o l v i n g predicate letters f r o m P , t h e n T\~ CLOSUREST). |  T h i s result seems t o be r e l a t e d t o D o y l e ' s [1984] c o m m e n t s o n i m p l i c i t d e f i n a b i l i t y .  S i n c e the  t h e o r y a l r e a d y c o n t a i n s a definition f o r P, c i r c u m s c r i p t i o n c a n n o t f u r t h e r c o n s t r a i n P. A s it is generally undecidable whether  T  |—Vzt  $ 5 * for a p a r t i c u l a r $ (let alone a l l <£), it follows  Px=  t h a t one c a n n o t d e c i d e w h i c h predicates to c i r c u m s c r i b e .  Corollary 5.12 It is g e n e r a l l y u n d e c i d a b l e w h e t h e r CLOSURE^  T) is stronger t h a n T.  It is w i d e l y ( a n d c o r r e c t l y ) b e l i e v e d t h a t CLOSUREp(Qx.  |  Px}) }= 3/z. Px (i.e., there is a  u n i q u e P). T h e r e a p p e a r s t o be some m i s u n d e r s t a n d i n g a b o u t h o w this i s a c h i e v e d , however. After  some  CLOSUREp(Pa)  experimentation,  the  idea  of  skolemization  comes  to  mind  a n d , indeed,  \= 3'z- Px ( a c t u a l l y V z . Px = z = a). S k o l e m i z a t i o n , h o w e v e r , c a n c h a n g e the  - 77 set of m i n i m a l m o d e l s of a t h e o r y ( a n d hence the results of c i r c u m s c r i p t i o n ) .  T o see this, notice  t h a t i n E x a m p l e 5 . 1 , T has no m i n i m a l m o d e l s , but the s k o l e m i z e d f o r m of T is u n i v e r s a l a n d hence w e l l - f o u n d e d .  T h e r e has been a t e n d e n c y to believe that the s k o l e m i z e d f o r m of a t h e o r y T  is e q u i v a l e n t t o T, w h i c h is false. I n fact, s k o l e m i z a t i o n preserves satisfiability, not  derivability;  the existence of m o d e l s , not t h e set of m o d e l s . T h e a c t u a l c i r c u m s c r i p t i v e d e r i v a t i o n of 3-'z- Px f r o m 3 x. Px i n v o l v e s the s u b s t i t u t i o n of a binary predicate for <&x, viz x = u, where « is a v a r i a b l e d i s t i n c t f r o m x. The  s k e l e t o n o f a correct c i r c u m s c r i p t i v e d e r i v a t i o n  in a natural deduction system of  3 fx. Px f r o m 3 x. Px f o l l o w s : 1  [(3z.  2  j(3z. z = u) A (Vz. z = u D  3  (Vz. Px D z = u)] V u . [(3z. z = u) A (Vz. z = uz D Pz)]  $z)  A (Vz. $z D Px)} D (Vz. Px D $z)]  CL0SURE {3x.Px) {P)  Px)} 1,  Z>  [z = u/$z]  2, uniwcrsa/ generalization  15 (Vz. Pz O z = «)]  4  3 3. -Px-  y»«en  5  Pa  hypothesis  6  [(3z. z = a) A (Vz. z = a  D  Pz)]  2> (Vz. Pz D z = a)] 7  P a D (Vz. P n i  8  V z . Pz D z = a  9  Pa A (Vz. Pz D x=  3, universal instantiation  = a)  6, tautology 5,7, tautology 5,8, tautology  a)  10  3j/- Pj/ A (Vz. Pz D z = y)  4,5,9, existential  11  3'z-  10, definition  Ps  There have been implicit  [McCarthy  1980] a n d e x p l i c i t  generalization  [Genesereth a n d N i l s s o n 1987]  suggestions t h a t the w a y a r o u n d some of the l i m i t a t i o n s of p r e d i c a t e c i r c u m s c r i p t i o n m i g h t be to c i r c u m s c r i b e o n l y " r e l e v a n t " p o r t i o n s of the t h e o r y .  T h e i d e a is t h a t , b y w e a k e n i n g T ( $ ) - hence  e l i m i n a t i n g some o f the c o n d i t i o n s t h a t O m u s t satisfy - perhaps some m o r e useful results w i l l obtain.  O b v i o u s l y , one m u s t be c a r e f u l ;  V z . -<Pz.  c i r c u m s c r i b i n g P i n Pa, l e a v i n g o u t Pa w i l l p r o d u c e  T h i s , b e i n g inconsistent w i t h Pa, is perhaps too useful.  O n e i d e a is to d i s t i n g u i s h ,  a m o n g s t the p o s i t i v e l i t e r a l s i n each clause of the theory, one w h i c h the clause is s a i d to be "about".  T h e n o n l y those clauses " a b o u t " P are t a k e n i n t o a c c o u n t i n f o r m i n g T(4>). T h i s m a y  i n d e e d a l l o w p o s i t i v e facts t o be d e r i v e d . F o r e x a m p l e , c o n s i d e r V z . Bird(x) A ->Penguin(x) O Flies(x) Bird(Tweety), Penguin(Opus), Opus £ Tweety V z . Penguin(x) D Bird(x) If the first a x i o m is t a k e n t o be a b o u t Flies, t h e n we get $Opus A [Vz. $ z D P z ] D [Vz. P z D when  we  circumscribe  (in  this  $z]  fashion)  V z . Penguin[x) = x = Opus a n d Flies(Tweety)\  Penguin  in  T.  From  this  we  can  derive  T h e r e are t w o d r a w b a c k s w i t h this a p p r o a c h ,  - 78 -  however.  The  first  is t h a t its s e m a n t i c s are u n k n o w n .  T h e y are not those of p r e d i c a t e  cir-  c u m s c r i p t i o n , a n d there is no k n o w n m o d e l - t h e o r y o r " s o u n d n e s s " result c o r r e s p o n d i n g t o t h a t for predicate c i r c u m s c r i p t i o n .  T h u s it is not c l e a r w h a t this a p p r o a c h c o m p u t e s .  consistency is n o t necessarily preserved;  M o r e seriously,  the first a x i o m of T is also " a b o u t " Penguins, in the  sense t h a t T* = (T Li {->Flies( Tweety)}) |—Penguin(Tweety).  T a k i n g the above a p p r o a c h t o c i r -  c u m s c r i b i n g T* w i l l result i n a n i n c o n s i s t e n c y .  5.5.  W h a t to Circumscribe?  O n e o b v i o u s p r o b l e m w i t h u s i n g c i r c u m s c r i p t i o n i n a g i v e n s e t t i n g is k n o w i n g j u s t w h a t to circumscribe.  S o m e of o u r results p r o v i d e clues i n this d i r e c t i o n . ( C o r o l l a r y 5.12 shows t h a t clues  are the best t h a t c a n be h o p e d for, i n general.) T h e o r e m 5.5 tells us t h a t if we w i s h to use p r e d i cate c i r c u m s c r i p t i o n to conjecture - i P ( a ) i n some w e l l - f o u n d e d t h e o r y t h e n we m u s t i n c l u d e a m o n g t h e p r e d i c a t e s being c i r c u m s c r i b e d . T h e o r e m s 5.4 a n d 5.5 t e l l us t h a t predicate  P  cir-  c u m s c r i p t i o n w i l l not do at a l l i f we w i s h to conjecture P{ct), as is the case for most forms of d e f a u l t r e a s o n i n g , so t h a t we m u s t a p p e a l t o some other m e c h a n i s m , s u c h as M c C a r t h y ' s more g e n e r a l f o r m of c i r c u m s c r i p t i o n , discussed i n the n e x t c h a p t e r .  6  CHAPTER  Generalizations of Circumscription  6.1.  Formula Circumscription  McCarthy  [1986] h a s r e c e n t l y  called formula circumscription. first-order  formulated  a g e n e r a l i z a t i o n of p r e d i c a t e  circumscription,  T h i s g e n e r a l i z a t i o n provides for the m i n i m i z a t i o n of a r b i t r a r y  expressions r a t h e r t h a n s i m p l e p r e d i c a t e s . It also p r o v i d e s f o r t h e t r e a t m e n t of desig-  n a t e d p r e d i c a t e s as v a r i a b l e s of the m i n i m i z a t i o n . l i m i t a t i o n s o f T h e o r e m 5.5 n o l o n g e r a p p l y .  I n this v e r s i o n of c i r c u m s c r i p t i o n some of t h e  T h u s , as some of M c C a r t h y ' s e x a m p l e s show, it is  possible t o c i r c u m s c r i b e a p r e d i c a t e P , t r e a t i n g a n o t h e r predicate Q as v a r i a b l e , a n d d e r i v e n e w p o s i t i v e a n d n e g a t i v e g r o u n d instances of Q. I n p a r t i c u l a r , M c C a r t h y ' s n e w f o r m a l i s m appears a d e q u a t e f o r the t r e a t m e n t of some f o r m s of default r e a s o n i n g , as his e x a m p l e s s h o w . M a n y of the l i m i t a t i o n s of p r e d i c a t e c i r c u m s c r i p t i o n s t e m f r o m t h e fact t h a t o n l y those p r e d i c a t e s b e i n g m i n i m i z e d are a l l o w e d t o v a r y .  F o r m u l a c i r c u m s c r i p t i o n r e t a i n s m a n y of the  a t t r a c t i v e features o f its predecessor, w i t h o u t some of i t s l i m i t a t i o n s .  M c C a r t h y ' s d e f i n i t i o n of the  f o r m u l a c i r c u m s c r i p t i o n of E(P,x) i n the t h e o r y T\P) takes the f o r m o f t h e second-order a x i o m , (22),  A V*.  T(P) where  E(P,x)  T(*)  is  any  A [Vxt £($,x) well-formed  z>  E(P,x)]  [Vxt  D  expression  E(P,x)  whose  free  D  (22)  E{&,x)\  individual  variables  are  among  x*= Xi,...,x a n d i n w h i c h some o f the p r e d i c a t e v a r i a b l e s P = { P i , . . . , P , J o c c u r free; E(3>,x) is the h  result of r e p l a c i n g e a c h free occurrence of the p r e d i c a t e letters, P,-, i n E(P,x) w i t h p r e d i c a t e v a r i ables,  of t h e same a r i t y . N o t e v e r y o n e is c o n v i n c e d of t h e need for second-order logic f o r c i r c u m s c r i p t i o n [Perlis a n d  M i n k e r 1986].  A  first-order  s c h e m a v e r s i o n of f o r m u l a c i r c u m s c r i p t i o n , (23), is o b t a i n e d b y delet-  i n g the s e c o n d - o r d e r q u a n t i f i e r , T{P)  A  It*)  V$.  A [Vxt £(*,x)  D  E[P,x)\  W e w i l l s o m e t i m e s w r i t e CLOSURE(T;  D  [Vxt  E[P,x)  D  £(*,x)]  (23)  P ; E[P,xj) f o r either a x i o m (22) o r s c h e m a ( 2 3 ) , i n d i c a t -  i n g m i n i m i z a t i o n of t h e e x p r e s s i o n E[P,x), w i t h the predicates P  t r e a t e d as v a r i a b l e , i n the  t h e o r y , T. McCarthy  presented  only  a  syntactic  characterization  of  formula  circumscription.  M o t i v a t e d b y a belief i n the i m p o r t a n c e o f s e m a n t i c c h a r a c t e r i z a t i o n s for r e a s o n i n g s y s t e m s , a n d b y the s t r i k i n g consequences of e x p l o r i n g the s e m a n t i c s of predicate c i r c u m s c r i p t i o n , w e e x p l o r e d the p o s s i b i l i t y t h a t a n a p p r o p r i a t e g e n e r a l i z a t i o n of the m i n i m a l - m o d e l s e m a n t i c s of p r e d i c a t e  - 79 -  - 80 -  circumscription w o u l d characterize formula circumscription.  1  T h i s l e d us t o a f o r m of the general-  i z e d m i n i m a l - m o d e l s e m a n t i c s w h i c h has since been used i n the e x p l i c a t i o n of a v a r i e t y o f closedw o r l d reasoning f o r m a l i s m s (see §2.1). T h e precise details are g i v e n below.  Definition:  Let  M < ^p-^Af'  T"(P) be a  finitely  a x i o m a t i z e d (first- o r second-order) theory, some ( b u t n o t neces-  s a r i l y all) o f whose predicates are those i n P ; let E(P,x) be a f o r m u l a whose free v a r i a b l e s are a m o n g ~x = x ...,x , a n d i n w h i c h some of t h e predicate v a r i a b l e s P = { j - i , . . . , . ? „ } v  o c c u r free;  n  a n d l e t M , M' be m o d e l s of T. W e say M i s a n E(P,x)-submodel o f A f ' (writ-  ten A f < np^M') iff (i) |Af| = \M'\ , (ii) If t i s a t e r m , t h e n \ t\ = \ M  ,  (iii) I f Q £ P is a predicate letter of T, t h e n \ Q\ = M  (iv)  Definition:  \E(P,x)\ Q M  I^P.x)^.  \Q\iJ , a n d  I  i?(P,z)-Minimal Model  A m o d e l , Af, of T is E(~P,xj-minimal iff T has n o m o d e l , M*, s u c h t h a t M' < ^ ^ M a n d P  T h a t this i s the correct s e m a n t i c s is suggested b y T h e o r e m s 6.1 a n d 6.2. a p p l i c a b l e o n l y t o the first-order-schema v e r s i o n o f f o r m u l a c i r c u m s c r i p t i o n ;  T h e o r e m 6.2 is  T h e o r e m 6.1 applies  b o t h t o t h a t a n d t o second-order f o r m u l a c i r c u m s c r i p t i o n .  T h e o r e m 6.1 — S o u n d n e s s CLOSURE(T;  P ; E(P,x)) is satisfied b y every ^ P . z J - m i n i m a l m o d e l of T.  |  T h e o r e m 6.2 — F i n i t a r y C o m p l e t e n e s s (Perlis and Minker) If a l l models of T have finite extensions f o r each P G P ( m o d u l o e q u a l i t y ) , t h e n M satisfies e v e r y i n s t a n c e o f CLOSURE(T) P ; E(P,xj) o n l y if A f is a n £ ( P , i ) - m i n i m a l m o d e l o f T. |  Lifschitz [1985, personal communication] argues that the model-theory for second-order logic provides sufficient semantics for the generalized forms of circumscription. While this may be true, the explicit notion of minimality leads to useful insights, as is indicated in the sequel. 1  - 81 P e r l i s a n d M i n k e r [1986] a c t u a l l y p r o v e a s l i g h t l y stronger result, a p p l i c a b l e i f a l l m o d e l s for P ; E(P,x)) h a v e finite extensions for each P S P . O f course, n o general c o m p l e t e -  CLOSURE(T;  ness result c o u l d be f o r t h c o m i n g .  T h e r e is a u n i q u e ( u p t o i s o m o r p h i s m ) m i n i m a l m o d e l for the  s t a n d a r d a x i o m a t i z a t i o n o f the n a t u r a l n u m b e r s , b u t there is n o r e c u r s i v e  first-order  axiomatiza-  t i o n w h i c h u n i q u e l y c h a r a c t e r i z e s this m o d e l . If c i r c u m s c r i p t i o n were c o m p l e t e , it c o u l d be used to c o n j e c t u r e s u c h a  first-order  axiomatization.  It is w o r t h w h i l e d e t e r m i n i n g w h i c h of the differences b e t w e e n p r e d i c a t e c i r c u m s c r i p t i o n a n d f o r m u l a c i r c u m s c r i p t i o n are r e a l l y necessary.  A s M c C a r t h y has suggested, t h e m i n i m i z a t i o n of  a r b i t r a r y expressions is n o t .  Theorem 6.3 T h e a b i l i t y t o m i n i m i z e a r b i t r a r y expressions, 2?(P,x), i n s t e a d of s i m p l e sets of p r e d i cates, is a n i n e s s e n t i a l e x t e n s i o n , p r o v i d e d predicates other t h a n those being m i n i m i z e d are a l l o w e d t o v a r y . |  T h e o r e m 6.3 tells us t h a t i t suffices t o c i r c u m s c r i b e predicates. T o see this, observe t h a t one c a n s i m p l y e x t e n d t h e l a n g u a g e w i t h a new p r e d i c a t e s y m b o l , \p a n d a d d t h e a x i o m : V x . rj>x= £ ( P , x ) to the t h e o r y .  C i r c u m s c r i b i n g ^>5Tin the e x t e n d e d t h e o r y w i t h P U {ip} v a r i a b l e results i n a con-  servative extension ( n o n e w theorems o v e r the o r i g i n a l language are d e r i v a b l e ) of the c i r c u m s c r i p t i o n of E{P,x) i n the o r i g i n a l theory.  6.2. Generalized Circumscription M c C a r t h y ' s f o r m u l a c i r c u m s c r i p t i o n has l a t e l y been generalized b y L i f s c h i t z [1984], e x p l o i t i n g pre-orders, as discussed i n §2.1.5.2. L i f s c h i t z ' generalized f o r m is: T p C ) A V X ' . T ( X ' ) A ( X ' < j{X) where  <  R  z> ( X  < *X')  (24)  denotes the pre-order o n tuples of (predicate, f u n c t i o n , a n d i n d i v i d u a l )  variables  i n d u c e d b y a r e f l e x i v e , t r a n s i t i v e r e l a t i o n , R. W e c a l l t h i s generalized circumscription, a n d w r i t e CLOSURE(T;  X ; R) f o r (24) o r the c o r r e s p o n d i n g  first-order  schema.  T h i s formulation allows for  a r b i t r a r y o r d e r i n g r e l a t i o n s t o d r i v e the m i n i m i z a t i o n , a n d provides f o r the d e n o t a t i o n s of terms ( c o n s t a n t a n d f u n c t i o n letters) t o be affected b y the m i n i m i z a t i o n process. T h e e x t e n d e d m i n i m a l - m o d e l s e m a n t i c s o u t l i n e d above is a m e n a b l e t o this f u r t h e r generalization.  T h e m o s t significant change f r o m the f o r m s we have seen t o t h i s p o i n t is t h a t t h e d e n o t a -  tions of some c o n s t a n t a n d f u n c t i o n t e r m s m a y change between a m o d e l a n d i t s s u b m o d e l s . T h e a p p r o p r i a t e d e f i n i t i o n s are:  - 82 -  Definition:  (X,R)M'  M <  L e t r(P)  be a f i n i t e l y a x i o m a t i z e d (first- or second-order) theory, whose p r e d i c a t e , f u n c -  t i o n a n d c o n s t a n t letters i n c l u d e ( b u t need not be l i m i t e d to) those i n X ; r e l a t i o n o n t u p l e s of type X ;  let  < R be the p r e - o r d e r i n d u c e d b y R;  m o d e l s of T. T h e n M i s a n (X.,R)-submodel of M' (i) (ii)  | M | = \M'\  (written M <  ^R)M')  let R be a b i n a r y a n d let M , M'  be  iff  ,  If t is a t e r m a n d t £ X , t h e n \t\ = [tl^ , M  X is a p r e d i c a t e l e t t e r of T, t h e n | Q | A /  (iii)  If Q  (iv)  < | X | , | X | ^ > eR. M  Definition:  1  =  \Q-\M > 1  a n <  i  I  (X,i2)-Minimal Model  A m o d e l , M , of T is (X.,R)-minimal iff T has no m o d e l , M ' , s u c h t h a t M * < (x,iJ)A^ -(M< ,*)M') . I  a n <  i  ( X  W e h a v e s h o w n t h a t g e n e r a l i z e d c i r c u m s c r i p t i o n is s o u n d vis-a-vis the set of m i n i m a l m o d e l s specified b y t h i s m o d e l t h e o r y .  T h e o r e m 6.4 — S o u n d n e s s CLOSURE(T;  X ; R) is satisfied b y e v e r y ( X , i ? ) - m i n i m a l m o d e l of T.  |  W e do not k n o w w h e t h e r there is a n analogue of T h e o r e m 6.2 ( f i n i t a r y completeness) for g e n e r a l ized circumscription. T h e p r o v i s i o n for v a r i a b l e t e r m s leads to some s u r p r i s i n g results.  T h e s e i n c l u d e new p o s i -  tive e q u a l i t y s t a t e m e n t s , a n d the p r o v a b i l i t y of new p o s i t i v e o r n e g a t i v e g r o u n d facts i n p r e d i cates not i n c l u d e d a m o n g those specified as v a r i a b l e .  P r o p o s i t i o n 6.5 If t e r m s are a l l o w e d t o v a r y , t h e n new g r o u n d e q u a l i t y s t a t e m e n t s m a y result from generalized circumscription. |  P r o p o s i t i o n 6.6 If t e r m s are a l l o w e d to v a r y , t h e n new g r o u n d facts i n v o l v i n g p r e d i c a t e s Q (£ X m a y r e s u l t f r o m CLOSURE^ T\ X ; R). |  - 83 -  E x a m p l e 6.1  C o n s i d e r t h e t h e o r y T= PaAPb/\Qb/\ Instantiation  {Pa, Pb, Qb }. V $ . V u . [$tt A  {P, a}; {P}) is  A ( V z . $ z D Pz)] D ( V z . P z D $ z )  w i t h [z = 6 / $ z ] a n d [6/u] gives V z . P z 3 z = 6, f r o m w h i c h we c a n infer  a = 6 a n d hence Q a .  6.3.  CLOSURE[T;  |  Well-Founded Theories  A s w i t h a l l of t h e f o r m s of m i n i m a l - m o d e l s e m a n t i c s w e h a v e discussed i n t h i s thesis, t h a t for g e n e r a l i z e d c i r c u m s c r i p t i o n p r o v i d e s f o r c e r t a i n elements to differ between a m o d e l a n d i t s s u b m o d e l s w h i l e others r e m a i n  fixed.  D e s p i t e T h e o r e m s 6.2-6.4, it is n o t necessarily c l e a r t h a t t h e  s y n t a c t i c m a n i p u l a t i o n s of g e n e r a l i z e d (or f o r m u l a ) c i r c u m s c r i p t i o n respect the i n t e n t expressed by  this  semantic characterization.  It  is c o n c e i v a b l e t h a t  a l l m o d e l s reflecting  c o n f i g u r a t i o n of s u p p o s e d l y fixed a t t r i b u t e s m i g h t h a v e n o m i n i m a l s u b m o d e l s .  a  particular  T h e semantics  t h e n fails t o guarantee t h a t c i r c u m s c r i p t i o n w i l l not affect these s u p p o s e d l y " i n v i o l a b l e " facets.  It  is n a t u r a l t o q u e s t i o n w h e t h e r there is a n y p r o p e r t y analogous to the w e l l - f o u n d e d n e s s p r o p e r t y we d i s c u s s e d f o r p r e d i c a t e c i r c u m s c r i p t i o n , w h i c h w o u l d address this c o n c e r n . I n fact, as w e s h a l l see, there is s u c h a n o t i o n .  L e t us redefine the t e r m " w e l l - f o u n d e d " as follows:  Definition — Well-Foundedness T h e t h e o r y , T, is well-founded with respect to ( X , i i ) iff e v e r y m o d e l of T has a n ( X , R)-minimal submodel. |  T h i s d e f i n i t i o n is s l i g h t l y w e a k e r t h a n t h a t g i v e n i n c h a p t e r 5, where w e r e q u i r e d t h a t every m o d e l of T h a v e a P - m i n i m a l s u b m o d e l f o r e v e r y  finite  tuple of predicates, P .  T h i s weaker  d e f i n i t i o n , r e l a t i v i z e d to ( X , P ) , is sufficient for d e c i d i n g w h e t h e r a p a r t i c u l a r c i r c u m s c r i p t i o n is well-behaved.  T h e m o r e d i r e c t g e n e r a l i z a t i o n of the d e f i n i t i o n of c h a p t e r 5 is so s t r o n g t h a t it  excludes a l l theories.  P r o p o s i t i o n 6.7  (Lifschitz)  U n i v e r s a l theories are n o t necessarily w e l l - f o u n d e d i f constants are allowed to v a r y . |  - 84 -  E x a m p l e 6.2  (Lifschitz)  T h e n a t u r a l - n u m b e r e x a m p l e of E x a m p l e 5.1, w i t h the e x i s t e n t i a l l y specified i n d i v i d u a l r e p l a c e d b y the c o n s t a n t ' 0 ' :  NO A V z . N z D succ(x) £ 0  V z . N z D Nsucc(z) Vzy. succ(z) = succ(y)  D  z= y  is not w e l l - f o u n d e d w i t h respect to m i n i m i z a t i o n of N  w i t h { N , 0} v a r i a b l e .  S i n c e the  d e n o t a t i o n of 0 is a l l o w e d t o c h a n g e f r o m m o d e l t o s u b m o d e l , the i n f i n i t e c h a i n s of m o d e l s p r e s e n t e d i n E x a m p l e 5.1 serve to s h o w t h a t t h i s t h e o r y has no m i n i m a l m o d e l s .  |  P r o p o s i t i o n 6.8 No class of theories is w e l l - f o u n d e d w i t h respect to a l l pre-orders.  |  E x a m p l e 6.3 Consider the theory w i t h no p r o p e r a x i o m s , a n d m i n i m i z e the expression E(P,x) = Px A [Vz. -iPz] A @ z - Px A ->Psx\. C o n s i d e r a m o d e l i n w h i c h P is i n t e r p r e t ed b y the n a t u r a l n u m b e r s , a n d s b y the successor f u n c t i o n . C l e a r l y any n o n - e m p t y i n i t i a l subset of the n a t u r a l n u m b e r s produces a p r o p e r s u b m o d e l , b u t the m o d e l w i t h the e m p t y i n t e r p r e t a t i o n for P m a k e s E t r u e e v e r y w h e r e . | P r o p o s i t i o n 6.8 a n d E x a m p l e 6.3 c a n best be u n d e r s t o o d i n terms o f T h e o r e m 6.3.  Minimization  of E(P,x) i n T is e q u i v a l e n t to m i n i m i z a t i o n of yjx, w i t h {rp, P} v a r i a b l e , i n  T' = | V z . ybx = JPZ A [Vz. --Pz] A 0 i . Pz A --Paz]  j  w h i c h does not b e l o n g t o a n y o f the k n o w n classes of w e l l - f o u n d e d theories (because P occurs p o s i t i v e l y w i t h i n the scope of e x i s t e n t i a l quantifiers).  I n some sense, a l l o w i n g a r b i t r a r y pre-orders  enables one to " i m p o r t " a r b i t r a r y a x i o m s i n t o the theory. W i t h these e x a m p l e s i n m i n d , we w i l l restrict o u r a t t e n t i o n i n the sequel to the case of s i m ple m i n i m i z a t i o n of some of the p r e d i c a t e s of X .  In other w o r d s , we w i l l c o n s i d e r a g e n e r a l i z a t i o n  of j o i n t p r e d i c a t e c i r c u m s c r i p t i o n , i n w h i c h o t h e r predicates a n d t e r m s m a y be a l l o w e d to v a r y . W e will write  < (x,p) f °  r  *  n e  pre-order d e t e r m i n e d b y the j o i n t m i n i m i z a t i o n of each of the p r e d i -  cates i n P , a l l o w i n g t h e predicates a n d t e r m s of X to v a r y .  ( X is assumed to c o n t a i n a l l of the  predicate symbols of P . ) T h e q u e s t i o n r e m a i n s , " A r e there a n y theories w h i c h are w e l l - f o u n d e d ? " answer is " Y e s " .  F o r t u n a t e l y , the  ( T h i s result h a s been p r o v e d i n d e p e n d e n t l y (using r a t h e r different  b y L i f s c h i t z [1985].)  techniques)  - 85 T h e o r e m 6.9 If T is a u n i v e r s a l theory, a n d X , P are finite tuples of predicate letters, t h e n T is w e l l - f o u n d e d w i t h respect t o < (x,P) • B  T h e existence of w e l l - f o u n d e d theories p r o v e d most distressing i n the c o n t e x t o f predicate circumscription.  W h a t are  the repercussions o f T h e o r e m 6.9 for g e n e r a l i z e d c i r c u m s c r i p t i o n ?  C e r t a i n l y , t h e y are less pessimistic. G e n e r a l i z e d c i r c u m s c r i p t i o n affords m u c h greater c o n t r o l over w h i c h aspects o f m o d e l s m u s t r e m a i n fixed w h e n c o n s t r u c t i n g submodels.  T h i s m e a n s t h a t gen-  e r a l i z e d c i r c u m s c r i p t i o n is not d r i v e n , w i l l y - n i l l y , to a v o i d conclusions w h i c h l e a d to the d e r i v a tion of new positive information.  T h u s , for w e l l - f o u n d e d theories, g e n e r a l i z e d c i r c u m s c r i p t i o n  a l l o w s useful c o n c l u s i o n s t o be d r a w n w i t h o u t sacrificing a clear s e m a n t i c i n t u i t i o n of e x a c t l y w h a t is o p e n to conjecture.  A l s o on the p o s i t i v e front, we have C o r o l l a r y 6.10:  C o r o l l a r y 6.10 If T is c o n s i s t e n t a n d w e l l - f o u n d e d w i t h respect to ( X , P ) , t h e n CLOSURE(T; X ; P ) is consistent. |  It is n a t u r a l t o q u e s t i o n the extent to w h i c h the negative results of c h a p t e r 5 a p p l y t o gene r a l i z e d c i r c u m s c r i p t i o n . It is clear t h a t , i n the case where o n l y the m i n i m i z e d predicates  are  a l l o w e d t o v a r y , t h a t a l l the results i n c h a p t e r 5 c o n t i n u e to hold, since i n this case generalized c i r c u m s c r i p t i o n reduces to predicate c i r c u m s c r i p t i o n . F u r t h e r m o r e , T h e o r e m 5.4 a n d a n a p p r o p r i ate v e r s i o n of T h e o r e m 5.5 c o n t i n u e t o h o l d , even w i t h v a r i a b l e predicates.  T h e o r e m 6.11 If T is w e l l - f o u n d e d w i t h respect to ( X , P ) ; P S P is an n-ary predicate; p r e d i c a t e letters; a n d a . . a ) are n-tuples o f g r o u n d terms; t h e n 1 /  CLOSURE(T;  v  X a set of  k  X ; P ) f- Pa\ V . . . V Pa  k  T (- Pa\ V . . . V Pa  <=•  k  .  |  T h e o r e m 6.12 If T is w e l l - f o u n d e d w i t h respect t o ( X , P ) ; X is a set of predicate letters; is a n n - a r y p r e d i c a t e ; a n d a ...,a are n-tuples o f g r o u n d terms; t h e n 1)  (i)  CLOSURE^  (ii) GLOSURE(  P ^ P U X  k  T; X ; P ) f - Pc7 V . . . V Pa <=> t  k  T; X ; P ) | - -iPo^ V . . . V ^tt  k  T (— P a  <=>  x  V . . . V tt , k  and  T (- - P o ^ V . . . V --P&? . t  |  - 86 -  T h e fact t h a t the m o d e l - t h e o r y o u t l i n e d i n §6.2 for generalized c i r c u m s c r i p t i o n (even w i t h v a r i a b l e terms) r e s t r i c t s t h e s u b m o d e l r e l a t i o n s h i p t o m o d e l s w i t h i d e n t i c a l d o m a i n s suggests t h a t g e n e r a l i z e d c i r c u m s c r i p t i o n ( a n d a fortiori f o r m u l a c i r c u m s c r i p t i o n ) c a n n o t be used t o conjecture d o m a i n closure a x i o m s . F o r w e l l - f o u n d e d theories, this is the case.  Theorem 6.13 If T is w e l l - f o u n d e d for ( P , . R ) a n d T has a m o d e l w i t h d o m a i n , D, t h e n so does CLOSURE(T(P);P;R). |  T h u s n e i t h e r g e n e r a l i z e d c i r c u m s c r i p t i o n w i t h o u t v a r i a b l e t e r m s n o r f o r m u l a c i r c u m s c r i p t i o n subsumes d o m a i n c i r c u m s c r i p t i o n . E q u a l i t y a p p e a r s t o r e m a i n p r o b l e m a t i c i f o n l y predicates are v a r i a b l e , b u t w e h a v e not prov e n a n a n a l o g u e of T h e o r e m 5.7. T h e o r e m 5.6 c o n t i n u e s to a p p l y even if terms are a l l o w e d t o vary.  Theorem 6.14 If T is a first-order t h e o r y c o n t a i n i n g a x i o m s w h i c h define the e q u a l i t y p r e d i c a t e , = , t h e n T f - CLOSURE( T , X , { = } ) . |  It appears t h a t u n i q u e names a x i o m s are d e r i v a b l e (for theories w i t h finite d o m a i n s ) g i v e n v a r i a b l e t e r m s , h o w e v e r [Lifschitz 1984]. problematic.  U n f o r t u n a t e l y , we h a v e seen that v a r i a b l e terms c a n be  T h e g e n e r a l f o r m u l a t i o n of c l o s e d - w o r l d reasoning about e q u a l i t y u s i n g g e n e r a l i z e d  c i r c u m s c r i p t i o n w i t h v a r i a b l e terms r e m a i n s a n open q u e s t i o n . A l s o o p e n axe t h e questions of analogues of T h e o r e m s 5.4 a n d 5.5 vis a vis a r b i t r a r y p r e orders a n d / o r v a r i a b l e terms.  B e c a u s e of the failure o f well-foundedness f o r these forms of c i r -  c u m s c r i p t i o n , t h e tools we h a v e used i n this c h a p t e r a n d i n c h a p t e r 5 d o not a p p l y t o these more general problems.  P r o p o s i t i o n 6.6 suggests t h a t s u c h analogues m a y not be f o r t h c o m i n g .  CHAPTER 7  D o m a i n Circumscription  I n c h a p t e r 2, we discussed the m o t i v a t i o n f o r a n d one r e a l i z a t i o n of d o m a i n c i r c u m s c r i p t i o n . I n t h i s c h a p t e r , w e i n v e s t i g a t e the f o r m a l i s m more t h o r o u g h l y . D o m a i n c i r c u m s c r i p t i o n [ M c C a r t h y 1977, 1980; D a v i s 1980] is i n t e n d e d t o be a s y n t a c t i c r e a l i z a t i o n of the m o d e l - t h e o r e t i c d o m a i n - c l o s u r e a s s u m p t i o n . It p r o v i d e s a m e c h a n i s m f o r c o n j e c t u r i n g d o m a i n - c l o s u r e a x i o m s , e l i m i n a t i n g the need to e x p l i c i t l y state t h e m .  T o circumscribe  the d o m a i n o f a sentence, A, the s c h e m a : Axiom(<&) A A * D V x. <J>(z) is  added  to  A.  Axiom($)  (25)  is the c o n j u n c t i o n  of $ a  for e a c h  constant  symbol  a and  V ...x . [ $ z A—A 3>zJ D <&fx ...x for each n-ary f u n c t i o n s y m b o l / . A® is the result of r e w r i t i n g Xl  n  x  l  n  A, r e p l a c i n g e a c h u n i v e r s a l or e x i s t e n t i a l quantifier,  'Vz.' o r 'Ejz.', i n A w i t h 'Vx.4>z D ' o r  ' 3 z . $ z A ', respectively.  7.1. A Revised Domain Circumscription Axiom Schema A s w a s n o t e d i n §2.1.5.3, the a p p r o p r i a t e  model-theoretic  characterization for domain-  closure i n v o l v e s r e s t r i c t i o n of models to progressively s m a l l e r d o m a i n s , p r e s e r v i n g agreement over c o m m o n terms. ture".  T h i s n o t i o n o f s u b m o d e l corresponds r o u g h l y to the s t a n d a r d n o t i o n o f " s u b s t r u c -  It i s s l i g h t l y stronger, however, i n the sense t h a t substructures are not r e q u i r e d to be  models of t h e t h e o r y i n q u e s t i o n . D a v i s [1980] shows t h a t every instance of (25) is true i n a l l m i n i m a l m o d e l s of t h e o r i g i n a l sentence A. T h i s result is correct for m o s t theories.  H o w e v e r , i n c o n s i s t e n c y results w h e n cir-  c u m s c r i b i n g u n i v e r s a l theories (theories whose prenex n o r m a l forms c o n t a i n n o l e a d i n g e x i s t e n t i a l quantifiers) w i t h n o c o n s t a n t s y m b o l s . F o r e x a m p l e , consider the r e l a t i o n a l t h e o r y : A = { Vz. Px }. B e c a u s e there are n o c o n s t a n t o r f u n c t i o n s y m b o l s , Axiom($>) is e m p t y , so t h e d o m a i n c u m s c r i p t i o n s c h e m a f o r A is:  M e r c e r [1984, p e r s o n a l c o m m u n i c a t i o n ] h a s noted t h a t s u b s t i t u t i n g ->Px for $ z gives:  - 87 -  cir-  - 88 -  Vi.  PIJ  D  Vi.  ->PX  w h i c h is c l e a r l y i n c o n s i s t e n t w i t h A. T h e r o o t of t h i s p r o b l e m is t h a t , for s u c h theories, $ c a n be chosen to be u n i v e r s a l l y false. M o d e l s of first-order theories m u s t h a v e at least one d o m a i n element, so the conjecture e v e r y t h i n g is a $ ( a n d hence there is nothing) is inconsistent.  that  H a v i n g i s o l a t e d the p r o b l e m , we  h a v e d e v e l o p e d a s i m p l e , easily m o t i v a t e d s o l u t i o n . S i n c e m o d e l s m u s t h a v e n o n - e m p t y d o m a i n s , those $ ' s w h i c h are i d e n t i c a l l y false m u s t be e x c l u d e d . T o a c h i e v e this, the c o n j u n c t ^x. <&i is a d d e d to the l e f t - h a n d - s i d e of the c i r c u m s c r i p t i o n s c h e m a (25), g i v i n g : 3i-  $i  A Axiom{$) A A* D V i . $ ( x )  (26)  D a v i s ' p r o o f is easily c o r r e c t e d a n d a m e n d e d to a p p l y to this r e v i s e d s c h e m a .  S c h e m a s (25) a n d  (26) are e q u i v a l e n t i n a l l b u t the p r o b l e m a t i c cases o u t l i n e d a b o v e . If A c o n t a i n s a c o n s t a n t s y m b o l , a , t h e n $ a o c c u r s o n the left of (25), a n d t h i s entails 9z. e x i s t e n t i a l q u a n t i f i e r s , t h e n ^x. $ z a l r e a d y occurs i n (25).  $x.  S i m i l a r l y , if A has a n y l e a d i n g  In those cases w h e r e ^x. $ i  is not  e n t a i l e d b y the left-hand-side of (25), (25) results i n i n c o n s i s t e n c y . T h e r e v i s e d s c h e m a m a y s t i l l take a consistent t h e o r y  w i t h no m i n i m a l models  to a n inconsistent c i r c u m s c r i p t i o n (for a n  e x a m p l e , see [ D a v i s [1980]), b u t so l o n g as A has a m i n i m a l m o d e l , (26) preserves c o n s i s t e n c y .  Theorem 7.1 — Soundness E v e r y i n s t a n c e of theory. |  s c h e m a (26)  is  true  in  every  minimal  model  of  the  original  7.2. Some Properties of Domain Circumscription In t h i s s e c t i o n we c o n s i d e r some properties of d o m a i n c i r c u m s c r i p t i o n . consequences w i t h respect assumption.  to  using domain  circumscription  to  W e examine their  f o r m a l i z e the  domain-closure  T o b e t t e r i l l u s t r a t e the properties of d o m a i n c i r c u m s c r i p t i o n , we refer to the f o l l o w -  ing example.  Example 7.1 Let  T=  {Pa,Pc, Qb,Qc}. T has the f o l l o w i n g m i n i m a l models.  b o l d f a c e l e t t e r for t h e i n t e r p r e t a t i o n s e q u i v a l e n c e classes {a,  M  i :  \M,\ =  {a, b, c}  1 1 ^ = {a. c}  c},  {b,  c},  ( W e use the c o r r e s p o n d i n g  of c o n s t a n t t e r m s , a n d a , 8,  a n d {a,  b, c},  respectively.)  a n d 7 represent  the  - 89 -  | = | « i = { ( « , * ) , ( b , b ) , (c,c)}  Mi  | M | = {a, b} 2  l<?k= {b, « } HM,=  { ( > ) > ( > ) > ( > )> ( > ) . ( > ) } A  A  B  B  C  C  A  C  C  A  A 4 : | M | = { a , 0} 3  \P\MS =  { a , 0}  l<?k= {/?} HM,  = {( >»)> ( b , b ) , ( c , c ) , ( b , c ) , ( c , b ) } a  M : | M | = {7} 4  4  \Q\M,=  {I}  I=|M = { ( a , a ) , (b,b), (c,c), (a,b), (b,a), (a,c), (c,a), (b,c), (c,b)} 4  S e v e r a l i m p o r t a n t features are evident i n t h e above e x a m p l e . one of M  t  — M  4  as a m i n i m a l s u b m o d e l .  |  F i r s t , every m o d e l of T has  A s w i t h other f o r m s of c i r c u m s c r i p t i o n a n d t h e i r  c o r r e s p o n d i n g n o t i o n s of m i n i m a l i t y , i t is interesting to k n o w w h e t h e r there is a class of theories each of whose m o d e l s has a m i n i m a l s u b m o d e l (i.e., w e l l - f o u n d e d theories).  It is f o r s u c h theories  t h a t d o m a i n c i r c u m s c r i p t i o n corresponds most closely w i t h one's i n t u i t i o n s .  I n t h e case o f d o m a i n  c i r c u m s c r i p t i o n , t h e m a t h e m a t i c a l logic l i t e r a t u r e p r o v i d e s a sufficient c o n d i t i o n ( c . / . [Barwise 1977, p 62]).  P r o p o s i t i o n 7 . 2 (tyo5-Tarski T h e o r e m ) U n i v e r s a l theories (possibly w i t h f u n c t i o n symbols) are w e l l - f o u n d e d f o r d o m a i n circumscription. I  It i s also c l e a r t h a t theories w i t h o n l y finite models are w e l l - f o u n d e d . S e c o n d , because t h e d o m a i n c i r c u m s c r i p t i o n s c h e m a is satisfied b y every m i n i m a l m o d e l , d o m a i n c i r c u m s c r i p t i o n does n o t p r o d u c e a n y new g r o u n d - t e r m e q u a l i t i e s o r i n e q u a l i t i e s , f o r w e l l f o u n d e d theories.  ( T h e s a m e l i m i t a t i o n also a p p l i e s t o p r e d i c a t e a n d f o r m u l a c i r c u m s c r i p t i o n . )  - 90 -  Theorem  7.3  If T is a w e l l - f o u n d e d t h e o r y w h i c h c o n t a i n s a x i o m s w h i c h define the e q u a l i t y p r e d i c a t e , = , a n d ay...,cx„ By...,8 &re g r o u n d t e r m s , t h e n n  (i)  T\-(Va = i  t=l  (ii) T  |—  8,)*=* DC(T)[-(\/a =  ( V a,- £ B,) <=>  i  B,)  E=l  DC(T)  «=i  \- ( V a , £ /?,) t=i  I  T h e a u t o m a t i c g e n e r a t i o n of a l l possible g r o u n d t e r m inequalities to c a p t u r e the u n i q u e - n a m e s a s s u m p t i o n [Reiter 1980b] r e m a i n s a t h o r n y issue i n knowledge r e p r e s e n t a t i o n . T h i r d , the a m b i g u i t y of the u s u a l s t a t e m e n t of the d o m a i n - c l o s u r e a s s u m p t i o n is r e v e a l e d . O n l y M4 has the m i n i m u m number of i n d i v i d u a l s necessary t o satisfy T (i.e., 1), yet e a c h of M  —  x  M 4 has o n l y i n d i v i d u a l s n a m e d ( a n d hence r e q u i r e d to exist) b y T. D o m a i n c i r c u m s c r i p t i o n c a p tures  a weak  sense of the d o m a i n - c l o s u r e a s s u m p t i o n w h i c h does not  interpretations.  decide b e t w e e n  these  B a s e d o n c o m m o n a p p l i c a t i o n s of the d o m a i n - c l o s u r e a s s u m p t i o n ( t y p i c a l l y  in  c o n j u n c t i o n w i t h some f o r m of u n i q u e - n a m e s a s s u m p t i o n ) , this w e a k sense appears to be the preferred sense. W h i l e new g r o u n d e q u a l i t y s t a t e m e n t s are not generally f o r t h c o m i n g , the results of d o m a i n c i r c u m s c r i p t i o n d o i n t e r a c t w i t h the e q u a l i t y t h e o r y i n interesting w a y s .  T h e c i r c u m s c r i p t i o n of  T i n E x a m p l e 7.1 e n t a i l s a — b A 6 = c D •Hxiy. x = y, for e x a m p l e .  The circumscription  {3x.  Px,  3z- Qx} e n t a i l s 3x. Px f\ Qx  O  EJzVy. x = y. S u c h f o r m u l a e seem to precisely c a p t u r e  the difference between the v a r i o u s m i n i m a l m o d e l s of the o r i g i n a l t h e o r y . result for d o m a i n c i r c u m s c r i p t i o n c a n be o b t a i n e d . only  finite  In fact, a completeness  T h i s result guarantees t h a t , for theories w i t h  m o d e l s ( a m o n g others), the set of m i n i m a l models of the o r i g i n a l t h e o r y  e x a c t l y the set of m o d e l s of the c i r c u m s c r i b e d theory. encouraging.  of  constitutes  S u c h a precise c h a r a c t e r i z a t i o n is v e r y  T h e proof of this result is analogous to P e r l i s a n d M i n k e r ' s [1986]  finitary  complete-  ness p r o o f for predicate a n d f o r m u l a c i r c u m s c r i p t i o n .  T h e o r e m 7.4 — F i n i t a r y C o m p l e t e n e s s  If T is a finitely a x i o m a t i z a b l e theory, a n d every m o d e l of T is m i n i m a l m o d e l s of T satisfy e v e r y instance of s c h e m a (26) for T.  finite,  then only  the  fl  In the s t a t e m e n t of T h e o r e m 7.4, the r e q u i r e m e n t t h a t a l l of T°s m o d e l s be finite is stronger t h a n necessary. The t h e o r e m holds e v e n if o n l y the m o d e l s w h i c h satisfy s c h e m a (26) are  Corollary  finite.  7.5  If T is a finitely a x i o m a t i z a b l e t h e o r y , a n d e v e r y m o d e l o f T U s c h e m a (26) is t h e n o n l y t h e m i n i m a l m o d e l s of T satisfy every instance of s c h e m a (26) for T.  finite, |  - 91 7.3.  Related Formalisms  McCarthy  [1980] c l a i m s t h a t d o m a i n c i r c u m s c r i p t i o n is a s p e c i a l case of p r e d i c a t e  cir-  c u m s c r i p t i o n , i n t h a t the d o m a i n c i r c u m s c r i p t i o n s c h e m a for a theory, A, c a n be d e r i v e d b y p r e d i cate c i r c u m s c r i p t i o n of a t h e o r y , A', w h i c h is a c o n s e r v a t i v e e x t e n s i o n of A. m i g h t a p p e a r t h a t interest  i n d o m a i n c i r c u m s c r i p t i o n is pointless.  I n v i e w of t h i s , it  A p a r t f r o m the fact  that  d o m a i n c i r c u m s c r i p t i o n is a m o r e d i r e c t a n d s o m e w h a t s i m p l e r a p p r o a c h to d o m a i n - c l o s u r e , a n d t h a t the m o d e l t h e o r y of d o m a i n c i r c u m s c r i p t i o n perhaps better captures our i n t u i t i o n s a b o u t the conjectures  involved,  there  is  another  reason  to  reject  this  argument  for  abandonment.  M c C a r t h y ' s d e m o n s t r a t i o n of this s u b s u m p t i o n a c t u a l l y rests o n a s t r e n g t h e n e d f o r m of predicate c i r c u m s c r i p t i o n w h i c h a l l o w s a x i o m s of the o r i g i n a l t h e o r y to be i g n o r e d d u r i n g the c i r c u m s c r i p t i o n process. A s we n o t e d i n c h a p t e r 5, this f o r m of c i r c u m s c r i p t i o n does not a l w a y s preserve c o n s i s t e n c y , e v e n for theories w i t h m i n i m a l models.  O r d i n a r y predicate c i r c u m s c r i p t i o n cannot, i n  general, y i e l d the d o m a i n c i r c u m s c r i p t i o n s c h e m a .  I n fact, this is f o r t u n a t e , since the f o r m of  d o m a i n c i r c u m s c r i p t i o n M c C a r t h y was t r y i n g to e m u l a t e i n t r o d u c e d inconsistencies i n t o some theories w i t h m i n i m a l m o d e l s . O u r r e v i s e d f o r m o f d o m a i n c i r c u m s c r i p t i o n , w h i c h preserves c o n s i s t e n c y for m o d e l a b l e theories, is s t i l l not  obtainable  using predicate c i r c u m s c r i p t i o n .  minimally  In c h a p t e r 5, we  s h o w e d t h a t p r e d i c a t e c i r c u m s c r i p t i o n is too w e a k to conjecture d o m a i n - c l o s u r e a x i o m s .  Since  d o m a i n c i r c u m s c r i p t i o n c a n conjecture s u c h a x i o m s , it follows t h a t it is not s u b s u m e d b y its predicate cousin.  I n c h a p t e r 6, we s h o w e d t h a t n e i t h e r f o r m u l a c i r c u m s c r i p t i o n n o r generalized  c i r c u m s c r i p t i o n w i t h o u t v a r i a b l e terms subsumes d o m a i n c i r c u m s c r i p t i o n , i n general. O u r s e m a n tic c h a r a c t e r i z a t i o n suggests t h a t it is u n l i k e l y t h a t a n y f o r m of g e n e r a l i z e d c i r c u m s c r i p t i o n c a n conjecture d o m a i n - c l o s u r e a x i o m s .  It appears, therefore, t h a t d o m a i n c i r c u m s c r i p t i o n  t o fill a n i c h e a m o n g the v a r i o u s m e c h a n i s m s for c l o s e d - w o r l d r e a s o n i n g .  continues  CHAPTER 8  Connections Between Default Logic and Circumscription  In c h a p t e r 3, we o b s e r v e d t h a t the model-set s e m a n t i c s for default logic bears a s u p e r f i c i a l r e s e m b l a n c e to t h e m i n i m a l - m o d e l s e m a n t i c s o f the v a r i o u s forms o f c i r c u m s c r i p t i o n . a n d 6 c o n s i d e r e d t h e f e a s i b i l i t y o f d o i n g default reasoning using c i r c u m s c r i p t i o n .  Chapters 5  W e now con-  sider the r e l a t i o n s h i p s b e t w e e n default logic a n d c i r c u m s c r i p t i o n i n m o r e d e t a i l . T h e n a t u r a l q u e s t i o n is w h e t h e r either f o r m subsumes the other.  Is there a d i r e c t c o r r e s p o n -  dence b e t w e e n d e f a u l t theories a n d c i r c u m s c r i p t i o n , o r vice versa?  Proposition 8.1 D e f a u l t logic c a n r e a c h c o n c l u s i o n s w h i c h c a n n o t be o b t a i n e d by generalized circumscription without variable terms.  |  Example 8.1  T h e default  theory  has a u n i q u e e x t e n s i o n , c o n t a i n i n g a J= b.  In  c h a p t e r 6, we s h o w e d t h a t g e n e r a l i z e d c i r c u m s c r i p t i o n w i t h o u t v a r i a b l e t e r m s c a n n o t c o n jecture new inequalities.  |  T h e c o n v e r s e o f p r o p o s i t i o n 8.1 is a p p a r e n t l y false. A s s u m i n g t h a t CLOSURE^T;  X ; R) is  consistent, t h e t h e o r y  o b v i o u s l y p r o d u c e s t h e r e q u i r e d results.  P e r h a p s t h i s is n o t w h a t one h a s i n m i n d w h e n one asks  if default l o g i c c a n c a p t u r e c i r c u m s c r i p t i o n , however! tions.  - 92 -  W e w i l l r e t u r n t o this q u e s t i o n i n l a t e r sec-  - 93 -  8.1.  " T r a n s l a t i o n " f r o m Default Logic to C i r c u m s c r i p t i o n  In v i e w o f p r o p o s i t i o n 8.1, the t i t l e of this section m i g h t seem p a r a d o x i c a l . T h e r e has been some w o r k o n p a r t i a l t r a n s l a t i o n s , however.  G r o s o f [1984] presents t w o e q u i v a l e n t  translation  schemes for n o r m a l default theories, one i n v o l v i n g 'ab' predicates (discussed i n §2.2.2), the other i n v o l v i n g m i n i m i z i n g a r b i t r a r y expressions. W e discuss t h e former. T h e t r a n s l a t i o n scheme carries the  i Pi Pi  first-order  a x i o m s , W, o v e r u n c h a n g e d . F o r e a c h closed  a :  n o r m a l default, — - — ,  the a x i o m a,- A —A'  3  °H*)  m  added.  T h e n ab is c i r c u m s c r i b e d in the  r e s u l t i n g t h e o r y , v a r y i n g ab a n d e a c h p r e d i c a t e w h i c h occurs i n a n y of the P-s. G r o s o f observes t h a t t h i s " t r a n s l a t i o n " a c t u a l l y differs f r o m default logic i n a n u m b e r of respects. e q u a l i t y p r e d i c a t e is not affected b y the c i r c u m s c r i p t i v e theory. defaults a b o u t e q u a l i t y t o r e m e d y t h i s , b u t t h i s is insufficient. w i l l not behave " c o r r e c t l y " in the c i r c u m s c r i p t i v e t h e o r y .  theory  A f u r t h e r difference is t h a t the cir-  a  a  D  n ' p  s  u  i  c  e  T h e m u l t i p l i c i t y o f extensions  are reflected i n d i s j u n c t i v e s t a t e m e n t s i n the t r a n s l a t e d t h e o r y .  G r o s o f s t r a n s l a t i o n of the n o r m a l default '  G r o s o f proposes to exclude  A n y default w h i c h affects equality  cumscriptive theory inherits circumscription's " c a u t i o u s " nature. o f a default  a  F i r s t , the  Finally,  ® a c t u a l l y more closely corresponds t o the default  the t r a n s l a t i o n a l l o w s the conjecture of -<a f r o m ->P, s o m e t h i n g G r o s o f appears not  to have n o t i c e d .  E v e n a l l o w i n g for these discrepancies, G r o s o f presents no more t h a n i n t u i t i v e  a r g u m e n t s a n d e x a m p l e s i n s u p p o r t of the correctness o f the t r a n s l a t i o n scheme. I m i e l i n s k i [1985] takes the c o m p l e m e n t a r y t a c k o f defining a t r a n s l a t i o n scheme to be adequate if the t h e o r y a n d its t r a n s l a t i o n p r o d u c e precisely the same c o n c l u s i o n s , a n d f u r t h e r m o r e the t r a n s l a t i o n scheme is " m o d u l a r " . first-order  M o d u l a r i t y requires t h a t the t r a n s l a t i o n of the defaults a n d  facts m u s t be i n d e p e n d e n t .  I m i e l i n s k i v i e w s the t r a n s l a t i o n of a set of defaults t o consist of a c o l l e c t i o n of first-order facts a n d a p r e - o r d e r r e l a t i o n .  B o t h o f these m u s t be d e t e r m i n e d f r o m the defaults a l o n e , w i t h o u t  reference t o the specific facts at h a n d .  T h i s is a desirable p r o p e r t y , since one does n o t w i s h to  h a v e to r e c o m p u t e one's r e p r e s e n t a t i o n of k n o w l e d g e (in a d d i t i o n to the necessary a d j u s t m e n t s to the set of one's conjectures) every t i m e a new fact is learned. G i v e n these s t r i c t u r e s , I m i e l i n s k i is able to prove t h a t even n o r m a l defaults are not m o d u l a r l y t r a n s l a t a b l e t o g e n e r a l i z e d c i r c u m s c r i p t i o n . T h e r e are some defaults w h i c h do h a v e m o d u l a r t r a n s l a t i o n s , h o w e v e r . T h e s e are the s e m i - n o r m a l defaults w i t h o u t prerequisites (e.g.,  ——j^—)-  T h e s e r e s u l t s h i g h l i g h t the necessity of the f u n d a m e n t a l d i s t i n c t i o n between the model-setr e s t r i c t i o n s e m a n t i c s of default logic (see c h a p t e r 3) a n d the m i n i m a l - m o d e l s e m a n t i c s of circumscription.  T h e p r e r e q u i s i t e s of t h e d e f a u l t s are r e q u i r e d to be p r o v a b l e .  T h i s is a g l o b a l  c h a r a c t e r i s t i c o f the set of m o d e l s . T h e s u b m o d e l r e l a t i o n , h o w e v e r , is o n l y able to c o n s i d e r pairs of m o d e l s . P r e r e q u i s i t e - f r e e defaults prerequisite-free.  fit  n i c e l y i n t o c i r c u m s c r i p t i o n precisely because they are  T h e r e are no (global) p r o v a b i l i t y r e q u i r e m e n t s , o n l y c o n s i s t e n c y r e q u i r e m e n t s .  C o n s i s t e n c y c a n be d e t e r m i n e d b y the existence of a single m o d e l , so c a n be l o c a l l y d e t e r m i n e d .  - 94 -  T h e r e r e m a i n s the q u e s t i o n of w h e t h e r the r e q u i r e m e n t of i d e n t i c a l sets of theorems is too strong.  I m i e l i n s k i ' s t h e o r e m , p r o v i n g t h a t n o r m a l default theories are not m o d u l a r l y t r a n s l a t a b l e ,  A : B rests o n the fact t h a t a n y m o d u l a r t r a n s l a t i o n of the default — ' - — , where the sets {A, B} a n d B {A, ~<B} are b o t h c o n s i s t e n t , w i l l necessarily y i e l d A D B as a t h e o r e m ( a s s u m i n g  W \f—>B).  W h i l e t h i s m a y be true, if a n e x t e n s i o n c o n t a i n s A o r B, it w i l l also c o n t a i n A Z> B.  It appears  t h a t the offending i m p l i c a t i o n is offensive o n l y i n those cases where it c a n n o t be u s e d to deduce anything "useful".  M o r e c o n v i n c i n g l y , we h a v e n o t e d t h a t default l o g i c is a " b r a v e " reasoner  w h i l e c i r c u m s c r i p t i o n is " c a u t i o u s " .  It seems reasonable t o expect t h a t a c i r c u m s c r i p t i v e t r a n s l a -  t i o n of d e f a u l t l o g i c w o u l d reflect this c a u t i o u s n a t u r e , perhaps r e t u r n i n g those facts true i n all extensions.  F i n a l l y , c i r c u m s c r i p t i v e conjectures a p p l y to a l l i n d i v i d u a l s , whereas those r e s u l t i n g  f r o m o p e n d e f a u l t s a p p l y o n l y to i n d i v i d u a l s w i t h n a m e s i n the l a n g u a g e . It m i g h t be reasonable to e x p e c t t h a t c i r c u m s c r i p t i v e versions of default  theories w i t h o p e n defaults w o u l d  therefore  p r o v e s t r o n g e r conjectures (at least for theories w i t h o u t d o m a i n closure a x i o m s ) . T h e s e c o n s i d e r a t i o n s suggest t h a t I m i e l i n s k i ' s results m i g h t be t a k e n as a " w o r s t  case"  s c e n a r i o , l e a v i n g o p e n the p o s s i b i l i t y of a c c e p t a b l e t r a n s l a t i o n schemes for defaults w i t h prerequisites, g i v e n a w e a k e r n o t i o n of " a c c e p t a b l e " . W e do not f u r t h e r c o n s i d e r this p o s s i b i l i t y here.  8.2.  Translations from Circumscription to Default Logic T h e o t h e r side of the c o i n we h a v e been e x a m i n i n g is w h e t h e r default l o g i c c a n be used to  p e r f o r m c i r c u m s c r i p t i o n (in a n y b u t the t r i v i a l sense m e n t i o n e d at the b e g i n n i n g of this c h a p t e r ) . T h e p r e v i o u s s e c t i o n o u t l i n e d a n u m b e r of the v e r y different c a p a b i l i t i e s of the t w o f o r m a l i s m s : b r a v e vs c a u t i o u s , effects o n e q u a l i t y , g l o b a l ( p r o v a b i l i t y )  vs l o c a l (consistency) c o m p a r i s o n s i n  the m o d e l - t h e o r y ( p r o o f - t h e o r y ) , a n d s t a t e m e n t s about " u n n a m e d " i n d i v i d u a l s .  I n a l l b u t the last  of these categories, default logic c a m e out o n the stronger e n d . T h i s suggests t h a t the search for a direct i m p l e m e n t a t i o n of c i r c u m s c r i p t i o n i n default logic m i g h t be m o r e successful t h a t the c o n verse a t t e m p t .  T h e a n s w e r to this is, " Y e s , a n d n o . " .  T h e r e is one facet of g e n e r a l i z e d c i r -  c u m s c r i p t i o n w h i c h is c o m p l e t e l y absent f r o m default l o g i c . T h a t is the a b i l i t y to specify w h i c h predicates are to be a l l o w e d t o v a r y d u r i n g the c i r c u m s c r i p t i o n process.  In default l o g i c , there is  no w a y t o r e s t r i c t the repercussions of the defaults to some p a r t i c u l a r set of p r e d i c a t e s ( a n d / o r individuals).  T h u s we h a v e T h e o r e m 8.2.  Theorem 8.2 If  T |— V i . I = &i V . . . V x — ct a n d T |— a,- j= ctj, for i  and X  n  j for  ground terms  ct ...,a ; lt  n  i n c l u d e s a l l of the p r e d i c a t e s of L; t h e n those f o r m u l a e true i n every e x t e n s i o n of  •  - 95 -  C o r o l l a r y 8.3  If E is a n e x t e n s i o n of A , t h e n e v e r y m o d e l of E is a n ( X , { . P } ) - m i n i m a l m o d e l o f T. |  C o r o l l a r y 8.4  If M i s a n ( X , { P } ) - m i n i m a l for some e x t e n s i o n of A .  m o d e l of T, t h e n M i s a m o d e l |  C o r o l l a r y 8.5  A c a p t u r e s t h e b r a v e c i r c u m s c r i p t i o n of P i n T w i t h every p r e d i c a t e v a r i a b l e .  N o t i c e t h a t T h e o r e m 8.2 requires t h a t closure a x i o m s .  If we d r o p t h e r e q u i r e m e n t  |  T have u n i q u e - n a m e a x i o m s as w e l l as d o m a i n for u n i q u e - n a m e a x i o m s , t h e n t h e d e f a u l t  theory  becomes s t r o n g e r t h a n the c i r c u m s c r i p t i v e theory, i n the sense t h a t C o r o l l a r y 8.3 c o n t i n u e s to h o l d b u t T h e o r e m 8.2 a n d C o r o l l a r y 8.4 do not.  W e h a v e not y e t d e t e r m i n e d w h e t h e r these  results generalize t o t h e j o i n t m i n i m i z a t i o n of s e v e r a l predicates.  B e c a u s e of the l i m i t a t i o n  of  o p e n defaults to n a m e d i n d i v i d u a l s , none of the results generalize to theories w i t h o u t d o m a i n closure a x i o m s .  P r o p o s i t i o n 8.6  If T does not e n t a i l a d o m a i n - c l o s u r e a x i o m , a n d T \/- Vz. ->Px, t h e n e v e r y e x t e n s i o n for A has m o d e l s w h i c h are not ( X , { P } ) - m i n i m a l . |  E v e n more p e s s i m i s t i c is the result t h a t fixed predicates preclude s u c h a s t r a i g h t f o r w a r d  transla-  t i o n of c i r c u m s c r i p t i o n to default logic, e v e n for c l o s e d - d o m a i n , u n i q u e - n a m e theories.  T h e o r e m 8.7  There  are  theories,  T,  such  that  T (— Vz. z = a  x  V...Vjc = a „ a n d T  for t j= j a n d y e t no c o m b i n a t i o n of the extensions of A c h a r a c t e r i z e s the ( X , { P } ) - m i n i m a l m o d e l s of T.  |  =  a,- £ a y , precisely  1  - 96 -  We  experimented  with  an extended  s p e c i f i c a t i o n o f " f i x e d " predicates.  version  of default  logic  which  allowed  for the  A l t h o u g h w e were able t o s h o w t h a t t h e results i n [ R e i t e r  1980a, c h a p t e r s 2 a n d 3] h o l d f o r this logic, a n d - f o r finite theories - the o b v i o u s g e n e r a l i z a t i o n o f t h e model-set r e s t r i c t i o n s e m a n t i c s of c h a p t e r 3 applies, w e a b a n d o n e d this a p p r o a c h w h e n i t p r o v e d i n c a p a b l e of y i e l d i n g a n analogue f o r T h e o r e m 8.2 i n the presence of fixed p r e d i c a t e s . ( T h e best t h a t c o u l d be g u a r a n t e e d w a s t h a t those g r o u n d literals i n P c o n t a i n e d i n a l l extensions were t r u e i n a l l m i n i m a l m o d e l s . T h i s is s i g n i f i c a n t l y w e a k e r — sufficiently so t h a t w e d o u b t t h a t the ( a b u n d a n t ) e x t r a m a c h i n e r y r e q u i r e d is w o r t h w h i l e .  E x a m p l e 8.2  L e t T be { V z . i = « V i = i , o/= 1, - . P a A ~^Pb Z> Qa} a n d let Q be fixed. T h e P - m i n i m a l m o d e l s of T are (loosely represented): { Pa, - . P 6 , ^Qa} {-.Pa,  Pb, -.<?a}  { - . P a , - . P 6 , Qa} T h e r e are n o g r o u n d literals i n P t r u e i n every P - m i n i m a l m o d e l . H o w e v e r , CLOSURE{T;  {P}; {P}) \- ( 3 z . P z = Qa) A ( - P a V - P 6 ) .  I n o t h e r w o r d s , one c a n c i r c u m s c r i p t i v e l y conjecture t h a t there is e x a c t l y one P i f Qa, a n d none o t h e r w i s e . |  G e l f o n d a n d P r z y m u s i n s k a [1985] p r o v e the weak result a l l u d e d t o above f o r t h e i r v e r s i o n of M i n k e r ' s g e n e r a l i z e d c l o s e d - w o r l d a s s u m p t i o n , w h i c h allows fixed predicates.  Gelfond, Przymu-  s i n s k a , a n d P r z y m u s i n s k i [1985] prove a m u c h stronger result for t h e i r e x t e n d e d c l o s e d - w o r l d assumption.  P r o p o s i t i o n 8.8 (Gelfond, Przymusinska, and Przymusinski)  A s t r u c t u r e , M, is a m o d e l for ECWA(T)  iff it is a m i n i m a l m o d e l for T.  |  A t first g l a n c e this m i g h t suggest t h a t there s h o u l d be some analogous result f o r some default theory.  It a p p e a r s t h a t t h e E C W A a c t u a l l y achieves t h i s p o w e r b y the subterfuge discussed n e a r  the b e g i n n i n g o f t h i s c h a p t e r , b y a d d i n g e v e r y instance o f the c i r c u m s c r i p t i o n s c h e m a . c e r t a i n l y t h e case i n t h e absence of v a r i a b l e predicates.  T h i s is  - 97 -  P r o p o s i t i o n 8.9  If there are no v a r i a b l e predicates (Z={ of t h e c i r c u m s c r i p t i o n s c h e m a .  It  }), then ECWA(T)  a d d s to T e v e r y instance  |  seems t h a t a n y g e n e r a l i z e d t r a n s l a t i o n f r o m c i r c u m s c r i p t i o n to default logic (for  finite  theories) - if s u c h a t h i n g exists, short of a d d i n g defaults for e a c h instance of t h e c i r c u m s c r i p t i o n schema -  requires m o r e p o w e r t h a n the c l o s e d - w o r l d default  appropriate translation remains open.  provides.  T h e existence of a n  CHAPTER 9  Open Problems  D o n ' t c o n f r o n t m e w i t h m y failings ... I have n o t f o r g o t t e n t h e m . - Jackson Browne  T h r o u g h o u t t h e thesis, a catalogue of o p e n p r o b l e m s has been c o m p i l e d . R a t h e r t h a n r e c a p i t u l a t e this list o f specific p r o b l e m s , t h i s c h a p t e r addresses a broader, p h i l o s o p h i c a l p e r s p e c t i v e . W e c o n s i d e r a g e n e r a l research p r o g r a m m e , i n s t e a d of a l i t a n y of isolated potholes i n need of filling. A l t h o u g h there h a s been considerable a c t i v i t y i n the area o f n o n - m o n o t o n i c r e a s o n i n g , a l o n g w i t h some r e m a r k a b l e successes, v e r y l i t t l e a t t e n t i o n has been focussed o n t h e d y n a m i c s of n o n monotonicity.  A s t h i s promises t o be a p a r t i c u l a r l y f r u i t f u l avenue of i n v e s t i g a t i o n , this c h a p t e r  addresses t w o aspects o f t h i s p r o b l e m :  h o w new i n f o r m a t i o n is a s s i m i l a t e d i n t o a t h e o r y i n v o l v i n g  a s s u m p t i o n s , a n d h o w n o n - m o n o t o n i c inference rules are a c q u i r e d a n d e m p l o y e d . T h e s e t w o areas are i n t i m a t e l y  related.  A m a j o r g o a l f o r future research s h o u l d be to  d e v e l o p a u n i f y i n g f r a m e w o r k w h i c h m a k e s t h e i r i n t e r r e l a t i o n s h i p s more a p p a r e n t .  T h i s p o i n t of  v i e w m a y be e x p e c t e d t o p r o v i d e n e w i n s i g h t s i n t o b o t h n o n - m o n o t o n i c reasoning a n d updates. F u r t h e r m o r e , m u c h o f the w o r k t h a t h a s been done t r e a t i n g these p r o b l e m s i n i s o l a t i o n c a n , h o p e f u l l y , be r e i n t e r p r e t e d t o a d v a n t a g e f r o m t h i s more general s t a n d p o i n t .  9.1. Principles of Non-Monotonic Reasoning T h e i m p o r t a n t issue of n o n - m o n o t o n i c i t y w h i c h r e m a i n s unaddressed is n o t p r i m a r i l y h o w c o n c l u s i o n s are o b t a i n e d g i v e n some facts a n d some n o n - m o n o t o n i c inference r u l e s .  R a t h e r , the  q u e s t i o n is h o w n o n - m o n o t o n i c rules are f o r m u l a t e d , d e t e r m i n e d to be a p p l i c a b l e , a n d a p p l i e d . T h i s q u e s t i o n c a n be i l l u s t r a t e d b y c o n s i d e r i n g the c i r c u m s c r i p t i v e e x a m p l e s o f §2.1.5.2. G i v e n a r e p r e s e n t a t i o n of t h e facts a b o u t t h e " w o r l d " , c e r t a i n predicates m u s t be c i r c u m s c r i b e d , other predicates specified as v a r i a b l e , a p p r o p r i a t e s u b s t i t u t i o n s d i s c o v e r e d , a n d t h e n t h e r e q u i r e d c o n jectures are obtained.  A s m u c h of the p r o b l e m lies i n these " a n c i l l a r y " tasks o f d e c i d i n g w h a t  a n d h o w t o c i r c u m s c r i b e as i n the c l o s e d - w o r l d r e a s o n i n g a c h i e v e d b y a c t u a l l y p e r f o r m i n g t h e c i r cumscription.  T o d a t e , m o s t o f the w o r k i n n o n - m o n o t o n i c reasoning ( i n c l u d i n g this thesis) has  focussed m o r e o n d e v e l o p i n g m e c h a n i s m s f o r p e r f o r m i n g c e r t a i n s p e c i a l i z e d r e a s o n i n g tasks t h a n  - 98 -  - 99 -  on u n d e r l y i n g p r i n c i p l e s o r even a n u n d e r s t a n d i n g of w h e n a n d how to e m p l o y the m e c h a n i s m s once t h e y are d e v e l o p e d . T h e c e n t r a l q u e s t i o n is: c a n we d i s c o v e r w a y s to make n o n - m o n o t o n i c r e a s o n i n g a u t o m a t i c a n d / o r goal-directed?  I.e., are there features of p a r t i c u l a r p r o b l e m s w h i c h c a n guide the c o m p l e -  t i o n of a n i n c o m p l e t e knowledge-base, w i t h o u t e x t e r n a l i n t e r v e n t i o n , t o solve those p r o b l e m s ?  A  first a p p r o x i m a t i o n to a theory of n o n - m o n o t o n i c t h e o r y c o n s t r u c t i o n was o u t l i n e d b y R e i t e r [1978a]. H e e x p l a i n e d n o n - m o n o t o n i c reasoning i n t e r m s of the c l o s e d - w o r l d a s s u m p t i o n . R e i t e r ' s i d e a w a s t h a t reasoners m i g h t assume t h e i r k n o w l e d g e a b o u t r e l e v a n t aspects of the s i t u a t i o n to be c o m p l e t e . C l o s e d - w o r l d r e a s o n i n g s a n c t i o n s e x a c t l y those c o n c l u s i o n s true i n a w o r l d completely characterized b y w h a t is k n o w n . S u c h a c l e a r , s i m p l e , u n i f o r m c h a r a c t e r i z a t i o n of n o n - m o n o t o n i c r e a s o n i n g a p p e a l s to i n t r o s p e c t i v e i n t u i t i o n s a b o u t t h e s i m p l i c i t y a n d naturalness of commonsense r e a s o n i n g . U n f o r t u n a t e l y , it p r o v e d s i m p l i s t i c as w e l l as s i m p l e . N o t every knowledge state u n i q u e l y c h a r a c t e r i z e s a state of the w o r l d .  A s s u m i n g the r e a l w o r l d is t h a t w o r l d c h a r a c t e r i z e d b y w h a t is k n o w n is a d u b i o u s  step w h e n no w o r l d is so c h a r a c t e r i z e d ! R e s e a r c h since 1978 has focussed o n m e c h a n i s m s w h i c h a v o i d the s h o r t c o m i n g s of the n a i v e i n t e r p r e t a t i o n s of the C W A . L i t t l e effort has been d i r e c t e d to finding  a c o r r e s p o n d i n g i n t u i t i v e e x p l i c a t i o n of the u n d e r l y i n g p r i n c i p l e s . T h e m i n i m a l - m o d e l semantics w h i c h we h a v e discussed i n one f o r m or a n o t h e r t h r o u g h o u t  this thesis does n o t q u a l i f y as the i n t u i t i v e e x p l i c a t i o n we seek, for t w o reasons. T h e first - a n d p e r h a p s less c o m p e l l i n g - is t h a t not a l l theories have m i n i m a l m o d e l s , a n d it is u n d e c i d a b l e w h e t h e r a p a r t i c u l a r t h e o r y has a m i n i m a l m o d e l .  C e r t a i n theories - q u i t e u n e x p e c t e d l y -  turn  out not t o h a v e m i n i m a l m o d e l s . F o r e x a m p l e , we have s h o w n t h a t the theory:  3x. Nx A Vy. Ny Z> x j= succ(y) Vx. Nx D Nsucc{x) Vzj/. succ(x) = succ(y) D x = y , has no m i n i m a l m o d e l s . numbers, N .  T h i s is because a n y m o d e l has a c h a i n of i V s i s o m o r p h i c t o the n a t u r a l  B u t t h i s c h a i n has a s u b c h a i n , also i s o m o r p h i c to N , w h i c h satisfies the a x i o m s .  H e n c e every m o d e l has a p r o p e r s u b m o d e l , a n d there are no m i n i m a l m o d e l s . B u t , since every m o d e l c o n t a i n s a s e g m e n t i s o m o r p h i c to N , a n d since there are models e x a c t l y i s o m o r p h i c to N , surely c o m m o n s e n s e d i c t a t e s t h a t N is a n acceptable m i n i m a l m o d e l ?  M i n i m u m - m o d e l semantics  force the m i n i m i z a t i o n process to go b e y o n d the b o u n d s of commonsense i n this case. M o r e t e l l i n g l y , m i n i m a l - m o d e l s e m a n t i c s enter the picture after m u c h of the n o n - m o n o t o n i c reasoning process is c o m p l e t e . O n l y after it has been d e c i d e d w h a t expression is t o be m i m i m i z e d , a n d the c o n n e c t i o n s b e t w e e n the m i n i m i z e d expression a n d the rest of the w o r l d h a v e been determ i n e d so t h a t v a r i a b l e p r e d i c a t e s c a n be chosen, c a n the s e m a n t i c c h a r a c t e r i z a t i o n t e l l us w h a t w o r l d ( s ) the n o n - m o n o t o n i c t h e o r y c h a r a c t e r i z e s . T h e s e m a n t i c s sheds no l i g h t o n these other d i m e n s i o n s o f the c o m m o n s e n s e r e a s o n i n g process. H e n c e , it is not the c h a r a c t e r i z a t i o n we seek. W h a t e v i d e n c e is there that there is a n y u n d e r l y i n g p r i n c i p l e ? finding  M i g h t not the d i f f i c u l t y in  s u c h a p r i n c i p l e s t e m , i n p a r t , f r o m its non-existence? O f course, the o n l y guarantee t h a t  the p r i n c i p l e we seek e x i s t s w i l l be its d e m o n s t r a t i o n . T h e r e is evidence w h i c h suggests t h a t some sort of u n i f o r m rules m i g h t underlie commonsense reasoning. O n e i n d i c a t i o n is the existence of  - 100 -  a p p r o x i m a t i o n s w h i c h fill the role of the sought-after rule i n l i m i t e d cases. rule.  T h e C W A is one s u c h  O t h e r s i n c l u d e m i n i m a l - m o d e l s e m a n t i c s (for theories w i t h m i n i m a l models), the model-set-  r e s t r i c t i o n s e m a n t i c s f o r default l o g i c , a n d the i n f e r e n t i a l distance concept i n s e m a n t i c n e t w o r k reasoning s y s t e m s . A final e x a m p l e is " O c c a m ' s R a z o r " , a h y p o t h e s i s - r a n k i n g r u l e w h i c h suggests t h a t the s i m p l e s t e x p l a n a t i o n for a n y p h e n o m e n o n is the best. O f course, there m a y be no u n i f o r m u n d e r l y i n g p r i n c i p l e s . interesting.  S o be it.  T h a t is, i n  itself,  B e s i d e s , if h u m a n s use n o u n i f o r m procedures at a l l , we c a n s t i l l hope t o u n c o v e r  h e u r i s t i c s w h i c h c a n h e l p guide the task of commonsense reasoning. F o r e x a m p l e , e v e n a w a y to a u t o m a t i c a l l y d e t e r m i n e , for some class of theories, w h i c h expressions to c i r c u m s c r i b e a n d / o r w h i c h p r e d i c a t e s to v a r y based o n the g o a l at h a n d a n d the current k n o w l e d g e state w o u l d be a significant contribution.  9.2.  Update T h e p r o b l e m s of u p d a t i n g theories w i t h i n f o r m a t i o n inconsistent w i t h t h e i r c u r r e n t state are  o b v i o u s l y p r o b l e m s of n o n - m o n o t o n i c inference: s u c h new facts m u s t force the r e t r a c t i o n of p r e v i o u s l y a c c e p t e d facts if c o n s i s t e n c y is t o be preserved. A second m a j o r open p r o b l e m is to d e v e l o p a v i e w of u p d a t e s w h i c h integrates them' w i t h o t h e r forms o f n o n - m o n o t o n i c r e a s o n i n g . Instead of b l i n d a d d i t i o n a n d d e l e t i o n - w h i c h o b v i o u s l y w i l l not w o r k - o r the p r o l i f e r a t i o n of alternate theories - w h i c h increases u n c e r t a i n t y — it seems a p p r o p r i a t e to v i e w updates as new i n f o r m a t i o n w h i c h leads to the reasoned assertion or r e t r a c t i o n of facts. T h e e x a c t f o r m t h a t this research m i g h t t a k e is unclear.  T h e final result w i l l l i k e l y be  h e a v i l y i n f l u e n c e d b y w o r k i n five areas: 1) R e l e v a n c e L o g i c ( A n d e r s o n a n d B e l n a p 1975]: i n R e l e v a n c e L o g i c , c o n t r a d i c t i o n s do not a u t o m a t i c a l l y l e a d to chaos. T h e repercussions of the v a r i o u s facts i n a n inconsistent t h e o r y c a n be e x p l o r e d w i t h o u t i n t r o d u c i n g " a r t i f a c t s " of the i n c o n s i s t e n c y . T h i s seems l i k e a n i d e a l e n v i r o n m e n t f o r i n v e s t i g a t i n g t h e effects of c o n t r a r y updates. 2) C o u n t e r f a c t u a l s a n d H y p o t h e t i c a l [Rescher 1964, 1976; L e w i s 1973]: T h e s e b r a n c h e s of p h i l o s o p h y d e a l w i t h w h a t w o u l d be t r u e i n a w o r l d w h i c h differs f r o m the r e a l w o r l d i n t h a t (at least) c e r t a i n s p e c i f i e d facts h o l d . T h e u p d a t e p r o b l e m c a n easily be c o n s t r u e d i n these t e r m s . O n e m i g h t therefore expect this w o r k to s h e d l i g h t o n u p d a t i n g . 3) C h a n g e - r e c o r d i n g , c o r r e c t i n g , a n d k n o w l e d g e - a d d i n g updates: W i l k i n s [1983] a n d K e l l e r [& W i l k i n s 1984a, b] d i s t i n g u i s h different k i n d s of updates depending o n w h e t h e r the u p d a t e expresses a c h a n g e i n the state of the w o r l d , a n error i n the database, o r s i m p l y new k n o w l e d g e . I n a d a t a b a s e w i t h i n c o m p l e t e i n f o r m a t i o n , a n u p d a t e can be e x p e c t e d to have different s e m a n t i c s d e p e n d i n g o n to w h i c h o f these categories it belongs. 4) N o n - m o n o t o n i c r e a s o n i n g systems a p p e a r to p r o v i d e useful t h e o r e t i c a l tools for e x a m i n i n g the repercussions o f u p d a t e s . U p d a t e s c o n t r a r y to w h a t was inferred b y default c a n be m a d e to a u t o m a t i c a l l y e x c l u d e these offending defaults after the u p d a t e . R e i t e r [1980a] has considered u p d a t e s t o d e f a u l t theories i n l i m i t e d c i r c u m s t a n c e s . H e shows t h a t c e r t a i n classes of u p d a t e s are k n o w l e d g e - c o n s e r v i n g ; t h e y d o not force the rejection of a n y conclusions. 5) B e l i e f R e v i s i o n S y s t e m s : T h e a s s u m p t i o n - b a s e d a p p r o a c h t o belief r e v i s i o n [ M a r t i n s 1983, de K l e e r 1984] p r o v i d e s a n a t t r a c t i v e b o o k - k e e p i n g s y s t e m for d e a l i n g w i t h s t r a i g h t f o r w a r d repercussions of c h a n g i n g sets of a s s u m p t i o n s . R e i t e r a n d G r o s o f [1985, p e r s o n a l c o m m u n i c a t i o n s ]  - 101 -  h a v e e a c h w o r k e d o n f o r m a l i z i n g these systems i n default logic. N o n - m o n o t o n i c reasoning a n d u p d a t e are i n t i m a t e l y c o n n e c t e d : n o n - m o n o t o n i c r e a s o n i n g is n o n - m o n o t o n i c precisely because of its b e h a v i o u r w h e n c o n f r o n t e d b y updates. In fact, it is possible to v i e w w h a t we h a v e been c a l l i n g n o n - m o n o t o n i c reasoning as a m o n o t o n i c , v a l i d , f o r m of inference.  A n y u p d a t e w h i c h forces a s s u m p t i o n s to be r e t r a c t e d c a n be c o n s t r u e d as c o n t r a r y to  the o r i g i n a l k n o w l e d g e - b a s e (i.e., a s s u m p t i o n s are v i e w e d as e n t a i l e d b y the knowledge-base u n d e r a m o d i f i e d e n t a i l m e n t r e l a t i o n [Israel 1980, N u t t e r 1983].)  U n d e r this v i e w ,  non-monotonicity  becomes s t r i c t l y a p r o b l e m of d e a l i n g w i t h c o n t r a r y updates. T h e p r o b l e m of u p d a t e s is also i m p o r t a n t w i t h i n the c o n t e x t of n o n - m o n o t o n i c r e a s o n i n g . G i v e n a s y s t e m f o r d r a w i n g n o n - m o n o t o n i c inferences, one is faced w i t h the p r o b l e m o f a d a p t i n g to n e w i n f o r m a t i o n .  E v e n u p d a t e s w h i c h d o not represent a change i n the state of the w o r l d are  p r o b l e m a t i c w h e n n o n - m o n o t o n i c i t y is i n v o l v e d .  T h e o b v i o u s p r o b l e m is t h a t c o n t r a r y i n f o r m a -  t i o n m a y h a v e been p r e v i o u s l y i n f e r r e d b y default. I n s u c h cases, the conflict c a n p e r h a p s be detected. genesis)  T h e d e f a u l t inference c a n t h e n s i m p l y be r e v o k e d (if the s y s t e m r e m e m b e r s its default or various consistency restoration  techniques c a n be a p p l i e d to reject  some set  of  " o f f e n d i n g " beliefs. T h e u p d a t e p r o b l e m i n n o n - m o n o t o n i c theories is c o m p o u n d e d b y the fact t h a t inferences m a y h a v e been b a s e d o n the absence of w h a t is now being asserted. I n s u c h c i r c u m s t a n c e s , there m a y be n o i n c o n s i s t e n c i e s t o s i g n a l the necessity of belief r e v i s i o n . U n l e s s the a s s u m p t i o n s underl y i n g facts i n the knowledge-base c a n be e x a m i n e d for c o m p a t a b i l i t y w i t h u p d a t e s in the same w a y t h a t the facts t h e m s e l v e s are, n o t h i n g c a n prevent the knowledge-base f r o m b e i n g " c a t a p u l t e d " i n t o s e l f - s u p p o r t i n g - but otherwise u n j u s t i f e d - belief sets.  F o r e x a m p l e , the  default  theory:  leads t o the beliefs P a n d R. U n l e s s care is t a k e n , belief i n R m a y s u p p o r t belief i n P after Q is asserted, e v e n t h o u g h R w a s o r i g i n a l l y inferred because of a l a c k of belief i n Q. W o r k o n t r u t h m a i n t e n a n c e s y s t e m s [ D o y l e 1979;  D o y l e a n d L o n d o n 1980] has shed some l i g h t o n these p r o b -  lems. I n a r e l a t e d v e i n , there are issues of how k n o w l e d g e r e p r e s e n t a t i o n languages s h o u l d be d e s i g n e d t o address these issues. W o r k o n b o t h database theory a n d n o n - m o n o t o n i c i t y has t e n d e d t o d e a l w i t h tenseless languages, v i e w i n g the knowledge-base as a snap-shot of some state-ofaffairs. U p d a t e is seen as a n a t o m i c process of t r a n s f o r m i n g f r o m one snap-shot t o the next, w i t h the state of the k n o w l e d g e - b a s e defined o n l y before a n d after - not d u r i n g - the u p d a t e . work  Other  i n A l has e m b r a c e d t i m e - e i t h e r reservedly, b y a d o p t i n g " s i t u a t i o n s " o r " s t a t e s " a n d  " f l u e n t s " w h i c h t r a n s f o r m the w o r l d f r o m one state to a n o t h e r [ M c C a r t h y & H a y e s 1969; M o o r e 1979], o r w h o l e h e a r t e d l y , b y a d o p t i n g a f u l l - b l o w n t e m p o r a l logic [ M c D e r m o t t 1981; A l l e n 1984], or s o m e w h e r e i n b e t w e e n .  P e r h a p s the best w a y to d e a l w i t h n o n - m o n o t o n i c i t y is m o n o t o n i c a l l y ,  b y r e p r e s e n t i n g t h e state of a n a g e n t ' s beliefs at a p a r t i c u l a r t i m e .  CHAPTER  10  Conclusions  I d o n ' t u n d e r s t a n d it. I d o n ' t e v e n u n d e r s t a n d the people w h o u n d e r s t a n d it. — Queen J u l i a n a of T h e Netherlands  10.1.  Default Logic a n d Inheritance  W e p r e s e n t e d a correspondence b e t w e e n default theories a n d i n h e r i t a n c e n e t w o r k s exceptions,  analogous  to  that  outlined  exception-free i n h e r i t a n c e n e t w o r k s .  by  Hayes  [1977]  between  first-order  theories  with and  T h i s correspondence a l l o w e d us t o specify m i n i m u m correct-  ness c r i t e r i a for a n y i n h e r i t a n c e - d e t e r m i n i n g a l g o r i t h m , i d e n t i f y i n g the n o t i o n of correct inference w i t h t h a t of d e r i v a b i l i t y w i t h i n a single e x t e n s i o n of the corresponding default t h e o r y .  These cri-  t e r i a s h o w t h a t p r o p o s e d p a r a l l e l m a r k e r - p a s s i n g i m p l e m e n t a t i o n s of i n h e r i t a n c e n e t w o r k s w i t h e x c e p t i o n s are n o t feasible for g e n e r a l theories. C o r r e c t b e h a v i o u r w o u l d require t h a t severe ( a n d difficult to define) c o n s t r a i n t s be p l a c e d o n the s t r u c t u r e of the i n h e r i t a n c e n e t w o r k s t h e y c o u l d represent a n d reason w i t h . G i v e n a n o t i o n of correct inference, it became possible to q u e s t i o n w h e t h e r i n h e r i t a n c e netw o r k s w i t h e x c e p t i o n s are a l w a y s coherent, i n the sense of a l w a y s representing a reasonable set of beliefs.  I n h e r i t a n c e g r a p h s are t y p i c a l l y a c y c l i c . W e s h o w e d t h a t a c y c l i c n e t w o r k s are coherent  a n d , i n fact, t h a t w e a k e r c r i t e r i a are sufficient t o ensure coherence. T h i s l e d to a g e n e r a l i z a t i o n of the n o t i o n of a c y c l i c i t y w h i c h c a n be a p p l i e d to default theories, c a l l e d " o r d e r e d n e s s " .  The  o r d e r e d theories c o n s t i t u t e a n a t u r a l class of theories a l l of w h i c h have at least one e x t e n s i o n .  We  p r o v i d e d a n inference a l g o r i t h m for o r d e r e d i n h e r i t a n c e networks w i t h exceptions w h i c h is p r o v a b l y correct w i t h respect to t h i s concept of d e r i v a b i l i t y . O u r f o r m u l a t i o n suggests t h a t i t m a y n o t be possible to c o r r e c t l y realize m a s s i v e l y p a r a l l e l m a r k e r - p a s s i n g h a r d w a r e of the k i n d e n v i s a g e d b y N E T L w h i c h is a p p l i c a b l e to a r b i t r a r y i n h e r i tance graphs.  It appears t h a t the best t h a t c a n be a c h i e v e d for s u c h n e t w o r k s is a r e s t r i c t e d ,  q u a s i - p a r a l l e l inference a l g o r i t h m .  W e h a v e s k e t c h e d s u c h a n a l g o r i t h m , but h a v e s h o w n t h a t not  e v e r y set of c o n c l u s i o n s j u s t i f i e d b y the n e t w o r k is accessible t o it.  It r e m a i n s to be seen w h e t h e r  the l i m i t a t i o n s i m p o s e d b y the a l g o r i t h m are acceptable. F o r t u n a t e l y , these p e s s i m i s t i c o b s e r v a t i o n s d o n o t p r e c l u d e p a r a l l e l a r c h i t e c t u r e s for s u i t a b l y restricted n e t w o r k s .  W e have shown that  T o u r e t z k y ' s i n f e r e n t i a l d i s t a n c e a l g o r i t h m produces correct conclusions. T o u r e t z k y s h o w s h o w to r e s t r i c t a n e t w o r k so t h a t p a r a l l e l m a r k e r - p a s s i n g produces the same c o n c l u s i o n s as the i n f e r e n t i a l  - 102 -  - 103 distance algorithm.  W e c o n c l u d e t h a t , for such restricted n e t w o r k s , p a r a l l e l m a r k e r - p a s s i n g is  correct. W e h a v e s h o w n default logic t o be a useful t o o l for f o r m a l i z i n g the r e a s o n i n g processes i n v o l v e d i n A l systems.  S u c h a s p e c i f i c a t i o n provides a m e t h o d for e v a l u a t i n g correctness a n d a  m e t r i c b y w h i c h v a r i o u s a p p r o a c h e s c a n be m e a s u r e d a n d c o m p a r e d . A default logic s p e c i f i c a t i o n of a s y s t e m c a n p r o v i d e b o t h a m o r e c o m p l e t e v i s u a l i z a t i o n of h o w the s y s t e m performs a n d a guarantee t h a t t h a t p e r f o r m a n c e is coherent. n u m b e r o f results o n d e f a u l t logic. default  T o f a c i l i t a t e s u c h a p p l i c a t i o n s , we h a v e presented a  T h e s e i n c l u d e a s e m a n t i c s for a r b i t r a r y  single-justification  theories, a c h a r a c t e r i z a t i o n of a large class of theories for w h i c h coherent r e a s o n i n g is  a l w a y s possible (i.e., theories w h i c h a l w a y s have at least one e x t e n s i o n ) , a n d a t o t a l l y correct inference a l g o r i t h m for a subclass o f these theories. It m i g h t be — a n d has been — a r g u e d t h a t a d e c l a r a t i v e f o r m a l i s m s u c h as default logic is i n a d e q u a t e for the tasks of k n o w l e d g e r e p r e s e n t a t i o n a n d reasoning. W h i l e we c l e a r l y disagree w i t h this p o s i t i o n , we expect default logic to be useful even to " p r o c e d u r a l i s t s " .  E v e n i f some  s y s t e m were f u n d a m e n t a l l y m o r e t h a n the s u m of its d e c l a r a t i v e content, default l o g i c c o u l d be used to f o r m a l i z e t h a t d e c l a r a t i v e c o n t e n t . T h e n o n - d e c l a r a t i v e " c o n t r o l " i n f o r m a t i o n c o u l d t h e n be t r e a t e d as a n inference a l g o r i t h m for the r e s u l t i n g default theory. T h e correctness of the s y s t e m w o u l d be d e t e r m i n e d b y w h e t h e r this inference a l g o r i t h m was correct w i t h respect t o the proof t h e o r y of d e f a u l t l o g i c . D e f a u l t s , i n one f o r m or a n o t h e r ,  are e x t r e m e l y  common in A l .  R e i t e r [1978b,  1980a]  discusses a w i d e v a r i e t y of c o m m o n s i t u a t i o n s to w h i c h t h e y c a n be a p p l i e d , i n c l u d i n g s e v e r a l A l k n o w l e d g e r e p r e s e n t a t i o n schemes. M a n y of these m a y be a m e n a b l e t o a n a l y s i s u s i n g a n a p p r o a c h s i m i l a r to t h a t w h i c h we have used for i n h e r i t a n c e n e t w o r k s .  If some are not, t w o possibilities  arise: the features not so a m e n a b l e m a y p r o v e i n c o r r e c t o r inessential, or t h e y m a y p o i n t  out  s h o r t c o m i n g s of d e f a u l t l o g i c . E i t h e r result w o u l d raise interesting questions.  10.2. Predicate Circumscription Although together  a model-theory  w i t h an attendant  for  predicate  circumscription  has been a v a i l a b l e  since 1980;  soundness r e s u l t , v e r y l i t t l e was k n o w n a b o u t the strengths  and  weaknesses of p r e d i c a t e c i r c u m s c r i p t i o n u n t i l r e c e n t l y .  W e e x p l o r e d the c o n s t r a i n t s i m p o s e d b y  circumscription's  find  model-theory  a n d were s u r p r i s e d to  them very rigid indeed.  e x p e c t a t i o n s for p r e d i c a t e c i r c u m s c r i p t i o n h a d been v e r y h i g h ;  Previous  e x a m p l e s i n the l i t e r a t u r e h a d  p u s h e d the t e c h n i q u e b e y o n d the safety of its s e m a n t i c justifications, a n d this fact h a d gone u n n o ticed. P r e d i c a t e c i r c u m s c r i p t i o n ( a n d f o r m u l a c i r c u m s c r i p t i o n ) c a n lead to inconsistent conjectures w h e n a p p l i e d t o theories w i t h o u t m i n i m a l models. In retrospect, this is not s u r p r i s i n g , b u t it does n o t a p p e a r t o h a v e o c c u r r e d to a n y o n e u n t i l we d i s c o v e r e d a n e x a m p l e . T h i s is p e r h a p s a t t r i b u t a b l e t o the s c h e m a t i c n a t u r e o f p r e d i c a t e c i r c u m s c r i p t i o n .  N o t e v e r y s u b s t i t u t i o n produces i n c o n -  s i s t e n c y , so unless a n i n c o n s i s t e n t s u b s t i t u t i o n is d i s c o v e r e d , c i r c u m s c r i p t i o n o f theories w i t h o u t  - 104 -  m i n i m a l m o d e l s m a y a p p e a r s i m p l y ineffectual.  T h e existence of theories w i t h inconsistent cir-  c u m s c r i p t i o n s suggests t h a t one must be c a r e f u l t o c i r c u m s c r i b e o n l y those theories w i t h m i n i m a l m o d e l s . A l a s , i t is u n d e c i d a b l e w h i c h theories h a v e m i n i m a l models. W e h a v e c h a r a c t e r i z e d a class o f theories, w h i c h we c a l l well-founded, w h i c h a l w a y s have m i n i m a l m o d e l s . W e then e x p l o r e d the properties of predicate c i r c u m s c r i p t i o n vis-a-vis these w e l l - f o u n d e d theories. W e d i s c o v e r e d t h a t the s e m a n t i c c h a r a c t e r i z a t i o n of p r e d i c a t e c i r c u m s c r i p t i o n - so i n t u i t i v e l y a p p e a l i n g o n the surface - r i g i d l y c o n s t r a i n e d the effectiveness of c i r c u m s c r i p t i o n i n c o n j e c t u r i n g n e w g r o u n d facts. T h e o n l y g r o u n d facts w h i c h predicate c i r c u m s c r i p t i o n c a n c o n j e c t u r e are n e g a t i v e instances of one of the predicates being c i r c u m s c r i b e d - a n d t h e n o n l y insofar as s u c h conjectures p r o v i d e n o new i n f o r m a t i o n about the extensions of n o n - c i r c u m s c r i b e d predicates.  F u r t h e r m o r e , the e q u a l i t y p r e d i c a t e is somehow resistant t o p r e d i c a t e ( a n d formula)  circumscription.  Generalizations of Circumscription  10.3.  T h e success of o u r m o d e l - t h e o r e t i c i n v e s t i g a t i o n s i n t o predicate c i r c u m s c r i p t i o n (pessimistic t h o u g h the results were) suggested t h a t a s i m i l a r e x p l o r a t i o n of the v a r i o u s g e n e r a l i z e d forms of c i r c u m s c r i p t i o n m i g h t also p r o v e w o r t h w h i l e .  M c C a r t h y [1986] d i d not p r o v i d e a m o d e l - t h e o r y  for f o r m u l a c i r c u m s c r i p t i o n , however. T h e first task for this i n v e s t i g a t i o n , thus, w a s t o d e v e l o p a model-theory. tion,  T h e m o d e l - t h e o r y presented is a g e n e r a l i z a t i o n of t h a t of p r e d i c a t e c i r c u m s c r i p -  w i t h appropriate  changes to  accomodate  the  introduction  of v a r i a b l e  predicates.  The  m i n i m i z a t i o n of expressions, rather t h a n p r e d i c a t e s , also forces m o d i f i c a t i o n s to the definitions of submodel and m i n i m a l model.  T h e soundness ( a n d , for c e r t a i n classes of theories, completeness)  of f o r m u l a c i r c u m s c r i p t i o n w i t h respect to this m o d e l - t h e o r y has been p r o v e n . U n i v e r s a l theories a l w a y s h a v e m i n i m a l m o d e l s regardless of the predicates v a r i e d o r m i n i m ized.  F o r these theories, the c o n s i s t e n c y of generalized c i r c u m s c r i p t i o n is assured.  In fact, the  proof shows t h a t e v e r y m o d e l of a u n i v e r s a l theory has at least one m i n i m a l s u b m o d e l .  As a  c o r o l l a r y of t h i s , g e n e r a l i z e d c i r c u m s c r i p t i o n of u n i v e r s a l theories does not affect the extensions of a n y predicates not d e s i g n a t e d as v a r i a b l e .  F o r such theories, the repercussions of c i r c u m s c r i p t i o n  do not e x t e n d b e y o n d those predicates e x p l i c i t l y i n d i c a t e d as liable to change. L i f s c h i t z [1984, 1985a,b] has d e v e l o p e d extensions to c i r c u m s c r i p t i o n a l l o w i n g constants a n d f u n c t i o n s t o be t r e a t e d as v a r i a b l e s d u r i n g the m i n i m i z a t i o n process! a n d a l l o w i n g a r b i t r a r y preorders t o be specified; m i n i m i z a t i o n proceeds a c c o r d i n g t o this pre-order. to  the  generalized circumscription  model-theory,  Suitable modifications  w h i c h a c c o m m o d a t e these extensions, were  presented. L i f s c h i t z ' i n n o v a t i o n s were s h o w n t o be sound w i t h respect t o this m o d e l theory.  We  e x a m i n e d the effects of some of these f o r m u l a t i o n s on the existence of m i n i m a l m o d e l s , o n c o n s i s t e n c y , a n d o n the t y p e s of conjectures w h i c h c a n be o b t a i n e d .  - 105 10.4. Domain Circumscription McCarthy cumscription.  [1980] c l a i m s t h a t d o m a i n c i r c u m s c r i p t i o n is a s p e c i a l case of predicate  cir-  W e s h o w e d t h a t the d e m o n s t r a t i o n a c t u a l l y rests o n a s t r e n g t h e n e d f o r m of p r e d i -  cate c i r c u m s c r i p t i o n w h i c h does not a l w a y s preserve consistency, e v e n for theories w i t h m i n i m a l models.  W e s h o w e d t h a t none of p r e d i c a t e c i r c u m s c r i p t i o n , f o r m u l a c i r c u m s c r i p t i o n , o r general-  i z e d c i r c u m s c r i p t i o n w i t h o u t v a r i a b l e terms supercedes d o m a i n c i r c u m s c r i p t i o n , i n general.  We  c o n j e c t u r e d t h a t e v e n v a r i a b l e terms are u n l i k e l y to suffice t o m a k e g e n e r a l i z e d c i r c u m s c r i p t i o n subsume d o m a i n circumscription. In fact, the d o m a i n c i r c u m s c r i p t i o n s c h e m a presented b y M c C a r t h y [1980] a n d D a v i s [1980] is also too s t r o n g . C e r t a i n theories w i t h m i n i m a l models t u r n out to h a v e inconsistent d o m a i n circumscriptions.  After  isolating  the  p r o b l e m , we o u t l i n e d  a straightforward  correction  which  preserves the a p p e a l i n g s e m a n t i c c h a r a c t e r i z a t i o n presented b y D a v i s [1980], a n d p r o v e d  its  correctness. W e h a v e also n o t e d the a m b i g u i t y of the do m a i n - c l o s u r e a s s u m p t i o n , as i t is u s u a l l y s t a t e d . W e argue t h a t the m o s t c o m m o n d i s a m b i g u a t i o n agrees w i t h the results o b t a i n e d f r o m d o m a i n circumscription.  A l s o , we c o n j e c t u r e d t h a t the completeness of d o m a i n c i r c u m s c r i p t i o n for c e r t a i n  classes of theories m i g h t be p r o v a b l e .  10.5. Relations Between Circumscription and Default Logic W e h a v e c o n s i d e r e d the r e l a t i o n s h i p between default logic a n d c i r c u m s c r i p t i o n . t h a t , i n some cases, the c l o s e d - w o r l d default coincides w i t h c i r c u m s c r i p t i o n ; l a r l y useless w a y , default  logic subsumes c i r c u m s c r i p t i o n ;  W e showed  that, in a particu-  a n d t h a t default logic is c a p a b l e of  affecting the e q u a l i t y t h e o r y w h i l e p r e d i c a t e , f o r m u l a , a n d d o m a i n c i r c u m s c r i p t i o n are not. W e s h o w e d t h a t the i n t r o d u c t i o n of fixed predicates a n d a p p l i c a t i o n s t o o p e n d o m a i n s each p r o v i d e c i r c u m s c r i p t i o n w i t h c a p a b i l i t i e s not a v a i l a b l e using s i m p l e c l o s e d - w o r l d default theories. 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[1983], " M u l t i p l e I n h e r i t a n c e a n d E x c e p t i o n s " , U n p u b l i s h e d M a n u s c r i p t , D e p a r t m e n t of C o m p u t e r S c i e n c e , C a r n e g i e - M e l l o n U n i v e r s i t y . T o u r e t z k y , D . S . [1984aJ, The Mathematics Science, C a r n e g i e - M e l l o n University.  of Inheritance Systems, P h D T h e s i s , D e p t of C o m p u t e r  T o u r e t z k y , D . [1984b], " I m p l i c i t o r d e r i n g o f defaults i n inheritance s y s t e m s " , Proc. Amer. Assoc. for Artificial Intelligences^T o u r e t z k y , D . [1985], " I n h e r i t a b l e relations: a l o g i c a l extension to i n h e r i t a n c e h i e r a r c h i e s " , Proc. Theoretical Approaches to Natural Language Understanding, H a l i f a x , 28-30 M a y 1985, p p 5 5 60. W i l k i n s , M . W . [1983], " P a r t i a l I n f o r m a t i o n a n d A l t e r n a t i v e University, Stanford, C A . W i n o g r a d , T . [1980], " E x t e n d e d Inference ( A p r i l 1980), N o r t h - H o l l a n d , p p 5-26.  Sets", Technical Report,  Stanford  M o d e s i n R e a s o n i n g " , Artificial Intelligence 1S(1,2),  W o o d s , W . A . [1975], " W h a t ' s I n A L i n k ? " , Representation and Understanding, A c a d e m i c P r e s s , p p 35-82.  APPENDIX A Proofs of Theorems  Background  Information  T h e r e are a few d e f i n i t i o n s a n d results d u e t o R e i t e r [1980a] o n w h i c h w e d r a w freely i n the f o l l o w i n g proofs. W e r e p r o d u c e t h e m here f o r the reader's c o n v e n i e n c e .  1) T h e o r e m 0.1  [Reiter 1980a, T h e o r e m 2.1] oo  E is a n e x t e n s i o n f o r A = ( D , W ) i f a n d o n l y i f E =  U E ; , where i=0  E  0  = W , and for i > 0  E  i + 1  = T h ( E j ) U {w |; SLlL w  e  D , a e E , a n d ->B £ E }  1  t  2) T h e Generating/ Defaults for E w i t h respect t o A are defined as: G D ( E , A ) = {.2-L£  €  D I a € E , -.0 £ E }  CO  3) If D is a set o f d e f a u l t s , t h e n CONSEQUENTS ID) = {w I  CONSEQUENTS  (D) is defined, as one w o u l d e x p e c t , as:  e D} w  4) ; T h e o r e m 0.2  [Reiter 1980a, T h e o r e m 2.5]  If E is a n e x t e n s i o n for A = ( D , W ) , t h e n E = T h ( WU  CONSEQUENTS(GD(E,A))).  5) T h e o r e m 0.3  [Reiter 1980a, C o r o l l a r y 2.2]  If E is a n e x t e n s i o n f o r A = ( D , W ) , t h e n E is consistent if a n d o n l y if W is. In t h e proofs of results f r o m chapters 3 a n d 4, we w i l l u s u a l l y assume t h a t f o r m u l a e are i n c l a u s a l f o r m : i.e., expressed as a c o n j u n c t i o n o f d i s j u n c t i o n s of literals. W e define the f u n c t i o n s CLA USES (•) a n d LITERALS If P =  V ... V B )  CLAUSES(6) LITERALS  Um  = {(Ai (B) = {A,j  (•) as follows: then  A ... A ( A ^ V ... V B^nJ V ... V B ) Um  | 1 < i < m }  I 1 < i < m, 1 < j <  m  i  }  A b u s i n g t h e n o t a t i o n s o m e w h a t w e s o m e t i m e s use CLAUSES 1  Note the explicit reference to E in the definition of E i  - Ill -  + 1  .  (T), where T is a set of f o r m u l a e , t o  - 112 -  refer t o  U  CLAUSES(7).  76T  W e w i l l define o t h e r n o t a t i o n as i t is r e q u i r e d .  Definition: Satisfiability, admissibility, and applicability  L e t X be a set of m o d e l s ; T a set o f f o r m u l a e ; a, 0, a n d w f o r m u l a e , a n d 5 =  —  a  default. T h e n i)  a is X-satisfiable (X-valid) iff ] x £ X . x [= «  ii)  r is X-admissible (X permits T) iff V 7 € l \ 3  ( V x € X . x j= a ) X  € X . x^=7  iii) 6 is X-applicable iff a is X v a l i d a n d /? is X - s a t i s f i a b l e .  |  Definition: Result of a default L e t X , T , a n d 6 be as above. T h e n the result of S i n ( X , T) is: f  ( X , T) i f 6 is n o t X - a p p l i c a b l e a n d T is X - a d m i s s i b l e , 6{X, F ) =  { ( ( X - { N I N }= -.w}), ( r U { £ } ) ) if £ is X - a p p l i c a b l e a n d T is X - a d m i s s i b l e , a n d I  otherwise.  |  Definition: Result of a sequence of defaults L e t X a n d T be as a b o v e , a n d let <S(> be a sequence of defaults. « * i > ( X , r ) = ( n X , U T O where  T = T;  (X =X;  ;  0  \(x  i  +  1  ,  0  r  i  +  1  ) = ^(x, r j ,  Then and i > o .  I  Definition: Stability L e t Y be a n o n - e m p t y theory.  set of m o d e l s , T a set of f o r m u l a e , a n d A = ( D , W ) a d e f a u l t  T h e n ( Y , T ) is stable for A iff  (1)  ( Y , T ) = < $ i > ( X , { }) for X = { M I M j= W } , a n d some  (2)  V * € D . 6(Y, T ) = ( Y , T ) ,  and  C D ,  - 113 -  (3)  T is F - a d m i s s i b l e .  |  T h e o r e m 3.1 — S o u n d n e s s  If E is a n e x t e n s i o n f o r A , t h e n there is some set T such t h a t ( { M | M j = E}, T ) is stable f o r A .  Proof  Define  GD=  j<5 = "  :  8  G D | a € E, - . ; 9 ^ E J .  From  theorem  0.2,  we  have  E = T h ( W U G D ) . T h e r e are 2 cases: G D = { }: C l e a r l y < > ( X , { }) = ( Y , { }). C o n s i d e r 6 =  Then E = Th(W).  SLlL  G  D . If a is Y - v a l i d  w a n d 8 is Y - s a t i s f i a b l e , t h e n E )— a, E \f—<B, so S G G D , w h i c h i s a c o n t r a d i c t i o n .  Hence  5 ( X , { }) = ( Y , { } ) . C l e a r l y { } is Y - a d m i s s i b l e . H e n c e ( Y , { }) i s stable w i t h respect t o A . G D jfe { }: L e t { 6 . . } be a n y o r d e r i n g o f G D . D e f i n e S'i b y S'i = 6 where j is t h e s m a l l e s t integer s u c h l v  )t  t h a t 5j is <S'o...6'i_i>{X,{ Inapplicable, a n d Sj £ {<^'o)---j^'i-i}j where 0 < i < n . It  c a n easily  b e seen  that  this  i s well-defined,  a n d uses  a l l of  T = J U S T I F I C A T I O N S ^ ' ; } ) , t h e n S G D i m p l i e s 6{Y,T) = ( Y , r ) . <6\>(X,{ }) = ( Y , r ) .  <S{>.  It r e m a i n s  O b v i o u s l y , if t o show  that  I t i s easily p r o v e d t h a t <S'o...S'i>(X,{ }) = ( X ^ i ) , w h e r e X i s the set of ;  a l l m o d e l s f o r T h ( W U {w'o,...,u'i}). - where the w ' i ' s are the consequents of t h e r e s p e c t i v e 6 ' i ' s and r , = JUSTIFICATIONS({5'o,...,5'i}). H e n c e < 5 ' i > ( X , { })  = ( { M | M }= ( T h ( W U G D ) ) } , J U S T I F I C A T I O N S ( G D ) ) = ( { M | M J= E}, J U S T I F I C A T I O N S ( G D ) ) = (Y, JUSTIFICATIONS(GD)) .  C l e a r l y J U S T I F I C A T I O N S ( G D ) ) is Y - a d m i s s i b l e . H e n c e ( Y , r ) is stable f o r A .  Q E D T h e o r e m 3.1  T h e o r e m 3.2 - C o m p l e t e n e s s  If ( Y , T ) i s s t a b l e f o r A t h e n Y is t h e set o f m o d e l s f o r some e x t e n s i o n o f A . (I.e., {w | V y G Y . y | = w} is a n e x t e n s i o n f o r A . )  - 114 Proof S i n c e ( Y , r ) is stable, ( Y , r ) = < ^ > ( X , { }) where X = { M | M \= W } a n d {6J C D . W i t h o u t loss of  generality,  let  <^>  be i n f i n i t e .  f X o . r o ) = ( X , { }), a n d f o r i > 0, ( X  (If  ,r  i + 1  i + 1  finite,  replicate  ) = ^(X^).  5„).  Define  Then Y =  (X^I?;)  as  D X ; ,andV =  follows:  U  a : 0 S i n c e ( Y , T ) is stable, f o r a n y default, S = — € E D , either S is n o t Y - a p p l i c a b l e , o r w is Y u>  f  v a l i d a n d B S T. I n e i t h e r event, T is Y - a d m i s s i b l e .  i  a  :  A  Assume 6 = . L e t F be the set o f X j - v a l i d f o r m u l a e . W e show t h a t F = T h ( W ) a n d t h a t if a e F , a n d -.ft £ Fi» t h e n F = T h ( F i U {w }). O t h e r w i s e F = F . ;  {  ;  0  ;  i  +  1  ;  i  T h i s is t r i v i a l f o r Fo- A s s u m e i t is true f o r F ; , a n d consider F ; . + 1  is X p a d m i s s i b l e . X  i +  If a ; S F ; , t h e n a  i = X ; - { N | N \= ^ w ; } , a n d F  Let E =  is X v a l i d .  ;  r  +  1  S i n c e V is Y - a d m i s s i b l e , each T  If ->A ^ F;, then A is Xj-satisfiable.  = T h ( F i U ( w j ) . Otherwise X  i + 1  ;  i  +  1  = X ; , so F  i  +  1  ;  Hence  = F . ;  U F . C l e a r l y Y = { M | M \= E } . It r e m a i n s to show t h a t E is a n e x t e n s i o n for A . ;  i=l  Define E = W , a n d E 0  i + 1  = T h ( E i ) U {w |  e D . a e E ^ ^ E } .  W e show E =  w  U Ei C E =  UE . ;  i=0  U F : ;  C l e a r l y E C F C E . A s s u m e E C E , a n d consider w € E 0  0  ;  i + 1  . T r i v i a l l y , if w e T h ( E j ) , w e E .  a • B O t h e r w i s e , there is a default, 5 = —  €  D , s u c h t h a t a € E ; a n d ->0 & E . S i n c e a € E , a n d ;  E i C E , a is Y - v a l i d . S i m i l a r l y , 0 is Y - s a t i s f i a b l e . CO  E =  00  U F; C i=0  B y ( f ) , w is Y - v a l i d , so u 6 E .  UE : ;  i=0 00  00  00  C l e a r l y F = T h ( W ) C E C U E . A s s u m e F ; C U E a n d consider F i=0 i=0 0  x  ;  ;  i + 1  . Since  00 5  ;  i  F-„ whence w €  i=o  ;  U E;.  i=o  F j t h e n S{ is X p a p p l i c a b l e .  S i n c e ( Y , r ) is stable, T is Y - a d m i s s i b l e . 00  {A} £  ;  00  closed a n d F C U & „ it suffices t o show t h a t a-, € F , a n d ->0  If a ; S F j , a n d —<0  U E is fc=0  C T , so E \/~ -i0„ SO --A £ E . a £ F ; C U E i , so a ; € E j , f o r some j . {  i=0  T h u s E i s a n e x t e n s i o n f o r A , b y T h e o r e m 0.1.  But  - 115 -  T h e o r e m 3.2  QED  Lemma  3.3.1  If E ' (i > 0) is a n e x t e n s i o n for the default t h e o r y A = (D;, E ' ) a n d E _ 1  ;  _  1  = W , t h e n the  f o l l o w i n g are e q u i v a l e n t : (1) a S E  1  (2) E'f-ot (3)  (W U  U CONSEQUENTS(GD(E , r  A ))) j - a r  r=0  Proof  (1)  a € E' <=•  E f- a !  T h i s follows f r o m t h e fact t h a t E is a n extension a n d thus l o g i c a l l y closed. 1  (2)  E f— a  <=>  1  (W U  U CONSEQUENTS(GT>(E ,  A ))) f - a  T  r  r=0  If E is a n e x t e n s i o n for A , t h e n b y T h e o r e m 0.2 we k n o w t h a t E = T h ( W U CONSEQUENTS(GD(E, Hence E ' = T h ( E  w  A))).  U CONSEQUENTS(GD(E',  AJ))  = ^ ( T h f E ^ U CONSEQUENTS(GDfE " , 1  A )))  1  M  U CONSEQUENTS(GD(E ,  A )))  i  ;  = T h ( T h . . . ( W U CONSEQUENTS(GD(E°,  A ))> 0  U ... U CONSEQUENTS(GD(E\  Since T h ( T h ( A ) E  1  U  = Th(W U  B) = T h ( A U B), U CONSEQUENTS(GD(E , r  A ))) .  r=0  F r o m t h i s , the result follows b y the d e f i n i t i o n of T h .  QED  Lemma  3.3.1  AJ))  r  - 116 -  D e f i n i t i o n 3.3.2:  «:  and «C  L e t A = ( D , W ) be a closed, s e m i - n o r m a l d e f a u l t theory. a l l f o r m u l a e are i n c l a u s a l f o r m . T h e p a r t i a l relations,  W i t h o u t loss of g e n e r a l i t y , assume  < C a n d < C , o n Literals  X Literals, are  defined as f o l l o w s : (1)  If a € W t h e n a = ( a V ... V a j , f o r some n > 1 . x  F o r a l l a ; , ctj G {cti,...,<*„}, if a j= ctj , let ->aj <C a j . ;  (Since: ( « ! V . . . V a j = [ ( - . a j A... A - a j - i A -<Qr (2)  If £ G D t h e n £ = —'—— P  -.  Let a  it  ... a  T  j+1  - x * „ ) 3 « j ])  A-A  , p\, ... 0 , a n d f S  lt  ...  7  t  be the l i t e r a l s of the  c l a u s a l f o r m s of a, 0, a n d 7, r e s p e c t i v e l y . T h e n (i)  If a ; G {a ...,a } a n d 0; G {/?!,...,&} let a, ^  (ii)  If T, G {7i,-.7t}, Pj 6 { & > - > & } a n d  u  t  (iii) A l s o , 0 = 0i A ••• A  7 i  0 . }  £ { & , . . . , & } let - ,  7 i  V ... V 0-^  }  , where m ; > 1 .  T h u s if Aj , A,k G { ^ , 1 , - , ^ ^ } a n d Aj £ A,k I<* -Aj T h e e x p e c t e d t r a n s i t i v i t y r e l a t i o n s h i p s h o l d for <SC a n d < C . (i)  If a «  y9 a n d 0 «  7 then a ^  (ii)  II a «  0 and 0  1 then a <3C 7.  (iii) If a < C 0 a n d £ «  0- .  An j for some m > 1.  F o r e a c h i < m , A = (A,i  (3)  «  7 or a  « ft . k  JT.e.,  7.  ^ a n d /? <SC 7 t h e n a <C 7.  |  Definition 3.3.3: Orderedness  A s e m i - n o r m a l d e f a u l t t h e o r y is said to be ordered iff there is n o l i t e r a l , cx, s u c h t h a t a < C a .  |  D e f i n i t i o n 3.3.4: U n i v e r s e o f A  F o r a c l o s e d , s e m i - n o r m a l default t h e o r y , A = ( D , W ) , define the Universe of A , U ( A ) , as follows: U(A)  = {a I a G Literals a n d [ 3f. [ ( a V £) G CLAUSES(W  U  or [ ( - . a V £) G CLAUSES^ U {a  ;  I 3 a , A7.  U {-7i I 3 a , A T  G D a n d a G LITERALS t  a  '^  A  7  G D and  7 i  (a) }  G LITERALS  (7) }  CONSEQUENTS(D))] U COWStfQHENrSfD))])}  - 117 -  O b s e r v e t h a t £ m a y be the n u l l clause.  D e f i n i t i o i i 3 . 3 . 5 : / : U ( A ) (-•  |  N  F o r a c l o s e d , o r d e r e d , s e m i - n o r m a l default  theory, A = ( D , W ) , we define the f u n c t i o n  / : U ( A ) |-» N , as f o l l o w s : If a,0 G U ( A ) a n d a «  0 t h e n I (a) < I (0). If a <K 0 t h e n I (0) > I {oc) + l.  If 0 G U ( A ) a n d for n o a € U ( A ) is ( a <SC 0) o r ( a «  0) t h e n / (0) = 0.  If n G N , 0 G U ( A ) , a n d f (/9) > n t h e n 3 « G U ( A ) . ( a «  S i n c e A is o r d e r e d , / is w e l l defined.  p") a n d / (a) = n.  O b s e r v e t h a t / is a t o t a l f u n c t i o n o n U ( A ) w h i c h assigns a  n a t u r a l n u m b e r to e a c h l i t e r a l i n U ( A ) . 1(a) m a y be t h o u g h t of as the l e n g t h o f the longest c h a i n o f s e m i - n o r m a l defaults w h i c h c o u l d figure i n a n inference of a.  D e f i n i t i o i i 3.3.6:  |  /MAX. ' M I N  If 0 is a closed f o r m u l a , a n d the c l a u s a l f o r m of 0 is  V ... V 0hm) A ... A t h e n define  (0 i V ... m  V  / M A X ( 0 = M A X ( / (AJ) W  $  = M I N ( / (/Jy)) .  |  L e m m a 3.3.7  If A = ( D , W ) is a n o r d e r e d , closed, s e m i - n o r m a l default t h e o r y , t h e n there is a p a r t i t i o n , { D j } , for D i n d u c e d b y : Vf 6 DJ  =  a  -  PJ\1  a  n  d  P  1^(0)  = i iff  s  e  D . ;  Proof C l e a r l y LITERALS  (CONSEQUENTS  ({5 G D } ) ) C U ( A ) , a n d / is t o t a l o n U ( A ) .  - 118 -  Therefore:  1)  Vtf G D . Vi. Vj. (S G D  2)  V<5GD. 3 1 ( 5 e D J .  A 6 G Dj) i m p l i e s i = j .  {  Q E D L e m m a 3.3.7  C o r o l l a r y 3.3.8  If S G D o , t h e n 5 is a n o r m a l d e f a u l t .  Proof  If 5 =  Q  :  *'A 7 P  e  D  q  t h e n /  m  m  (  ^  >  / m a x (  ^) > o .  Q E D C o r o l l a r y 3.3.8  C o r o l l a r y 3.3.9  If i > 0 a n d D j j= { }, there is at least one n o n - n o r m a l (i.e., s e m i - n o r m a l ) d e f a u l t i n D . ;  Proof If  D  ;  contains  only  lhm(CONSEQUENTS(D^)  normal  defaults,  then  the  minimality  of  /  guarantees  that  < i, w h i c h is a c o n t r a d i c t i o n .  Q E D C o r o l l a r y 3.3.9  L e m m a 3.3.10 If T is c o n s i s t e n t , if / M A X H ) < J . 3  a  n  d  if / (7) is defined for a l l 7 € LITERALS  (T), t h e n  there is a l i n e a r r e s o l u t i o n r e f u t a t i o n of 8 f r o m V if a n d o n l y if there is a l i n e a r r e s o l u t i o n refutation of 8 from Proof  w h e r e * C T a n d rp G *  iff / M I N ( ^ ) < J-  - 119 -  (-0 T h e p r o o f is b y c o n s t r u c t i o n of s u c h a r e f u t a t i o n . S i n c e T is c o n s i s t e n t , i f there is a r e f u t a t i o n of B f r o m T , there is a r e f u t a t i o n w i t h t o p clause i n C L A U S E S (8).  I.e., Ro  ^ C  0  A  A K-k+l = D and R  0  e CLAUSES  (B), C  0  € V.  W e p r o c e e d b y i n d u c t i o n on the steps i n the r e f u t a t i o n , base A s s u m e 8 is i n c l a u s a l f o r m , i.e., 8 = B A ... A B r  (;->Ar) <  B y hypothesis, / , that C so / ( C  0 i l  0  and  n  , a n d t h a t C Q ^ resolves on  ) < j a n d / M I N ( C O ) < j • It f o l l o w s t h a t C  Vs. / (-,R  1<8  -.Co,; «  0  € *  C Q I , / (-.Co,;) < / ( C o , i ) <  A,i  t  x  A,  nil  => A i , i  Thus C Q !  produce R .  o  ••• V  v  0  0mo  Since for i > l ,  A = A,i  W i t h o u t loss of g e n e r a l i t y , assume t h a t R . =  J-  = C Q ! V ... V C  A = A,i V ... V A^ , for i = l,...,n .  .  j .  T h u s , if R  x  = R  M  V ... V R  l  t  then  ) < j .  step Assume C  r  that  R .= R ;  £ {R^.-.R^J.  {1  V ... V R ^  that  Vs. / (-.R^J  <  j,  and  that  Vr<i.  C o n s i d e r the r e s o l u t i o n of R[ w i t h C ; . C ; = C ^ V ... V C . x  or Q e { R o , . . . , R i } .  x  Hence f ( 0 ^ ) = / ( - . R ^ J  <  C  R  e *  or  . W i t h o u t loss  Um  of g e n e r a l i t y , assume C ^ = - i R ^ . C; e *  ,  j a n d so /MINCCO < j •  So  For r > l , / ( - C J < / ( C ) < j . Thus Vs. / (-R i, ) < j . M  B y i n d u c t i o n , f o r e v e r y clause, C ; , i n the r e f u t a t i o n of 8, C ; S  i +  s  o r C ; is a descendent of ^ U  T h u s , there is a l i n e a r r e s o l u t i o n r e f u t a t i o n of 8 f r o m "9.  (H T r i v i a l : S i n c e $ C f , t h e r e f u t a t i o n f r o m iff serves as a r e f u t a t i o n f r o m V.  {Al-  120 -  Q E D l e m m a 3.3.10  T h e o r e m 3.3 — C o h e r e n c e  If A = ( D , W ) is a n o r d e r e d , s e m i - n o r m a l default theory, then A has a n e x t e n s i o n .  Proof  If W is i n c o n s i s t e n t , t h e n A has the t r i v i a l extension, L.  Hence assume W is consistent.  W e p r o c e e d b y c o n s t r u c t i n g a n e x t e n s i o n , E for A . F i r s t , let {D;} be a p a r t i t i o n of D i n d u c e d b y R e c a l l t h a t b y C o r o l l a r y 3.3.8, i f 8 € D  / , as d e s c r i b e d i n L e m m a 3.3.7.  0  t h e n 8 is a n o r m a l  d e f a u l t , a n d t h a t b y C o r o l l a r y 3.3.9, for i > 0, D; m u s t c o n t a i n at least one s e m i - n o r m a l default, say 8  and  —  / AX(-'7) M  <  ,  'MTN(^)-  W e n o w c o n s t r u c t a n e x t e n s i o n for A .  Let A  = (Do, W ) .  0  Since A  0  is a n o r m a l default theory a n d W is consistent, A  0  has a consistent  e x t e n s i o n , say E ° .  F o r i > 0, c o n s t r u c t A ; as follows:  \ 2-^- e D; V SLlIJSSL  D ^ i ^ l A  D , -.7 £ E } i_1  €  S  (D^E- ) 1  l  =  Where E  1 _ 1  is a n e x t e n s i o n for A  w  .  S i n c e each A ; is a n o r m a l default t h e o r y , e a c h A  ;  has at least  oo  one e x t e n s i o n , E \  Let E =  U E ' . S i n c e W is consistent, so is E°, b y T h e o r e m 0.3. S i n c e E is a n 1  i=0  e x t e n s i o n f o r ( D / , E^ ), E ' is consistent i f E 1  1 _ 1  is, a n d E  1 - 1  C E . B y i n d u c t i o n E is consistent. 1  We  oo  n o w s h o w t h a t E is a n e x t e n s i o n for A . B y T h e o r e m 0.1, it is sufficient to show t h a t E =  U F; , i=0  - 121 -  where F  0  = W , a n d for  F  i + 1  i>0  = T h ( F j ) U {w  | SLLL  D , a G F , a n d -i/3 <£ E } .  e  ;  w oo  (1)  W e first s h o w t h a t  U F C E. ;  i=0  a) F  0  = W C E° C E.  b) A s s u m e F C E . W e show t h a t F ;  F i)  i + 1  = Th(Fj U  {0  SLLiASL  |  C E.  i + 1  g D , a 6 F j , (-•/? V -vy) $ E }  S i n c e F C E a n d E is l o g i c a l l y closed, T h ( F i ) C E . ;  ii) C o n s i d e r 0 G {0  \  ^  a  A  G D , a G F , {-.0 V  1  ;  S i n c e a G F j , a G E , a n d hence a G E for some j 1  S i n c e (-.0 V - . ) £ E , -.7 £ E ' " , so  G D/.  1  7  B u t - , / ? £ E , so -./? £ E* T h e r e f o r e , since E is a n e x t e n s i o n for A j = ( D / , E *)) a n d a G E , 0 G E . j  j  j  j  T h e r e f o r e j3 6 E . OO  B y induction,  U F C E. £  00  (2)  F i n a l l y , we s h o w t h a t E C  U F . ;  i=0  A) Consider w G E°.  E ° is a n e x t e n s i o n for A , so b y T h e o r e m 0.1 E ° = 0  U G , where ;  i=0  G  0  = W , a n d for  G  i + 1  i>0  a :w w  = Th(G0 U {w I  D , a G G , and - w £ E }. 0  G  0  ;  OO  It therefore suffices t o show t h a t  U G  C  x  U F  t  i=0  1=0 00  a) G  0  = W  = F  0  C  U P , . i=0  00  b) A s s u m e G C ;  U F ; , a n d consider w G G  i  +  1  .  i=0  G  i + 1  = Th(G0 U {w  j -2-Lii G D  , a G G , -w $ E } 0  0  ;  00  i)  00  If w € T h ( G ; ) t h e n w G U F b y hypothesis since U F ; is l o g i c a l l y closed. ;  i=0  ii) O t h e r w i s e w G {w  I  a  W  i=0  G D  0  , a G G : , ->UJ f£ E°>.  w But:  1)  If w G G  i  +  1  and E ° =  U G j then u G E ° C E . i=0  S i n c e E is consistent, ->w  E.  00  2)  If a G G j t h e n a G U F j b y hypothesis, so a G F i=0  k  for some k.  - 122 -  D  3)  D  C  0  OO  Thus w g F  k  +  1  C  U F . ;  i=0 oo  oo  U G ;C U F , .  B y induction,  i=o B) Assume E T C  i=0  U F , , a n d show E C  U F-,.  i=0  i=0  1  j  oo  C o n s i d e r w g E . E is a n e x t e n s i o n f o r A j = ( D / , E j  J  j _ 1  ) , so E =  U G , where  J  s  i=0  G  0  = H T , and for  G  i + 1  1  i > 0  = T h ( G j ) U {w | SLUL  €  D / , a € G ; , and  $ E^} .  w oo  a) B y h y p o t h e s i s , G = E  i _ 1  0  C  U F; . i=0  oo  b) A s s u m e G ; C  U F ; a n d consider w g G  i  +  1  .  i=0 oo  i)  If w g T h ( G i ) t h e n w g  oo  U F ; b y h y p o t h e s i s since U F j is l o g i c a l l y c l o s e d . i=0  ii) O t h e r w i s e  i=0  | — ' • — g D j ' , a g G , a n d ->u £ E*} . ;  CO oo  Since a g G j , we know that a g E a n d a g  U F j . A l s o , if w g G i=o  J  i  +  1  then w g E  1  so w 6 E . T h e r e f o r e -icu ^ E , since E is consistent. If 6 =  g D j , t h e n either  g D or d 7 -  w  !  - - € D . !  L  Thus there  to  w  are t w o cases: a • co a) E i t h e r — — g D , a g  °° °° U F , a n d -iw £ E a n d hence w g U F ,  :  CO  ;  ;  i=0  i=0  oo  b) O r  Q  :  "  A  7  g D, a g  U F ; , and i=o  w  £ E. oo  C l e a r l y , i f (-.7 V -.w) £ E t h e n w g  U F . ;  i=0  S i n c e co g E , i t c a n be s h o w n t h a t (-17 V ->w) g E iff -17 g E . W e s h o w t h a t -17 ^ E . C l e a r l y lyucdrl) < ' M I N M = j - A s s u m e -.7 g E . T h e n 3 r > j . (-.7 g E ) . r  By Lemma  3.3.1,  (W U  U CONSEQUENTS  ( G D ( E , A J ) ) j - -7. ;  i=0  T h u s there is a l i n e a r r e s o l u t i o n r e f u t a t i o n of 7 f r o m T = (W U  U CONSEQUENTS(GD(E\  AO)).  i=0  Observe  that  ' M I N ( C O N S E Q UENTS  tion of  7  from  so if  6 g GD(E', A;) (6)) = i . B y L e m m a  T, g i v e n  /MAX(  -  ,  7)  <  J> i m p l i e s  then 3.3.10,  6g Dj'  and  the existence o f a r e f u t a -  t h a t there is a r e f u t a t i o n f r o m  - 123 -  $ C T such t h a t ip € W *-* / M I N ( ^ ) < J- T h u s there is a r e f u t a t i o n f r o m # = (W U  U CONSEQUENTS(GD(E\  A;))).  i=0  Hence * then  j — -.7 a n d , b y L e m m a 3.3.1,  -.7 £ E  i _ 1  a n d so E  1  "  1  - 7  *  |  since E  .7 iff E J  1  i _ 1  ]  .7. B u t if 6 e  is l o g i c a l l y closed.  D/  H e n c e we  o b t a i n a c o n t r a d i c t i o n b y a s s u m i n g t h a t -17 € E , so -17 £ E . 00  T h u s (-.7 V ->w) £ E a n d so w e  U F;. j=o  00  W e see t h a t G Therefore E C j  i  +  C  1  00  00  U F , and by induction  U G; C  U F .  i=0  i=0  i=0  (  ;  U F; . i=0 00  B y induction, E C  U F . s  i=0 oo  T o g e t h e r , (1) a n d (2) s h o w t h a t E =  U F j , so E is a n e x t e n s i o n for A . i=0  Q E D T h e o r e m 3.3  B e f o r e p r e s e n t i n g the p r o o f of T h e o r e m 3.4, we repeat the d e f i n i t i o n of the p r o c e d u r e to generate e x t e n s i o n s g i v e n earlier.  S u p e r s c r i p t s have b e e n a d d e d w h i c h serve o n l y as reference points  i n the proofs. T h e y do not effect the c o m p u t a t i o n .  - 124 H  0  -  W;  j ^ O ;  repeat j « - j + 1; h  0  -  W;  GD 0  { };  i « - 0;  repeat  V> _ { !L±A. G  D  |( ; j_ n  a)t  ( h  i  ^  ( H  H  I/- -,/J) };  if - n u l l ( D / - G D J ) t h e n c h o o s e 6 f r o m (D> - GT>{); G D ^ - G D / h ^ i « - h/  U  U {8}; { C O N S E Q U E N T ^ } } ; endif;  i « - i + 1; until nullpj.! - G D ^ ) ; Hj =  hk  u n t i l Hj =  Lemma  Hj^  3.4.1  If A is a finite d e f a u l t theory, t h e n the a l g o r i t h m c a n f a i l to converge o n l y if one of the a p p r o x i m a t i o n s is r e p e a t e d . I.e., for some j a n d some k > j + 1 , H j = H  .  k  Proof  If A  is  finite,  there are o n l y a finite n u m b e r of different c o m b i n a t i o n s possible. T h u s there are  o n l y a finite n u m b e r of d i s t i n c t H^s w h i c h c a n be c o n s t r u c t e d . If H j = H j  + 1  , the a l g o r i t h m c o n -  verges.  QED Lemma  Lemma  3.4.1  3.4.2  If A is a finite, s e m i - n o r m a l d e f a u l t t h e o r y , a n d W is consistent, t h e n \- 0 -  ^  ^  \f- ->p.  Proof  A s s u m e H ]— 0, —>p. L e t r, s be t h e smallest integers s u c h t h a t h*)— P, hg 1 1  ;  >p. A s s u m e r < s,  - 125 -  \f- -iff. B y h y p o t h e s i s , h, j — 0, ->0. N o w h ' = h ^ U {w},  so  1  8  a  :  w  A  ^  e  D, a  e hU,  V - , ) , a n d h ^ \f-  H j - i y-  B u t if h * j — A -./9, t h e n ( h ^ U {w}) s  V -,7) .  7  )— 0, ->0 so h ^ )  where  .w a n d hence h ^ j — (-<u V -.7), w h i c h is  a c o n t r a d i c t i o n . T h e p r o o f is s i m i l a r i f s < r.  Q E D L e m m a 3.4.2  Definition 3.4.3: N e t w o r k Default  Theory  A d e f a u l t t h e o r y , A = ( D , W), is a network theory if it satisfies the f o l l o w i n g c o n d i t i o n s : W contains only:  (1)  a) L i t e r a l s (t.e., A t o m i c f o r m u l a e or t h e i r negations), or b) D i s j u n c t s of the f o r m ( a V 0) where a a n d 0 are l i t e r a l s . (2)  D c o n t a i n s o n l y n o r m a l a n d s e m i - n o r m a l defaults of the f o r m :  a:0  a :  —^—  0 A 7i  A ... A 7n  7.  or  0  0  where a, 0, a n d 7; are l i t e r a l s .  |  L e m m a 3.4.4  If A is a finite, o r d e r e d , n e t w o r k d e f a u l t theory, if W is consistent, a n d if 0 is a l i t e r a l , t h e n  f - 0 -+ H; \j-  Proof 00  A s s u m e Hj_i j — 0, a n d c o n s i d e r Hj =  U h}.  A s s u m e Hj )— —>0. T h e proof proceeds b y i n d u c t i o n .  i=o base h  0  = W. S i n c e H ^ j / -  c l e a r l y W \f- -i£ T h e r e f o r e h  0  \f- ->0.  step A s s u m e hj \f- ->0 a n d h ^ ] 1  "  :  i  A  w  e D,  W \-  a, hi  <0. hf  +1  = h} U {w},  V- ( - 7  where  V - w ) , and H  H  \f- ( - 7 V - w ) .  - 126 -  C l e a r l y , co j=- ->B o r else H ^ | <co. Note that: i)  H j c o n t a i n s o n l y d i s j u n c t i o n s o f t w o literals.  ii)  h J = W U CONSEQUENTS  {GD})  iii) GD{ Q D iv)  C  CONSEQUENTS(GD>)  Literals.  C o n s i d e r a l i n e a r r e s o l u t i o n r e f u t a t i o n o f B (i.e., a proof of ->B) f r o m h ^ i , w i t h t o p clause 8. W e c o n t i n u e b y i n d u c t i o n o n t h e s t r u c t u r e of this r e f u t a t i o n . / C  B  0  .A  D  base a n d co  co G Literals  ^B so C £ co. C l e a r l y , C £ 8. T h u s C € h . If C 6 W - W , t h e n j  0  C  0  € Literals.  C  0  € W . C l e a r l y C ^ Literals,  0  0  s  0  B u t t h e n C = ~<B w h i c h leads to the c o n t r a d i c t i o n t h a t h} )—->B. T h u s 0  0  as a b o v e . Hence C = ( / 3 V ^ ) , w i t h £ g Literals.  Thus  _ ,  0  R i = £ £ D. step Assume:  i) ii)  w £ {C  { C » , . , C J C W  iii) { R x Let  C ^ }  0  R } C n  R = fj € Literals.  Literals.  If C „ = w  n  t h e n w = - . I J so W U {w} \— ->B b u t W C H  H  and  H j ^ |— ft so H j ^ j — ft -ift w h i c h c o n t r a d i c t s L e m m a 3.4.2. C l e a r l y r] j= ->B, so C /= 6, or n  else W |— -1/9 w h i c h C  n  So:  is false.  T h u s C g W . C l e a r l y C ^ Literals,  = (-"7 V A) w i t h A e Literals. i)  u £ {C ,...,C }  ii)  {C ,...,C } C W  0  0  iii) { R i  n  Therefore R  n  n  +  1  as a b o v e , hence  = A j= D-  n  n  Rn+i} ^ Literals .  B y i n d u c t i o n , there is n o s u c h r e s o l u t i o n r e f u t a t i o n a n d the r e q u i r e d result is p r o v e d .  QED  L e m m a 3.4.4  - 127 L e m m a 3.4.5  If A is a finite, o r d e r e d , n e t w o r k d e f a u l t theory, a n d { o ^ , . . . , a „ } C Literals , t h e n H ; \— (o^ V ... V a j i f a n d o n l y i f W f— ( a V ... V c t j o r H ; ) — ctj , f o r some j . 1  Proof  (<-) T r i v i a l .  (—•) A s s u m e false, a n d consider a l i n e a r r e s o l u t i o n p r o o f o f ( « ! V ... V a j (i.e., a r e f u t a t i o n o f  (->cti A ••• A ""O^n))  f r o m H j , w i t h t o p clause R € {->ai,...,->a }. 0  n  Ro  C  0  Q  We  know  that  C £ Hi U 0  {-ia ...,->a }, a n d t h a t , f o r i > 0 , C € H i o r C e { R j | j < i } o r 1)  n  ;  ;  C € {-ia ,...,-ia }. W e p r o c e e d b y i n d u c t i o n . ;  1  n  base Without  loss  of generality,  W {j— (a V ... V a j , so C x  assume  R = -IO^.  Clearly  0  a j ^ {-^a ,...,-'a }, 1  I  o r else  {-!«!,...,,-ia,,}. C l e a r l y C £ aj, o r else H j f— a w h i c h c o n t r a d i c t s  0  0  x  o u r a s s u m p t i o n . H e n c e C = (a V 7) € W , for some 7 € Literals, 0  t  a n d so R = 7 f=- Q . x  step Assume a) { R b) Let R = n  our  0  {C  0  R „ } C Literals C ^ }  C  W .  rj e Literals. If C = ->fj e {-•a ,...,->a } t h e n W j — ( a ^ V ... V a j w h i c h c o n t r a d i c t s  hypothesis. If  hypothesis.  1  n  n  C =->i)eHiU { R . . . , R } then H ; j— a B  0 )  n  x  which  H e n c e C = (->rj V £) € W , w i t h £ €E Literals a n d R n  n  +  1  also c o n t r a d i c t s t h e  = £ ^= Q.  B y i n d u c t i o n , there i s n o s u c h r e s o l u t i o n r e f u t a t i o n , a n d the l e m m a is p r o v e d .  QED  L e m m a 3.4.5  - 128 -  L e m m a 3.4.6  If A is a finite, ordered, n e t w o r k default t h e o r y , a n d a G Literals, t h e n H i } — a if a n d o n l y if W \- a o r 33 £ L i t e r a l s . I (0) < I (a), j ? 6 H , a n d W ) - ( / 3 3 a ) . ;  Proof (<-) T r i v i a l . (—•) A s s u m e false a n d consider a l i n e a r r e s o l u t i o n proof of a (i.e., a r e f u t a t i o n of -io:) f r o m H i , w i t h t o p clause ->a. W e p r o c e e d b y i n d u c t i o n .  base C l e a r l y C J= a o r else a G H Q  C  ;  a n d / ( a ) < / (a) a n d W \— (a 3 a) w h i c h c o n t r a d i c t s the  = (a V 7) G W ,  hypothesis.  Hence  R  C l e a r l y W |— (-.7 3 a ) .  x  = 7 £ •.  0  for  7 G Literals.  By  definition,  / (-17) < / ( a ) .  step Assume:  a)  {CQJ.-JC^} C  b)  {Ro,...,R„}  c)  1 ( ^ R J < 1(a)  d)  Wj-(-.R Da)  R„ = r?.  Let  C  W  Literals  n  If  C  N  = ->tj G H  ;  then  H ; {-a,  -itj G H ; ,  / (-"^J = / (—Rn) — ' (<*) w h i c h c o n t r a d i c t s o u r a s s u m p t i o n . w h i c h is also a c o n t r a d i c t i o n . H e n c e C and /  N  N  ( - i R ^ ) = / (-.£) < / (-.77) = / ( - R J < 1(a). B y m o d u s ponens, W f - (-.£  L e m m a 3.4.6  and  = ->rj = —*a t h e n W J— a  = (->r] V £) G W , w i t h £ G Literals,  T h u s there is n o s u c h r e f u t a t i o n , a n d the result is p r o v e d .  QED  If C  W j — (-.77 3 a ) ,  3  R  n +  a).  i = £  Q,  - 129 Lemma 3.4.7 If A is a finite, o r d e r e d , n e t w o r k default theory, a n d a G L i t e r a l s , a £ H , , a ^ H j , a n d a G H for i < k < j , then k  30 e L i t e r a l s . (/ (0) < I (a)) a n d 0 6  U H AH . ;  r  i<r<j  Proof L e t j be t h e least j > k s u c h t h a t a £ H j . Define  D = {8 G D | 8=  "  1  a  A  "*  -  A  }  A  a oo  GDi = U G D ' n D r  a  r=0  Clearly  GDi"" j= 1  { } a n d GD£  =  { }. C o n s i d e r 8 G  GDJ" . 1  Since  5  £ GD£  three cases are possi-  ble: 1)  Hj_j By  | — (—Wx  V ... V  Lemma  n u j .  B y Lemma there  3.4.6.  is  there i s a n w , s a y w, s u c h t h a t H j _ j  3.4.5,  r  0 G Hj_  a  such  x  x  1(0)  that  <  I  iw.  (w)  and  W f - (0 D w). B u t t h e n I (0) < I (a). C l e a r l y 0 £ H j _ , so 0 is the r e q u i r e d l i t e r a l . 2  2)  H j j — (-"Wi V ... V -<u^. T h e a r g u m e n t f o r case 1 applies.  3)  H j \f~ 7 . B y r e c u r s i v e l y a p p l y i n g the foregoing arguments t o 7 , w e c a n c o n s t r u c t a set of 7 ' s w h i c h were i n H j _ a n d are n o t i n H j . T h e first of these t o go i n t o H j _ m u s t also go r  x  x  i n t o H j , unless H j ^ U H j c o n t a i n s a 0 <SC 7  r  a which was not i n H . ;  QED Lemma 3.4.7  Lemma 3.4.8 If A is a finite, o r d e r e d , n e t w o r k default theory, a n d a G L i t e r a l s , a G H ; , a 6 H j , a n d a ^ H for i < k < j , then either k  1)  30 G L i t e r a l s . (/ (0) < I (a)) a n d 0 G  U H;AH  r  ,  or  i<r<j  2)  30 G L i t e r a l s . (/ (0) < I (a)) a n d 0 G H j a n d 0 £ H ; .  Proof L e t k be the least k > i s u c h t h a t a ^ H . L e t j be the least j > k such t h a t a G H j . k  - 130 -  Consider 6 = C a s e s : 1) H  7  k  : a  )  A  ^ G G D * . C l e a r l y G D i jfe { }, a n d <S 0 G D * .  'ft H j \J— ->0. T h i s gives the first of the required c o n d i t i o n s , b y L e m m a s 3.4.5  a n d 3.4.6. 2)  H _ i ) — -i0, H j (/- -<0. T h e a r g u m e n t f o r case 1 a p p l i e s .  3)  H \f- % H j j — 7 . B y L e m m a 3.4.6, 3  k  k  C a s e s : a) 7  X  «  7l  a.  7  G Hj ,  l  7  l  £ H . k  ^ H . T h i s is t h e second of the r e q u i r e d c o n d i t i o n s . ;  b) 7 ! G H j . R e p e a t i n g the above a r g u m e n t s f o r 7 ^ y i e l d s a (possibly c h a i n of 7 ' s such that 7 G H _ ! , 7 £ H r  r  H _ i . It m u s t also go i n t o H k  k  k  r  k  cyclic)  . C o n s i d e r the first 7 t o go i n t o r  , w h i c h is a c o n t r a d i c t i o n .  QED Lemma 3.4.8  Theorem 3.4 — Convergence T h e p r o c e d u r e p r e s e n t e d above a l w a y s converges w h e n a p p l i e d to a finite, o r d e r e d , n e t w o r k default t h e o r y .  Proof By  L e m m a 3.4.1, n o n - c o n v e r g e n c e i m p l i e s there  H ; = H j a n d H j= H ;  Choose a e  i + 1  is a c y c l e .  I.e., for some i a n d some j > i ,  .  U ( H i A H j s u c h t h a t a G Literals a n d f o r every 0 G »<k<j  U ( H i A H J , ->(/ (0) < I (a)). i<k<j  T h u s a is the " l e a s t " l i t e r a l t o change state between H i a n d H j . T h e r e are t w o cases: (1)  If a ^ H ; a n d a G H  k  t h e n , b y L e m m a 3.4.7, 30 G U ( H i A H ) . I (0) < I (a), so a is not k  i<k<j  the least s u c h a, w h i c h is a c o n t r a d i c t i o n .  (2)  If a G H i a n d a £ H t h e n , b y L e m m a 3.4.8, either k  a)  30 G  U ( H i A H J . / (0) < I (a) i<k<j  so a is n o t the least s u c h a, w h i c h is a c o n t r a d i c t i o n , o r  b) 30. 0G Hj <md0(E Hj w h i c h i m p l i e s t h a t H i j=- H j w h i c h is also a c o n t r a d i c t i o n .  T h e r e f o r e , there is no c y c l e , a n d so t h e p r o c e d u r e converges.  - 131 -  Q E D T h e o r e m 3.4  T h e o r e m 3.5 — S t r o n g C o n v e r g e n c e  T h e p r o c e d u r e g i v e n a b o v e a l w a y s converges i m m e d i a t e l y w h e n a p p l i e d to a finite,  n o r m a l d e f a u l t t h e o r y A = ( D , W ) - i.e., T h ( H i ) is a n e x t e n s i o n .  Proof  E t h e r i n g t o n [1982] shows t h a t  = H  2  i f a n d o n l y if T h f i y  is a n e x t e n s i o n for A . If W is i n c o n -  sistent, t h e n T h ( H ] ) = L w h i c h is a n e x t e n s i o n for A . H e n c e assume W is consistent.  T o show  oo  t h a t T h ( H i ) is a n e x t e n s i o n for A , w e i n v o k e T h e o r e m 0.1 a n d show t h a t T h ( H i ) =  U E j , where i=0  E = 0  E,  + 1  W  = T h ( E 0 U {w  a) W e first s h o w t h a t  | £±2-  € D, a € E i ,  ^ Th(Hi)} .  OO  OO  U E ; C T h ( H i ) . R e c a l l t h a t Ej_ =  U h^.  i=0  i=0  base Clearly E  0  = W = h^ Q T h ( H ) . 1  step . Assume E ; C T h ( H ) and consider w S E 1  i)  i  +  1  .  If w e T h ( E i ) t h e n w € T h ( H i ) , b y h y p o t h e s i s a n d closure.  ii) O t h e r w i s e  w € {w  |  "  =  "  6 D , a £ E , -w (£ T h f f i J } .  Therefore  ;  U y~-<w. i  Hence  w H  \f—>OJ since H  0  0  = W C H i . A l s o , a € E , so a € T h ( H ) , b y h y p o t h e s i s . ;  1  b y [ E t h e r i n g t o n 1982, L e m m a 3.3] t h a t H i ]— w. Hence E  i + X  C  Th(Hi). OO  b) F i n a l l y , we show t h a t T h ( H i ) C  U E . r  r=l oo  oo  S i n c e U E is l o g i c a l l y closed, it suffices to show t h a t H r  r=l  x  C  U E . r  r=l  base Clearly h^ = W = E  0  C  U E . r  step oo  Assume that h; C 1  U E , a n d consider hj+i . r  It follows  - 132 -  = h ; U {w}, f o r some u G CONSEQUENTS  (D; ).  1  1  oo  Since h j C 1  oo  U E b y h y p o t h e s i s , w e need o n l y show t h a t w € r  i=l  UE . r  r=l  S i n c e u e CONSEQUENTS  g D , a <5 V,  (Bi ), f o r some 6 = 1  a; H  0  — i w , a n d h * j / - ->w.  B y h y p o t h e s i s , since a 6 t ' , a €  oo  U E , so a € E j f o r some j . r  r=l  Since w € h ; ! ^ C K  u  i t follows b y L e m m a 3.4.2 t h a t H x ]/—>w. oo  B u t then b y definition of E  j + 1  , w 6 E  j + 1  C U E . r  =i C o m b i n i n g (a) a n d ( b ) , w e h a v e the desired result.  Q E D T h e o r e m 3.5  - 133 -  T h e o r e m 4.1  A n y n e t w o r k i n w h i c h t h e s u b g r a p h of I S - A l i n k s a n d exceptions t h e r e t o is a c y c l i c c o r r e s p o n d s t o a n o r d e r e d t h e o r y .  Proof  The  l i n k s c o r r e s p o n d i n g t o a D —>B,  a :  -JB  a: -<B  and  A  ~>li A  -A  ->7i  g i v e rise t o a ^  -<B  a n d 7,- <K —>B. T h e r e a r e n o l i n k s w h i c h m a k e a t r a n s i t i o n f r o m negative t o p o s i t i v e or negative to n e g a t i v e , so s u c h l i n k s c a n n o t p a r t i c i p a t e i n a n y cycle leading t o w «  u f o r a n y w.  What  r e m a i n s are I S - A l i n k s a n d e x c e p t i o n s thereto.  Q E D T h e o r e m 4.1  T h e o r e m 4.5 In t h e absence of n o - c o n c l u s i o n l i n k s , a l l g r o u n d facts r e t u r n e d b y T o u r e t z k y ' s i n f e r e n t i a l d i s t a n c e a l g o r i t h m l i e w i t h i n a single e x t e n s i o n o f the default t h e o r y c o r r e s p o n d i n g t o t h e inheritance network i n question.  Proof  W e p r o v e t h a t a l l the g r o u n d facts i n a n y " g r o u n d e d e x p a n s i o n " o f the n e t w o r k l i e w i t h i n a single extension.  F r o m t h i s t h e result f o l l o w s .  A s a n o t a t i o n a l s h o r t c u t , w e w i l l use dtP t o s t a n d f o r +P  or -P (or, o c c a s i o n a l l y , f o r P o r ~<P). T h e i n t e n d e d m e a n i n g s h o u l d be clear f r o m c o n t e x t .  Let  F be a n e t w o r k  i n T o u r e t z k y ' s sense.  Let $  be a g r o u n d e d e x p a n s i o n f o r F .  Define  facts($) = { < + a , ± P > € C(4>) [ a i s a n i n d i v i d u a l t o k e n } , a n d facts'{$) - {Pa | < + a , + P > e facts($)} U {->Pa | <+a,-P> S facts{$)}.  If  <+a,±P>  definition. <+a,±P>  then  S facts($) Hence, e facts($).  by  f o r some  [Touretzky  Thus  P a  for t = l , . . . , n - l , a n d <+P ±P> a  x  e  P ...,P , w e have  c l a i m t h a t facts'($)  theorem  2.3],  P a, ±Pa € facts'($). n  n  <+a,+Pi>  i+1  and  <+a,+P >, n  Furthermore,  b y [ T o u r e t z k y 1984a, t h e o r e m 2.3]. Pfl : P x — Pi+iX  by  <+a,+Pi,...,+P ,±P> G  P^c : ±Px ±Px  is i n c o n s i s t e n t iff W is. B y d e f i n i t i o n ,  where a is a n i n d i v i d u a l t o k e n } . some a a n d T.  n  1984a,  T b y T o u r e t z k y 1984a, t h e o r e m 2.2 . H e n c e  We  v  <+P,-,+P, > € $ + 1  H e n c e t h e y are a l l i n  € D.  W — {±Ra \ <+a,±R>  ET,  T h e r e f o r e , W i s inconsistent iff <-ra,+R>, <+a,-R> G T , f o r  - 134 -  The  right-to-left  d i r e c t i o n of the c l a i m is t r i v i a l .  is i n c o n s i s t e n t .  facts'($)  F o r the left-to-right d i r e c t i o n , assume t h a t  T h e n Ra, -*Rct G facts'($)  so <+a,+R>, <+a,-R> G facts($), so  ay = <+a,j/ ,...,yy ,+R> a n d cr = <+a,x ...,x ,-R> G 1  2  inconsistent.  ll  So $ contradicts  k  H e n c e T is inconsistent, b y [ T o u r e t z k y 1984a, t h e o r e m 2.8].  o~i n o r cr is i n h e r i t a b l e i n  a n d cr , a n d <& is 2  F u r t h e r m o r e , neither  so b o t h are i n T, since $ is a grounded e x p a n s i o n o f T. B u t t h e n  2  /= k = 0, so < + a , + P > a n d < + a , - P > G I\  H e n c e , P a , — P a G W, so W inconsistent.  N o w if  facts'(<&) i n c o n s i s t e n t , W is i n c o n s i s t e n t , so A has a unique e x t e n s i o n , Th(L) 3 facts'(<&). I n t h e sequel, w e assume facts'(<&) consistent.  We D  ,  show  that  f P,<x : ±P a = < i+1  normal  is  E' = Th(facts'($))  an  . \ 1 < t<A:>.  <+a,+Pi,...,dcP > G k  default  theories,  there  D' C CLOSED-DEFAULTS(A)  will  extension  be  an  for  A ' = ( I > ' , W),  where  T h e n , b y the s e m i - m o n o t o n i c i t y of  extension,  ED  for  E'  A,  since  [Reiter 1980a, t h e o r e m 3.2]. OO  A s u s u a l , w e show t h a t E  =  U E. t  £=0  oo  E' D U E :  Consider  {  w = ±Ra G E =  < + a , ± P > G F C <£, so < + a , ± P > G /acfcs(<3>), so ±Ra G facts'($). assume  Ej C E',  w G {P ict i+  a n d consider Pfic : Pi+ia.  | 5=  P^-i"  <+a,+P ,...,±P > G 1  tii G  If  . G D , P,a G E  f o r some k >  J t  grounded expansion. So P , a , ± P ,  + 1  tii G Th(E ), t  Then  F o r the i n d u c t i v e step,  then  . a n d - P i + i a ^ £ }.  h  G T}.  W = {±Ra \ <+a,±R>  0  tii G i ? ' .  Since  Hence < + a , + P , > , < + a , ± P ,  + 1  Otherwise,  8 G D , we have > G <5, since $ is a  a G /acis'(<5).  oo  C  U E:  C o n s i d e r ±Ra G facts'($).  {  1984a, Ra x  theorem  2.3],  Then  ll  <+a,+P >  <+a,+Py>,  1  If < + a , + i 2 i > G $ , then  Rfc, ±Ra G facts'($).  G $.  <+a,+R ...,+Rp±R>  By  <+a,±R>  [Touretzky so  G facts(§),  <+a,+i?!> G F, by  [Touretzky  CO  1984a,  theorems  2 . 3 , 2.2], so  R a G W C U E,. y  F O F the  inductive  step,  assume  t=0  Rycx  oo  °°  R/fic G U E , f o r k<j. W e show t h a t P ^ a G U E . i=0 £=0 {  {  Now5 =  R a : Ru+ia k  Rk+l  .  ——GD .  a  oo  Since  P * a G U Ej,  P ^ a G i?,-,  for  some  t.  Since  <+a,+R ...,+R x> u  k+  G  t=0  <+a,P  f c + 1  > G C ( $ ) so < + a , P A . > G / a c t s ( $ ) , so R^a H  1  G facts'($).  B y the c o n s i s t e n c y of T,  oo  E' \f—'P^a,  so Pjfe+ia G ^ - i . S o P ^ a G U fc=o  for 1 < k < n, b y i n d u c t i o n .  ±Ra. oo  T h u s E' =  U 2?,. S o £ ' is a n e x t e n s i o n for A ' , b y T h e o r e m 0.1. «=o  Similarly for  - 135 QED Theorem 4.5  - 136 -  T h e p r o o f of T h e o r e m 5.1 follows i m m e d i a t e l y f r o m M c C a r t h y ' s proof of the soundness of p r e d i c a t e c i r c u m s c r i p t i o n a n d the d e f i n i t i o n o f well-foundedness.  T h e o r e m 5.2  U n i v e r s a l theories are w e l l - f o u n d e d .  Proof  T h e p r o o f is i d e n t i c a l t o t h a t of P r o p e r t y 1.3.2 i n [Bossu a n d S e i g e l 1985]. T h e d e f i n i t i o n of s u b m o d e l used there is less r e s t r i c t i v e t h a n t h a t used here, b u t this does not a l t e r t h e f o r m of the proof.  Q E D T h e o r e m 5.2  T h e o r e m 5.4 If T is a w e l l - f o u n d e d theory, ai,...,a are n-tuples of g r o u n d terms, a n d P s P , is a n n-ary k  predicate, then  \r- PSj. V . . . V P3  CLOSURE (T) P  <==> T \- PS* V . . . V Pa .  k  k  Proof  T h e r i g h t - t o - l e f t d i r e c t i o n is i m m e d i a t e . tion.  Assume that  W e prove t h e c o n t r a p o s i t i v e of the left-to-right d i r e c -  k  k  ( r - V P a , - a n d T \f- V P a , - .  CLOSUREp(T)  T h e n T has a m o d e l , Af, i n  w h i c h P a , - is false, f o r all t = l,...,k. S i n c e T is w e l l - f o u n d e d , there is a P - m i n i m a l s u b m o d e l , A f ' , 1  of Af: F u r t h e r m o r e , since the c i r c u m s c r i p t i o n is true i n all P - m i n i m a l s u b m o d e l s , P a , - is true i n 1  A f ' , f o r some 1 < i < k. B u t t h e n A f is n o t a P - s u b m o d e l of Af; a n d this c o n t r a d i c t s the fact t h a t A f ' is a P - m i n i m a l s u b m o d e l of M  T h e r e f o r e CLOSURE {T)  \f- Pa  P  l  V . . . V Pa . k  Q E D T h e o r e m 5.4  T h e o r e m 5.5 If T is a w e l l - f o u n d e d theory, a ,...,a l  k  are n-tuples o f g r o u n d terms, a n d P ^ P is a n n-ary  predicate, then  (0  CLOSURE^T)  (— P a ! V . . . V PS  k  T\-PS  1  V...V P a * , and  - 137 -  (ii)  CLOSUREp{T)  ( - - . P c ? ! V...V -,PS <==> T \- -^PS^ V...V -*P3 . k  k  Proof  (i) T h e  right-to-left  d i r e c t i o n is i m m e d i a t e .  W e prove the c o n t r a p o s i t i v e o f the  left-to-right  k direction.  A s s u m e T \f- V P a , - .  T h e n there is a m o d e l , Af, for T i n w h i c h Pc?,- is false, for a l l  *=1,...,k. S i n c e T is w e l l - f o u n d e d , there is a P - m i n i m a l s u b m o d e l , A f ' , of M.  B y the d e f i n i t i o n of  s u b m o d e l , the i n t e r p r e t a t i o n of P r e m a i n s the same i n A f a n d A f ' , since P ^ P . H e n c e P a , - is false i n A f ' , for a l l t=l,...,fc .  Since the c i r c u m s c r i p t i o n schema is satisfied b y a l l m i n i m a l models,  k CLOSUREp(  T)\f- V P a , - . T h e p r o o f for (ii) is s i m i l a r .  Q E D T h e o r e m 5.5 I n the proofs o f T h e o r e m s 5.6 a n d 5.7 we use the f o l l o w i n g n o t a t i o n a l conventions: 1.  is the c i r c u m s c r i p t i o n s c h e m a resulting from c i r c u m s c r i b i n g the predicates of  SCHEMA(T,T?) P i n T.  2.  CLOSURE{y(7)  =  T. ( T h e closure of T w i t h respect t o the  empty  set o f predicates  is  defined to be T itself.)  o  3.  If A f is a m o d e l ,  T h e o r e m 5.6  A  is t r u e i n Af. ( T h e e m p t y c o n j u n c t i o n is v a c u o u s l y t r u e i n a l l models.)  (Reiter)  If T is a n a r b i t r a r y ,  finitely-axiomatized  the e q u a l i t y predicate, = , t h e n T |—  t h e o r y c o n t a i n i n g a x i o m s w h i c h define CLOSURE^ . (T). =  }  Proof  C o n s i d e r the s c h e m a r e s u l t i n g f r o m c i r c u m s c r i b i n g ' = ' i n T: SCHEMA{T,{'='}) = [ T ( $ ) A V i y . * x y D i = y] D Vary, x = y D $ z y F i r s t , observe t h a t  \— ( V z . # z z ) D ( V z y . I = J D * z y ) for a n y predicate letter, \P. F u r t h e r m o r e ,  V z . ^ z z is one o f the c o n j u n c t s of T ( $ ) i n SCHEMA(T,{'=*}) since V z . z = z m u s t be a n a x i o m of a n y t h e o r y w i t h e q u a l i t y .  T h u s i f a n y instance o f T ( $ ) is true i n a m o d e l o f T, so is the  c o r r e s p o n d i n g instance of V z y . z = y D $ z y . i n e v e r y m o d e l of T, so T |—  Q E D T h e o r e m 5.6  H e n c e , every instance of SCHEMA(T,{'='}) is true  CLOSURE^iT).  - 138 -  5.7  Theorem  If T is a w e l l - f o u n d e d t h e o r y c o n t a i n i n g a x i o m s w h i c h define the e q u a l i t y p r e d i c a t e ; a n d a, ~$ are t u p l e s of g r o u n d t e r m s ; t h e n  (i)  CLOSURE^T)  ( - 5? =  (ii)  CLOSURE^T)\-c?  <=>  T\-S  £0*  and  = ^,  T \-~a j=0*.  Proof  (i)  T h i s is a c o r o l l a r y of T h e o r e m s 5.4 a n d 5.5(i).  (ii)  T h e r i g h t - t o - l e f t d i r e c t i o n is i m m e d i a t e .  c o m p o s i t i o n of P . P = {=},  If  T o prove the left-to-right d i r e c t i o n , we consider the  does not o c c u r i n P , the result follows d i r e c t l y f r o m T h e o r e m 5.5(ii).  the result f o l l o w s f r o m T h e o r e m 5.6.  F i n a l l y , consider P = P ' U {'—'), for a n a r b i -  t r a r y set of predicates P ' = {Pi,...,P^ not i n c l u d i n g e q u a l i t y . | - S j t ^  CLOSUREp'(T)  If  B y T h e o r e m 5.5(H),  T\-cl£~$  <r=>  W e show that CLOSURE (T)  \-aj=~$  p  <==> T\-ct  j=~$ .  W e have SCHEMA(  T",P')  3  3 t * i , - , * J A (.A (V*.  P^)j  n D  A (Vxy.  ¥xy  A s s u m e CLOSUREp( Any  model  of  CLOSUREp(T).  J2X*i»—A  =  SCHEMA(T,P)  T  A (Vx*. P? D <J>,x)  D x = y)j  D ^  (,A(Vz. (Vx.  every  instance  of  is also true. F i r s t observe t h a t F u r t h e r m o r e , Vx. *xx of  is true  SCHEMA(T,P)  H e n c e a j= ~$ is true i n t h a t m o d e l .  is true a n d a J= 0* is false.  | — (Vx.  true  in  a  model  of  SCHEMA(T,P)  x = y D tyxy) for a n y predicate letter, 1  n  i n SCHEMA(T,P).  n  T,  so is the  T h u s if a n y  corresponding  L e t A f b e a m o d e l of T where every instance o f SCHEMA(T,P')  sider a n i n s t a n c e , 7, of SCHEMA(T,P),  %  for  W e show t h a t i n every  is true, every instance of  3 (Vxy.  ^xx)  is also a m o d e l  F u r t h e r m o r e , there is some m o d e l of T i n  is one of the c o n j u n c t s of T(<& ,...,$ , * )  T[$i,...,$ jty) is  x = y D Vxy.  T) \f- a £  p  which  m o d e l of T i n w h i c h every instance of SCHEMA(T,P')  Vxy.  *xy  T) |— a £ /2> a n d T \f-~a j=~$. It follows t h a t CLOSURE i( in  w h i c h e v e r y i n s t a n c e of SCHEMA(T,P')  instance  x = y D  PflZ) $p) A Vxy.  w i t h the predicates  and  instance is true.  s u b s t i t u t e d for  of  Con-  and \P,  r e s p e c t i v e l y . T h e r e are t w o cases: n  1)  A (Vx*.  P&Z)  $,'x)  false i n M o r Vxy. n 2)  A (Vx.  P j X D $,'x)  is t r u e i n M. x = y D *'xy is  false  B y the o b s e r v a t i o n a b o v e , either r ( $ ' , . . . , $ „ ' , * ' ) x  is true. in  is  In e i t h e r case, lis true i n M. M  But  then  T($ ',...,$ ') 1  n  is  false  or  - 139 -  A (Vx*. t=i the  $,'x*D Pjiz) is false, since every instance of SCHEMA(T,P')  latter  case  /  [ r ( $ i ' , ,  is  also  true  in  M.  In  the  former  A Vxy. ty'xy Z> x = y] is false i n M , then / is true.  o b s e r v a t i o n a b o v e , Vxy.  x = j D ^ 'xy is true a n d , hence, so is Vxy.  T ( $ ' , . . . , $ ' , I ' ) is *  result of s u b s t i t u t i n g # '  1  1  is true i n M.  r  n e  I t  T ( $ ! ' , . . . , $ „ ' ) , so T ( 4 > i ' , . . . , $ „ ' , * ' )  case,  if  O t h e r w i s e , b y the x — y = \P 'xy.  But  for some of the occurrences of ' = '  is false, because  In  in  ' , . . . , $ „ ' ) is, a n d t h i s is a con-  tradiction. T h u s , for every m o d e l of T, i f SCHEMA(T,T') true i n every m o d e l of CLOSURE >[T). p  is true, so is SCHEMA(T,P).  H e n c e CLOSUREpi(T)  t i o n , since T \/-a £p*. W e c o n c l u d e t h a t CLOSUREp(T)  B u t then a j= p* is  |— a /= p\ w h i c h is a c o n t r a d i c -  \f-ct j=~$.  Q E D T h e o r e m 5.7  C o r o l l a r y 5.8 If T is a w e l l - f o u n d e d theory c o n t a i n i n g a x i o m s w h i c h define the e q u a l i t y predicate, P is a n n-ary p r e d i c a t e , a n d a is a n n-tuple of g r o u n d terms, then CLOSUREp( T (— a i= ~j$ for a l l g r o u n d n-tuples  T) |— -*Pa implies  such that T — ( P/3.  Proof  O t h e r w i s e CLOSUREp(T)  |— a f  /3 a n d T \f-~a j=~$ w h i c h c o n t r a d i c t s T h e o r e m 5.7.  Q E D C o r o l l a r y 5.8  T h e o r e m 5.9  If T is a w e l l - f o u n d e d t h e o r y ;  aj  a „ are g r o u n d terms;  a n d P is a set of some of the  predicate s y m b o l s of T\ t h e n CLOSURE {T) p  \-  Vx. x = a V...V x = «„<==>• T\-Vx.x=a x  V...V x = a „ .  l  Proof  The  right-to-left  T\f- Vx.x  = a  x  V...V  direction x = a„.  is  immediate.  For  the  left-to-right  T h e n T has a m o d e l w h i c h falsifies Vx.x  T is w e l l - f o u n d e d , t h i s m o d e l has a P - m i n i m a l s u b m o d e l  direction, = a  x  V...V  B u t Vx. x = ct^ V...V  assume x = a  n  x = a  n  that  . Since  is false i n  this s u b m o d e l , because the e x t e n s i o n of the e q u a l i t y predicate i n this s u b m o d e l m u s t be a subset of i t s e x t e n s i o n i n the o r i g i n a l m o d e l .  S i n c e the c i r c u m s c r i p t i o n is true i n a l l m i n i m a l models,  - 140 -  CLOSUREp(T)  \/-  Vi.i= V...Vi=a . Q l  n  Q E D T h e o r e m 5.9  T h e o r e m 5.10  If T is a w e l l - f o u n d e d theory, a n d T has a m o d e l w i t h some d o m a i n , D, t h e n s o does CLOSURE^  T).  Proof  McCarthy  [1980] shows t h a t  CLOSUREp(T)  is true i n a l l m i n i m a l models.  founded, every m o d e l has a m i n i m a l submodel.  S i n c e T is w e l l -  B y the d e f i n i t i o n of s u b m o d e l , the d o m a i n o f a  m i n i m a l s u b m o d e l o f M is the same as t h a t of M.  Q E D T h e o r e m 5.10  T h e o r e m 5.11 If T |— Vx*. Px = 3>af f o r some expression $5*, not i n v o l v i n g predicate letters f r o m P , t h e n T \-  CLOSURE (T)'. P  Proof  Tfi!),  o n the left-hand  side of t h e c i r c u m s c r i p t i o n  s c h e m a , i n c l u d e s Vx*. ^ 5 * = <I>z*.  B u t any  choice o f m o d e l , M, a n d p r e d i c a t e , ^ , w h i c h satisfies the L H S c l e a r l y a l r e a d y satisfies the R H S , V x . F x * D ^x*, since e v e r y m o d e l o f T satisfies Vx*. $ 5 * = P i * .  Q E D T h e o r e m 5.11  - 141 -  Definition:  The  Formula Circumscription  of the formula E(P,x) i n the theory T, w i t h the predicates P  circumscription  treated  as v a r i a b l e , is g i v e n by:  T ( P ) A V $ . T ( $ ) A | W . E{&,x) D E{P,x)} D [ V ? . E{P,x) 3 E{$,x)}  Definition:  Let  M <  ^-^M  T ( P ) be a  1  finitely-axiomatized  (first- or second-order) t h e o r y , some (but not neces-  s a r i l y all) of whose p r e d i c a t e s are those i n P ;  let E[P,x) be a f o r m u l a whose free v a r i a b l e s  are a m o n g x * = x ...,x , a n d i n w h i c h some of the predicate v a r i a b l e s P = v  o c c u r free;  (ii)  1  n  iff  |M| = |M'| , If t is a t e r m , t h e n  (iii)  If Q  (iv)  \E(P,?)\ C  Definition:  {P ,...,P }  a n d let M , M ' be m o d e l s of T. W e say M i s a n E(P,x^-submodel of M ' ( w r i t -  ten M < ^ p ^ M ' ) (i)  n  |<|JI/=  I'IA/  I  P is a p r e d i c a t e l e t t e r of T, t h e n M  |<2|A/= \Q\IJ  ,  and  I  |£(P,x*j|^.  2?(P,2^-Minimal Model  A m o d e l , M , of T i s £ ( P , i ) - m m t ' m a / iff T h a s n o m o d e l , M , 1  M ' < ^ P ^ M and - ( M < ^ M ' ) .  I  such that  T h e o r e m 6.1 — S o u n d n e s s  CLOSURE(  T; P ; 22(P,zj) is s a t i s f i e d b y e v e r y £ { P , z > m i n i m a l m o d e l of T.  Proof  The  proof follows M c C a r t h y ' s [1980] p r o o f o f the soundness of p r e d i c a t e c i r c u m s c r i p t i o n .  sider a m i n i m a l m o d e l , M , a n d a n i n s t a n t i a t i o n , w i t h some p r e d i c a t e ,  Con-  o f the s c h e m a (or  second-order a x i o m ) w h i c h m a k e s the l e f t - h a n d side true a n d the R H S false. T h e n b y the second c o n j u n c t of the L H S , | £ ( P , I ) | J ^ C [ £ ( $ , 1 ^ 1 ^ s t r u c t e d b y l e t t i n g P agree w i t h  Q E D Theorem  6.1  B u t t h e n a p r o p e r s u b m o d e l , M ' , c o u l d be c o n -  B u t this c o n t r a d i c t s the fact t h a t M i s m i n i m a l .  - 142 -  T h e o r e m 6.3 T h e a b i l i t y t o m i n i m i z e a r b i t r a r y expressions, E(P,x), i n s t e a d of s i m p l e sets of p r e d i cates, is a n i n e s s e n t i a l e x t e n s i o n , p r o v i d e d predicates o t h e r t h a n those being m i n i m i z e d are a l l o w e d to v a r y .  Proof  W e show t h a t the t h e o r y , d e f i n i t i o n Vx*. ^ 5 * =  T, c a n be e x t e n d e d b y a d d i n g a new predicate s y m b o l , St, a n d the  E[P,x), a n d t h a t c i r c u m s c r i b i n g \P in the e x t e n d e d theory, T', w i t h P v a r i -  able is e q u i v a l e n t to c i r c u m s c r i b i n g E{P,~x) i n the o r i g i n a l theory.  I.e., t h a t  A [Vi*. £ ( * , 2 ) 3 -E(P,x}]j 3 [Vx*. E ( P , x j D £($,xj]  T A  (27)  and T'  A [ T(*,rj>) A [Vx. rjix = £($,?)] A [Vx.  tfx[D  [Vx*. tfx  D  Vi]  (28)  are e q u i v a l e n t o v e r t h e language of T. T o see t h a t (27) e n t a i l s (28), let M be a m o d e l w h i c h satisfies (27). we c a n i n t e r p r e t *  as we choose. T h e r e f o r e , let | ¥ |  v e r s e l y , let M satisfy (28), a n d let  = |£(P)| . M  mention  C l e a r l y , M J = (28).  Con-  be a tuple o f predicate v a r i a b l e s satisfying the L H S of (28).  C l e a r l y , 7*' f— T, a n d T'($) \— T ( $ ) . M\=  M  Since (27) does not  B y s u b s t i t u t i o n of e q u i v a l e n t s , we get the rest of (27), so  (27).  Q E D T h e o r e m 6.3  Definition:  Let X  Generalized Circumscription  be a t u p l e of p r e d i c a t e , f u n c t i o n , a n d / o r c o n s t a n t s y m b o l s , a n d let R be a b i n a r y  r e l a t i o n o n t u p l e s of t y p e X . a c c o r d i n g t o the pre-order, r(x)  Definition:  <  R  T h e generalized circumscription  A V x ' . r(x') A (x' < *x)  M <  L e t T ( P ) be a  of X  D  (x <  jpc')  finitely  a x i o m a t i z e d (first- or second-order) theory, whose p r e d i c a t e , func-  t i o n a n d c o n s t a n t letters i n c l u d e (but need not be l i m i t e d to) those i n X ; let  < R be the p r e - o r d e r i n d u c e d b y R;  m o d e l s of T. T h e n M i s a n (X,iZ)-submodel of M ' ( w r i t t e n M <  (ii)  T,  (x,i?)M'  r e l a t i o n o n t u p l e s of t y p e X ;  (i)  i n the t h e o r y ,  , i n d u c e d b y R is g i v e n by:  |M| = |W| , If t is a t e r m a n d t  X , t h e n |t|ji/ = \t\fj  ,  let IE be a b i n a r y a n d let M , M'  (x,j?)M') iff  be  - 143 -  X is a p r e d i c a t e letter of T, t h e n | Q | A / = \ Q-\M »  (iii)  If Q  (iv)  <|XU, |X| > e i 2 .  1  ( x  D  (X,i?)-Minimal Model  A m o d e l , M , of T is (X.,R)-minimal Af* <  N  I  V  Definition:  A  iff T has no m o d e l , A f ' , such t h a t  , i j ) M a n d - . ( A f < (X,J?)M').  |  T h e o r e m 6.4 — Soundness CLOSURE(T;  X ; i?) is satisfied b y e v e r y ( X , J ? ) - m i n i m a l m o d e l of T.  T h e proof is s i m i l a r t o t h a t o f T h e o r e m 6.1, except t h a t the i n t e r p r e t a t i o n s o f each of t h e v a r i a b l e t e r m s m u s t also be set.  Definition:  I  Well-Foundedness  T h e t h e o r y , T, is well-founded with respect to ( X , i i ) iff every m o d e l of T has an (X,i2)-minimal submodel. |  T h e o r e m 6.9 If T is a u n i v e r s a l t h e o r y , a n d X , P are finite tuples of predicate letters, t h e n T is w e l l - f o u n d e d w i t h respect t o < (x,p) •  Proof  W e s h o w t h a t a n y c h a i n of s u b m o d e l s o f a m o d e l of T has a lower b o u n d a m o n g t h e s u b m o d e l s o f that model.  It f o l l o w s b y Z o r n ' s l e m m a t h a t e v e r y m o d e l has a m i n i m a l s u b m o d e l .  L e t Mo,... b e a c h a i n of m o d e l s of T , o r d e r e d u n d e r t h e s u b m o d e l r e l a t i o n .  If the c h a i n i s finite, i t  has a l o w e r b o u n d , hence assume i t i s i n f i n i t e .  L e t {dj,...} be t h e e l e m e n t s o f |A^>|. E x t e n d t h e language of T, L, t o L' b y a d d i n g a n e w c o n s t a n t symbol,  i,  d, {  MifcPl).  for each  d,-.  Let  T' = T U {Pi  | f o r a l l i, Afj-1= Pet}  U { - P c f | f o r some  - 144 -  A s s u m e T' i s i n c o n s i s t e n t .  T h e n , b y c o m p a c t n e s s , so is a finite subset. B u t t h e n some M,- m u s t  set e a c h P e t i n this finite set a c c o r d i n g l y , so M,- \f= T, w h i c h is a c o n t r a d i c t i o n , since the c h a i n {Mi} is o r d e r e d . H e n c e T' is consistent, so T' h a s a m o d e l , M'.  N o w w e c a n a d d the d i a g r a m s (over a l l g r o u n d t e r m s of L') o f the e q u a l i t y p r e d i c a t e a n d a l l fixed p r e d i c a t e s f r o m MQ t o T' to get T".  B y the a b o v e a r g u m e n t , T"  must be consistent.  there is a n M " s u c h t h a t M"  B y v i r t u e of the fact t h a t M"  satisfies the d i a g r a m of the  j= T".  e q u a l i t y p r e d i c a t e f r o m MQ , we c a n i s o m o r p h i c a l l y e m b e d the d o m a i n of M T"  i n t o M".  0  Hence  (Because  c o n t a i n s t h e d i a g r a m s o f the e q u a l i t y p r e d i c a t e o v e r a l l g r o u n d terms of L ' , it is clear t h a t  the r e s u l t i n g s u b s t r u c t u r e is closed u n d e r a n d preserves the functions.) F i n a l l y , since T" M"  j = T.  Since M<  3 T,  T is a u n i v e r s a l t h e o r y , the r e s t r i c t i o n , M, of M" (x,p)M,- > f °  Q E D Theorem  Theorem  r  to | M | is a m o d e l o f T. 0  Clearly  a l l * i so M i s the lower b o u n d w e require.  6.9  6.11  If T is w e l l - f o u n d e d w i t h respect to ( X , P ) ; P € P is a n n-ary p r e d i c a t e ; p r e d i c a t e letters; a n d a\,...,a a r e n-tuples of g r o u n d terms; t h e n  X is a set of  k  CLOSURE{T;  Theorem  T f— Pa  X ; P ) f - Pa^ V . . . V Pb? ^=> k  1  V . . . V Pa  k  I  .  6.12  If T is w e l l - f o u n d e d w i t h respect t o ( X , P ) ; X is a set of predicate letters; is a n n - a r y p r e d i c a t e ; a n d ay...,a are n-tuples o f g r o u n d terms; t h e n  P j ^ P U X  k  (i) (ii)  Theorem If  CLOSURE{  T; X ; P ) | - P d ? ! V . . . V PoT  CLOSURE(T;  fc  X ; P ) (- - P a \ V . . . V - P o ? *  T \— Paty V . . . V P3  K  , and  T |— - P c ^ V . . . V - P a  f c  .  I  6.13  T is w e l l - f o u n d e d f o r ( P , P ) a n d T has a m o d e l w i t h d o m a i n D, t h e n so does  CLOSURE( Theorem  r(P);P;P).  I  6.14  If T is a first-order t h e o r y c o n t a i n i n g a x i o m s w h i c h define the e q u a l i t y p r e d i c a t e , = , then T \- CLOSURE( T ; X ; { = } ) . |  - 145 -  T h e proofs of T h e o r e m s 6.11, 6.12, 6.13, a n d 6.14 are essentially a l p h a b e t i c v a r i a n t s of those of T h e o r e m s 5.4, 5.5, 5.10, a n d 5.6, r e s p e c t i v e l y .  W e do not repeat t h e m here.  - 146 -  T h e o r e m 7.1 — S o u n d n e s s E v e r y i n s t a n c e of the r e v i s e d d o m a i n c i r c u m s c r i p t i o n s c h e m a for a theory, T, is true i n a l l m i n i m a l m o d e l s o f T.  Proof  T h e proof is i d e n t i c a l to t h a t presented i n [ D a v i s 1980, p75], except t h a t , i n the p r o o f of the l e m m a , t h e r e v i s e d s c h e m a guarantees t h a t D  0  is n o n - e m p t y a n d hence N is w e l l - d e f i n e d .  Q E D T h e o r e m 7.1  T h e o r e m 7.3 If T is a w e l l - f o u n d e d t h e o r y w h i c h c o n t a i n s a x i o m s w h i c h define the e q u a l i t y p r e d i c a t e , = , a n d ai,...,a , are g r o u n d terms, t h e n n  (i)  T |— ( V a,- =  ft)  DC( T) — [ ( V a,- =  «=i  ft)  t=i  (ii) T ( - ( V a , - £ ft) < = •  DC(T)\-  (Va  {  £ ft)  Proof  E v e r y m o d e l of a w e l l - f o u n d e d t h e o r y has a m i n i m a l s u b m o d e l .  L e t M ' be a m o d e l of T. L e t  M' be a m i n i m a l s u b m o d e l of M?.  T h u s M a n d M! agree o n a l l g r o u n d terms, a n d M i s the  r e s t r i c t i o n of M ' t o a s m a l l e r d o m a i n .  B u t t h e n c l e a r l y they must h a v e the same set of g r o u n d  M<  (in)equalities, since n e w equalities i m p l y t h a t M is not a r e s t r i c t i o n of M ' , a n d new i n e q u a l i t i e s i m p l y t h a t | M | does not c o n t a i n the i n t e r p r e t a t i o n of some of the g r o u n d terms (since M i s a rest r i c t i o n of M * ) , w h i c h is false.  Q E D T h e o r e m 7.3  T h e o r e m 7.4 — F i n i t a r y C o m p l e t e n e s s If T is a finitely a x i o m a t i z a b l e theory, a n d every m o d e l of T is finite, then o n l y the m i n i m a l m o d e l s of T satisfy every instance of the d o m a i n c i r c u m s c r i p t i o n s c h e m a for T, DC(T).  Proof  A s s u m e e v e r y m o d e l of T is  finite.  C o n s i d e r some n o n - m i n i m a l m o d e l , M .  W e assume t h a t every  instance of DC( T) is t r u e i n M a n d a r r i v e at a c o n t r a d i c t i o n . M is finite, w i t h m e l e m e n t s i n its d o m a i n . S i n c e M is not m i n i m a l , there is a s u b m o d e l , N <  M,  - 147 -  w i t h n < m d o m a i n elements. instantiate  these x,'s i n M  L e t <&x be x = x V . . . V x = x where the x / s are v a r i a b l e s . W e c a n  3x. $ x is true, as is AXIOM($). \Px.  x  n  to be the n elements w h i c h s u r v i v e the s u b m o d e l i n g to N. A* m u s t be true, as follows:  Clearly  C o n s i d e r a n a r b i t r a r y expression,  V x . tyx Z) [ V x . $ X D \ P X ] , a n d the e x i s t e n t i a l s g i v e n b y T m u s t be satisfied i n N (since TV is a  model).  F u r t h e r m o r e , $ is true for a l l of \N\. T h u s [3x.  w i l l be t r u e i n M.  G T w i l l m e a n t h a t 3x. $ z A * z  B u t since n < m, V x . $ x is c l e a r l y false i n M, so we have a falsifying  instance  of the schema.  Q E D T h e o r e m 7.4  C o r o l l a r y 7.5 If T is a finitely a x i o m a t i z a b l e theory, a n d e v e r y m o d e l of D C ( T ) is finite, t h e n o n l y the m i n i m a l m o d e l s of T satisfy e v e r y instance of DC[T).  Proof  DC(T) is t r u e i n a l l m i n i m a l models, so there are no infinite m i n i m a l m o d e l s .  DC(T) false i n a l l  infinite models, so o n l y finite n o n - m i n i m a l models r e m a i n to be e l i m i n a t e d .  E v e r y finite m o d e l  has a m i n i m a l s u b m o d e l (there c a n ' t be a n infinite c h a i n of p r o p e r s u b m o d e l s ) . T h e o r e m 7.4 serves to r u l e out n o n - m i n i m a l finite models.  Q E D C o r o l l a r y 7.5  T h e a r g u m e n t for  - 148 Theorem 8.2 If  T |— V z . x = a  and X A  =  x  V...V  x = a „ a n d T |— a,• j= a y ,  for i j= j for g r o u n d  terms  cty...,a ; n  i n c l u d e s a l l of the predicates of L; t h e n those formulae true i n every e x t e n s i o n of : -Px'  are precisely those e n t a i l e d b y CLOSURE(T\  iPz  X; {P».  Proof L e m m a 8.2.1 shows t h a t e v e r y m o d e l for a n y e x t e n s i o n of A is ( X , { P } ) - m i n i m a l . shows t h a t e v e r y ( X , { P } ) - m i n i m a l  L e m m a 8.2.2  m o d e l of T is a m o d e l for some e x t e n s i o n of A .  F r o m these  the result f o l l o w s .  QED Theorem 8.2  Lemma  If  8.2.1  T |— V z . z = cti V...V  x = a  n  for g r o u n d terms a , . . . , a ; x  and X  n  i n c l u d e s a l l of  ' : -.Pz'  p r e d i c a t e s of L; t h e n a n y m o d e l of a n y extension of A -  the  is a n ( X , { P » -  -.Pz  m i n i m a l m o d e l for T.  Proof A n y model, M minimal.  for a n e x t e n s i o n , E, for A has d o m a i n |A4| =  Then  \P\M=  there  is  an  M'<M  Without  0<k < n}, a n d \P\M = {a ...,a \ 0 < v  T  N o w , g i v e n t h e existence of M ' , it is c l e a r t h a t E \j- Pa  k  is a c o n t r a d i c t i o n .  U {Jail^. loss  r<k).  Assume that M i s  of  (k>0  generality, or there  assume  is n o  M'<M)  so —Pajt m u s t be in E, so M \f= E, w h i c h  Hence, M i s minimial.  QED Lemma 8.2.1  Lemma 8.2.2 If  T |— V z . x = oti V...V  and X  x= a  n  a n d T \— a,- j= a y ,  for t ^= /  for g r o u n d  i n c l u d e s a l l of the predicates o f L; t h e n a n y ( X , { P } ) - m i n i m a l i  m o d e l of s o m e e x t e n s i o n of A  =  . ^Pz  not  terms  m o d e l for  a^.-.^a^ T is a  - 149 -  Proof  W e c o n s t r u c t the e x t e n s i o n , E, f r o m the m i n i m a l m o d e l , M . : -.Pa, — i n GD(E,A).  O b v i o u s l y , TU  C l e a r l y M\=  CONSEQUENTS(GD(E,A))  T. If M \= - . P a , - , p u t  t h e n entails P a , for e a c h a ,  -.Pa,s u c h t h a t Pa <£ guarantees E=  Th{TU  QED  ( O t h e r w i s e M i s not m i n i m a l ) .  CONSEQUENTS(GD(E,A)).  }  that  E \f- P a , -  for  CONSEQUENTS(GD[E,A))  the  a,-'s  which  is a n e x t e n s i o n for A .  make  T h e existence of up  M  Thus  GD.  C l e a r l y M j= E.  L e m m a 8.2.2  P r o p o s i t i o n 8.6  If T does not e n t a i l a d o m a i n - c l o s u r e a x i o m , a n d T J / - V z . ->Px, t h e n every e x t e n s i o n for A has m o d e l s w h i c h are not ( X , { P } ) - m i n i m a l .  T h e p r o o f of this p r o p o s i t i o n lies i n the o b s e r v a t i o n t h a t one c a n a l w a y s sei - Pa for some d o m a i n element a w h i c h does not c o r r e s p o n d t o a n y term- i n the language.  Since T does not e n t a i l a  d o m a i n closure a x i o m , a m o d e l w i t h s u c h a n element w i l l a l w a y s exist.  |  T h e o r e m 8.7  There  are  theories,  T,  such  that  T f— V z . z = a  x  for i j= j a n d yet no c o m b i n a t i o n of the extensions of  V . . . V JL = a A  n  a n d T' i— a J= a -, {  3  precisely  c h a r a c t e r i z e s the ( X , { P } ) - m i n i m a l m o d e l s of T.  T h e proof of this t h e o r e m follows f r o m E x a m p l e 8.2.  P r o p o s i t i o n 8.9  If there are no v a r i a b l e predicates (Z = { }), t h e n ECWA(T) of the c i r c u m s c r i p t i o n s c h e m a .  T h e p r o o f of this follows d i r e c t l y P r s y m u s i n s k i ' s [1985] t h e o r e m 1.  a d d s to T every instance  f r o m of the t h i r d c o r o l l a r y t o G e l f o n d , P r s y m u s i n s k a , a n d |  APPENDIX B  Dictionary of Symbols  Symbol G ^ U n { } A  h ¥= 3  A V 3 V  D  Definition Set membership Set non-membership Set union Set intersection T h e e m p t y set Set difference: $ - r = { a | a £ * a n d a (£ T} S y m m e t r i c set difference: I A T = ( f - T ) U ( f First-order provability First-order non-provability Logical entailment Logical non-entailment Logical implication Logical negation Logical and L o g i c a l or L o g i c a l equivalence Existential quantifier U n i v e r s a l quantifier P r e c e d i n g q u a n t i f i e r ' s scope extends over 1st enclosing f o r m u l a .  Th  T h e n u l l clause Contradiction L o g i c a l closure o p e r a t o r " I t follows t h a t " or " I m p l i e s " If a n d o n l y if  <sC <i )-»  S t r o n g precedence r e l a t i o n o n Literals x Literals W e a k precedence r e l a t i o n o n Literals x Literals Function mapping  iff,  L N Literals |  T h e first-order l a n g u a g e (i.e., a l l w e l l - f o r m e d formulae) T h e set of a l l N a t u r a l n u m b e r s T h e set of a l l a t o m i c f o r m u l a e a n d t h e i r n e g a t i o n s M a r k s end of d e f i n i t i o n , e x a m p l e , or t h e o r e m  - 150 -  APPENDIX C Useful Logical Definitions  C l a u s e - A clause is a finite d i s j u n c t i o n of literals. C l o s e d F o r u m u l a - A f o r m u l a is closed iff it c o n t a i n s no free v a r i a b l e s . G r o u n d - A n e x p r e s s i o n ( l i t e r a l , t e r m , or f o r m u l a ) is ground iff it c o n t a i n s no v a r i a b l e s . Herbrand  Universe  H(T) =  -  If  is a u n i v e r s a l theory,  T  {r[h,...,t^ | f  1  then  the  Herbrand  Universe  is a n n-ary f u n c t i o n - l e t t e r of T, a n d ty...,tne H(T)}.  of  T  is  ( T h i s is w e l l -  defined because the O-ary f u n c t i o n - l e t t e r s (or constants) p r o v i d e the base f o r the recursion.)  Herbrand  Base  -  If  is  T  H(T) = {P"(t ...,Q | F is 1  u  Herbrand Interpretation  a  universal  theory,  then  the  Herbrand  Base  of  T  is  a n n^ary p r e d i c a t e - l e t t e r of T, a n d t ...,t e H[T)}. n  u  - If T is a u n i v e r s a l theory, t h e n a Herbrand Interpretation, I, of T is  a subset of Ts H e r b r a n d base, H(T).  T h o s e a t o m i c f o r m u l a e P (t ,...,t^ G I are i n t e r p r e t e d n  1  as true i n I, a l l others are i n t e r p r e t e d as false.  Herbrand Model -  If  T is a u n i v e r s a l theory, t h e n a Herbrand  Model of T is a H e r b r a n d  i n t e r p r e t a t i o n of T w h i c h satisfies e v e r y f o r m u l a i n T, a c c o r d i n g to the u s u a l d e f i n i t i o n of satisfaction b y an interpretation.  H o r n - A set of clauses, T, is Horn iff e v e r y clause i n T c o n t a i n s at most one p o s i t i v e l i t e r a l .  L i t e r a l - A literal is a n a t o m i c f o r m u l a or the n e g a t i o n of an a t o m i c f o r m u l a .  S k o l e m i z e d f o r m - T h e Skolemized form of a theory is the t h e o r y o b t a i n e d b y c o n v e r t i n g to prenex-normal  form  then  progressively, f r o m  the  right-most  quantifier,  replacing each  e x i s t e n t i a l l y q u a n t i f i e d v a r i a b l e b y a new f u n c t i o n - s y m b o l t a k i n g as a r g u m e n t s each of the v a r i a b l e s c a p t u r e d b y quantifiers o c c u r r i n g f u r t h e r t o the left. s k o l e m i z e d f o r m o f a t h e o r y is c a l l e d skolemization.  - 151 -  T h e process of o b t a i n i n g the  

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