Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1986_A1 E83.pdf [ 8.14MB ]
Metadata
JSON: 831-1.0051930.json
JSON-LD: 831-1.0051930-ld.json
RDF/XML (Pretty): 831-1.0051930-rdf.xml
RDF/JSON: 831-1.0051930-rdf.json
Turtle: 831-1.0051930-turtle.txt
N-Triples: 831-1.0051930-rdf-ntriples.txt
Original Record: 831-1.0051930-source.json
Full Text
831-1.0051930-fulltext.txt
Citation
831-1.0051930.ris

Full Text

R e a s o n i n g 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 I n v e s t i g a t i o n s o f N o n - M o n o t o n i c R e a s o n i n g By DAVID WILLIAM ETHERINGTON B.Sc , The University of Lethbridge, 1977 M . S c , The 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 T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E S T U D I E S (Depar tment of C o m p u t e r Science) We accept this thesis as conforming to the required s tanda rd . T H E U N I V E R S I T Y O F B R I T I S H 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 ther ing ton , 1986. In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t c o p y i n g or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date J a n u a r y 29 , 1987 E R R A T A 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: LITERALS (a) should be LITERALS (a A 7). Page 131/: Every occurrence of (J should be [J. r=1 r =0 Page 148: The three occurrences of -Px • _/> r should be 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 Abstract In te l l igent behav iou r rel ies heav i l y on the ab i l i t y to reason in the absence of comple te infor-m a t i o n . U n t i l recent ly , there has been l i t t le work done on deve lop ing a fo rma l unders tand ing of how such reason ing can be per formed. W e focus on two aspects of th is p rob lem: defaul t or proto-t y p i c a l reason ing , and c losed-wor ld or c i r cumscr ip t i ve reasoning. A f t e r su rvey ing the work in the field, we concent ra te on Re i te r ' s defau l t log ic a n d the v a r i -ous c i r cumsc r i p t i ve fo rma l i sms deve loped by M c C a r t h y and others. T a k i n g a large ly semant ic app roach , we deve lop a n d / o r ex tend model - theoret ic semant ics for the fo rma l i sms i n quest ion. These and other tools are then used to char t the capab i l i t ies , l im i ta t i ons , and in ter re la t ionsh ips of the va r ious approaches. It is a rgued tha t the f o r m a l systems cons idered, wh i le in terest ing i n thei r own r ights , have an i m p o r t a n t role as spec i f i ca t i on /eva lua t i on tools vis-a-vis exp l i c i t l y c o m p u t a t i o n a l approaches. A n app l i ca t i on of these pr inc ip les is g iven i n the fo rma l i za t i on of inher i tance ne tworks i n the pres-ence of except ions, us ing defaul t logic. ii Table of Contents Abs t rac t i i T a b l e of Con ten ts i i i L i s t of F igures v i i i L i s t of T a b l e s ix Acknow ledgements x 1 C h a p t e r 1 - Incomplete In format ion 1 1.1 O v e r v i e w of the thesis 3 2 C h a p t e r 2 - App roaches to Incomplete Know ledge 5 2.1 C l o s e d - W o r l d Reason ing 6 2.1.1 N a i v e C losure 7 2.1.2 Nega t ion A s Fa i l u re T o Der ive 9 2.1.3 Database C o m p l e t i o n 9 2.1.4 Genera l i zed Rea l i za t ions of the C W A 13 2.1.5 C i r c u m s c r i p t i o n 15 2.1.5.1 P red i ca te C i r c u m s c r i p t i o n 15 2.1.5.2 F o r m u l a C i r c u m s c r i p t i o n 18 2.1.5.3 D o m a i n C i r c u m s c r i p t i o n 20 2.1.6 Res t r i c t i ng C l o s e d - W o r l d Inferences 22 2.1.7 Semant i c Interconnect ions 23 2.2 Defau l t or P r o t o t y p i c a l Reason ing 24 2.2.1 Defau l t Log i c 24 2.2.1.1 Defau l t Theor ies 24 2.2.1.2 C l o s e d De fau l t Theor ies and T h e i r Ex tens ions 26 2.2.1.3 G e n e r a l Defau l t Theor ies 27 2.2.1.4 In teract ing Defau l ts 28 2.2.2 M i n i m i z i n g A b n o r m a l i t y 30 2.2.3 N o n - M o n o t o n i c Log i c 32 2.2.4 Au toep i s tem ic Log ic 34 2.2.5 K F O P C 35 2.2.6 Ob jec t ions to N o n - M o n o t o n i c Fo rma l i sms 40 3 C h a p t e r 3 - Defau l t Log ic 42 3.1 The Semant i cs of Defau l t Theor ies 42 Def in i t i on : Sat is f iab i l i t y , admiss ib i l i t y , and app l i cab i l i t y 44 Def in i t ion : Resu l t of a default 44 De f in i t i on : Resu l t of a sequence of defaults 44 Def in i t i on : S tab i l i t y 44 T h e o r e m 3.1 - Soundness 45 T h e o r e m 3.2 - Comple teness 45 E x a m p l e 3.1 45 E x a m p l e 3.2 46 3.2 Coherence of De fau l t Theor ies 47 3.3 Orde red Defau l t Theor ies 49 Def in i t ion : « : and « : 50 iii Def in i t i on : Orderedness 51 T h e o r e m 3.3 - Coherence 51 3.4 Cons t ruc t i ng Ex tens ions 51 De f in i t i on : Ne twork Defau l t T h e o r y 53 T h e o r e m 3.4 - Convervence 54 T h e o r e m 3.5 - S t rong Convergence 54 4 C h a p t e r 4 - Inher i tance Ne tworks w i t h Excep t ions 55 T h e o r e m 4.1 61 Co ro l l a r y 4.2 61 Co ro l l a r y 4.3 61 Co ro l l a r y 4.4 61 4.1 P a r a l l e l Ne two rk Inference A lgo r i t hms 61 4.2 T h e o r y Preference 65 T h e o r e m 4.5 66 5 C h a p t e r 5 - P red ica te C i r c u m s c r i p t i o n 69 5.1 F o r m a l P re l im ina r ies 69 5.2 O n the Cons is tency of P red ica te 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 inconsistent c i rcumscr ip t ion 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 5.3 W e i l - F o u n d e d Theor ies and Pred ica te C i r cumsc r i p t i on 71 T h e o r e m 5.4 71 T h e o r e m 5.5 72 5.4 E q u a l i t y 73 5.4.1 T h e U n i q u e - N a m e s A s s u m p t i o n 73 T h e o r e m 5.6 (Rei ter) 74 T h e o r e m 5.7 74 C o r o l l a r y 5.8 74 5.4.2 T h e D o m a i n C losure A s s u m p t i o n 75 T h e o r e m 5.9 76 T h e o r e m 5.10 76 5.4.3 Some Misconcep t ions 76 T h e o r e m 5.11 76 C o r o l l a r y 5.12 76 5.5 W h a t to C i r cumsc r i be? 78 6 C h a p t e r 6 - Genera l i za t ions of C i r cumsc r i p t i on 79 6.1 F o r m u l a C i r c u m s c r i p t i o n 79 De f in i t i on : M< ^ p ^ A f * 80 Def in i t i on : £ ( P , i ) - M i n i m a l M o d e l 80 T h e o r e m 6.1 - Soundness 80 T h e o r e m 6.2 - F i n i t a r y Comple teness (Per l is and M inke r ) 80 T h e o r e m 6.3 81 6.2 Genera l i zed C i r c u m s c r i p t i o n 81 Def in i t ion : M< (x,fl)Af' : 82 iv Def in i t i on : ( X , . R ) - M i n i m a l M o d e l 82 T h e o r e m 6.4 - Soundness • 82 P ropos i t i on 6.5 82 P ropos i t i on 6.6 82 E x a m p l e 6.1 83 6.3 W e l l - F o u n d e d Theor ies 83 Def in i t ion - We l l -Foundedness 83 P ropos i t i on 6.7 (L i fsch i tz) 83 E x a m p l e 6.2 (L i fsch i tz ) 84 P ropos i t i on 6.8 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 7 C h a p t e r 7 - D o m a i n C i r c u m s c r i p t i o n 87 7.1 A Rev i sed D o m a i n C i r c u m s c r i p t i o n A x i o m Schema 87 T h e o r e m 7.1 - Soundness 88 7.2 Some Proper t ies 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 ropos i t i on 7.2 (J ios-Tarsk i Theorem) 89 T h e o r e m 7.3 90 T h e o r e m 7.4 - F i n i t a r y Comple teness 90 Co ro l l a r y 7.5 90 7.3 Re la ted F o r m a l i s m s 91 8 C h a p t e r 8 - Connec t ions Be tween Defau l t Log i c and C i r cumsc r i p t i on 92 P ropos i t i on 8.1 92 E x a m p l e 8.1 92 8.1 " T r a n s l a t i o n " f r om Defau l t Log i c to C i r cumsc r i p t i on 93 8.2 T rans la t ions f r o m C i r c u m s c r i p t i o n to Defau l t Log ic 94 T h e o r e m 8.2 94 Co ro 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 ropos i t i on 8.6 95 T h e o r e m 8.7 95 E x a m p l e 8.2 96 P ropos i t i on 8.8 (Ge l fond , P r z y m u s i n s k a , and P rzymus insk i ) 96 P ropos i t i on 8.9 97 9 C h a p t e r 9 - O p e n P rob lems 98 9.1 P r i nc ip les of N o n - M o n o t o n i c Reason ing 98 9.2 U p d a t e 100 10 C h a p t e r 10 - Conc lus ions 102 10.1 Defau l t L o g i c and Inher i tance 102 v 10.2 P red ica te C i r c u m s c r i p t i o n 103 10.3 Genera l i za t i ons of C i r c u m s c r i p t i o n 104 10.4 D o m a i n C i r c u m s c r i p t i o n 105 10.5 Re la t ions Be tween C i r c u m s c r i p t i o n and Defau l t Log i c 105 A A p p e n d i x A - Proofs of Theorems I l l B a c k g r o u n d In fo rmat ion I l l De f in i t i on : Sa t is f iab i l i t y , admiss ib i l i t y , a n d app l i cab i l i t y 112 Def in i t ion : Resu l t of a default 112 Def in i t i on : Resu l t of a sequence of defaults 112 Def in i t i on : S t a b i l i t y 112 T h e o r e m 3.1 - Soundness 113 T h e o r e m 3.2 - Comple teness 113 L e m m a 3.3.1 115 Def in i t i on 3.3.2: <K and <K 116 De f in i t i on 3.3.3: Orderedness 116 Def in i t i on 3.3.4: Un iverse of A 116 De f in i t i on 3.3.5: I : U ( A ) j - * N 117 De f in i t i on 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 Co ro 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 - Coherence 120 L e m m a 3.4.1 124 L e m m a 3.4.2 124 Def in i t i on 3.4.3: Ne two rk Defau l t T h e o r y 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 - Convergence 130 T h e o r e m 3.5 - S t rong Convergence 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 (Rei ter) 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 • 140 De f in i t i on : F o r m u l a C i r c u m s c r i p t i o n 141 Def in i t i on : M< j ^ M * 141 vi Def in i t i on : £ ( P , 5 ) - M i n i m a l M o d e l 141 T h e o r e m 6.1 - Soundness 141 T h e o r e m 6.3 142 De f in i t i on : Gene ra l i zed C i r cumsc r i p t i on 142 Def in i t i on : M< (x,/j)Af' 142 De f in i t i on : ( X , . R ) - M i n i m a l M o d e l 143 T h e o r e m 6.4 - Soundness 143 Def in i t i on : We l l -Foundedness 143 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 - Soundness 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 Comple teness 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 ropos i t i on 8.6 - 149 T h e o r e m 8.7 149 P ropos i t i on 8.9 149 B A p p e n d i x B - D i c t i ona ry of S y m b o l s 150 C A p p e n d i x C - Use fu l L o g i c a l Def in i t ions 151 vii List of Figures Figure 4.1 — F ragmen t of a t axonomy 56 F igure 4.2 — L i n k s w i t h except ions 58 F igure 4.3 — Ne two rk representat ion of our knowledge about Mo l l uscs 59 F igure 4.4 — N E T L - l i k e representat ions of our knowledge about Mo l l uscs 59 F igure 4.5 — P r o b l e m s for l oca l inher i tance a lgor i thms 62 F igure 4.6 — A mu l t i - l eve l inher i tance graph 63 F igure 4.7 — A genuinely ambiguous inher i tance g raph 67 viii List of Tables T a b l e 4.1 — Leve ls of l i terals 64 T a b l e 4.2 — Poss ib le outcomes using different p ropagat ion schemes 64 be Acknowledgements M y superv isor , R a y Re i te r , sparked m y interest i n A l , and par t i cu la r l y i n f o r m a l approaches thereto, then fostered i t w i t h bo th h is insight and his resistance to inadequate ly reasoned ideas (as we l l as w i t h financial and mora l suppor t ) . H e , together w i t h Rober t Merce r , engaged me i n hun -dreds of f ru i t fu l (and not-so-frui t fu l ) discussions w h i c h have shaped bo th the fo rm and content of m y research career. Pa r t s of chapters 4, 5, a n d 7 appear i n [Ether ington a n d Re i te r 1983], [E ther ing ton, Merce r , and R e i t e r 1985], and [Ether ing ton a n d M e r c e r 1986]. I gratefu l ly acknowledge Merce r ' s and Re i te r ' s s igni f icant con t r ibu t ions to th is thesis. M a n y thanks are due to A n d r e w A d l e r , A l e x Bo rg i da , P a u l G i l m o r e , A k i r a K a n d a , and D a v i d T o u r e t z k y for the i r generosi ty w i t h the i r t ime, the i r ideas, a n d the i r encouragement, and to R o n B r a c h m a n for a l l tha t and more. F i n a n c i a l l y , I have benef i t ted f r om predoc tora l scholarships f r o m the N a t u r a l Sciences and Eng ineer ing Research C o u n c i l of C a n a d a and the Isaac W a l t o n K i l l a m trust , as w e l l as a research assistantship sponsored b y N S E R C grant A7642 to R a y m o n d Re i te r . I a m also thank fu l for A T & T B e l l Labora to r ies ' suppor t , w h i c h I received as a v is i t ing scient ist i n M u r r a y H i l l , dur ing the fa l l of 1985. Las t and most , thanks to Jan ine for shar ing m y victor ies, commisera t ing over m y defeats, and tak ing o n the task of exp la in ing non-monoton ic i t y to our f r iends; and to those fr iends for lots of tea and love. x C H A P T E R 1 Incomplete Information T h e perverse m a x i m that whatever y o u can get away w i t h is r igh t has i ts coun -terpar t i n the c l a i m that whatever works is clear. I might not understand the devices I emp loy i n mak ing usefu l computa t ions or pred ic t ions any more t han [one] understands the car [one] dr ives to br ing home the groceries. T h e u t i l i t y of a no t ion testif ies not to i ts c la r i ty , but ra ther to the ph i losoph ica l impor tance of c lar i fy ing i t . — Ne lson G o o d m a n [1955]. H u m a n common-sense reasoning appears to re ly heav i l y upon the ab i l i t y to use general rules subject to except ions; wha t has been ca l led pro to typ ic or defaul t i n fo rmat ion . V i r t u a l l y none of the decisions one makes everyday are made w i t h complete cer ta in ty . W i t h l i t t le effort, an endless supp ly of more or less probable scenarios can be const ruc ted w h i c h cont ra ind icate any chosen course. Y e t people are not para lyzed by indec is ion; they cont inue to act and to decide i n spite of a l l this uncer ta in ty . Science fiction fans w i l l recognize " Insuf f ic ient D a t a " as a favour i te c l i che : computers are f requent ly charac ter ized as para lyzed by not hav ing enough in fo rmat ion to arr ive at a log ica l ly sound conc lus ion . If computers are restr ic ted to sound modes of reasoning based on complete in fo rmat ion then A r t i f i c i a l Intel l igence is a fut i le goal . F o r a va r ie ty of reasons, " In te l l i gence" (whatever i t m a y be) must invo lve the ab i l i t y to funct ion w i thou t complete in fo rmat ion about the wor ld . In the first p lace, complete in fo rmat ion is hard to come by, even in the most con t r i ved s i tuat ions. Cons ider , for example, a s imple " b l o c k s - w o r l d " s i tua t ion i n w h i c h there are three b locks of k n o w n d imensions, masses, and locat ions, a n d a robot man ipu la to r a r m w i t h a k n o w n l i f t ing capac i t y , effective rad ius, a n d pos i t ion. If a l l of the b locks are of a size a n d mass w i t h i n the tolerances of the a r m , can the a r m be used to stack the blocks? A t first g lance, the answer seems an obv ious " y e s " . Ref lec t ion shows that th is might be hasty. O u r in fo rmat ion about the s i tua t ion is incomple te . There m a y be th ings we know no th ing about w h i c h m a y interfere. F o r example , the a r m m a y be broken. ( T h i s argument m a y not conv ince those who say, " I f so, the ac tua l l i f t ing capac i t y of the a r m (Og) was not rea l ly known!" . ) G r a n t i n g th is , there m a y s t i l l be a w a l l between the a r m a n d the b locks — we do not know. W e can improve ou r speci f icat ion of the p rob lem to avo id such incomplete in fo rmat ion b y say ing that there is no th ing between the a r m and the b locks (not even a i r? ) , bu t we s t i l l cannot une-qu ivoca l l y answer the quest ion. A monkey m a y be ho ld ing back the a r m - the perverse m i n d can - 1 -- 2 -generate possible reasons for fa i lure indef in i te ly . W i t h o u t more in fo rmat ion , these cannot be ru led out. T h e next s tep is to a d d the in fo rmat ion that noth ing prevents the a r m f rom get t ing to and l i f t ing the b locks . N o w we c a n safely decide that the a r m can lift the b locks. O f course, if noth-ing prevents the goa l , we do not need any knowledge of b locks and arms to answer the quest ion: we have g iven too m u c h away . P u t t i n g a finer po in t on our knowledge, we might say " n o t h i n g prevents the a r m f rom func-t ion ing accord ing to spec i f i ca t ion" . W e w i l l be char i tab le , for the moment , and assume that this precludes monkeys . C a n the a r m l i f t the b locks now? W e l l , the b locks m a y be too s l ippery, may explode w h e n touched, o r any of a number of th ings " t o o r id icu lous to consider" m a y happen. It seems tha t , short of being exp l i c i t l y to ld - or ac tua l ly t ry ing - we can never know enough to decide whether a n a t tempted lift w i l l succeed. E v e n i n s i tuat ions where one in tu i t i ve l y should have complete knowledge, incompleteness m a y resul t from the imprac t i ca l i t y of represent ing a l l of the re levant in fo rmat ion . F o r example , an a i r l ine database w h i c h records flights and the ci t ies w h i c h they connect w o u l d be overwhe lmed i f i t had to keep t rack of a l l of the pairs of c i t ies not connected by each flight. If this "nega t i ve " in fo rmat ion is not exp l i c i t l y s tored, however, how can we decide whether P W 8 1 9 , w h i c h connects V a n c o u v e r and G u y a m a s , connects T o k y o and H o n g K o n g ? T h e t rad i t i ona l approach to this p rob lem has been to invoke the Closed-World Assumption. If we assume that we have complete knowledge about a l l of the posi t ive facts true of the wor ld , we can infer that any th ing we do not know to be true - such as CONNECTS(PW819,Tokyo,HongKong) - is false. If ou r knowledge about any aspect of the wo r l d m a y be incomplete , however, th is assump-t ion is obv ious ly suspect. Suppose, for example , that we want 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 area, bu t do not ye t know whether i t w i l l ac tua l ly go to N e w Y o r k or Newark . Pe rhaps the database also stores in fo rmat ion about flights' " h o m e por t " for main tenance pur-poses. W e m a y wan t to enter F i c t i c t i ous A i r l i nes 001, w i t h home port Vancouve r , so that the main tenance depar tment c a n gear up for the ex t ra aircraf t . T h e c losed-wor ld assumpt ion wou ld then a l l ow us to in fer that F A L 0 0 1 connects V a n c o u v e r to nei ther N e w Y o r k nor Newark (nor anywhere else, for that mat ter ) . T o prevent such unwar ran ted inferences, we must ret ract our assumpt ion of complete knowledge. T h u s we can no longer use the c losed-wor ld assumpt ion . A s a side-effect, ou r uncer-ta in ty about F A L 0 0 1 re int roduces uncer ta in ty about whether P W 8 1 9 connects T o k y o and H o n g K o n g . In th is case, we migh t decide to manage the uncer ta in ty by exp l i c i t l y s ta t ing for wh i ch flights we have comple te knowledge. T h e c losed-wor ld assumpt ion can then be used where i t is appropr ia te , a n d avo ided elsewhere. T h e c losed-wor ld assumpt ion is of ten made even when i ts app l i cab i l i t y cannot be guaranteed. O n e can imagine s i tuat ions - i n domains less s t ruc tured t han a i r l ine databases - i n w h i c h it m a y not be k n o w n whether the in fo rmat ion at h a n d is complete. Phys i c i s t s , for example , per iod ica l ly bel ieve tha t they have t racked down the f u l l sui te of subatomic par t ic les, and work using this assumpt ion . So far, no-one has been able to say how we w i l l know when a l l such par t i -cles have been d iscovered. In such s i tuat ions, the best course of ac t ion is of ten to act as though - 3 -one has comple te in fo rmat ion u n t i l one has reason to suspect otherwise. T h e quest ion of when to suspect otherwise then becomes qui te impor tan t . T h e pr inc ip les w h i c h guide this type of reasoning appear d i f f icul t to e luc idate. C e r t a i n l y , knowing noth ing is reason to doubt that one knows every-th ing , bu t where does one d raw the l ine? C losed -wor l d reasoning takes posi t ive facts as g iven, and sanct ions negat ive conclusions. Commonsense reasoning often requires a different sort of assumpt ion to be made. Because of the need to act , and the pervasiveness of incomplete in fo rmat ion , humans are usua l l y w i l l i ng to assume - of ten qui te unconsc ious ly - vast numbers of " n o r m a l c y " condi t ions w i thou t exp l ic i t jus t i f i ca t ion . In p lann ing to get to the a i rpor t by going out the f ront door, get t ing in to one's car, and d r i v i ng , one assumes that the door w i l l open, the car w i l l s tar t , the a i rpor t hasn ' t moved , and that one's usua l route is s t i l l passable. These assumpt ions rare ly reach the conscious leve l , unless c i rcumstances m a k e i t l i ke ly tha t they w i l l be v io la ted . F o r example, at - 4 0 ' C , one m igh t make cont ingency p lans for the car 's fa i l ing to start. S h o u l d subsequent i n fo rmat ion or ref lect ion v io la te any of these " i m p l i c i t " assumpt ions, ad justments are made; bu t the absence of v io la t ion need not be proven before assumpt ions are made. T h e k inds of assumpt ions w h i c h are made to dea l w i t h the var ious forms of incomplete in fo rmat ion cannot be sound, i n the usua l sense of never leading f rom true premises to false con-clusions. T h i s is d isappo in t ing to the pur is t . Un fo r tuna te ly , if one wants to get any th i ng done, cer ta in assumpt ions must be made. If we are w i l l i ng to forsake soundness, how do we avo id embrac ing i r ra t iona l i t y? T h e best one can hope for is some fo rm of " j us t i f i ca t i on " for one's assumpt ions; pr inc ip les w h i c h a l low gaps in one's knowledge to be filled and w h i c h guarantee that — most of the t ime - these assumpt ions w i l l not lead too w i l d l y as t ray . Dec id i ng what const i tutes the " n o r m a l " state-of-affairs and when to assume tha t th ings are indeed " n o r m a l " are impor tan t problems. C l e a r l y , one must be very good at de tec t ing a b n o r m a l cond i t ions before assuming that every th ing is no rma l . Fu r the rmore , once such assumpt ions have been made, one must be prepared to detect a n d d e a l w i t h any conf l ic t ing (or apparen t l y conf l ict ing) in fo rmat ion w h i c h turns up . 1.1. Overview of the thesis T h e thesis a t tempts to pu l l together a number of threads — aspects of var ious approaches to reasoning w i t h incomple te in fo rmat ion . T h e results presented fa l l in to two m a i n categories: those w h i c h ex tend our unders tand ing of the capabi l i t ies and l im i ta t ions of par t i cu la r approaches, and those w h i c h explore the in terconnect ions, s imi lar i t ies, and differences between approaches. (Hope-fu l ly the la t ter category is subsumed by the former.) C h a p t e r 2 presents a deta i led survey of a number of impor tan t systems for non-monoton ic reasoning. W e d raw together a number of resul ts f r om the l i terature a n d some o r ig ina l observa-t ions. T h e emphas is is on present ing a cohesive p ic ture of the field. T h e p resen ta t ion . thus a t tempts to stress the commona l i t i es a n d essent ia l differences of the var ious approaches. T h e reader shou ld be ab le t o come away w i t h an unders tand ing of bo th the prob lems and state of the - 4 -art of the field. C h a p t e r 3 consists of invest igat ions of the propert ies of a pa r t i cu la r f o rma l sys tem, Re i te r ' s l o g i c 1 for defaul t reasoning. W e present a general semant ics for default theories, a n d show how this semant ics h igh l ights the essent ia l s imi lar i t ies and d iss imi lar i t ies between defaul t logic and other non-monoton ic systems. W e then character ize a broad class of default theories w h i c h are we l l -behaved, i n the sense of preserving the coherence of the under ly ing wor ld -descr ip t ion . W e tu rn , i n chap te r 4, to a n invest iga t ion of inher i tance networks w i t h except ions. W e develop a correspondence between such networks and default theories. W e then use this correspondence to prove a number of interest ing results, i nc lud ing suff icient condi t ions for the correctness of a network inference a lgo r i thm a n d for the coherence of an inher i tance network representat ion of a body of knowledge. W e conclude b y showing that T o u r e t z k y ' s [1984a] " i n fe ren t i a l d is tance" a lgo r i t hm satisfies these cr i te r ia . Chap te rs 5 th rough 7 tu rn f rom default logic to discuss a qui te different approach to i ncom-plete in fo rmat ion , the var ious forms of m i n i m a l enta i lment or c i r cumscr ip t ion . In chap te r 5, we discuss a number of semant i ca l l y -mot i va ted pessimist ic results concern ing the capabi l i t ies of predicate c i r cumsc r ip t i on . C h a p t e r 6 looks at a genera l izat ion of predicate c i r cumscr ip t i on , ca l led fo rmu la c i r cumscr ip t i on . Mode l - theor ies are presented for some var ian ts of th is approach , and a number of results (bo th posi t ive a n d negat ive) are p roved concerning the i r power. T h e " l o n g - d e a d " d o m a i n c i r cumscr ip t i on fo rma l i sm is " resu r rec ted" in chapter 7. W e argue that this app roach prov ides an impor tan t capab i l i t y for common-sense and database reasoning systems. W e uncover and correct an error i n the or ig ina l presentat ion, and we show that a niche remains for d o m a i n c i r cumscr ip t ion by refut ing subsumpt ion c la ims made in favour of predicate (and formula) c i r cumscr ip t i on . W e conc lude the chapter w i t h some resul ts concern ing d o m a i n c i rcumscr ip t ion 's capab i l i t i es and l imi ta t ions. In chapter 8, we re tu rn to defaul t logic, this t ime in the context of ou r d iscuss ion of cir-cumscr ip t i on . W e present a number of resul ts deta i l ing the re la t ionsh ip between these rather d isparate fo rmal isms, showing the i r po ints of correspondence and the i r (unfor tunate ly) more-frequent po in ts of d ivergence. T h e thesis conc ludes w i t h a lengthy discussion of some impor tan t open prob lems and interest ing research d i rect ions, i n chapter 9, and a summary and eva lua t ion of the signif icance of the work presented, i n chapter 10. E v e r y a t tempt has been made to make the thesis as sel f -contained as possible. A fami l ia r i t y (at t imes, in t imate) w i t h first-order log ic is assumed throughout , however . (See [Mendelson 1964] for an in t roduct ion. ) T o preserve con t inu i t y , the proofs of the theorems have been re legated to A p p e n d i x A , wh i le Append i ces B and C con ta in no ta t iona l convent ions a n d def in i t ions of log ica l terms assumed elsewhere in the thesis. T h e in ten t ion has been to keep the degree of log ica l sophis-t i ca t ion requ i red to read the bu lk of the thesis to a m i n i m u m . 1 Some have objected to the use of the term " logic" (and even "formal") for the systems we discuss here. Bather than debate this issue, we encourage those who find the terminology objectionable to substi-tute whatever term(s) they find appropriate. C H A P T E R 2 Approaches to Incomplete Knowledge T r a d i t i o n a l logics suffer f rom the ' M o n o t o n i c i t y P r o b l e m ' — D r e w M c D e r m o t t In t rad i t i ona l log ica l systems, ex tend ing a set of ax ioms can never prevent the der iva t ion of conclus ions der ivab le f r om the or ig ina l set. M o r e fo rma l l y , i f 5 and S' are arb i t ra ry sets of fo rmu-lae then : S C S' — {w | 5 | - w} C { « ; | S' f - w}.1 T h e add i t i on of formulae to a set monotonically increases what can be proved f rom that set; hence such logics are somet imes ca l led monotonic. Recen t l y , i t has been noted [ M c C a r t h y 1977, M i n s k y 1975] that monoton ic logics seem inadequate to capture the tentat ive nature of h u m a n reasoning. S ince people 's knowledge about the wo r l d is necessar i ly incomple te , there w i l l a lways be t imes when they w i l l be forced to d raw conclus ions based on an incomple te speci f icat ion of per t inent detai ls of the s i tua t ion . U n d e r such c i rcumstances, assumptions are made ( imp l i c i t l y or exp l ic i t l y ) about the state of the u n k n o w n fac-tors. Because these assumpt ions are not i r refutable, they m a y have to be w i t h d r a w n at some later t ime shou ld new evidence prove t hem inva l i d . If th is happens, the new evidence w i l l prevent some assumpt ions f r o m being used; hence a l l conclus ions w h i c h can be a r r i ved at on ly i n con junc t ion w i t h those assumpt ions w i l l no longer be der ivable . T h i s causes any sys tem w h i c h a t tempts to reason cons is tent ly using assumpt ions to exh ib i t non-monoton ic behav iour . Common-sense conc lus ions are often based on bo th suppor t ing evidence and the absence of con t rad ic to ry ev idence. T r a d i t i o n a l logics cannot emulate this fo rm of reasoning because they lack any means for cons ider ing the absence of knowledge. A number of systems have been developed to address th is shor tcoming, by augment ing a t rad i t iona l first-order logic w i t h some mechan ism for p red ica t ing conclus ions on the absence of specif ic knowledge. In A l , log ic-based a t tempts to solve the prob lems presented by incomplete in fo rmat ion have fa l len in to two categories. ( F o r the purposes of th is thesis, we ignore " p r o b a b i l i s t i c " approaches.) T h e first category inc ludes those w h i c h assume tha t a l l o f the relevant positive in fo rmat ion (e.g., w h i c h ind i v idua ls ex is t , w h i c h predicates are sat isf ied by w h i c h ind iv iduals) is known . F r o m this 1 S |— w means w is provable from premises S. - 5 -- 6 -assumpt ion , i t fo l lows that any th ing wh ich is not " k n o w n * to be true must be false. Negat i ve f a c t s 2 can thus be omi t ted , since they can be inferred f rom the absence of the i r pos i t ive counter-parts. S u c h assumpt ions of complete posi t ive knowledge underl ie P L A N N E R ' S " T H N O T " [Hewit t 1972] and re lated negat ion operators i n A l p rogramming languages, semant ic networks, and databases [Rei ter 1978a, b], as w e l l as more fo rma l reasoning techniques such as predicate comp le t ion [C lark 1978], and c i r cumscr ip t ion [ M c C a r t h y 1980, 1986]. In contrast , m a n y have wan ted to represent a n d use what wou ld general ly descr ibed as " de fau l t " or " p r o t o t y p i c " in fo rmat ion . Defau l ts are used to fill gaps i n knowledge. In the absence of specific evidence, they a l low a sys tem to make (hopeful ly) en l ightened "guesses" , ins tead of reserving judgement or assuming that whatever is u n k n o w n is false. Non -mono ton i c logic [ M c D e r m o t t & Doy le 1980], default logic [Reiter 1980a], t ru th main tenance systems [Doyle 1979, M c A l l e s t e r 1978, 1980], and var ious network- a n d f rame-based procedura l knowledge representa-t ion schemes [Qu i l l i an 1968, M i n s k y 1975] a l l embody this idea. T h e two approaches are not mu tua l l y exclus ive - each of these reasoning techniques has been used to ach ieve the other. Compar i sons of the power of the two parad igms are most notable for the i r absence f rom the l i terature, however. T h e discussion in the rema inder of th is chapter does not p rov ide such a compar ison, a l though some points of correspondence are ind ica ted . 2.1. C l o s e d - W o r l d R e a s o n i n g Negat i ve facts - those w h i c h state wha t is not true about the wo r l d - vas t l y ou tnumber posi t ive facts. F o r example , i n a d iscussion at a suff iciently h igh level , every th ing w h i c h is at some place is not at every o ther p lace. S im i l a r l y , i f T u m n u s is a cat , he is not a dog, a fish, or a tree (among other th ings) . T h e amount of negat ive in format ion about a wor ld increases geometr ica l ly w i t h the size of the H e r b r a n d Un ive rse . O n e wou ld l ike to avo id hav ing to exp l i c i t l y represent a l l such in fo rmat ion . T h e in fo rmat ion must somehow be ava i lab le , however - at some po in t it m a y become useful to know that T u m n u s is not a dog. In cer ta in s i tuat ions, it is reasonable to assume that one knows a l l of the re levant t ruths. F o r example , i t is reasonable to assume that a company 's inventory database l ists a l l par ts sup-p l ied by that c o m p a n y , that one's T 4 s l ips l ist a l l deduct ions f rom one's income, and that one's e lect r ic i ty w i l l not be cu t off tomor row. S u c h assumpt ions are just i f ied ei ther by the design and in tended func t i on of the ins t rument i n quest ion or, as i n the la t ter example , by the imp l i c i t bel ief tha t i f a fact were impor tan t enough - such as the impend ing cessation of one's electr ic services -one w o u l d p resumab ly have heard about i t . If one assumes " t o t a l knowledge about the doma in being represented" , i t is no longer neces-sary to exp l i c i t l y represent negat ive in fomat ion . Negat i ve facts m a y s imp l y be in ferred f r om the absence of the i r pos i t ive counterpar ts . Re i te r [1978a] cal ls th is assumpt ion the Closed- World Assumption ( C W A ) , since i t impl ies a c losed doma in i n w h i c h a l l t ru ths are known . T h e c losed-3 A fact is negative iff a l l of the literals in its clausal form are negative. - 7 -wor ld assumpt ion o n a knowledge-base, KB, corresponds rough ly to an inference ru le of the form: If KB \f- Pthen infer -<P appl icab le to pos i t i ve facts, P. T h i s ru le can be paraphrased as " I f P is not provab le f rom the knowledge-base, assume ->P." 2.1.1. N a i v e C l o s u r e Re i te r prov ides the fo l lowing syntac t ic rea l iza t ion of the C W A , w h i c h we w i l l c a l l naive clo-sure (NC)? Def ine EKB, the negat ive extens ion of K B , as fol lows: EKB = { ->Pa\P is an n-ary predicate letter, a is an nrtuple of g round terms, and KB \f- Pet. }. T h e n the na ive closure of KB is def ined as those formulae provable f r om KB U EKB. W e wr i te KB \— N C . It is impor tan t to not ice that na ive closure extends the knowledge-base b y add ing a set of g round l i terals. U n i v e r s a l s tatements cap tur ing the C W A for par t i cu la r predicates do not gen-era l ly fo l low f r o m the naive closure of the knowledge-base. 4 F o r example , if KB = { Penguin(Opus) }, t hen EKB = { ->Penguin(Tweety), ->Penguin(Fred), ... } but KB \/r NC Vx- x fr Opus D -'Penguin(x). T o see this, not ice that we can construct a mode l for KB U EKB w i t h a doma in e lement, say a, w h i c h does not cor respond to any named by KB o r EKB and set Penguin{ct) true. A semant ic charac te r iza t ion of th is type of c losed-wor ld reasoning can be g iven i n terms of m i n i m a l He rb rand models , as ou t l ined below. W e int roduce the not ion of m i n i m a l mode l i n greater general i ty than is immed ia te l y requi red. T h i s w i l l help s impl i fy subsequent d iscussion and c lar i fy the re la t ionsh ips between the var ious formal isms we w i l l be discussing. In general , i f we are g iven an order ing re la t ion , < o n some class of in terpretat ions, I, we say that IG I is minimal i n I iff V/ ' € I. -<(/' < I) or ( / ' = J). F o r our present purposes, let P , Q , a n d Z be d is jo int sets of predicate- let ters w h i c h jo in t l y exhaust the supp ly of predicate- let ters of the language. W e can define an order ing < on sets of He rb rand in terpretat ions as fo l l ows : 5 ' 6 h < h = VP e P . \P\h C l i ^ , and V<? £ Q . \Q\h = IQ^ . In other words, the extensions of predicates i n Q are ident ica l , a n d those i n P are (not necessari ly 8 A confusion prevalent in the literature conflates 'the C W A ' wi th what we are calling 'naive closure'. 4 To 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. 6 We wil l use \I\ to represent the domain of the interpretation, / , and \P\p \t\jto represent the in-terpretation in / of the predicate, P, and term, t, respectively. - 8 -proper) subsets. Observe that no th ing is sa id about the in terpretat ions of the predicates i n Z . R e t u r n i n g to the semant ics of the C W A , it can be shown [van E m d e n and K o w a l s k i 1976] that the na ive c losure of KB corresponds to m i n i m a l i t y i n the set of He rb rand models of KB, i n the fo l lowing sense. Le t P be the set of a l l predicate symbols of L (hence Q = Z = { }). T h e n , if KB is H o r n and consistent , there is a unique m i n i m a l e lement, M, i n the class of H e r b r a n d models of KB ( in fact , M= D { M | M i s a He rb rand mode l of KB }). Fu r the rmore , for g round clause, L, KB U EKB \- L iff M\= L. T h e class of H e r b r a n d models of a theory is interest ing for common-sense reasoning because each H e r b r a n d mode l conta ins precisely the ind iv idua ls for w h i c h the theory prov ides names. In tu i t i ve ly , th is is a t t rac t ive for c losed-wor ld reasoning, since one w o u l d imagine tha t a closed wor ld wou ld con ta in no spur ious ind iv idua ls . Un fo r tuna te ly , as we have seen, i n general both KB and KB U EKB m a y have models w i t h ind iv idua ls not corresponding to any name. T h i s accounts for the fact that , wh i l e KB U EKB agrees w i t h i f B ' s m i n i m a l He rb rand mode l , M, for g round clauses, there m a y be facts true in M w h i c h do not fo l low f rom the na ive closure of KB (speci f ical ly, those w h i c h enta i l there being exact ly the n a m e d ind iv idua ls ) . If the knowledge-base entai ls that there are on ly finitely-many ind iv idua ls , it can be shown that M i s the on ly mode l (up to isomorphism) for KB U EKB. Despi te i ts at t ract iveness as a means of imp l i c i t l y represent ing negat ive knowledge, c losed-wor ld reasoning is not w i thou t shor tcomings and pi t fa l ls . T h e most obv ious of these is that there is no room for genuinely incomplete knowledge under the C W A - any th ing w h i c h is not known w i l l be assumed false. T o see the prob lems presented by incomplete in fo rmat ion , consider a da ta -base consis t ing of on ly BLOCK(A) V BLOCK(B). S ince i t is possible to der ive nei ther BLOCK(A) no r BLOCK(B), naive closure al lows the der i va t ion of ^BLOCK(A) and ->BLOCK(B). It is easy to see that such s i tuat ions lead to inconsistent conclus ions. T h e fact that some c lass ica l ly consistent databases are not consistent w i t h na ive closure leads to the quest ion , " U n d e r what c i rcumstances can na ive closure be consistent ly emp loyed?" There is no comple te charac ter iza t ion of sui table databases, a n d the on ly k n o w n suff icient cond i -t ion is that the database be H o r n and consistent. P u r e l y negat ive in format ion (clauses w i thou t posi t ive l i terals) p lays no par t in c losed-wor ld query eva lua t ion for such databases. Since nega-t ive in fo rmat ion can be reconstructed using the C W A , i t can be ignored w i thou t loss of deduct ive power [Rei ter 1978a]. A more subt le d rawback is that the " \f- " re la t ion is not effect ively computab le , since first-order p rovab i l i t y is on ly semi-dec idab le . T h u s , even where na ive c losure preserves consistency, it m a y be imposs ib le to even enumerate a l l of i ts consequences. W h i l e first-order log ic is semi -dec idable, a n d i ts theorems recurs ive ly enumerable , nei ther of these ho lds for first-order logic + na ive c losure. - 9 -2.1.2. Negation As Failure To Derive A l p rog ramming languages (e.g., P R O L O G [Roussel 1975], P L A N N E R [Hewit t 1972]) have often addressed the prob lem of negat ive knowledge by adopt ing a weakened f o r m of the C W A . T h e y represent on ly posi t ive in fo rmat ion , assuming that whatever cannot be shown to be true must be false. S u c h systems embody a weakened fo rm of the C W A because they do not fu l ly imp lement the " )f- " re la t ion. A der iva t ion of ->P t yp i ca l l y consists of an unsuccessful exhaust ive search for a der i va t ion of P. T h i s technique is ca l led negation as failure ( N A F ) . Because the search-space m a y not be finite, the search for a der iva t ion of P m a y never fa i l , even when P t ru l y does not fo l low f rom the knowledge-base. T h u s , N A F m a y not be able to find a l l of the negat ive facts i m p l i e d by the C W A . In P R O L O G , an a t tempt to prove the l i te ra l , ->P, consists of (recursively) a t tempt ing to prove P. If th is fai ls, hav ing exhausted the po ten t ia l proofs for P, then the proof of ->P succeeds. T h i s is the on ly inference ru le for negat ion, and it is appl icable on ly when P is a pos i t ive ground l i teraL C l a r k [1978] just i f ies this approach to negat ion by showing that the inference of ->P f rom a database, D B , by N A F corresponds to a proof of ->P f rom an extended database w h i c h is imp l i -c i t l y g iven by D B . (Th i s extended database is discussed in de ta i l i n the next section.) C l a r k shows that N A F can be v iewed as a der ived inference rule, a heurist ic for der iv ing negat ive facts wh i ch are (under the C W A ) imp l i c i t i n the database. Because of the requi rement of finite fa i lure, the syntac t ic fo rm of the database, as we l l as i ts log ica l content , can p lay a role i n what can be der ived by N A F . F o r example, Sheperdson [1984] points out that wh i le the databases: DB1 = {Pa} and : DB2 = { --Pa Z> Pa } are log ica l ly equ iva lent , P R O L O G can prove Pa on ly f rom DBV A n at tempt to prove Pa f rom DB2 leads to a n inf in i te proof tree: T h e subgoal ->Pa is set up , leading to a fur ther subgoa l of (fai lure to prove) Pa, ad infinitum. A l t h o u g h the a t tempt to prove Pa obv ious ly fai ls, i t does not finitely fa i l , so the fa i lure proof never returns. O f course, both databases log ica l ly en ta i l Pa, and the C W A funct ions cor rec t ly i n either. 2.1.3. Database Completion T h e C W A al lows a sys tem to act on the assumpt ion that " t h e objects that can be shown to have a cer ta in proper ty by reasoning f rom cer ta in facts are a l l the objects tha t have tha t pro-per ty " [ M c C a r t h y 1980]. It does not, however, a l l ow the reasoner to der ive this assumpt ion . Such systems c a n never be "consc ious * of the under ly ing pr inc ip les wh i ch they are imp l i c i t l y assuming. C l a r k [1978] remedies th is shor tcoming by m a k i n g the completeness assumpt ions exp l ic i t i n the database. A l l of the in fo rmat ion about a par t i cu la r re la t ion i n the database, D B , is gathered - 10 -together a n d a completion axiom is added w h i c h states tha t a par t i cu la r tup le satisfies the re la t ion on ly i n those cases where D B says it must . A p p l y i n g this process to a l l of the re lat ions i n D B yie lds the completed database ( C ( D B ) ) . T h i s comp le t ion of the database makes exp l ic i t the assumpt ions of t o ta l wo r ld knowledge. T h e database is v iewed as a set of clauses, each w i t h at most one d is t ingu ished posi t ive l i tera l . A c lause is sa id to be about the predicate occurr ing i n i ts d is t inguished pos i t ive l i te ra l . A l l of the clauses i n D B about each n-ary predicate, P, are gathered together and conver ted to equivalent imp l i ca t i ons w i t h i ' ( z 1 / . . . , z n ) as the i r consequents. T h i s imp l i ca t i ve f o rm makes c lear a l l of the cond i t ions w h i c h D B gives as suff icient for P. P red ica te comp le t ion asserts that these condi t ions are also necessary, thus y ie ld ing a definition for P. If Ei(x),...,Ek(x) are the lef t -hand sides of a l l of the imp l i ca t i ons for P{x\ in D B , then the completion axiom for P i n D B is: Vz*. P(t) D [^(5) V...V Ek(tj]. If there are no ax ioms about a predicate, the comp le t ion a x i o m says that that predicate is univer-sal ly false. T h e completed database, C ( D B ) , is the or ig ina l database, together w i t h the comple t ion ax ioms for each predicate. F o r example, the theory: Bird[ Tweety) ( l ) V s . Penguin(x) D Bird(x) (2) V z . Bird(x) A ->Penguin(x) D Flies(x) (3) gives rise to the fo l lowing imp l ica t ions about Bird: V z . x = Tweety D Bird(x) ( l ' ) V z . Penguin(x) Z> Bird(x) (2*) (Bird does not occur pos i t i ve ly i n (3)). T h u s , the comple t ion a x i o m for Bird, g iven these ax ioms is: V z . Bird(x) D x = Tweety V Penguin(x) . (4) S im i l a r l y , the comp le t i on a x i o m for Flies is: V z . Flies(x) Z> Bird(x) A ->Penguin(x) , (5) and the comp le t ion a x i o m for Penguin is: V z . -<Penguin(x) . (6) ( W e have assumed that (3) is about Flies.) Hence, C( { (1 ) , (2), (3)}) = { ( l ' ) , ( 2 ' ) , (3)-(6)}, w h i c h says the on ly b i rds are Tweety and the penguins, a n d a l l non-pengu in b i rds fly. Fu r the r -more, there are no penguins, so a l l (and only) b i rds fly. Bes ides the o r i g i na l theory and the comple t ion ax ioms, C l a r k adds " U n i q u e N a m e s A x i o m s " [Reiter 1978a]. These are inequa l i ty ax ioms s ta t ing that different names denote different objects. T h u s , for example , i f we a d d : Penguin(Opus) to the D B ( l ) - ( 3 ) , we get the new comple ted database: -11 -C(DB') = { ( l ' ) , ( 2 ' ) , (3) - (5) , Vx. Penguin(x) = x = Opus, Opus f Tweety }, wh ich enta i ls Flies(Tweety) a n d ->Flies(Opus). W i t h o u t the unique names a x i o m , C(DB') wou ld enta i l ne i ther Flies(Tweety) no r ->Flies(Opus). W h e n rest r ic ted to H o r n databases, w h i c h have at most one posi t ive l i te ra l , database c o m -p le t ion preserves consistency. However , if clauses are a l lowed to have more t han one posi t ive l i te ra l problems m a y resul t . F o r example , the c lausa l fo rm of (3), above, is: ->Bird(x) V Penguin(x) V Flies(x) . W e arb i t ra r i l y dec ided that (3) was about Flies (because it i l lus t ra ted ou r po in t ) , bu t we cou ld as easily have chosen Penguin. It is easy to see that our choice makes the comple t ion of: DB= { ( l ) - ( 3 ) , ->Flies(Tweety) } inconsistent . Because (3) is taken to be about Flies, i t is not taken in to account when ca lcu la t ing the comp le t ion of Penguin, even though i t c a n be used to infer Penguin( Tweety). Hence , the com-p le t ion a x i o m s ta t ing that there are no penguins can s t i l l be der ived, even though i t is now incon-sistent w i t h DB. Database comp le t i on c a n somet imes be consistent ly extended to non -Horn theories b y treat-ing a clause w i t h posi t ive l i terals, Lv...,Lk, as k clauses, each about a different . T h i s may a l low database comp le t ion to be app l i ed to databases conta in ing incomplete in fo rmat ion w i thout in t roduc ing inconsistencies. F o r example , the database: BLOCK(A) V BLOCK(B), w h i c h is not H o r n a n d is inconsistent w i t h na ive closure, can be rewr i t ten as: V i . \-^BLOCK\A) A i = B o BLOCK{x)},\ \fx. [-^BLOCK(B) A x = A D BLOCK(x)} j These resul t i n the consistent comp le ted database: {Vx. [BLOCK(x) = (-^BLOCK(A) A x = B) V (-iBLOCK(B) /\x= A)}, A B } , or equ iva lent ly , {[Vx. BLOCK(x) = x = A] V [Vx. BLOCK(x) = x = B], Aj= B) . Not ice tha t the comp le ted database states that there is exac t ly one b lock, and it must be ei ther A or B. T h e d is junc t ion i n the or ig ina l database, w h i c h d i d not exc lude the poss ib i l i t y of two b locks, has become " e x c l u s i v e " i n the comple ted database. T h i s app roach has two drawbacks . F i r s t , the pr ice pa id for preserv ing consistency is weak-ened conjectures. F o r example , i f a x i o m (3) is t reated as also being about penguins, the comple-t ion a x i o m for Penguin i n ( l ) - ( 3 ) becomes: Vx. Penguin(x) 3 Bird(x) A ->Flies(x) , and the comp le ted database no longer a l lows us to conc lude tha t Tweety does not fly. In fact , i t is a s imple coro l la ry o f resul ts by Re i t e r [1982] and those in chapter 5 tha t predicate comple t ion cannot be used to conjecture posi t ive facts (such as Flies(Tweety)) w i thou t r isk of inconsis tency. - 12 -A more serious d rawback , however, is that th is weakened fo rm s t i l l does not guarantee con-sistency. Shepherdson [1984] shows that the database: P[a)yP(a) (7) has an inconsistent comp le t ion , namely : Vs. P{x) = x = a A ~>P(a) . T h i s is especia l ly d is tu rb ing , since (7) is equiva lent to the t r i v i a l database, P(a). Pe rhaps con-sistency can be guaranteed by rest r ic t ing databases to some no rma l f o rm w h i c h precludes (7), but exc lud ing a l l p rob lemat i c cases w o u l d presumably require a sophis t icated a lgo r i t hm, capable of de termin ing when one set of clauses subsumes another. Such an a lgo r i thm wou ld lose some of the advantages of s imp l i c i t y and directness wh ich predicate comp le t ion enjoys. N o r m a l forms aside, the precise l im i t s of the consistent app l i cab i l i t y of predicate comp le t ion are as yet unknown . T h i s i l lus t ra tes w h a t is s imu l taneous ly a s t rength and a weakness of database comple t ion . T h e man ipu la t i ons i nvo l ved i n comple t ing the database are determin is t ic syn tac t ic t ransforma-t ions. A n y database c a n thus be effect ively comple ted w i t h re la t i ve ly l i t t le effort. T h i s same fact, however, means tha t log ica l ly equiva lent databases m a y have different complet ions. Thus , the syn tac t i c forms of fo rmulae take o n semant ic s igni f icance, w h i c h is foreign to most log ica l sys-tems. Besides somet imes leading to inconsistency, th is seems to argue against C l a r k ' s v iew that the comp le t ion ax ioms are somehow imp l i c i t i n the database. Re i te r [1984] explores the effects of add ing comple t ion ax ioms to no rma l re la t i ona l da ta -bases. He demonstrates app l ica t ions of these techniques to prob lems invo lv ing some types of incomplete i n fo rma t ion common l y encountered i n the database field, such as n u l l va lues and dis-junc t i ve in fo rmat ion . Database comp le t ion is more power fu l than a first-order sys tem augmented by N A F . C l a r k shows that , for P R O L O G programs, the structure of a fa i lure proof is i somorph ic to that of a first-order proof f r o m the comple ted database. Converse ly , the comp le t ion of the database: DB = {Penguin(Opus)} is: C(DB) = {Vs. [Penguin(x) = x = Opus}} (8) f rom w h i c h Vs. [s J= Opus D ->Penguin[x)] fol lows by first-order reasoning. F o r any par t i cu la r s j= Opus, N A F app l i ed to D B can show -<Penguin(x), but the un iversa l summary (8) is beyond i ts capab i l i t ies . ( T h i s fo l lows f rom the fact that N A F is weaker t han naive closure and naive c lo-sure cannot der ive the un ive rsa l summary . ) Database comp le t i on does not a v o i d a l l of the prob lems of N A F s imp ly because a l l of the deduct ions are first-order. There w i l l s t i l l be proposi t ions wh ich are not dec ided b y the completed database - for example , proposi t ions corresponding to those for w h i c h the exhaust ive search for a fa i lure proof never terminates. Cons ide r the database: DB = { Penguin(Opus), Vs. Penguin(father(x)) D Penguin(x) } - IS -w h i c h says that the proper ty of be ing a penguin is handed down f rom father to son. N A F cannot prove ->Penguin(Bruce) because the search for a der i va t ion of Penguin(Bruce) w i l l search forever for a pengu in among Bruce'8 pa terna l ancestors. T h e completed database, C(DB) = { V x . Penguin(x) = x = Opus V Penguin(father(x)), Bruce ^= Opus } also fai ls to en ta i l -iPenguin(Bruce). Because of the c i rcu la r i t y i n the def in i t ion for Penguin, i t cannot prove the nonpengu in 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 t h e C W A T h e C W A is the assumpt ion of complete knowledge about w h i c h posi t ive facts are true in the wor ld . A s we have seen, th is assumpt ion is not a lways appropr ia te , a n d can lead to incon-sistency if made i n s i tua t ions where knowledge is genuinely incomplete . T h i s has led a number of researchers to develop more sophist icated knowledge-c losing operators w h i c h are able to handle incompleteness i n cer ta in aspects of the K B w i thou t comple te ly ret reat ing to the " O p e n - W o r l d A s s u m p t i o n " that wha t is k n o w n is precisely wha t fol lows f rom what is exp l i c i t l y s ta ted . T o speci fy the generalized closed-world assumption (GCWA), M i n k e r [1982] also uses m i n i m a l models to character ize what fo l lows f rom the closure of the database. Res t r i c t i ng his a t ten t ion to c lausa l databases (hence to un iversa l theories) w i t h a finite set of terms, M i n k e r con-siders the set of m i n i m a l He rb rand models of the database. (Fo r n o n - H o r n theories, there m a y not be a un ique m i n i m a l H e r b r a n d model.) T h e G C W A augments the database w i t h the negat ions of a l l the l i terals w h i c h are false i n a l l of i ts m i n i m a l H e r b r a n d models. It can be shown that the resul t ing extended database is con-sistent iff the o r ig ina l database is, and that no new posi t ive clauses are der ivab le f rom the aug-mented database. T o i l lus t ra te the idea, consider the theory {BLOCK(A) V BLOCK(B), BLOCK(C) V ->BLOCK(D)}. T h i s database has nine He rb rand models: Mi = {BLOCK(A), BLOCK(B), BLOCK(C), BLOCK(D)} M2 = {BLOCK(A), BLOCK(B), BLOCK(C), -^BLOCK(D)} M3 = {BLOCK(A), BLOCK(B), ^BLOCK(C), ->BLOCK(D)} A * 4 = {BLOCK(A), -<BLOCK(B), BLOCK(C), BLOCK(D)} Ms = {BLOCK(A), --BLOCK(B), BLOCK(C), -iBLOCK(D)} A^= {BLOCK(A), ->BLOCK(B), --BLOCK(C), ->BLOCK(D)} M7= {-^BLOCK(A), BLOCK(B), BLOCK(C), BLOCK(D)} M8 = {^BLOCK(A), BLOCK(B), BLOCK(C), -^BLOCK(D)} Mg = {->BLOCK{A), BLOCK(B), ^BLOCK(C), ~>BLOCK(D)} of w h i c h M$ and M9 are m i n i m a l . Acco rd i ng l y , the G C W A sanct ions -•BLOCK(C) and ->BLOCK(D), s ince they are bo th false i n a l l m i n i m a l He rb rand models, bu t y ie lds no conclusions about w h i c h of A a n d B are b locks. T h u s , where the database cou ld consistent ly be const rued as c losed, the G C W A closes i t , but where i t is k n o w n to be incomple te (i.e., BLOCK(A) V BLOCK(B)), no conc lus ion is d rawn . - 14 -Because H o r n theories have un ique m i n i m a l He rb rand models, it is easi ly seen that this def in i t ion of the G C W A corresponds to na ive closure for H o r n theories. T h e G C W A has the advantage that i t does not overcommi t i tsel f to the pr inc ip le that a l l posi t ive in fo rmat ion is known . F a c e d w i t h a s i tua t ion where some posi t ive in fo rmat ion is c lear ly not known , judgement is reserved, ra ther t h a n b lunder ing in to inconsis tency. M i n k e r also prov ides a syn tac t ic def in i t ion of the G C W A , w h i c h he proves corresponds to the semant ic charac te r i za t ion g iven above. T h e database, DB, is extended by add ing EDB, the set of negat ions of g round a tomic formulae occur r ing i n m i n i m a l pos i t ive clauses der ivab le f rom DB. Spec i f i ca l l y : EDB - { ->Pc | MK. DB \/- (PcV K), where K is a disjunction of 0 or more positive literals such that DB \f- K } It is easi ly seen that , for H o r n theories, th is reduces to: EDB= EDB — { ->Pc* \DB\f-Pc} - the closure set generated by naive closure - since for H o r n DB and a posi t ive clause, K, DB |— (PcV K) iff DB ( - P c * o r DB \- K. M i n k e r proves the G C W A preserves consistency - DB U EDB is consistent iff DB is - and in t roduced no new posi t ive in fo rmat ion - i f K is a posi t ive clause, then DB U EDB |— i f iff DB |— K. These facts, together w i t h the fact that the G C W A subsumes naive closure indicate that the G C W A is an interest ing extension. O f course, the G C W A is even less t ractab le than naive closure (to the extent that either can be said to be t ractab le) , since it involves mu l t i p le \f-tests for each l i te ra l . T h i s suggests that naive closure might be preferred in those cases (Horn theories) where i t is app l i cab le . Ge l f ond and P r z y m u s i n s k a [1985] have developed an extension of the G C W A a n d naive clo-sure. T h e i r " c a r e f u l closure procedure" differs f rom the G C W A (and naive closure) i n that the effects of c los ing the wor ld can be const ra ined by ind ica t ing precisely w h i c h predicates m a y be affected. T h e predicates of the theory are d i v i ded in to three sets, P , Q , and Z . P consists of those aspects of the w o r l d w h i c h are to be c losed; Q conta ins the predicates w h i c h are not to be affected by the c losure; a n d the predicates i n Z m a y be affected in any way (consistent w i t h the knowledge-base) necessary to achieve m a x i m u m "c losed-mindedness" about P . T h i s ar rangement a l lows greater flexibility i n c los ing the wor ld . F i r s t l y , b y requ i r ing that cer ta in predicates not be affected by the c losure (those i n Q ) , one can avo id inadver ten t l y mak ing conclusions about , for examp le , the pr ice of tea i n C h i n a whi le one's in ten t ion was to conclude tha t the ava i l ab i l i t y of tea at the l oca l supermarket has not changed. Second ly , a l l ow ing the predicates i n Z to v a r y weakens the G C W A / n a i v e closure res t r ic t ion tha t no new pos i t ive facts be der ivab le f r om the closure of the database. T h i s means that i f one is conf ident that one has a l l the posi t ive i n fo rmat ion about P , but knows on ly cer ta in constra ints on the re la t ionsh ip between P a n d Z , then Z can v a r y as necessary to establ ish the m i n i m a l extensions for P . - 15 -T h e " c a r e f u l c losure" of DB w i t h respect to (P, Q , Z ) is def ined as DB* — DB U EDB, where EDB= { - .Pc* | V{Lh...,Ln} C (P+ U Q + U Q " ) . DB \f- Lx V...V Lm P?£ {L,}, or3k<n. DB\-LX V...V Lk}7 In tu i t i ve ly , one can assume ->Pc' unless this wou ld a l low the de r i va t ion of new facts about Q a n d / o r posi t ive P. T h e semant ic def in i t ion of carefu l closure again involves a var ian t of the no t ion of m i n i m a l H e r b r a n d m o d e l ou t l i ned earl ier, th is t ime i n i ts fu l l general i ty ( P , Q , and Z may a l l be non-empty ) . Ge l f ond and P r z y m u s i n s k a show that, for a un iversa l knowledge-base, KB, every m i n i m a l H e r b r a n d mode l of KB satisfies KB , and that KB is consistent iff KB is. It is easy to see that if Q = Z = { } then the above semant ic charac ter iza t ion is the same as that for the G C W A . Fur the rmore , if the knowledge-base is also H o r n , the same is true for naive closure. S ince G e l f o n d and P r z y m u s i n s k a do not require that the knowledge-base be func-t ion free nor have a finite set of constants, th is observat ion shows that these restr ic t ions g iven in the deve lopment of the G C W A were unnecessary. 2.1.5. 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, 1986] has presented a number of rules of conjecture for c losed-wor ld reasoning. These rules are based on syntact ic manipu la t ions , rather than consistency. Instead of the undec idab i l i t y of appeals to non-provab i l i t y on wh ich some approaches to non-monoton ic rea-soning are based, these " c i r c u m s c r i p t i v e " formal isms s imp ly add new ax ioms (conjectures). These conjectures force m i n i m a l , " c l o s e d - w o r l d " , interpretat ions on par t i cu la r aspects of the under ly ing 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 T h e most w ide ly s tud ied of these rules of conjecture is "p red ica te c i r cumscr ip t i on " [ M c C a r t h y 1980]. P red ica te c i r cumscr ip t i on a l lows expl ic i t completeness assumpt ions, s im i la r to C l a r k ' s comp le t ion ax ioms, to be conjectured as they are requi red. T h i s provides a means for closing off the w o r l d w i t h respect to a par t icu lar predicate at a par t i cu la r t ime. A schema for a set of first-order sentences is generated. T h i s schema is then ins tant ia ted by subst i tu t ing sui table predicates for the predicate var iab les i t conta ins. T h e par t i cu la r subst i tut ion(s) chosen determine w h i c h i nd i v idua ls are conjectured to compr ise the entire extension of the c i rcumscr ibed predicate. T h e semant ic i n tu i t i on under ly ing predicate c i rcumscr ip t ion is the now- fami l ia r no t ion that c losed-wor ld reasoning about one or more predicates of a theory corresponds to t ru th i n a l l models 7 If R is a get of predicates, we use R + and R~, respectively, to indicate the positive and negative ground literals over prediatea in R . - 16 -of the theory w h i c h are m i n i m a l in those predicates. Speci f ica l ly , let T ( P 1 , . „ , P J be a first-order theory, some (but not necessari ly all) of whose predicates are those in P = { P 1 ) . . . , P n } . A mode l M o f T is a P-submodel of a mode l M ' of T (wr i t ten M < p M ' ) iff the extension of each P,- i n M i s a subset of i ts extens ion i n M ' , and M a n d M ' are otherwise ident ica 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 iden t i ca l to M F o r finitely ax iomat i zab le theories, ! T ( P 1 , . . . , P n ) , M c C a r t h y [1980] proposes rea l i z ing pred i -cate c i r cumsc r i p t i on syn tac t i ca l l y by add ing the fo l lowing ax iom schema to T: 21(*i,...,*J A A [Vx*. (S^D Pg)\ Here are predicate var iab les, w i t h the same ari t ies as Plt...,Pm respect ive ly . T($lr...,$„) is the sentence ob ta ined by conjo in ing the sentences of T, then rep lac ing every occurrence of Pi,...,Pn i n T b y $ ! , . . . , $ „ , respect ive ly . T h e above schema is ca l led the (joint) circumscription schema of P1,...,Pn in T. L e t CLOSUREp(T) - the closure of T with respect to P = { P D . . . , P J -denote the theory consist ing of T together w i t h the above a x i o m schema. M c C a r t h y fo rma l l y identi f ies reasoning about T under the c losed-wor ld assumpt ion w i t h respect to the predicates P w i t h first-order deduct ions f rom the theory CLOSURE?(T). M c C a r t h y [1980] shows that any instance of the schema resul t ing f r om c i r cumscr ib ing a s in-gle predicate P i n a sentence T(P) is true i n a l l { P } - m i n i m a l models of T. T h i s general izes d i rec t ly to the jo in t c i r cumscr ip t ion of mu l t ip le predicates. A n argument due to D a v i s [1980] can be used to show that no general "comple teness" result can be ob ta ined ident i fy ing the " c i r -cumscr ip t i ve theorems" w i t h precisely those formulae true in a l l m i n i m a l models of the theory. M i n k e r and Pe r l i s [1983, 1984a] prove a " f i n i t a r y " completeness resul t , however. Spec i f ica l ly , if the o r ig ina l theory (or the c i rcumscr ibed version) entai ls that the m i n i m i z e d predicates have finite extensions, the m i n i m a l models of the o r ig ina l theory are a l l (and only) the models of the cir-cumscr ibed theory. M c C a r t h y considers the b locks-wor ld example , discussed prev ious ly , i n w h i c h a l l that is k n o w n is: BLOCK(A) V BLOCK(B)8 (9) If the predicate va r iab le , 0 , i n the c i rcumscr ip t ion of (9): [ 0 ( A ) V © ( B ) ] A Vx. \e(x) D BLOCK(x)\ D Vx. \BLOCK{x) D O(x)] is rep laced successively by the predicates x = A and x = B, the conjecture: Vx. [BLOCK(x) D x = A] V Vx. \BLOCK{x) D x = B\ (10) can be der ived . A s d i d the comple ted database, (10) says that there is only one b lock: A or B . A g a i n , the conjecture closes the wor ld and puts the " e x c l u s i v e " in terpreta t ion on the o r ig ina l d is-j unc t i on . Recall that this theory is N O T consistent with its naive closure. - 17 -T h e choice of subst i tuends is c ruc ia l in determin ing what can be ob ta ined by c i rcumscr ip -t ion . It is not c lear, i n general , how these subst i tuends are to be chosen. M c C a r t h y suggests that the desired goa l d i rects the choice of appropr ia te subst i tut ions. It remains to be seen whether this can be t rans la ted in to genera l rules. T h e re la t ionships between predicate c i rcumscr ip t ion and the var ious forms of c losed-wor ld reasoning are on ly par t ia l l y understood. Re i te r [1982] shows that predicate c i r cumscr ip t ion can somet imes be used to der ive the database comple t ion ax ioms. M c C a r t h y c i r cumscr ip t i ve l y der ives the induc t ion a x i o m for a r i thmet ic , w h i c h shows that predicate c i r cumscr ip t i on is more power fu l than database comp le t ion . Doy le [1984] has observed that c i r cumscr ip t ion is re lated to the idea of imp l i c i t def inab i l i ty as it occurs i n M a t h e m a t i c a l Log i c . A set of ax ioms, A, implicitly defines a predicate, P, i f A forces a " u n i q u e " in terpretat ion for P, or, more fo rmal ly , if A{$) Z> [VF. PF = *2] is v a l i d for each expression, of the same ar i ty as P. It is easy to see that th is schema impl ies the c i r cumscr ip t i on schema. Be th ' s De f inab i l i t y T h e o r e m [Beth 1953] guarantees that i f A imp l i c i t l y defines P then A explicitly defines P. T h a t is, A |f- VF. Px = $x where <j> is some expression using on ly symbols of A (exclusive of P). T h i s result is much-s tud ied in logic, and the k n o w n consequences inc lude methods for finding an appropr ia te <f>. In those cases where the c i r cumscr ip t ion schema ac tua l ly imp l i c i t l y defines P, these techniques can be used to reduce the schema to an exp l ic i t def in i t ion ax iom. C i r c u m s c r i p t i o n does not a lways result in an imp l ic i t def in i t ion for P. In general , i t is not even dec idable whether P is imp l i c i t l y def ined. In the Block(A) V Block(B) example c i ted above, for examp le , a l l that is obta inab le is a disjunctive definition, [Vz. Block(x) = x = A] V [Vz. Block(x) = x = B] . There are techniques for finding d is junct ive def ini t ions w i t h k d is juncts, where such def in i t ions exist , but i t is undec idab le in general whether a d is junct ive def in i t ion (or a d is junct ive def in i t ion of size k) exists. Doy le suggests that there m a y be prof i t in searching the M a t h e m a t i c a l Log ic l i terature (and enqu i r ing of ma thema t i ca l logicians) for results w h i c h m a y shed l ight on such quest ions as: 1) W h e n does c i r cumscr ip t i on imp l i c i t l y define PI D is junc t i ve ly? W h e n does it fa i l? A r e there in terest ing cases w h i c h can be character ized? Recognized? 2) W h a t does c i r cumscr ip t ion do when i t fai ls to define PI 3) W h e n are new ax ioms i r re levant to p r io r c i rcumscr ip t ions? T h a t is, when is the add i t i on of new in fo rmat ion guaranteed not to inva l ida te c i rcumscr ip t i ve ly der ived exp l ic i t def in i t ions? 4) H o w can the rev is ion of c i rcumscr ip t i ve conclusions in the face of new in fo rmat ion be mechan-ized? - 18 -W e have d iscovered a number of surpr is ing l imi ta t ions on the app l i cab i l i t y and efficacy of predicate c i r cumscr ip t i on . These are deta i led in chapter 5. 2.1.5.2. Formula Circumscription M a n y of the l im i ta t ions of predicate c i rcumscr ip t ion stem f rom the fact that on ly those predicates be ing m i n i m i z e d are a l lowed to va ry . M c C a r t h y [1986] has developed a general ized fo rm of c i r cumscr ip t i on w h i c h addresses this p rob lem. T h i s new fo rma l i sm, fo rmu la c i rcumscr ip -t ion , retains m a n y of the a t t rac t ive features of i ts predecessor, w i thout some of i ts l im i ta t ions . T h e fo rmu la c i r cumscr ip t i on ax iom looks l ike : V$. T{$) A [V?. E($,x) D E[P,x)] Z> [Vz*. E(?,X) D E@,X)} where E(P,x) is any wel l - formed expression whose free i nd i v i dua l var iab les are among ~x = xx,...,xk , and in w h i c h some of the predicate var iab les P = P1,...,Pn occur free; E(3>,x) is the result of rep lac ing each free occurrence of the predicate var iables, F,- , i n E(P,x) w i t h predicate var iab les , , of the same ar i ty . The re are three m a i n differences between the predicate c i rcumscr ip t ion schema and the for-m u l a c i r cumscr ip t i on a x i o m . F i rs t , the former is a f irst-order ax iom schema, wh i le the la t ter is a second-order a x i o m . M c C a r t h y suggests that th is is advantageous because it a l lows the results of one c i r cumscr ip t i on to par t ic ipate i n subsequent c i rcumscr ip t ions. However , this feature is not essential ; the second-order a x i o m can be rep laced w i t h a first-order schema. A l t h o u g h weaker, the first-order-schema va r ian t appears adequate for many appl icat ions [Perl is and M i n k e r 1986]. Invest igat ions in to the re la t ive advantages and disadvantages of second-order ax ioms vs first-order schemas for c i r cumscr ip t i on are s t i l l con t inu ing , and the quest ion of the va lue of adopt ing a second-order logic remains undec ided. T h e second new feature of fo rmu la c i rcumscr ip t ion is that a rb i t ra ry predicate expressions, ra ther t han s imple predicates, m a y be m in im i zed . M c C a r t h y [1984, personal communicat ion] sug-gests that th is is an inessent ia l change, since the same effect cou ld be ind i rec t ly ob ta ined by in t ro-duc ing new predicates, w i t h ax ioms def ining these predicates as equivalent to the requi red expres-sion. W h i l e this is true for fo rmu la c i rcumscr ip t ion , we show in chapter 5 that predicate cir-cumscr ip t i on cannot dea l w i t h such def in i t ions. W e also discuss add i t i ona l mechan isms wh ich are somet imes used to augment predicate c i r cumscr ip t ion w h i c h a l low def in i t ions to be c i rcumscr ibed. These mechan isms do not a lways preserve consistency, however. T h e th i rd , and most s igni f icant, i nnova t ion is that the predicates a l lowed to v a r y are no longer ident i f ied w i t h those being m in im i zed . T h i s is reflected in the fact that P [al ternately, $ ] m a y con ta in predicate var iab les not occur r ing in E(P,x) [respectively, i ? ($ ,z ) ] (and vice versa). T h i s separat ion a l lows c i r cumscr ip t ion to operate in r i ch ly connected wor lds. P r o v i d e d predicates w h i c h wou ld be a l tered by the m i n i m i z a t i o n of the expression i n quest ion are among those ident i f ied as " v a r i a b l e " , c i r cumscr ip t ion can have the desired effect. C h a p t e r 6 describes a mode l theory we have developed for fo rmu la c i r cumscr ip t i on , a long the l ines of M c C a r t h y ' s [1980] semant ics for predicate c i rcumscr ip t ion . F o r fo rmu la - 19 -c i rcumscr ip t ion , the appropr ia te no t ion of submodel is one in w h i c h the extensions of the var iab le predicates are a l lowed to expand or contract , p rov ided that the extension of E(P,x) contracts. T h e extensions of the predicate parameters (those predicates w h i c h are not among the predicates designated as var iab le) must be iden t i ca l in a mode l and its submodels . A mode l is m i n i m a l if i t has no proper submodels . It is shown that fo rmu la c i rcumscr ip t ion is sound w i t h respect to this mode l theory; any th ing der ivab le f rom the c i rcumscr ibed theory is true in a l l m i n i m a l models of the o r ig ina l theory. Per l i s and M i n k e r 1986] consider the completeness of the f i rs t -order-schema va r ian t of for-m u l a c i r cumscr ip t i on w i t h respect to th is mode l theory. T h e y present results analogous to their finitary completeness results for predicate c i r cumscr ip t ion [M inke r and Per l i s 1983, 1984a]. These results pa r t i a l l y answer some of Doy le ' s [1984] quest ions about the re lat ionship between cir-cumscr ip t ion and exp l i c i t / d i s junc t i ve def inabi l i ty , a t least i nasmuch as they establ ish exp l ic i t and d is junct ive def inab i l i t y as suff icient cond i t ions for the completeness of fo rmu la c i r cumscr ip t ion . These resul ts have yet to be extended to the case of second-order fo rmu la c i r cumscr ip t ion . L i f sch i tz [1984] has s tud ied second-order fo rmu la c i r cumscr ip t i on and der ived cer ta in cond i -t ions under w h i c h the second-order c i rcumscr ip t ion a x i o m can be reduced to an equ iva lent first-order a x i o m . S u c h equivalences improve the usefulness of fo rmula c i rcumscr ip t ion , in some cases, by e l im ina t ing bo th the need for a second-order logic and the p rob lem of finding the " r i g h t " sub-st i tu t ions. L i f sch i tz defines a fo rmu la to be separable in P iff i t can be wr i t t en in the fo rm: V A [Vz. Ei{x) z> PfS) ] A [V?. P(i) D Fffl) where C,-, Et, and F,- are P-free formulae. Essent ia l l y , a formula is separable i f i t is not recursive in P. L i f sch i t z proves that the second-order fo rmu la resul t ing f rom c i rcumscr ib ing P i n a separ-able fo rmu la , A, a l low ing on ly P to va r y is equiva lent to a first-order fo rmu la w i t h about the same log ica l comp lex i t y as A. In itself, th is result is not ve ry exc i t ing , since second-order c i r cumscr ip t ion of P w i t h only P var iab le is subject to the same l im i ta t ions chapter 5 out l ines for predicate c i r cumscr ip t ion . L i fs -ch i tz also shows, however , that the c i rcumscr ip t ion of P in A w i t h P and Y var iab le is equivalent to the c i r cumscr ip t i on of P in [3 Y. A] w i t h only P var iab le . Fu r the rmore , if A is separable i n Y, then [3 Y. A] is equ iva lent to a first-order fo rmu la w i t h comp lex i t y lower than A. W h i l e these t ransformat ions do not a lways preserve separabi l i ty [Reiter, personal communica t ion ] , i t appears that these techniques m a y be useful for e l im ina t ing the second-order quant i f iers in t roduced by for-m u l a c i r cumscr ip t i on - w i thou t re- in t roduc ing the awkwardness of a x i o m schemata and " r i g h t " subst i tu t ions. A n o t h e r i nnova t ion due to L i fsch i tz is to m in im ize accord ing to a rb i t ra ry pre-orders (ref lexive, t rans i t ive b ina ry re lat ions) , ra ther than s imple subset re la t ions. Spec i f ica l ly , i f X is an n-tuple of predicate, func t ion , a n d / o r constant letters of T, and X ' is an n-tuple of predicate, func t ion , a n d / o r i n d i v i d u a l var iab les of corresponding types and ar i t ies, then the general ized cir-cumscr ip t ion a x i o m has the fo rm: - 2 0 -? p q A V x ' . T p c ' ) A ( x ' < R x ) D ( x < K x ' ) where < ^ is an appropr ia te pre-order. T h e use of pre-orders a l lows a number of interest ing and potent ia l l y useful extensions to cir-cumscr ip t i on . F o r example , the pre-order X < RY defined by (Vz. XlX Z> YlX) A ((Vz. YlX D XlX) Z> (Vz. X2X D K2z)) al lows the jo in t m i n i m i z a t i o n of the unary predicates Xx and X% w i t h the m i n i m i z a t i o n of X± hav-ing a " h i g h e r p r i o r i t y " . T h e effect of a l low ing X to inc lude constant and funct ion letters is to a l low constants and funct ions to va r y du r ing the m i n i m i z a t i o n process. It appears that - for languages w i t h finite sets of constants - i t is possible to c i r cumscr ip t i ve l y conjecture new facts about equa l i t y , inc lud ing unique names ax ioms, by a l lowing constants to va ry . Unfor tunate ly , L i f sch i tz ne i ther mot ivates nor discusses the va r i ab i l i t y of terms i n de ta i l . A semant ic exp lanat ion of the process invo lved has yet to appear . In chapter 6, we show that a l lowing c i r cumscr ip t i ve ly var iab le terms corresponds to weakening the def in i t ion of submode l i n the semant ic charac te r iza t ion of fo rmula c i r cumscr ip t i on by d ropp ing the requi rement that a mode l and i ts submodels share ident ica l in terpreta t ions o f constant and funct ion symbols . W e also show that this approach can lead to some unexpected consequences. 2 . 1 . 5 . 3 . D o m a i n C i r c u m s c r i p t i o n In database and commonsense reasoning, it is often necessary to assume that the on ly i nd i -v idua ls whose existence is re levant to some task are those requi red to exist b y wha t is known about the task. In such s i tuat ions, the domain-closure assumption is made [Reiter 1980a]. Th i s is the assumpt ion tha t the " w o r l d " conta ins on ly ind iv idua ls whose existence is requ i red by the ava i lab le in fo rmat ion . Re i te r observes that this assumpt ion is imp l i c i t in re la t iona l database theory, where it is enta i led by the manner in wh i ch un iversa l queries are t reated. T h u s , for exam-ple, in the educat ion database: Teacher(Smith) Student(Brown)< Teacher(Jones) Student(Black) Teacher(Plato) Student(Aristotle) w i t h an in tegr i ty const ra in t speci fy ing that the sets of teachers and students are d is jo int , even the s imple query , " W h o are a l l of the teachers?" cannot be answered w i thout imp l i c i t l y assuming that the d o m a i n consists of on ly the l is ted ind iv idua ls . In cases where there are on ly finitely many ind iv idua ls , th is assumpt ion can be s ta ted using domain-closure aiioms. These are ax ioms of the fo rm: Vz. z = r 1 V . . . V z = tn (11) where the t,- are g round terms. A n y mode l sat is fy ing (11) w i l l have at most n d i s t inc t ind iv idua ls i n i ts d o m a i n , those corresponding to the Re i te r [1980a, 1984] shows that domain-c losure - 21 -ax ioms have an impor tan t role in log ica l ly fo rmal iz ing the theory of re la t iona l databases. E v e n when the doma in cannot be enumerated to fo rm a d o m a i n closure a x i o m , useful res-t r ic t ions can somet imes be put on the size and compos i t ion of the doma in by conjectur ing that i t coincides w i t h the extens ion of some predicate or func t ion whose extens ion is (part ly) known . F o r example, in the educa t ion database discussed above, i f i t is known is that teachers are employees and students are not, assuming doma in closure a l lows one to conjecture that teachers are the on ly employees. B y conjectur ing that the d o m a i n consists on ly of teachers and students (i.e., Vx. Teacher(x) V Student(x)), i t becomes possible to deduce that there are no non-teacher employees (regardless of whether a l l of the teachers and students are known) . Domain-c losure ax ioms are also impor tan t w i t h respect to a var ie ty of c losed-wor ld reason-ing formal isms. Pe r l i s and M i n k e r [1986], for example , show that the effects of pred icate and for-m u l a c i r cumscr ip t i on [ M c C a r t h y 1980, 1986] c a n be more precisely charac ter ized i n con junc t ion w i t h c losed-domain theories. S im i l a r l y , C l a r k [1978] requires domain-c losure ax ioms i n the development of his predicate comple t ion approach. G i v e n the impor tance of domain-c losure ax ioms, the quest ion arises: W h y not exp l i c i t l y add them to theories? P r o b a b l y the most impor tan t reason is that the appropr ia te domain-c losure a x i o m m a y not be obv ious. T h e repercussions of choosing too strong or too weak a n a x i o m ( inconsistency or loss of useful conjectures, respect ively) argues in favour of a more au tomat i c approach. Fu r t he rmore , as the state of the wo r l d (or the system's knowledge) changes to br ing more ent i t ies in to cons iderat ion, the same mechan ism cou ld be used to generate new doma in -closure ax ioms. In cer ta in cases, d o m a i n c i rcumscr ip t ion provides such an au tomat ic mechan ism. A c t u a l l y the first of the c i rcumscr ip t ive formal isms, doma in c i r cumscr ip t ion [ M c C a r t h y 1977, 1980; D a v i s 1980] is in tended to be a syntac t ic rea l iza t ion of the model - theoret ic doma in -closure assumpt ion . It prov ides a mechan ism for conjectur ing doma in closure ax ioms, e l im ina t ing the need to exp l i c i t l y state them. T o c i rcumscr ibe the doma in of a sentence, A, M c C a r t h y proposes add ing the schema: Axiom($) A A * D V i . $(x) (12) to A. Axiom(<&) is the con junc t ion of $ a for each constant s y m b o l a and Vzi-.-in. [$Z i A---A 3>x„] 3> $/xi...in for each »-ary funct ion symbo l / . A * is the result of rewr i t ing A, rep lac ing each un iversa l or ex is tent ia l quant i f ier, 'Vx.' or '3z.', i n A w i t h 'Vz.$x Z) ' or '3x.$x A ', respect ive ly . T h i s a x i o m schema represents the conjecture that the doma in of discourse is no larger than it must be g iven the sentence A. F o r any predicate, i f $ is true for a l l i nd i v idua ls whose existence is g iven by the constant terms, th rough funct ion app l i ca t ion , or by ex is tent ia l quant i f i ca t ion , a n d i f a l l i nd iv idua ls in $ ' s extension satisfy a l l of the un iversa l ly quant i f ied for-mulae, then $ is assumed to con ta in the ent ire d o m a i n . If the extension of some predicate meet-ing these requi rements is known , then the doma in is (assumed to be) comple te ly known . T h e semant ic i n tu i t i on under ly ing d o m a i n c i r cumscr ip t ion is minimal entailment: on l y those models w i t h m i n i m a l domains shou ld be considered i n determin ing the consequences of the g iven in fo rmat ion . In th is connect ion , a mode l , M, of a sentence is said to be a submodel of another - 22 -model , N, i f M is the rest r ic t ion of N to a subset of ffs doma in . A mode l is sa id to be minimal if it has no proper submodels. Dav is [1980] shows that every instance of (12) is true in a l l m i n i m a l models of the o r ig ina l sentence A. T h i s result is correct for those theories w i t h at least one con-stant symbo l . In chapter 7, however, we show that inconsistency results when c i rcumscr ib ing theories whose prenex no rma l forms conta in no leading ex is tent ia l quant i f iers and no constant symbols . W e also present a s imple , easi ly mot i va ted so lut ion. T h i s leads to a rev ised vers ion of doma in c i r cumscr ip t i on w h i c h is shown to preserve consistency. 2.1.6. Restricting Closed-World Inferences O n e m a y wan t to d o c losed-wor ld reasoning to f o rm conjectures about the under l y ing p r inc i -ples govern ing a s i tua t ion . In this case, one is mak ing un iversa l ( induct ive) conjectures about the state of the wo r l d . T h i s is the type of reasoning w h i c h is i nvo lved in deduc ing laws, such as " a n unsuppor ted object drops when re leased" . In many cases, however, c losed-wor ld reasoning y ie lds stronger conjectures t han m a y be desirable. F o r example, i t is often sufficient to conc lude that the s i tua t ion immed ia te l y at hand does not have cer ta in propert ies. In day- to -day reasoning, one is usual ly interested in fo rming par t i cu la r conjectures in a id of comple t ing a pa r t i cu la r deduct ion. These conjectures shou ld be of as l im i ted scope as possible whi le s t i l l s t rong enough to a l low the desired goa l to be ach ieved. Thus , for example , i f we knew that Twee ty is a b i rd and that a l l b i rds except penguins fly, we migh t wan t to conjecture that T w e e t y cou ld fly (and hence that Twee ty is not a penguin) . It is un l ike ly that we wou ld want to conjecture that there are no penguins at all, however. " 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 inker & Per l i s 1984b] provides one means for de l im i t i ng the effects of c losed-wor ld reasoning. T o prevent the c i rcumscr ip t ion of P i n a theory, A, f rom conjec-tur ing that iS*s are not P°s, the predicate, S, is protected by weakening the c i r cumscr ip t ion schema to: A($) A [Vz*. ($z*A -Si) 3 Pxl D [Vi*. (Px A ~>Sx) D *x] . T h e conclusions of protected c i rcumscr ip t ion app ly on ly to those ind iv idua ls that do not satisfy the protected predicate. T h u s , for example , M c C a r t h y [1984, personal communicat ion] has sug-gested that one m a y w ish to conclude on ly that there are no penguins present. A s s u m i n g that there is a predicate, Present(x), w h i c h says that an i nd i v i dua l is in the immedia te v i c in i t y , pro-tect ing -<Present wh i le c i rcumscr ib ing Penguin w i l l resul t i n conjectures wh ich say noth ing about those penguins w h i c h are not present. Us ing fo rmu la c i r cumscr ip t ion , the scope of conjectures can be l im i ted by con jo in ing a pro-tec t ing pred icate w i t h the expression to be m in im i zed , and not a l low ing the pro tec t ing predicate to va ry . F o r example , to m in im ize present penguins w i t h respect to a theory, A, wh i le protect ing possible " a b s e n t " penguins, the fo l lowing c i r cumscr ip t i on 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 no th ing new about 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 , and P r z y m u s i n s k i [1985] have extended the " c a r e f u l c losure" not ion of G e l f o n d and P r z y m u s i n s k a [1985], by a l lowing the theory to be augmented w i t h the negat ions of a rb i t ra ry formulae meet ing admiss ib i l i t y c r i te r ia . T h i s is more power fu l than add ing on ly nega-t ions of g round a tomic formulae. Ge l f ond , P r z y m u s i n s k a , and P r z y m u s i n s k i restr ict the i r a t ten-t ion to fixed-domain theories, those w i t h ax ioms s ta t ing that there are f in i te ly many ind iv idua ls , and that each t e r m of the language denotes a unique i nd i v idua l . L e t P , Q , and Z be as in sect ion 2.1. T h e n a fo rmu la , K, not i nvo lv ing l i terals f rom Z , is free for negation iff there is no ground clause, B, made up of l i tera ls i n P + U Q + U Q ~ such that T \- K V B and T \f- B. T h e n the extended CWA for T is denned as: ECWA(T) = T U { ^K\K is free for negat ion i n T } . Us ing the same par t ia l -order re la t ion on models as Ge l fond and P r z y m u s i n s k a [1985] (see sect ion 2.1.4), Ge l f ond , P r z y m u s i n s k a , and P r z y m u s i n s k i c l a i m that the set of formulae free for negat ion in T are precisely those whose negations are true in every m i n i m a l mode l of T. (Here we refer to m i n i m a l i t y over a l l , not jus t Herb rand , models.) Thus , for consistent, funct ion-free, f i xed-domain theories, T, ECWA[T) is cons is ten t , 9 and corresponds precisely to the formulae true in a l l m i n i m a l models of T. It fo l lows that the free-for-negation formulae character ize the results of for-m u l a c i r cumscr ip t i on for such theories. In fact, because of the fixed-domain property , one need on ly consider those K w h i c h are con-junc t ions of l i tera ls f rom P + U Q + U Q ~ It can be shown that ECWA(DB) corresponds, syntac-t i ca l l y , to the carefu l closure of DB if DB is a fixed-domain theory. T h e semant ic correspondence fol lows f rom the fact that every m o d e l of a fixed-domain theory is isomorph ic to a He rb rand mode l , and hence every m i n i m a l mode l to a m i n i m a l He rb rand mode l . B y su i tab ly ma tch ing the model - theory to the proof- theory, i t is possible to show that , for fixed-domain theories, predicate c i rcumscr ip t ion corresponds to the G C W A and , for H o r n theories, to naive c losure. These observat ions show how cent ra l the not ion of m i n i m a l mode l is to the var ious formal isms for c losed-wor ld reasoning. T h e two forms of m i n i m i z a t i o n - of extensions of predicates and of the doma in of the mode l (hence produc ing a fixed-domain model) - suffice to connect t hem a l l . 0 Every consistent, finite-domain theory has at least one minimal model. - 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 Never ut ter these words: 'I do not know this, therefore it is false. ' O n e must s tudy to know, know to unders tand, unders tand to judge. — A p o t h e g m of Ne ruda A l l of the approaches discussed so far prov ide ways of becoming more " c l o s e d - m i n d e d " . E a c h funct ions by restr ic t ing the set of models for the g iven ax ioms. T h e goal has been to al low on ly m i n i m a l models, i n w h i c h on ly a m i n i m a l set of predicate instances or d o m a i n elements necessary to sat isfy the ax ioms is a l lowed. T h e complementa ry approach also involves restr ic t ing the set of models considered. Ra the r t han focussing on m i n i m a l i t y , the systems discussed in the sequel p rov ide more flexibility in deter-m in ing w h i c h models are considered " i n t e res t i ng " . 2.2.1. D e f a u l t L o g i c Re i te r [1978a, 1980a] addresses the prob lem of incomplete in fo rmat ion by a l low ing new inference rules to be added to a s tandard f i rst-order logic. These rules sanct ion thei r conclusions prov ided that the set of beliefs satisfies the condi t ions out l ined i n thei r premises. U n l i k e s tandard logic, these premises are a l lowed to refer both to what is known and to what is not known . T h e lat ter p roper ty a l lows rules to be added that specify inferences that w i l l be made on ly when specif ic in fo rmat ion is miss ing. These inferences can be used to ta i lo r the comple t ion of par t ia l knowledge, un l i ke c losed-wor ld reasoning, wh i ch invo lves a un i form comple t ion st rategy. 2.2.1.1. D e f a u l t T h e o r i e s A default is any expression of the f o r m : 1 0 A ( x ) : frffl,..., Bm{x) VJ{X) where A(x), J3,(i*), a n d UJ(X) are a l l formulae whose free var iab les are among those i n 5*= xi,...,xn. A, Bfy and w are ca l led the prerequisite, justifications, and consequent of the default , respect ively. If none of A, Bj, a n d w con ta in free var iab les, the defaul t is said to be closed. If the prerequisi te is empty , i t m a y be taken to be any tauto logy. T w o classes of defaul ts hav ing on ly a single jus t i f ica t ion, B(x), are d is t ingu ished. Those w i t h B(x) = u>(~x), are sa id to be normal, whi le those w i t h B[x) = w(x) A G{x), for some C(x), are ca l led semi-normal. V i r t u a l l y a l l of the defaults 1 0 This notation differs from Reiter'a in 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. - 25 -occur r ing i n the l i terature fa l l in to one of these two categories, (tyukaszewicz [1985] argues that the remain ing class of s ingle- just i f icat ion defaults, where B(x) \f= w{x) are i l l -mo t i va ted , and we know of no app l i ca t i on for mul t i - jus t i f i ca t ion defaults.) Defau l ts serve as rules of inference or conjecture, augment ing those no rma l l y p rov ided by first-order logic. U n d e r cer ta in condi t ions, they sanct ion inferences w h i c h cou ld not be made w i t h i n a s t r i c t l y first-order f ramework. If the i r prerequisi tes are k n o w n and thei r just i f icat ions are "cons is ten t " (i.e., the i r negat ions are not p rovab le) , then thei r consequents can be inferred. T h u s the te rm " j us t i f i ca t i on " is seen to be somewhat mis lead ing, since just i f icat ions need not be known , mere ly cons i s ten t . 1 1 T h e consequent 's status is ak in to that of a belief, subject to rev is ion should the just i f icat ions be den ied at some future t ime. It is th is character is t ic w h i c h induces the non-monoton ic behav io r of defaul ts. Defau l t rules can be seen to have a great dea l i n c o m m o n w i t h many prev ious ly ment ioned approaches. F o r example , the C l o s e d - W o r l d A s s u m p t i o n states: If \f- VJ infer ->w w h i c h can be represented i n default logic by: In fact, (13) w i l l la ter 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 assignments w h i c h can be a t tached to f rame slots i n K R L [Bobrow & W i n o g r a d 1977] also appear to be re lated. K R L prov ides a mechan ism for ob ta in ing a va lue for a slot i n the absence of a " be t t e r " va lue . A K R L defaul t va lue , d, for a slot , s, in a frame instance, /, can be v iewed as: I f ¥- s(f) J= d i n f e r «W = d or, i n defaul t logic, as: :s(f) = d s(f) = d S i m i l a r mechan isms are ava i lab le in many other f rame-based knowledge representat ion schemes [M insky 1975]. A c losely re la ted approach is Sandewa l l ' s [1972] "Unless" operator. "Unless[P)" is inter-preted as " \f- P", and "Unless" terms are a l lowed in the const ruct ion of wffs, w i t h results l ike: A A Unless(B) D C w h i c h corresponds rough ly to: A :^B C ' "Unless" was or ig ina l l y proposed as a so lu t ion to the f rame p rob lem [Hayes 1973]. Ra the r t han hav ing to have exp l i c i t ax ioms s ta t ing tha t the propert ies of objects rema ined inva r ian t f rom s i tuat ion to s i tua t ion unless exp l i c i t l y changed, Sandewa l l suggested that these " f r a m e ax i oms" be 1 1 In a modal logic with the operator K (know) the justifications 5,- might appear as ->K->Bi. - 2 6 -rep laced by a frame inference rule l ike: IS[obje c t,prope rty, situation) Unless(ENDS(obiectpropertv.Successor( situation.act))) IS(object,property,Successor(situation,act)) w h i c h can be in terpre ted: If an object has a proper ty i n a s i tuat ion, it can be conc luded to re ta in that proper ty in the successor s i tua t ion resul t ing f rom performing 'ac t ' , unless it can be shown otherwise. N o fo rmat ion ru les were p rov ided for "Unless", however, so quest ionable formulae such as: A 3 Unless(B) can be const ruc ted. T h e semant ics of such formulae are, at best, di f f icult to determine. Sandewa l l also fai ls to prov ide any f o rma l unders tand ing of the impac t of the " Unless" ru le on the under ly-ing logic. Defau l t logic has, to some extent , remedied these shortcomings. 2 . 2 . 1 . 2 . C l o s e d D e f a u l t T h e o r i e s a n d T h e i r E x t e n s i o n s A default theory, A , is an ordered pair , (D, W). D is a set of defaults; W is a set of first-order formulae. Re i te r [1980a] describes the extensions of a default theory as "accep tab le sets of beliefs that one m a y ho ld about an incomple te ly specif ied wor ld , W°. D is v iewed as extending the first-order knowledge of W i n order to prov ide in format ion not der ivable f r om W. Since defaul ts a l low reference to wha t is not provable i n the de te rmina t ion of wha t is prov-able, the " t h e o r e m s " of a default theory are not so easy to generate as are those of a f irst-order theory. W h a t is provab le bo th determines and is determined by wha t is not provable . T o avo id this apparent c i r cu la r i t y , the theorems of a default theory are defined by a f ixed-point construc-t ion . A n extens ion, E, for A is required to have the fo l lowing propert ies: W C E ThL{F) = E „ , , .. , A: Bi,...,B F o r each defaul t , G D, if A G E, and - i 5 , , . . . , - . B m & E w then w e E. These propert ies state that E must con ta in a l l the k n o w n facts, that E must be closed under the |— re la t ion , and that the consequent of any defaul t whose prerequisi te is sat isf ied by E, and whose just i f icat ions are consistent w i t h E, must a lso be in E. Re i te r defines an extension for a closed defaul t theory to be a m i n i m a l fixed-point of a n operator hav ing the above character is t ics. T h e extensions of a default theory select restr ic ted subsets of the models of the under ly ing first-order theory , W. A n y mode l for an extension of A w i l l also be a mode l for W, but the con-verse is general ly not t rue. Defau l t theories need not a lways have extensions, even when W is consistent. The re are, however, cer ta in classes of theories for w h i c h the existence of at least one - 27 -extension is guaranteed. Theor ies w i t h on ly no rma l defaul ts have been shown a lways to have extensions [Reiter 1980a]. In chapter 3, we prove the same result for cer ta in classes of theories w i t h semi -no rma l defaul ts. Re i te r [1980a] presents an i terat ive mechan ism for decid ing whether a set of fo rmulae forms an extens ion for a theory, A . T h e method is, unfor tunate ly , not sui table for const ruc t ing exten-sions. T h i s is because a n oracle is requi red wh ich can decide whether a par t i cu la r fo rmu la 's nega-t ion w i l l be in the set. Re i te r [1980a] and E the r ing ton [1982] also present const ruct ive mechan isms app l icab le to n o r m a l theories and to a rb i t ra ry finite theories, respect ively. Some examples of defaul ts were presented in the preceeding sect ion. T h e fo l lowing example i l lustrates the extensions induced by the c losed-wor ld default on the theory: W= {BLOCK(A) V BLOCK(B)}. T h e c losed-wor ld defaul t is real ly a defaul t schema w h i c h is app l icab le to any pos i t ive g round l i te ra l . In th is case, it results in the fo l lowing set of no rma l defaults: ( -.-iBLOCK(A) :^BLOCK(B) ) \ ->BLOCK(A) ' ^BLOCK(B) j T h e theory, (D, W), has two extensions, E1 and E^. Et = Th({^BLOCK(A), BLOCK(B)}) E2 = Th({BLOCK(A), -^BLOCK(B)}) Note that E= Tk({BLOCK(A), BLOCK(B)}) is not an extension. L i ke database comple-t ion and c i r cumscr ip t i on , the c losed-wor ld default sanct ions the exclusive in terpre ta t ion of d is-junc t ions to w h i c h it is app l ied . In tu i t ive ly , th is is because the defaul ts force as many things to be false as possible, resul t ing i n extensions whose models m a y be m i n i m a l models for W. M o r e precisely, E is not an extension because it v io lates the m in ima l i t y cond i t ion of the def in i t ion of extensions. (Were W also to con ta in both BLOCK(A) and BLOCK(B), E wou ld be the on ly extension.) No t i ce how the extensions Ex and E2 manifest W ' s inconsistency w i t h the C W A . T h e incon-sistent assignments for BLOCK(A) a n d BLOCK(B) are s t i l l obta inable, bu t they are separated in to o r thogona l , self-consistent extensions. In fact, Re i te r has shown that the extensions of any default theory w i l l a lways be self-consistent p rov ided that the first-order theory W is consistent, and that a l l the extensions of a no rma l default theory w i l l be (pairwise) mu tua l l y inconsistent . 2.2.1.3. General Default Theories In cont rast to c losed defaul ts, an open defaul t is one in w h i c h at least one of A ( z ) , B^x), or w(x) con ta in free var iab les in z*. A n open default is in terpreted as s tanding for the set of c losed defaul ts ob ta inab le by rep lac ing i ts free var iab les by ground terms. If the set of g round terms is in f in i te th is resul ts i n a defaul t theory w i t h an inf in i te set of defaults. - 28 -M o s t interest ing default theories are not c losed. Cons ider what , by now, must be the arche-t y p a l defaul t theory : Vz. Penguin(x) ~D Bird(x), Vz. Penguin(x) D ->Can-Fly(x), Vz. Dead-Bird(x) D Bird(x), W= 4 Vz. Dead-Bird(x) Z> -> Can-Fly (x), 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 defaul t , w h i c h is not c losed, might be interpreted as " I f z is a b i rd , and it is consistent that x can fly, conc lude that it c a n " . T h i s theory a l lows one to conclude, for an a rb i t ra ry b i r d (e.g., Twee ty ) , that i t can fly - unless one is to ld that i t cannot , or that i t is a penguin , an ost r ich, or dead. T h e conc lus ion m a y la ter have to be revoked should Twee ty tu rn out to be a pengu in , but c o m m o n sense seems to sanc t ion the same conc lus ion. T h i s is par t l y because people tend to assume that they have the relevant in fo rmat ion i n most s i tuat ions (cf. l ingu is ts ' use of G r i c e ' s Conve rsa t i ona l Impl ica tures [Griee 1975]: one of these is that a l l in fo rmat ion necessary to inter-pret an ut terance is expected to be conta ined in the utterance.) 2.2.1.4. Interacting Defaults T h e i r b road app l i cab i l i t y and the guarantee of coherence makes no rma l defaul ts a t t rac t ive for knowledge representat ion and reasoning. The re are, however, some types of knowledge w h i c h n o r m a l defaul ts cannot comple te ly character ize. F o r example , Re i te r and Cr i scuo lo [1983] have not iced that defaul ts somet imes in teract w i t h one another, and that no rma l defaul ts cannot ade-quate ly cons t ra in these in teract ions. O n e mani fes ta t ion of this occurs when two defaul ts w i t h dis-t inct bu t not mu tua l l y exclus ive prerequisi tes have cont rad ic to ry consequents. In such cir-cumstances i t is not a lways c lear w h i c h default shou ld be app l ied. Commonsense reasoning usu-a l ly prefers one of the compet ing defaul ts by v i r tue of i ts prerequisi te being more speci f ic, mak ing the defaul t app l icab le for on ly a subset of those ind iv idua ls for w h i c h the compet ing defaul t is app l icab le . T h i s preference cannot be enforced using on ly no rma l defaul ts. F o r example , assume we are g iven : T y p i c a l adu l ts are emp loyed . T y p i c a l h igh-school dropouts are adul ts . T y p i c a l h igh-schoo l dropouts are not emp loyed . T h i s m a y be expressed b y the fo l lowing n o r m a l defaults: - 2 9 -{ Adult(x) : Employed(x) Dropout(x) : Adult(x) Dropout(x) : ->Employed(x) Employed(x) ' Adult(x) ' -<Employed(x) F o r a g iven a dropout , th is theory can be seen to have two extensions w h i c h differ on h i s / h e r state of emp loymen t . In tu i t ion d ic tates that we assume s /he is unemp loyed . C a r e f u l cons idera-t ion shows that the conf l ic t arises because t yp i ca l dropouts are not typical adul ts ; th is a t yp i ca l i t y should b lock the t rans i t i v i t y f rom Dropout through Adult to Employed. T h e first defaul t incor-porates no exp l i c i t reference to these except iona l c i rcumstances w h i c h shou ld b lock i ts app l i ca -t ion . O n e w a y to address th is p rob lem is to require that the case under cons iderat ion not be a k n o w n except iona l case. T h i s requi rement is then added to the jus t i f i ca t ion. T h u s the first defaul t above becomes: Adult(x) : Employed(x) A -'Dropout(x) w h i c h is not app l icab le to k n o w n dropouts. S e m i - n o r m a l defaul ts can be used to resolve the ambigui t ies resu l t ing f rom the in teract ions between defaul ts . T h i s is done by m a k i n g in teract ions exp l ic i t , as except ions to the app l i cab i l i t y of defaul ts. The re are three major object ions to th is approach, however. F i rs t , the comp lex i t y of theories w i t h semi -norma l defaults is subs tan t ia l l y greater than 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 may force conclusions ob ta ined f rom pre-v ious ly app l ied defaul ts to be ret racted. T h i s phenomenon, wh i ch cannot occur w i t h no rma l default theories, precludes the type of s t ra ight forward proof theory developed by Re i te r [1980a] for n o r m a l theories. Second ly , it is possible to so overconst ra in the interact ions between defaults that the resul t -ing theory has no extens ion. C h a p t e r 3 explores ways of guaranteeing that this does not happen, but , for comp l i ca ted theories w i t h m a n y in teract ions, i t m a y be di f f icul t to detect such overcon-st ra in ing . F i n a l l y , in teract ions must be not iced and exp l i c i t l y dealt w i t h at the t ime new knowledge is g iven to the sys tem. In a large, comp l i ca ted , sys tem this is l i ke ly to be an enormous task. T h e cont r ibutors of new knowledge m a y not be aware of a l l possible in terat ions between their cont r i -but ions a n d the rema inder of the knowledge base. In secur i ty-conscious env i ronments , con t r i bu -tors m a y not even be a l lowed access to some of the in format ion w h i c h interacts w i t h thei r con t r i -bu t ion . T o u r e t z k y [ l984a,b] argues tha t exp l ic i t con t ro l of in teract ions i n defaul t theories is inap-propr ia te, for the reasons ou t l ined above and because many of the ambigu i t ies in t roduced by such in teract ions can be reso lved using more general pr inc ip les. In semant ic network systems, w h i c h can be v iewed as corresponding to default theories (see chapter 4), the s tandard such pr inc ip le is the "shor tes t -pa th heu r i s t i c " , w h i c h resolves ambigui t ies by preferr ing wh ichever conc lus ion can be reached by t ravers ing the smal lest number of network arcs. E the r i ng ton [1982] shows how to const ruct networks w h i c h defeat the shortest-path heur is t ic and other s imp le -m inded amb igu i t y reso lut ion techniques. T o u r e t z k y [1984a] presents a more sophis t icated amb igu i t y resolut ion device, the inferential distance topology, w h i c h appears to capture the in tent ion of the shor test -path heur is t ic w i thout i ts Employed(x) - 30 -naive rea l i za t ion . H e explo i ts the subc lass/superc lass relat ions, wh i ch are one of the raisons d'etre for semant ic networks, to arb i t ra te between r i v a l conclusions. In the " D r o p o u t " example, above, since Dropout is a (default) subclass of Adult, the inferent ia l d istance order ing perfers con-clusions associated w i t h Dropout (i.e., unemployed) over those associated w i t h Adult (i.e., employed) , i n accord w i t h our in tu i t ions. In chapter 4, we discuss T o u r e t z k y ' s approach in more deta i l , and show its re la t ionsh ip to defaul t logic. In spi te of the fact that i t is app l icab le on ly i n subclass/superc lass hierarchies, the success of T o u r e t z k y ' s app roach in agreeing w i t h the in tu i t i ve ly acceptable conclusions (a vague cr i te r ion , to be sure) suggests that i t m a y be possible to e lucidate some set of general pr inc ip les w h i c h avo id the necessity of ad hoc man ipu la t i ons of the knowledge base. F i n d i n g and eva luat ing such p r inc i -ples remains a n impor tan t open p rob lem. 2.2.2. Minimizing Abnormality Defaul t reasoning can invo lve con jectur ing bo th posi t ive and negat ive instances of predi -cates. T h i s w o u l d seem to preclude the use of any of the c losed-wor ld o r c i r cumscr ip t i ve fo rma l -isms, discussed ear l ier , in s i tuat ions where general default reasoning is requ i red. (In chapter 5, th is is shown conc lus ive ly i n the case of predicate c i rcumscr ip t ion. ) E x p a n d i n g on an idea first presented (to our knowledge) by Levesque [1982], M c C a r t h y [1986] and Groso f [1984] have exp lored the poss ib i l i t y of using fo rmu la c i rcumscr ip t ion for default reasoning. Essent ia l l y , the idea i nvo l ved is that i f defaults represent the propert ies of " n o r m a l " ind iv idua ls , then there is " s o m e t h i n g a b n o r m a l " about an i n d i v i d u a l who does not fit the defaul t patern. B y appropr ia te ly ax iomat i z ing abnorma l i t y , it is possible to do defaul t reasoning by c i rcumscr ib ing abnorma l i t y . A n i n d i v i d u a l m a y be no rma l in some respects and abno rma l i n others; few, i f any , are ever to ta l ly " t y p i c a l " . T h u s , some al lowance must be made for these di f fer ing aspects of abnorma l i t y . M c C a r t h y exp l i c i t l y in t roduces these aspects into his onto logy, speak ing of the (ab) no rma l i t y of par t i cu la r aspects of an i nd i v idua l . Grosof , preferr ing not to prol i ferate objects undu ly , instead has a var ie ty of abno rma l i t y predicates, each corresponding to abnorma l i t y of a pa r t i cu la r aspect in M c C a r t h y ' s no ta t ion . A n example helps to c lar i fy the me thod . W e fol low M c C a r t h y ' s no ta t ion : - 31 -Vz. Thing(x) A ->ab(aspectl(x)) D -*Fly(x) Vz. Bird(x) D ab(aspectl(x)) Vz. Bt'rd(z) A ->ab(aspect2(x)) D F/y(z) Vz. Penguin(x) D a6(a5pect2(z)) Vz. i>enou»*n(z) 3 B»'rd(z) Vz. Penjtttn(z) A -<ab(aspect3(x)) D ->F/y(z) 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&(aspect5(z)) D ->.Fty(z) aspecti(x)-aspect^x) are the aspects, and a t is the abnorma l i t y predicate on aspects of i nd i v i du -als. G i v e n Thing(Theodore), c i r cumscr ib ing ab(x) va ry ing ab and Fly a l lows us to conclude ->ab(aspecti(Theodore)) and hence ~>Fly{Theodore). G i v e n Bird(Tweety), c i r cumsc r ip t i on w i l l y i e l d ab(aspect1(Tweety)), ->ab[aspect2XTweety)), and Fly[Tweety). If Opus is a penguin , the con-jectures w i l l be ab(aspect1(Opus)), ab(aspect2{Opus)), ->ab(aspect3(Opus)), and -<Fly(Opus). T h i s re fo rmu la t ion of default reasoning as c losed-wor ld reasoning about abno rma l i t y can dea l w i t h m a n y of the prob lems of in teract ing defaults that forced the cons iderat ion of semi-no rma l default theories. The direct defaul t representat ion of the above example looks l ike : Thing(x) : ->Fly[x) Ostrich(x) : -<Fly(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) In order to preserve a unique extens ion, the comp l i ca ted interact ions between the defaul t state-ments must be exp l i c i t l y ref lected in the rules. In t roduc ing abnorma l i t y a l lows a normal defaul t representat ion consis t ing of the f irst-order ax ioms (14), together w i t h the single c losed-wor ld defaul t : : ->ab(x) -iab(x) Groso f [1984] has developed a t rans la t ion scheme, using abnorma l i t y predicates, w h i c h he c la ims produces representat ions of n o r m a l default theories in a fo rm sui table for c i rcumscr ip t i ve default r e a s o n i n g . 1 2 H e is cur rent ly seeking a way of ex tend ing th is approach to arb i t ra ry semi-n o r m a l defaul t theories. T h e re lated p rob lem - whether 06 can be used to reduce semi -norma l defaul t theories to n o r m a l defaul t theories - also remains open. In fact, the representation correctly translates only prerequisite-free normal defaults. - 32 -M c C a r t h y [1984, personal communica t ion] has discovered that i t is possible to r u n into in te rac t ion problems w i t h 06 predicates. A u g m e n t i n g (14) wi th : Vz. Canary(x) A ->ab(aspectG(x)) D Bird(x) Vz. Gangster-Canary[x) Z> Canary(x) Vz. Gangster-Canary(x) D ab(aspect6(x)) , to a l low the poss ib i l i ty that " c a n a r y " may be used i n the sense of o ld gangster-movies, m a y result i n a m b i g u i t y . Dinsdale the Canary must be abnormal w i t h respect to ei ther aspect^ or aspect^ C i r c u m s c r i b i n g ab can on ly conjecture ei ther that Dinsdale flies (because he is a b i rd and hence a b n o r m a l i n aspect^, or that he is an abnorma l canary ( in aspect^. M c C a r t h y [1986] has pro-posed a var ian t of fo rmula c i rcumscr ip t ion , p r io r i t i zed c i rcumscr ip t ion , w h i c h a l lows several expressions to be s imul taneously m i n i m i z e d according to some par t i cu la r precedence. T h i s can e l iminate undesirable interactions, bu t at the cost that the precedence must be e x p l i c i t l y worked out before c i r cumscr ib ing . T h e cr i t ic i sms appl ied to semi-normal default representations, that interact ions must be k n o w n a n d accommodated when knowledge is represented, a p p l y equal ly to the p r io r i t i zed c i r cumsc r ip t ion of abnormal i ty representations. W h e t h e r such interact ions can be dealt w i t h w i thou t des t roying the conceptual c la r i ty and naturalness of the ab representat ion scheme is u n k n o w n . 2.2.3. Non-Monotonic Logic M c D e r m o t t a n d Doy le [1980, M c D e r m o t t 1982] propose a fo rmal i sm complementa ry to default logic, w h i c h they c a l l non-monotonic logic ( N M L ) . U n l i k e default logic , w h i c h uses the no t ion of consistency on ly at the " m e t a " level ( in the inference rules), N M L centres a round the in t roduc t ion of consistency in to the object language. T h e first incarna t ion of N M L [ M c D e r m o t t & D o y l e 1980] consists of a s tandard first-order logic, augmented w i t h an " A f operator, roughly equivalent to the fami l i a r " \f-—1 " . T h e set of theorems is defined as the intersect ion of a l l of the fixed-points of an operator, NM. Essent ia l ly , NM produces the log i ca l closure o f the o r ig ina l theory together w i t h as m a n y assertions of the form Mq as possible. T h e set o f theorems can be contrasted w i t h the extensions of a default theory, each of wh ich is a fixed-point. T h i s indicates that non-monotonic theoremhood is, i n some sense, a more conservative or restr ict ive concept than extension membership . M o o r e [1983a] suggests that this difference can be understood by v iewing fixed-points as sets of beliefs an agent might come to ho ld g iven his premises, whi le the intersect ion of the fixed-points determines wha t an outside observer cou ld infer about the agent's beliefs k n o w i n g on ly his premises. In fact, the extensions of default theories and the fixed-points of non-monotonic theories are incomparable i n general. T h e two formal isms often agree, as w o u l d i n t u i t i v e l y be expected, g iven that any default: A : Bv...,Bm w can be app rox ima ted i n N M L by: - 33 -A A MBX A -A MBm D w. There are, however , defaul t theories w h i c h have extensions even though the corresponding non-monoton ic theories have no fixed-points, and vice versa (see [Reiter 1980a] for examples) . D a v i s [1980] suggests that i t m igh t be impossible to assign a reasonable semant ics to the M operator were it i nc luded in the object language. M c D e r m o t t a n d Doy le point out that MP, in tu i -t i ve ly read as "P is cons is ten t " , is not necessari ly inconsistent w i t h -<P\ Moo re [1983a] observes that th is is caused by the lack of any p roh ib i t i on , i n the fixed-point const ruc t ion , against ->P and MP being con ta ined in a single fixed-point. Thus , i n Moore ' s terms, -<P may be be l ieved w i thou t the s ta tement "->P is b e l i e v e d " (->MP or L->P) being bel ieved. T h i s a l lows weaker in terpretat ions to be p laced on M t h a n the in tended " i s cons is tent " . These and other problems led to the recast-ing of the theory i n terms of a more c lass ica l m o d a l logic [McDermo t t 1982] . 1 3 T h e resul t ing non-monoton ic S5 is un for tunate ly redundant , since it is no more power fu l than S5. Because of this, M c D e r m o t t suggests fa l l ing back to non-monoton ic S4 or non-monoton ic T . T h i s suggestion is pecul iar , s ince M c D e r m o t t acknowledges that the character is t ic a x i o m of S5 (->LP D L->LP) - if P is not be l ieved, it is be l ieved not to be bel ieved - seems appropr iate for any bel ief sys tem. How-ever, the col lapse of non-monoton ic S5 was seen to force this retreat. Moo re [1983a, b] argues that th is retreat is i l l -mot iva ted . H e goes on to show tha t the co l -lapse of non-monoton ic S5 is ac tua l l y due to the a x i o m LP Z> P, wh i ch says that whatever is be l ieved is true. W h i l e this ax i om is appropr ia te for knowledge, Moore c la ims that a non-monoton ic sys tem is ac tua l l y deal ing w i t h belief, and that an ax iom stat ing the in fa l l ib i l i t y of an agent shou ld be expected to lead to pecu l ia r consequences. As ide f rom the quest ion of the i r appropr iateness, M c D e r m o t t presents no proofs of the con-s istency of non-monoton ic T and S4. Such proofs are a necessary step i n the deve lopment of non-monoton ic T and S4. In the second paper on N M L , M c D e r m o t t [1982] acknowledges the restr ic t iveness of bel iev-ing on ly those formulae in the intersect ion of a l l the fixed-points of a theory. He proposes a " b r a v e robot " w h i c h wou ld bel ieve a l l of the formulae of some par t i cu la r fixed-point. Such an approach is requi red in order to prov ide an in tu i t i ve ly sat isfactory semant ics for Mp: "p is con-sistent w i t h what is be l i eved" . T h e ava i l ab i l i t y of the " A f terms in the language has advantages and d isadvantages. F o r example, it can be shown that sentences of the form: p 3 Mq where p and q are a rb i t ra ry formulae, are ei ther redundant or inconsistent [E ther ington & Mercer , 1982, unpub l i shed notes]. (Th is fol lows because the " theo rems" of any N M L theory must inc lude a l l fo rmulae Mp w h i c h are not inconsistent.) Such sentences cannot be fo rmed i n defaul t logic, but are read i ly ava i lab le i n N M L (as they are i n Sandewal l ' s fo rmal ism) . 1 3 A discussion of modal logics is beyond the scope of this proposal. See [Hughes and Cresswel 1972] for an introduction. - 3 4 -O n the posi t ive side, the defaul t rules can be man ipu la ted by the theory. F o r example , in the no rma l defaul t theory w i t h no ax ioms and the defaul ts: impl ies M B and MB Z> B, f rom wh ich B can be inferred. T h i s appears to be more in accord w i t h no rma 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 moda l logic ians wou ld agree that "p is prov-ab le" impl ies " p is t rue " , the converse is usual ly not accepted. Hughes and Cresswe l l [1972, p28] conclude that " n o i n tu i t i ve l y p lausib le m o d a l sys tem" wou ld have such a thesis. T h i s indicates that there may be fundamen ta l problems w i t h N M L . 2 . 2 . 4 . A u t o e p i s t e m i c L o g i c M o o r e [1983a, b] prov ides a de ta i led c r i t i c i sm and reconst ruct ion of N M L . He begins by dis-t inguish ing between defaul t reasoning and "au toep i s tem ic " reasoning. T h e lat ter is def ined to be what goes on in an idea l ly ra t iona l agent reasoning about her own beliefs. It is th is type of reason-ing - not defaul t reasoning - that N M L at tempts to mode l , accord ing to Moore . Moo re sees a N M L ax iom of the fo rm: as say ing not " T y p i c a l b i rds can fly", as M c D e r m o t t and Doy le interpret it, but ra ther " T h e on ly b i rds w h i c h do not fly are those known not to fly". R e a d in this way , ax ioms such as (15) become statements about the state of an agent 's knowledge, not about t yp i ca l i n d i v i d u a l s . 1 4 H a v i n g made th is d is t inc t ion , Moo re points out that default and autoepis temic reasoning are nonmonoton ic for different reasons. Defau l t reasoning is tentat ive, and thus defeasible. It pro-v ides p lausib le grounds for ho ld ing cer ta in beliefs, but these beliefs m a y have to be ret racted shou ld those grounds prove to have been merely p lausib le, rather than true. Au toep is temic rea-soning makes on ly v a l i d inferences. P r o v i d e d that the premises are true, the conclusions fo l low w i t h a l l of the force of logic beh ind them. Non-mono ton ic i t y enters because autoepis temic state-ments are context-sensitive or indexical. T h e y exp l i c i l y refer to the entire knowledge context that conta ins them. T h u s , thei r meaning changes depending on what is known . Obv ious l y , what fo l -lows f rom not know ing a w i l l ho ld when a is not known , but may not ho ld if a is learned. M o o r e argues that the possible sets of beliefs an ideal ly ra t iona l agent can ho ld based on a consistent set of premises, A, are those sets, T, such that noth ing can be inferred about B. T h e corresponding non-monoton ic theory: {A A MB D B, ->A A MB D B} Vx.Bird(x) A M[Can-Fly(x)) D Can-Fly(x) (15) " Similar arguments can be applied to default logic and other consistency-based non-monotonic for-malisms. - 3 5 -T = Th(A U {LP\Pe T) U {^LP\P<£ T}) , where LP means "P is be l i eved" . These sets he cal ls the stable expansions of A. A stable expan-sion inc ludes the premises, accurate ly character izes what is and what is not be l ieved, and inc ludes no beliefs not suppor ted by the premises. M o o r e shows that stable expansions con ta in a l l and on ly those formulae w h i c h are true in every in terpreta t ion wh ich satisfies a l l of the premises and makes LP true for every fo rmu la , P , i n the expans ion. T h e impor tan t th ing to not ice here is that i f the premises (wh ich m a y con ta in imp l i ca t ions f r om wha t i s / i s not bel ieved) are true, and the set of beliefs corresponds to the beliefs con ta ined i n a par t i cu la r expans ion, then a l l of (and only) the formulae i n that expans ion can be t rue. T h i s in tu i t i ve ly corresponds to the idea that the different conclusions one can d raw f rom incomple te ly specif ied knowledge w i l l be comple te ly de termined by what one chooses to bel ieve. U n l i k e N M L , autoepis temic logic ( A E L ) is a propos i t iona l m o d a l logic. N o p rov is ion is made for i n d i v i d u a l var iab les o r quant i f iers. M o o r e [1984, personal communica t ion ] suggests tha t i t shou ld be a re la t ive ly easy mat te r to ex tend A E L to i ts first-order counterpar t , p rov ided that the M operator is app l ied on ly to c losed formulae. T h i s means that no M occurs w i t h i n the scope of a quant i f ier , so the prob lems of " q u a n t i f y i n g i n " to the scope of a moda l i t y are avo ided . Unfor-tunate ly , m a n y of the statements one wou ld l ike to make using a first-order vers ion of A E L invo lve quan t i f y ing in . F o r example, to say " A l l of the a's are k n o w n " seems to require an a x i o m of the form: Vx. a(x) D La(x) ( in N M L , th is w o u l d be wr i t ten as Vx. Af->a(x) Z> ->a(x)), but the free 'x' i n 'La (x) ' is captured by the un iversa l quant i f ier outside the m o d a l context . T h e problems of quan t i f y ing in are an impor -tant topic i n the s tudy of moda l logics (c./. [L insky 1971]), but the p rob lem has yet to be s tud ied in dep th f rom the perspect ive of non-monoton ic i t y . L i k e m a n y other non-monoton ic reasoning systems, A E L was presented non-const ruc t ive ly [Moore 1983a, b]. Ne i the r the semant ic basis nor the syntact ic rea l iza t ion of that semant ics pro-v i ded a mechan ism for enumerat ing the theorems of a g iven theory. Moore [1984] has recent ly developed an a l ternat ive semant ic charac te r iza t ion based on the fami l i a r K r i pke - s t y l e possible-wor lds structures. In th is semant ics, i t is possible to enumerate a l l of the in terpretat ions (each mode l is finitely specif iable and , if the language is finite, there are on ly finitely m a n y ) , decide w h i c h of these are mode ls , and determine what is true in those. Moo re [1984, personal c o m m u n i -cation] po in ts out that , for theories w i t h few propos i t iona l constants, th is is an easy task. One migh t hope for a more di rect means of a r r i v ing at the theorems of a n autoepis temic theory, but th is remains an open p rob lem. 2 . 2 . 5 . K F O P C Levesque [1982, 1984] approaches the p rob lem of incomplete in fo rmat ion di f ferent ly. Instead of immed ia te l y addressing the task of comple t ing the incompleteness, he considers what an incomple te knowledge-base m igh t be expected to know about i ts own knowledge, and wha t one - 3 6 -might reasonably ask or te l l such a knowledge base. T h i s entai ls quest ions of what const i tutes a reasonable answer to a query under incompleteness, and what an incomplete knowledge base can be expected to know after being to ld a par t i cu la r fact. A f t e r developing a log ica l f ramework in w h i c h to ta lk about ( incomplete) knowledge, Levesque discusses ways in w h i c h th is f ramework can be app l i ed to comple t ing knowledge bases. In fo rma l i z ing knowledge (or " r a t i o n a l b e l i e f ) , Levesque [1982] makes three basic assump-t ions about the nature of knowledge bases. These are: 1) Cons is tency : T h e knowledge in a knowledge base is self-consistent. There is some possible state-of-affairs tha t makes every th ing that is known true. 2) Compe tence : E v e r y unknown sentence is false i n some wor ld compat ib le w i t h every th ing that is known. I.e., a l l l og ica l consequences of what is k n o w n are known . 3) C losu re : T h e knowledge base has complete and accurate self-knowledge. A n y sentence wh ich deals on ly w i t h the state of the knowledge base w i l l be known to be t rue or false accord ing as i t is t rue or false, respect ive ly , for the knowledge base. These assumpt ions lead to a first-order moda l logic of knowledge ( K F O P C ) , rough ly s imi la r to " w e a k " S5 [Stalnaker 1980]. T h i s logic character izes the beliefs of an idea l ly ra t iona l agent capable of ref lect ing on her own beliefs. A n y sentence wh ich is not k n o w n is k n o w n to be unk-nown (and vice versa), knowledge is log ica l ly c losed, and the agent bel ieves i n the ve rac i t y of her beliefs. T h i s last s tatement does not m e a n that every belief is true, mere ly that the agent believes a l l of her beliefs to be t rue. One of the most surpr is ing aspects of this system is that, a l though K F O P C al lows one to te l l / ask the knowledge base th ings that cannot general ly be phrased in first-order logic, the query and update mechan isms, and the knowledge base itself, are a l l first-order representable. Levesque presents a func t ion that t ranslates any update or query into an equivalent first-order sentence w i t h respect to a par t i cu la r knowledge base. Unfo r tuna te ly , the mapp ing is not a pa r t i a l recur-sive func t ion : In the genera l case, choosing first-order representabi l i ty means that effectiveness must be t raded for some heur is t ic component . Levesque does not explore whether there are effective subcases. Non -mono ton i c i t y enters K F O P C in two ways. T h e first, most obv ious, comes f rom the assumpt ion of closure. T h a t is, i f KB does not know P, it knows this (i.e., KB \—K--KP). C l e a r l y , though , KB U {P} \- KP, and hence KB U {P} \j- K^KP. A s Levesque points out, ^KP means on ly that P is not cur rent ly known, not that i t w i l l never be known . Levesque's so lu t ion to this p rob lem is to have sel f-knowledge come on ly f rom in t rospect ion: f rom imp l i c i t , rather than exp l ic i t , s tatements. S u c h a KB w i l l never con ta in statements of the fo rm ->KP: " . . . any assert ion w i l l be a statement about the wo r l d and not the KB. If the assert ion ta lks about wha t is k n o w n ... i t is on ly do ing so to help make a statement about the wor ld . T h u s , there is no w a y to te l l a KB about itself... . T h i s is to be expected, howev-er, s ince a KB has been assumed to have complete and accurate knowledge of i tself at any t ime . " T h e non-mono ton ic i t y of s tatements about lack of knowledge is not par t i cu la r l y p rob lemat ic when t reated in th is way . - 37 -A second fo rm of non-monoton ic i t y arises when lack of knowledge is used as a premise in deduct ions. F o r examp le , the " F l y i n g B i r d s " defaul t can be expressed in K F O P C as: V*. Bird(x) A ^K-^Fly(x) D Fly(x) (16) - any b i r d not k n o w n not to fly c a n fly. S u c h statements a l low conclusions about wha t is t rue i n the wor ld to be based on what is not k n o w n by the current knowledge base. S ince the state of the knowledge base can change, non-monoton ic i ty can ex tend to encompass more t han pure ly intros-pect ive statements. T h i s representat ion of defaults in K F O P C is subject to the prob lems of in te rac t ion that plague n o r m a l defaul t theories [Reiter & Cr i scuo lo 1983]. F o r example, the fact that A u s t r a l i a n b i rds are non-f l iers by default can be wr i t ten : Vz. Australian-bird(x) D Bird(x) Vz. Australian-bird(x) A ->KFly(x) ^  ->Fly(x) but A u s t r a l i a n b i rds can also be conjectured to fly by v i r tue of being b i rds, th rough (16). A g a i n , the logic prov ides no means of decid ing between these al ternat ives. A more serious p rob lem arises w h e n the knowledge base knows that some u n k n o w n i n d i v i -d u a l is a t yp i ca l . F o r example , the knowledge base: Bird(Bl), Bird(B2), ^Fly{Bl) V ^Fly{B2) is inconsistent w i t h the defaul t (16) because there is a b i rd (51 o r B2), not k n o w n not to fly, w h i c h nonetheless does not fly. In fact, this type of reasoning is not real ly defaul t reasoning at a l l , but wha t M o o r e [1983a] cal ls " 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 rea l ly i nvo l ved is a v a l i d form of inference: f r o m the premise that a l l except iona l cases are known to the conc lus ion that an i nd i v i dua l , not k n o w n to be except iona l , is t yp i ca l . If a l l of the except iona l cases are not known, as i n the above examp le , then the " d e f a u l t " is s imp ly false. T h i s means that defaul ts cannot s im-p ly be s ta ted as ax ioms , since the logic is too r ig id to a l low for occas ional v io la t ions by par t i cu la r i nd iv idua ls . Levesque addresses these problems by "p rep rocess ing " the defaults. T h i s is done by means of a mapp ing on defaul ts that rejects any wh ich are cont rad ic ted by the knowledge base or wh ich , g iven the current state of the knowledge base, conf l ic t w i t h another default . All conf l ic t ing defaul ts are rejected. T h i s leads to a s t rongly conservat ive in terpre ta t ion of defaults. If two defaul ts cannot be app l ied together because they are m u t u a l l y inconsistent , nei ther w i l l be app l ied . N o t hav ing any grounds for dec id ing between them, K F O P C chooses to reject bo th . T h i s is s im i la r to non-monoton ic logic [ M c D e r m o t t & Doy le 1980], wh ich sanct ions on ly those beliefs in every f ixed-po in t of i ts theories. In contrast , defaul t logic [Reiter 1980] and auto-epis temic logic [Moore 1983a] sanc t ion mu l t i p l e sets of beliefs, one suppor t ing each of the a l ternat ives. C i r cumsc r i p t i ve defaul t reasoning conjectures the d is junc t ion of the two a l ternat ives. T h e on ly defaul ts assumed in K F O P C are those w h i c h are assumable independent ly of a l l other defaul ts. - 3 8 -Levesque discusses some interest ing techniques for c i rcumvent ing these di f f icul t ies i n cer ta in cases. T o a v o i d the p rob lem of in teract ing defaul ts wi thout requi r ing defaults to exp l i c i t l y a l low for excep t iona l cases, Levesque suggests a representat ion scheme invo l v ing a " t y p i c a l " - p r e d i c a t e -forming operator , \7:k. If P is a predicate letter, S?:kP(x) is in terpreted as say ing that x* is t y p i c a l w i t h respect to the ktl> aspect of Pness. T h e propert ies of t yp i ca l i nd iv idua ls are s ta ted as first-order ax ioms. F o r example , the KB: Vz. V : l5 i ' rd ( z ) 3 Fly(x) Vz. \7:lBird(x) 3 -<Australian-bird(x) Vz. Australian-bird(x) 3 Bird(x) Vz. V:1 Australian-bird(x) 3 -iFly(x) says that a l l b i rds t y p i c a l i n aspect 1 of " B i r d n e s s " fly and are not Aus t ra l i an -b i rds , whi le S7:lAustralian-birds are bi rds wh ich do not fly. S u c h ax ioms do not state defaul t propert ies of classes of ind iv idua ls . Ra the r , they state propert ies that a l l typical i nd iv idua ls of those classes must (or must not) have. Defau l t reasoning is performed by conjectur ing that i nd iv idua ls are t yp i -ca l . F o r example , the statements: Vz. Bird(x) A -.if->V:l5i'r<i(z) 3 S7:lBird(x) (17) Vz. Australian—bird(x) A ->K->S7:lAustralian-bird(x) Z> \/:\Australian-bird(x) (18) say that - unless k n o w n otherwise - b i rds and A u s t r a l i a n birds are t y p i c a l i n the specif ied aspects. If Tweety is a b i rd not known not to fly (nor be otherwise a t yp i ca l i n aspect 1), (17) says that she is t y p i c a l i n aspect 1, and hence flies. A n A u s t r a l i a n b i rd , Oscar, on the other hand , is k n o w n to be an a t y p i c a l b i rd in aspect 1, so (17) is inapp l icab le . T h e default (18) is app l icab le , however, so ^:lAustralian—bird(Oscar) can be conjectured, and hence -<Fly(Oscar). T o prevent defaul ts w h i c h are con t rad ic ted for par t icu lar i nd iv idua ls (or classes) f rom being rejected out r ight , ax ioms can be added wh ich exp l i c i t l y state that those ind iv idua ls (or members of those classes) are a t y p i c a l i n the relevant aspects. F o r example, g iven the knowledge base: Quaker(john), Vz. S7Quaker(x) 3 Pacifist(x), Republican(george), Vz. VRepublican(x) 3 -iPacifist(x), Quaker(nixon) A Republican(nixon), Vz. Quaker(x) A -<K->J7Quaker(x) 3 VQuaker(x) (19) Vz. Republican(x) A -<K-i\7Republican(x) 3 VRepublican(x) (20) the defaults, (19) and (20), s ta t ing that Repub l i cans and Quakers are t y p i c a l Repub l i cans and Quakers , respect ive ly , are not app l icab le for any i nd i v i dua l because they are mu tua l l y cont rad ic -to ry for nixon. nixon v io la tes the defaults: he is not known to be an a t yp i ca l Quake r , so (19) sanct ions VQuaker(nixon); s im i la r l y , (20) sanct ions S7Republican(nixon); but these conclusions are mu tua l l y inconsistent . U n d e r the conservat ive in terpretat ion of defaul ts, K F O P C rejects both (19) and (20) for this knowledge-base. Hence , John and george cannot be conc luded to be typ ica l , and so Pacifist(john) and ->Pacifist(george) cannot be conc luded. T o remedy th is, ax ioms stat ing that the t y p i c a l R e p u b l i c a n is not a Q u a k e r (and vice versa) can be added: - 3 9 -Vx. VQuaker(x) D -iRepublican(x) Vx. VRepublican(x) O -<Q,uaker(x) . W i t h these add i t i ona l facts, nixon no longer const i tu tes a v io la t ion of the defaults, since K->S7Republican(nixon) A i f - ' V Quaker(nixon) fo l lows f rom the knowledge base. Typ ica l -p red ica tes can also be used to specify a precedence-hierarchy among mul t ip le , po tent ia l l y -conf l i c t ing , defaul ts. F o r example , the ax ioms: 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 the defaul ts: 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 theory that w i l l assume students are undergrades i f possible, otherwise MSc's i f possible, and otherwise PhD's i f possible. Levesque gives numerous examples showing that these strategies can be comb ined to obta in remarkab ly subt le con t ro l of the in teract ions between defaults, w i thout mod i fy ing the st ructure of the defaul ts themselves. A l l that is requi red is the add i t i on of new ax ioms. H e cites three advan-tages of th is fo rma l i za t i on of default reasoning: 1) A defaul t need not be d iscarded and rep laced when a subclass that t yp i ca l l y fa i ls to sat isfy the defaul t is d iscovered. A d d i t i o n a l ax ioms can be added, s ta t ing the inapp l i cab i l i t y of the default for members of that subclass. 2) T h e knowledge g iven to the knowledge base is more s t ructured. Instead of a rb i t ra ry defaults, propert ies of t y p i c a l i nd iv idua ls are l is ted. 3) O n l y a single type of default (e.g., (21)) need be considered, and on ly one of these for each typ ica l -p red ica te , S7:kP. Levesque 's use of typ ica l -pred icates as a representat ion scheme for defaul ts corresponds d i rec t ly to M c C a r t h y ' s subsequent use of abnormal i ty -pred icates . There is a s t ra ight forward map-p ing between Levesque 's ax iomat i za t ions using V -p red i ca tes and M c C a r t h y ' s using ab-predicates. T h e defaul ts, Vx. Px A ~<K->S7'-kPx D *7:kPx, then correspond to m in im iza t ions of the correspond-ing abnormal i ty -p red ica tes . It is not yet known whether this mapp ing const i tutes a t rans la t ion, whether the two approaches lead to the same conjectures for corresponding default theories. There are s t r i k ing s imi lar i t ies , however. F o r example , tangled hierarchies, where in members of one class m a y t yp i ca l l y — but not a lways - be members of another, are p rob lemat ic i n bo th para-d igms. M c C a r t h y ' s " G a n g s t e r a n d C a n a r i e s " example, discussed ear l ier , requires add i t i ona l K F O P C rules, beyond those requi red by the st ra ighforward a b n o r m a l i t y / t y p i c a l i t y representat ion, to express pr ior i t ies o r preferences for par t i cu la r k inds of a typ ica l i t y . C a p t u r i n g these pr ior i t ies in K F O P C appears to invo lve a loss of c la r i t y and naturalness of representat ion s im i la r t o that incur red by M c C a r t h y ' s i n t roduc t ion of pr ior i t ies in to c i rcumscr ip t i ve abno rma l i t y theories. r - 4 0 -In spi te of Levesque 's insights in to represent ing default knowledge, defaul t reasoning in K F O P C rema ins largely unexplored. S im i la r l y , the app l i ca t ion of Levesque's ideas on t yp i ca l -(abnormal-) predicates to default reasoning based on other formal isms has on ly begun. B o t h of these areas promise to prov ide impor tan t insights in to reasoning about incomple te ly specif ied wor lds, and deserve fur ther exp lo ra t ion . 2 . 2 . 6 . O b j e c t i o n s t o N o n - M o n o t o n i c F o r m a l i s m s K r a m o s i l [1975] c la ims to have shown that any formal ized theory w h i c h a l lows unp rovab i l i t y as a premise in deduct ions must ei ther be "mean ing less ' ' , or no more power fu l than the corresponding first-order theory w i thou t rules invo lv ing such premises. H e presents two "p roo fs " to suppor t h is c l a i m . C a r e f u l examina t ion shows that the first result fol lows f rom a def in i t ion of " f o r m a l i z e d theory" w h i c h expressly excludes any theory wh ich exh ib i ts the types of behavior c o m m o n to non-monoton ic theories. T h e second result is based on an incorrect def in i t ion of "p roo f " and hence of " t h e o r e m h o o d " and is i tself meaningless. A s the paper stands, it shows on ly that non-monoton ic theories must behave dif ferent ly than monoton ic theories in those cases where the former c a n der ive results unobta inab le using the lat ter. K r a m o s i l was not the on ly one to be uncomfor tab le w i t h opening the " P a n d o r a ' s B o x " of non-monoton ic i t y . Sandewa l l [1972] notes that the " U n l e s s " operator has " s o m e d i r t y log ica l pro-per t ies" . Cons ide r ing the example : A A A Urdess(B) ^  C A A Utdess(C) D B he observes that e i ther B a n d C can be theorems, but , in general , not bo th s imul taneous ly . Re i te r [1978b] makes a s im i l a r observat ion in an ear ly paper, s ta t ing that : S u c h behav io r , [is] c lear ly unacceptable; A t the very least, we must demand of a default theory tha t i t sat isfy a k i n d of 'Chu rch -Rosse r ' property : N o mat te r wha t the order i n w h i c h the theorems of a theory are der ived, the resul t ing set of theorems w i l l be un ique. It appears tha t the Chu rch -Rosse r property is a necessary casua l ty i f non-monoton ic i ty is accepted'. A fur ther p rob lem w h i c h must be faced by those embrac ing consistency- or unprovab i l i t y -based approaches to non-monoton ic i t y is that the non-theorems of a first-order theory are not recurs ive ly enumerab le . T h i s means that the rules of inference in theories i nvo l v ing the •)/- opera-tor cannot be effective i n generaL It fo l lows that the theorems are not recurs ive ly enumerable. B y contrast , in mono ton ic logics,, the ru les of inference M U S T be effective and the theorems M U S T be recurs ive ly enumerab le . F i n a l l y , the ve ry non-monoton ic i t y wh i ch makes such theories interest ing means that " t h e o r e m s " m a y have to be re t racted i f the assumpt ions on w h i c h they are based are refuted (either by new knowledge or changes i n the state of the wor ld ) . T o be useful , a non-monoton ic - 41 -reasoning sys tem must be able to remember w h i c h assumpt ions under ly each theorem a n d be able to unw ind the po ten t ia l l y comp lex cha in of deduct ions founded on re t rac ted just i f icat ions. C H A P T E R 3 Default Logic If the whee l is f ixed, I w o u l d s t i l l take a chance. If we' re t reading o n th in ice, T h e n we might as we l l dance. — Jesse W inches te r In th is chapter , we explore defaul t logic i n some deta i l . W e present a model- theoret ic semant ics for a rb i t ra ry default theories, thus rect i fy ing a major def ic iency. T h e rema in ing sec-t ions invest igate the causes of incoherence in cer ta in default theories. T h i s leads to a strong suff icient (a l though not necessary) syntac t ic cond i t ion for the existence of extensions for par t i cu la r theories. 3 . 1 . T h e S e m a n t i c s o f D e f a u l t T h e o r i e s In his deve lopment of defaul t logic, Re i te r p rov ided a fixed-point charac te r iza t ion of the extensions of a defaul t theory, but no model- theoret ic semant ics for the logic. E the r i ng ton [1982, 1983] observes that the semant ics can be v iewed in terms of restr ict ions of the set of models of the under ly ing theory, tyukaszewicz [1985] formal izes this in tu i t ion for n o r m a l defaul t theories. Because of the we l l -behaved nature of these theories, this is re la t ive ly s t ra ight forward. T h e resu l t ing semant ic charac te r iza t ion amounts to consider ing the T a r s k i a n semant ics of each of the pa r t i a l extensions const ruc ted by proceding monoton ica l l y toward an extens ion by sat is fy ing, at each step, the next appl icable n o r m a l default (according to some arb i t ra ry order ing of the defaults) by m a k i n g i ts consequent true. If, after each default i n the sequence has been con-s idered, no more defaul ts f rom D are app l icab le , the resul t ing set, together w i t h the first-order theory, W, y ie lds an extension. S ince each step aff irms a fo rmu la consistent w i t h those af f i rmed prev ious ly , the set of models contracts monoton ica l l y . T h e intersect ion of the sets of models f r om each stage is precisely the set of models of the extension. T h i s semant ics can perhaps best be envisaged as a t rans i t ion network, whose nodes are sub-sets of M , the set of a l l models of W, w i t h arcs label led by defaults, as fo l lows: F r o m the node corresponding to a set of models N , for every 6 — a * ^  € D, an arc label led 6 leads (i) back to N P i f no mode l i n N satisfies f$ or some sat isfy - i a , or (ii) to the node corresponding to the set: - 42 -- 4 3 -{ N | N €E N and N |= p }, otherwise. E a c h leaf - a node a l l of whose ou tbound l inks loop back -reachable f r om M corresponds to the set of models of some extension of A . Fu r the rmore , the set of models of each extens ion of A corresponds to such a leaf node. T h e set of arc- labels for every pa th f rom root to leaf gives the generat ing defaults for the extension corresponding to the node. T h i s approach does not app ly d i rec t ly to non-norma l defaul ts, since the proper ty of semi-mono ton i c i t y w h i c h guarantees its success holds on ly for no rma l defaul ts [Reiter 1980a, theorem 3.2]. J iukaszewicz pa r t i a l l y addresses this p rob lem b y present ing a t rans la t ion scheme f rom non-n o r m a l defaul ts to n o r m a l defaul ts. He argues that , of s ingle- just i f icat ion defaul ts, on ly no rma l and sem i -no rma l defaul ts have reasonable interpretat ions. Non-semi -no rma l defaults are therefore t rans la ted to semi -no rma l defaul ts by conjo in ing the consequent to the jus t i f i ca t ion: a •• P _ « : M 7 7 7 T h e t rans la t ion f rom semi -norma l to no rma l , w h i c h is somewhat more cont rovers ia l , involves rep lac ing the consequent w i t h the jus t i f i ca t ion: a : P A 7 _ <* = P A 7 7 ~* P/\l T h i s makes sense, fyukaszewicz argues, so long as a ' s wh i ch are also •y's are t yp i ca l l y /?'s. T h a t is, so long as one cou ld reasonably augment the theory w i th a A 7 : P P O n e can imagine s i tuat ions where this is not appropr ia te. F o r example , a system for legal reasoning might want to have a ru le suggesting that those w i t h mot ives who might be gu i l ty shou ld be suspects: has-motive(x) : guilty(x) suspect{x) It is c lear ly reasonable to t ranslate this to: has-motive(x) : suspect(x) A guilty(x) suspect(x) ' a l low ing that there m a y be reasons not to inc lude someone on the l ist of suspects even wi thout know ing the i r innocence. It is not reasonable to fo l low through by assert ing the gui l t of a l l suspects: has-motive(x) : suspect(x) A guilty(x) ^  suspect(x) A guilty(x) Thus , wh i le the semant ics tyukaszewicz out l ines covers many cases, there is reason to want a semant ics w h i c h covers more than n o r m a l defaul ts. T o this we now tu rn . Because of the fa i lure of semi -monoton ic i ty for non-norma l theories, s i m p l y app ly ing one defaul t after another w i l l not, i n general , lead to extensions. It is necessary to ensure that the app l i ca t ion of each defaul t does not v io la te the just i f icat ions of a l ready-app l ied defaul ts. If we augment ^ukaszew icz ' s semant ics b y encoding some in format ion i n each state about the set of - 44 -defaul ts w h i c h led to a par t i cu la r state, we can determine whether a node is on a v iab le pa th toward an extens ion. T h e precise detai ls are these: D e f i n i t i o n : Sat is f iab i l i t y , admiss ib i l i t y , and app l i cab i l i t y a : B Le t X be a set of models; T a set of formulae; a, B, and w formulae, and 8 = — a w defaul t . T h e n i) a is X-satisfiable (X-valid) iff ^x € X. x j= a ( V i G X. x \= a) ii) r is X-admissible (X permits T) iff V 7 e" I\ 3 z € X . x {= 7 i i i) 8 is X-applicable iff a is X v a l i d and B is X-sat is f iab le . | D e f i n i t i o n : Resu l t of a default Le t X, T , and 8 be as above. T h e n the result of 8 in (X, T) is: {( X , T) i f 8 is not X -app l i cab le and T is X -admiss ib le , \{X-{N\ N\= nw}) , ( r U {6})) i f 8 is X -app l i cab le and T is X -adm iss ib l e , and I otherwise. | D e f i n i t i o n : Resu l t of a sequence of defaults Le t X and T be as above, and let <£,-> be a sequence of defaults. T h e n the result of <£,-> is: <5 ,> (X , T) = ( D X,-, U r t ) where J X 0 = X ; r 0 = T; and \(Xi+1, Ti+1) = S , { X „ I \ ) , i > 0 . I D e f i n i t i o n : S t a b i l i t y Le t Y be a non-empty set of models, T a set of formulae, and A = (D, W) a defaul t theory. T h e n (Y, T) is stable for A iff (1) (Y, T) = <8i>(X, { }) for X = {M\ Mf= W), and some {£,} C £>, (2) W e D . <5( K, T) = ( 7, T) , and (3) r is F-admiss ib le . | In o ther words, a set of models and a set of constra ints is stable for a default theory, (D,W), i f they are the resul t of some sequence of defaul ts in D app l ied to the set of models of W and no const ra in ts , i f no defaul t in D produces any change i n th is resul t , and the const ra in ts are satisf ied - 45 -by the set of models. Note that cond i t ion (2), together w i t h the def in i t ion of " r e s u l t " means that cond i t ion (3) is redundant . W e inc lude it for conceptua l c lar i ty . T h e soundness and complete-ness results for th is semant ics are g iven by Theorems 3.1 and 3.2, respect ive ly . Theorem 3.1 — Soundness If E is an extens ion for A , then there is some set T such that R e t u r n i n g to the t rans i t ion network analogy, the nodes are now pairs consis t ing of a subset of M and a subset of the just i f icat ions of the defaults i n D. N o w A ' s extensions correspond to those leaf nodes, (X, T), where X permi ts I\ W e say that such nodes are viable. If a l l leaf nodes are the theory has no extensions. A g a i n , the generat ing defaul ts for the extens ion Th(X) are those defaul ts labe l l ing arcs on any pa th f rom ( M , { }) to (X, T ), for any 1 . Example 3.1 Cons ide r the defaul t theory: ({M\M\= E), Y) is stable for A . Theorem 3.2 — Completeness If (X, T) is stable for A then X is the set of models for some extens ion of A . (I.e., Th(X) is an extens ion for A . ) | T h i s produces the fo l lowing t rans i t ion network. ({Af | Af j= A}, { }) ({M| M(= A,B), {BA-iC}) ({M\ M(= A,C\, {Cf\^B}) B o t h leaves are v iab le , so the theory has two extensions, Th({A, B}) and Th({A, C}). | - 4 6 -Example 3.2 T h e incoherent theory : { } gives rise to: ({M\M\=py^p},{ }) ( { M | M\= A}, {^A}) i n w h i c h the leaf is not v iab le . Hence th is theory has no extension. I It is ins t ruc t i ve to compare this model-set res t r ic t ion semant ics w i t h the m i n i m a l - m o d e l semant ics of c losed-wor ld reasoning presented i n chapter 2. There , the semant ics of the closure of a theory was def ined i n terms of a rest r ic t ion of the set of models of the under l y ing theory, accord ing to the pr inc ip le of m in im i za t i on . T h e model-set rest r ic t ion semant ics for defaul t logic s im i la r l y prov ides a pr inc ip le for determin ing subsets of the models of a first-order theory w h i c h character ize acceptable belief-sets, on the basis of m a x i m a l sat isfact ion of the set of defaults. There are severa l s igni f icant differences, however. F i r s t l y , ra ther than an order ing o n i n d i v i d u a l models, this semant ics imposes an order ing on sets of models. Secondly , the order ing is defined in terms of access ib i l i ty v i a a sequence of defaults, ra ther than st r ic t ly i n terms of in t r ins ic features of the models themselves. F i n a l l y , each extension is determined by a single ex t remum of the ord-er ing, ra ther t han by the set of a l l ex t rema. T h e first of these differences resul ts because the extensions of a defaul t theory — un l ike the models of a f i rst-order theory - are not complete. T h e y do not decide every fo rmu la . Because they incomple te ly speci fy the wor ld , sets of models - ra ther than single models - are requi red to a l low for undec ided formulae. U s i n g s i tuat ions (Barwise and Per ry 1983] - incomple te mode l -descr ipt ions - ins tead of sets of models m igh t lead to a closer correspondence. In tu i t i ve ly , cer-ta in ly , one can s imp l y v iew the model-sets as pa r t i a l model-descr ipt ions w i thou t i l l effect. T h e second dev ia t i on resul ts f rom the fact that defaults are general inference rules. Conse-quent ly , the submodel(-set) re la t ion is potent ia l ly more complex for defaul t log ic . L i f sch i t z ' [1984] recent work a l lowing arb i t rary pre-orders as we l l as s imple subset order ings m a y v o i d th is difference, bu t the quest ion remains open. T h e fact tha t i n d i v i d u a l ex t rema determine extensions is the resul t of the " b r a v e " ( in M c D e r m o t t ' s [1982] termino logy) character of defaul t logic. Re i te r ' s presentat ion of defaul t logic def ined each extens ion as an acceptable set of beliefs, w i t h the in tent ion that a reasoner w o u l d somehow " c h o o s e " a single extens ion w i t h i n w h i c h to reason about the wo r l d . O t h e r non -monoton ic fo rma l i sms (see chapter 2) are based on " c a u t i o u s " approaches w h i c h accept a default - 47 -conc lus ion on ly if i t occurs i n all acceptable sets of beliefs. One can easi ly const ruct a va r ian t of default logic w h i c h pursues a " c a u t i o u s " course. (The converse is not obv ious ly true for a l l " c a u -t ious" systems, as we see i n chapter 8.) S u c h a system wou ld define the theorems of a default theory to be those formulae true in a l l extensions, w i t h the obvious change to the semant ics: the theorems w o u l d then be def ined as those formulae true i n a l l models of a l l v iab le leaves. 3.2. Coherence of Default Theories Extens ions p lay a fundamenta l role in defaul t logic. A n extens ion is a set of beliefs w h i c h are i n some sense " j u s t i f i e d " or " r e a s o n a b l e " i n l ight of wha t is known about a wor ld . F o r m a l l y , extensions are a t t rac t ive because they are both grounded and complete: A fo rmu la enters an extens ion, E, on ly if i t is i n W, i f it is p rovab le f rom other formulae i n E, or i f i t is the consequent of a defaul t whose prerequisi tes are in E and whose just i f icat ions are not denied by E; further-more, every f o rmu la w h i c h meets these requi rements is in E. T h e first of these restr ic t ions prevents extensions f rom conta in ing spur ious, unsuppor ted beliefs. T h e second ensures that just i f ied beliefs are not ignored. These restr ic t ions are analogous to those w h i c h define the theorems of a first-order theory. S ince the i n d i v i d u a l extensions of a defaul t theory are bo th grounded and complete, it is qui te na tu ra l to require any default inference system to restr ict i ts conclusions to a single common extension. If no extens ion of a theory conta ins a fo rmula , then it is not in any acceptable set of beliefs associated w i t h that theory. If conclusions are d rawn f rom different extensions, they m a y be incompat ib le . Cons ide r the b locks-wor ld example f rom the previous chapter . In that example , bo th ->Block(A) and ->Block(B) are reasonable assumpt ions. T h e y are d rawn f rom different exten-sions, however , and conc lud ing bo th leads to inconsistency. S ince reasonable conclusions must reside i n a n extension of the default theory under con-s idera t ion , it is c lear ly impor tan t to know whether every theory has extensions. S i m p l y put , the answer is " N o " . F o r example , the theory: has no ex tens ion. S u c h theories are incoherent ; they suppor t no reasonable set of bel iefs about the wor ld . B e y o n d po in t ing out the existence of incoherent theories, the most useful answer wou ld inc lude a syn tac t i c charac te r iza t ion of w h i c h theories have or do not have extensions. W h i l e no such charac te r i za t ion is known , there are sufficient condi t ions w h i c h guarantee extensions. W e present three such cond i t ions below, i n order of increasing u t i l i t y . A theory , ({ },W), w i t h no defaul ts has a un ique extension, Th( W), the log ica l closure of the under l y ing first-order theory. O f course, th is is a t r i v i a l default theory. W e men t ion it on ly to emphasize that , since defaul t log ic is a superset of first-order logic, the requi red resul ts ob ta in for the area of over lap . W={} - 4 8 -T h e d is t inc t ions between c o m m o n l y encountered types of defaults lead to more enl ightening results. A n y defaul t of the fo rm: a : p P is sa id to be normal. N o r m a l defaults are sufficient for knowledge representat ion and reasoning i n m a n y na tu ra l l y occur r ing contexts . In fact, they can express any rule whose app l i ca t ion is subject on ly to first-order prerequisi tes a n d the consistency of its conclus ion w i t h the rest of what is be l ieved. R u l e s l i ke : " A s s u m e a b i r d can fly unless y o u know o therwise . " , or " A s s u m e a th ing is not a b lock unless it is requ i red to b e . " t ranslate easi ly in to no rma l defaul ts: Bird(x) : Can-fly(x) ^ : -iBlock(x) Can-fly(x) -iBlock(x) T h e consequent of a no rma l defaul t is equivalent to i ts jus t i f icat ion. In tu i t i ve ly , th is makes the default i napp l i cab le where the consequent has been denied. S u c h defaults cannot in t roduce inconsistencies, they cannot refute the just i f icat ions of other, a l ready app l ied , n o r m a l defaul ts, nor can they refute the i r own just i f icat ions. T h i s gives rise to wel l -behaved theories. A n y theory i nvo l v ing on ly n o r m a l defaul ts (a normal theory) must have at least one extension [Rei ter 1980a]; A n y defaul t of the fo rm: <* = P A 7 P is sa id to be semi-normal. Sem i -no rma l defaults differ f rom no rma l defaults by hav ing just i f icat ions w h i c h en ta i l but are not en ta i led by thei r consequents. The assurances of we l l -behavedness assoc ia ted w i t h no rma l theories do not car ry over to theories w i t h semi -no rma l defaults. F o r example , the theory: W={} / :A A - B :B A : C / \ - > A \ (1) \ A ' B ' C j has no extens ion. T h i s appears to be a somewhat ar t i f ic ia l example, i nasmuch as we have been unable to find a na tu ra l s i tua t ion w h i c h fits this pat te rn . W h i c h semi -norma l theories, then, are assured of extensions? D o till " n a t u r a l " theories have extensions? Perhaps pa tho log ica l examples are mere ly f o r m a l cur iosi t ies? W e do not purpor t to answer these quest ions — pa r t l y because of the d i f f icu l ty of de l im i t ing the class of " n a t u r a l " theories. There is, however, a large class of semi -no rma l theor ies w h i c h are coherent. W e character ize this class, w h i c h appears to be sufficient for m a n y c o m m o n app l ica t ions , i n the next sect ion. - 49 -3 . S . O r d e r e d D e f a u l t T h e o r i e s There appears to be a un i fy ing character is t ic among default theories w i thou t extensions. Cons ider aga in the theory: W={} w h i c h has no ex tens ion. T h e on ly reasonable candidates are Ey = Th({ }) or E2 = Th({->A}). A is consistent w i t h Ex, so to be an extension Ex must con ta in -<A, w h i c h it does not. S im i l a r l y , A is inconsistent w i t h E2, so E2 cannot con ta in ->A. T h e p rob lem is that the defaul t 's jus t i f i ca t ion is denied by i ts consequent; not app ly ing the default forces i ts app l i ca t ion , a n d v ice versa. Re tu rn ing to the semi -no rma l theory ( l ) , we see that app ly ing any one default leaves one other app l icab le . A p p l y i n g any two, however, results i n the den ia l of the non-norma l par t of the just i f icat ions of at least one of them. A n y set sma l l enough to be an extens ion is too sma l l ; any set large enough is too large. T h i s behav iour is character is t ic of theories w i t h no extension; the requ i rement that extensions be c losed under the defaul t rules forces the app l i ca t ion of defaul ts whose consequents lead to the den ia l of jus t i f i ca t ions of other app l ied defaults. T h e exact source of the p rob lem can be fur ther isolated by recal l ing that a l l n o r m a l theories have extensions. S ince the jus t i f ica t ion and consequent of no rma l defaults are iden t i ca l , no app l i -cable defaul t can refute the just i f icat ions of an a l ready app l ied default : app l ied n o r m a l defaul ts have a l ready asserted thei r just i f icat ions. T h i s means that any no rma l default capable of refut ing those just i f icat ions is inapp l icab le , since its just i f icat ions have a l ready been refuted. It fol lows that that par t of the jus t i f ica t ion w h i c h dist inguishes non-norma l defaul ts f rom no rma l defaults is in tegra l ly i nvo l ved i n mak ing a theory incoherent. Res t r i c t ing our a t tent ion to semi -norma l default theories, we see that once a defaul t has been app l ied, on ly those conjuncts of its jus t i f i ca t ion not enta i led by i ts consequent are suscept ib le to re fu ta t ion by other defaul ts. These conjuncts p lay a key role in the d iscussion below. T h e conf l ict between closure under defaul ts and consistency of just i f icat ions can occur on ly i f some fo rmu la depends on the absence of another and at the same t ime m a y serve to suppor t the inference of tha t fo rmu la . In the theory ( l ) above, for example, A depends on the absence of B, B o n that of C, a n d C on that of A. Hence inferr ing A wou ld b lock the inference of C, a l lowing the inference B, w h i c h w o u l d inva l ida te the inference of A, and s imi la r ly for B and C. T h e examples presented so far have invo lved defaul ts i n thei r s implest fo rm: a : p\ A ... A 0n where a, u> and ft are a l l l i tera ls (i.e., a tomic formulae or negat ions of a tomic formulae) . The p rob lem of de termin ing dependencies is more compl ica ted when a, w and ft are a l lowed to be a rb i t ra ry first-order formulae. F o r example , the consequent of a defaul t m a y be an imp l i ca t i on ; app ly ing that defaul t wou ld in t roduce new dependencies. T h e essent ial idea remains the same, - 50 -however: determine whether the dependencies invo lve potent ia l ly unresolvable c i rcu lar i t ies . T h e fo l lowing def in i t ions out l ine a syntac t ic method for determin ing whether such c i rcu lar i t ies exist w i t h i n a sem i -no rma l theory. D e f i n i t i o n : <c and Le t A = (D, W) be a c losed , 1 semi -norma l defaul t theory. W i t h o u t loss of general i ty , assume a l l fo rmulae are i n c lausa l fo rm. T h e pa r t i a l re lat ions, and <§; , on Literals X Literals, are defined as fo l lows: (1) If a € W then a = (ay V ... V a j , for some n > 1 . F o r a l l ait aj € {a1,...,an}, i f =^ aJt let -la; at . CK : 3 A 7 (2) If 8 e D then 8 = —'—— . L e t o^, ... a r , p\, ... Bs, and 7^ ... 7 t be the l i tera ls of the P c lausa l forms of a, fi, and 7, respect ive ly . T h e n (i) If a { G { « ! , . . . , a r } and A € {A,-A} let o t « A . (ii) If 7i S {7i,-,7t}. A € {& , . . . ,&} and 7, £ {&,..,&} let - 7 i « A . (ii i) A l s o , B = At A ••• A An 1 f ° r some m > 1. F o r each i < m, A = (Ai V ... V A,nJ , where mj > 1 . T h u s if flj , A,k e {A^.-vA^mJ a n d A,j £ A,k let -Aj ^  A,k-(3) T h e expected t rans i t i v i t y re lat ionships hold for <K and <C . I.e., (i) If a ^ /? and /9 « 7 then a ^ 7. (ii) If a <K 8 and /3 <SC 7 then a <§; 7. (ii i) li a <K B and /? <C 7 or a «C /9 and /3 <K 7 then a <C 7. | T h e def in i t ion is complex , but the in ten t ion is that a <£. B or a /3 i f there is any w a y tha t a cou ld figure i n an inference of B i n the theory as i t stands. T h e i n tu i t i on beh ind par ts ( l ) and (2. i i i) is that any d is junct ion of n l i tera ls can be in terpre ted as an imp l i ca t i on of any one of those l i terals. E.g., ( a x V...V a j = [(-"a*! A---A ""Xj-i A " 'C ' j+ i A---A -,"n) ^> ct\ ]• T h e spec ia l prominence we have a l luded to for the conjuncts i n a jus t i f ica t ion not enta i led by the consequent is ref lected i n par t (2.ii) by the use of the d is t inguished " <§: " re la t ion. T h e negat ion, ->7; , occurs in par t (2.ii) since it is not know ing -17; w h i c h makes 7; consistent. 1 The definition is readily extensible to open theories using a technique given in [Reiter 1980a]. - 51 -Definition: Orderedness A semi -no rma l default theory is said to be ordered if and only if there is no l i te ra l , a, such that a « a . | A n ordered theory has no potent ia l l y unresolvable c i rcu la r dependencies. T h e theory i n example (1) is not ordered, since B <s: A, C <K B, and A <£. C; hence A « i . T h e theory : W = { } (2) / j _ A _ / W B : B/\^D : (C D D) A ->A \ \ A * B ' ( C ^ D ) j is also not ordered. T h e defaul ts give rise to the fo l lowing re lat ionships: {B<s:A}, {D<KB}, and { C « D, - i £ > « - > C , A <sz ->C, A < ^ D } , respect ive ly . Hence A « O « B « A . T h e signi f icance of orderedness for semi -norma l default theories is shown by T h e o r e m 3.3. Theorem S.S — Coherence If a semi -no rma l default theory is ordered, then it has at least one extension. | N o r m a l theories are c lear ly ordered, since on ly non-normal defaults give rise to " <SC " re la-t ionships. T h u s the coherence of a l l no rma l theories is a coro l lary of T h e o r e m 3.3. T h i s is encouraging inasmuch as it suggests that orderedness is not merely a spec ia l purpose g i m m i c k but, rather , it subsumes an ex is t ing, w ide ly appl icable charac ter iza t ion . It is impor tan t to notice that orderedness is only a sufficient cond i t ion for existence of extensions. Non-o rdered theories have potent ia l ly unresolvable c i rcu lar i t ies but , for one reason or another, these c i rcu lar i t ies do not a lways interfere. T h e theory (2) is not ordered, but it does have an extens ion: Th({B, (C D D)}). T h e c i rcu la r i t y wou ld cause problems, however, i f C were added to W: the resul t ing theory has no extensions. In other cases, two or more po ten t ia l c i rcu lar -i t ies m a y cance l each other out. A t present, we do not know whether the g iven cond i t i on can be strengthened to one w h i c h is bo th necessary and sufficient for the coherence of semi -norma l theories and ye t is s t i l l decidable. 3.4. Constructing Extensions H a v i n g de l ineated a large class of theories w h i c h have extensions, we tu rn to the p rob lem of generat ing extensions. Re i te r [1980a] shows that extensions need not be recurs ive ly enumerable, and that i t is not general ly semi-decidable whether a fo rmula is i n any extens ion of a theory. F a c e d w i t h such pess imism, fur ther exp lo ra t ion might seem point less. S t i l l , there are t ractab le - 52 -subcases. E t h e r i n g t o n [1982] presents a procedure w h i c h can generate a l l the extensions of an arbi -t ra ry finite default t h e o r y . 2 T h e procedure centres on a re laxa t ion style constraint propagat ion technique. Ex tens ions are constructed by a series of successive approximat ions . E a c h app rox ima-t ion , H}, is b u i l t up f rom the first-order components i n W b y app ly ing defaults, one at a t ime. A t each step, the default to be appl ied is chosen from those, not yet appl ied , whose prerequisites are " k n o w n " a n d whose just i f icat ions are consistent w i t h bo th the previous a p p r o x i m a t i o n and the current state of the current app rox ima t ion . W h e n no more defaults are appl icable , the procedure continues w i t h the next app rox ima t ion . If two successive approximat ions are the same, the pro-cedure is sa id to converge. T h e choice of w h i c h default to app ly at each step of the inner loop m a y int roduce a degree of non-de te rmin ism. G e n e r a l i t y requires th is non-determinism, however, since theories do not necessarily have unique extensions. De te rmin i s t i c procedures can be constructed for theories which have unique extensions, or i f fu l l general i ty is not required. a • 8 In the presentat ion of the procedure, below, CONSEQUENT^——) is defined to be 7. 7 r e p e a t * o - W\ GD0*-{}; i « - 0 ; r e p e a t A «- I ^ G D I h a), (^ \f- ^0), [Hhl \h ^ i f ->null(Di - GDi) t h e n c h o o s e S f r o m (£>; - GDj; GDi+1 <- GDi U {6}; hi+i«- hi U {CONSEQUENT^)}; e n d i f ; * ' « - t + 1; u n t i l n u / i ( P i _ 1 - GD^); H\ = h-i u n t i l H} = T o see how this procedure works , consider the theory: W= {A} 2 A finite theory is one with only finitely many variables, constant symbols, predicate letters, and de-faults. No 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 in-stances of open defaults. - 5 3 -/ A : B A : C B : D B : ~>D A ~>C \ \ B ' C ' D ' -Z? /' w h i c h has the unique extens ion, Th({A,B, C,D}). T h e procedure can generate any of the fo l lowing sequences of approx imat ions : Ho={A} HM^B^C) H^{A} H0={A} H^{A,B,Cf) H^i^C&D} H^{A,B,C,D} Hs={A,B,D,C) E^HX H^Ht H*=H3 (The fo rmulae in each app rox ima t i on are l is ted in the order in w h i c h they are der ived.) In the first sequence of approx imat ions , ->D occurs in Hx because it can be inferred in before C is inferred i n k3. Ethe r i ng ton [1982] proves: There is a converging computation such that Hn = and Th(H^) = E if and only if E is an extension for the default theory (D,W). In other words , the procedure can re tu rn every extension, and on ly extensions are re tu rned. T h i s result fal ls short in two respects: F i r s t , wh i le the procedure can converge on every extens ion, there are appeals to non-provability. In general , such tests are not computab le , since a rb i t ra ry first-order formulae are invo lved . The re are computab le subcases, however. If the set: W belongs to a decis ion class for first-order p rovab i l i t y , extensions are computab le . P ropos i t i ona l theories a n d funct ion-free, monad ic theories fa l l in to this class, as do finite theories, p rov ided W is also finite. T h e second shor tcoming is that some finite theories admi t non-converg ing computa t ions . T h e procedure m a y never terminate even though the theory has an extension and each step is computab le . In such cases, the procedure cycles forever between two or more d is t inct #j's. Fo r -tunate ly this cyc l i c behav iour seems to be caused by features s im i la r to those w h i c h make theories incoherent . W e have character ized cer ta in classes of ordered theories for w h i c h the procedure is more we l l -behaved. T h e o r e m 3.4 shows that one such class is the class of ordered, network theories. Definition: Network Default Theory A defaul t theory , A = (D, W), is a network theory iff i t satisfies the fo l lowing condi t ions: (1) W conta ins on ly : a) L i te ra ls (i.e., A t o m i c formulae or thei r negat ions), and b) D is junc ts of the fo rm (a V 8) where o; and B are l i terals. (2) D conta ins on ly no rma l and semi -no rma l defaults of the form: a : B a : B A li f\ ... f\ la —TT— or P P - 54 -where a, B, and -y; are l i terals. I Theorem 3.4 — Convergence F o r finite, ordered, network theories, the procedure g iven above a lways converges on an extens ion. | W e w i l l have more to say about network theories in the next chapter . W e conjecture that T h e o r e m 3.4 can be general ized to app ly to a rb i t ra ry ordered semi -n o r m a l theories, but we have no proof. T h e proof m a y require a more restr ic t ive def in i t ion of in the procedure, viz: but it can be shown that a l l the results of [Ether ington 1982] and those of th is chapter s t i l l ho ld for the stronger vers ion, so th is should present no prob lem. F o r n o r m a l theories, an even stronger result can be proved: Theorem 3.5 — Strong Convergence F o r finite n o r m a l theories, the procedure g iven above a lways converges on an extens ion immed ia te l y — i.e., Th(Hi) is a lways an extension. | instead of: C H A P T E R 4 Inheritance Networks with Exceptions A cent ipede was happy , qui te, U n t i l a frog, i n fun, Sa id , " P r a y , w h i c h leg comes after w h i c h ? " T h i s ra ised his m i n d to such a p i t ch He lay d is t rac ted i n a d i t ch , Cons ider ing how to run . O n e of the prob lems w i t h the non-monoton ic formal isms we have discussed to this point is thei r i n t rac tab i l i t y . Defau l t logic, i n the general case, is not even semi-dec idable. Because of the need to b u i l d systems wh ich have good compu ta t i ona l propert ies, many researchers have sacr i f iced f o rma l prec is ion. W h i l e this has somet imes led to very fast " i n fe rence" mechanisms, there has often been l i t t le more than vague in tu i t ions about exac t ly what these mechanisms infer. A s the field matures and systems capable of assuming responsib i l i ty for such th ings as nuclear reactors and med ica l diagnosis are touted as " o n the ho r i zon " , it becomes increasingly impor tan t tha t i t be understood what such systems "cons ide r " just i f iable inferences. T h e argument has long been made that, because of the general in t rac tab i l i t y of fo rma l sys-tems, i t is unreasonable to consider them for p rac t i ca l appl icat ions. T h i s is taken as suppor t for the use of systems such as semant ic networks wh ich , a l though not comple te ly understood, can compute qu ick l y . T h i s argument fal ls down o n two points. The first is that most of these fast inference a lgor i thms are appl icab le to a l im i ted class of problems. It cou ld we l l be that - for these prob lems - f o rma l systems such as defaul t logic are just as t ractable, and fast imp lementa t ions m a y be possible. Second ly , even if f o rma l systems are not imp lemented d i rec t ly in an inference system, they m a y be useful as speci f icat ion tools. In th is way , an imp lementa t ion cou ld ei ther be shown a lways to reach just i f ied conclusions or, at the very least, to dev iate in wel l -understood ways from jus t i f ied conclus ions. In the former case, the fast a lgor i thm cou ld ac tua l l y be v iewed as an implementation of an appropr ia te ly - res t r ic ted vers ion of the general f o rma l sys tem; in the la t ter case, at least wou ld-be purchasers of such systems cou ld make enl ightened decisions about the r isks i nvo l ved . In th is chapter , we employ defaul t logic to out l ine a speci f icat ion for " i nhe r i t ance " reasoning i n the presence o f except ions. Semant i c networks have been wide ly adopted as a representat iona l mechan i sm for A l . In such networks, " i n f e r e n c e " is equated w i t h inher i tance of propert ies by nodes f rom the i r superiors. Recent work has considered the effects of a l lowing except ions to inher i -tance w i t h i n ne tworks [B rachman 1982; E the r i ng ton and Re i te 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]. Such except ions represent e i ther exp l i c i t or - 5 5 -- 5 6 -imp l i c i t cance l la t ion of the no rma l property inher i tance wh ich networks enjoy. In the absence of except ions, an inher i tance network is a taxonomy organized by the usual I S - A re la t ion , as i n F igu re 4.1. Schuber t [1976] a n d Hayes [1977] have argued that such networks correspond qu i te na tu ra l l y to cer ta in theories of f i rst-order logic. E.g., NAUTILUS(Fred) V z M 0 L L U S C t x \ D WVERTEBRATE(x) Vx. NAUTILUS(x) D CEPHALOPOD(x) Vx. CEPHALOPOD(x) D MOLLUSC(x) S u c h a correspondence can be v iewed as p rov id ing the semant ics wh i ch " s e m a n t i c " networks had prev ious ly l acked [Woods 1975]. I N V E R T E B R A T E I N S E C T M O L L U S C A R A C H N I D C E P H A L O P O D B I V A L V E N A U T I L U S C U T T L E F I S H T F r e d Figure 4-1 — F r a g m e n t of a taxonomy. T h e s igni f icant features of th is semant ics are these: (1) Inher i tance is a log ica l property of the representat ion. G i v e n that NAUTILUS(Fred), MOLLUSC(Fred) is provable f rom the g iven formulae. Inher i tance is the repeated app l i ca -t ion of modus ponens. (2) T h e node labels of such a network are unary predicates: e.g., NAUTILUS(*), INVER-TEBRATE^). (3) N o except ions to inher i tance are possible. If F r e d is a naut i lus, he must be an inver tebrate, regardless of any other propert ies he enjoys. Un fo r tuna te ly , th is correspondence no longer appl ies when except ions to inher i tance are a l lowed. T h e log ica l propert ies of networks change drast ica l ly when except ions are permi t ted . F o r example , consider the fo l lowing facts about elephants: (1) E lephan ts are gray, except for a lb ino elephants. (2) A l l a lb ino elephants are elephants. Common-sense reasoning about " e l e p h a n t s " a l lows one, g iven an i nd i v i dua l e lephant not k n o w n to be a n a lb ino , to infer tha t she is gray. Subsequent d iscovery — perhaps by observat ion — that she is a n a lb ino e lephant forces the re t rac t ion of the conclus ion about her grayness. Thus , common-sense reasoning about except ions is non-monoton ic ; new in fo rmat ion can inva l ida te pre-v ious ly der i ved facts. T h i s non-monoton ic i t y precludes the use of first-order representat ions, l ike - 57 -those used for taxonomies , for fo rmal iz ing networks w i t h except ions. W e establ ish a correspondence between networks w i t h except ions and network defaul t theories. T h i s correspondence prov ides a fo rma l semant ics and a no t ion of correct inference for such networks. A s was the case for taxonomies, inher i tance emerges as a log ica l feature of the representat ion. Those propert ies Pi,...,Pn w h i c h an i nd i v i dua l , 6, inher i ts prove to be precisely those for w h i c h P 1 (6 ) , . . . ,P n (6 ) a l l belong to a c o m m o n extension of the defaul t theory. Shou ld the theory have mu l t i p le extensions — an undesirable feature, as we sha l l see — then b m a y inher i t different sets of propert ies depending on wh ich extension is chosen. W e cons ider two rad ica l l y different remedies for th is prob lem. T o see how defaul ts m igh t be used to represent networks w i t h except ions, consider the elephant examp le , w h i c h can be represented by the default theory: W — |vz. Albino-Elephant(x) D Elephant(x) jy _ f Elephant(x) : Gray(x) A ->Albino—Elepkant(x) \ Gray(x) It is easy to see that i f we are to ld on ly Elephant(Fred) then, so far as we know, Gray(Fred) A ->Albino-Elephant(Fred) is consistent; hence Gray(Fred) m a y be inferred. G i v e n on ly Albino—Elephant(Sue) one can conclude Elephant(Sue) using first-order knowledge, but Albino-Elephant(Sue) " b l o c k s " the app l i ca t ion of the default, prevent ing the der iva t ion of Gray(Sue), as required 1 . W e adopt a network representat ion w i t h seven l ink types. O the r approaches to inher i tance m a y omi t one or more of these, but our fo rma l i sm subsumes these. T h e seven l ink t ypes , 1 w i t h the i r t rans la t ions to default logic, are: (1) St r ic t I S - A : A. • . 5 : A ' s are a lways B's. S ince this is un iversa l ly t rue, we ident i fy it w i t h the f i rst-order fo rmu la : Vx. A(xj Z> B(x). (2) M e m b e r s h i p : ao • . A : T h e i nd i v i dua l a belongs to the class A. W e represent this w i t h the first-order fact A(a). (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 , th is is a un iversa l s tatement, ident i f ied w i th : Vx. A ( z ) Z> -<B(x). (4) Non -membersh ip : ao | j | >• .A: T h e i n d i v i d u a l a does not belong to the class A. W e represent th is w i t h the first-order fact -iA(a). (5) Defau l t I S - A : A. >.B: N o r m a l l y A's are B's, but there m a y be except ions. T o prov ide for except ions, we ident i fy this w i t h a default : 1 Note that strict and default links are distinguished graphically by solid and open arrowheads, respec-tively. - 58 -A(x) : B(x) B(x) (6) Defau l t ISN'T-A: A. \ \ \ >.B: N o r m a l l y A's are not B's, but except ions are a l lowed. Ident i f ied w i t h : A(x) : -iB(x) ^B(x) (7) E x c e p t i o n : A. > T h e except ion l ink has no independent semant ics; i t serves on ly to make exp l i c i t the excep-t ions, i f any, to the above defaul t l inks . There must a lways be a defaul t l ink at the head of an except ion l ink; the except ion then alters the semant ics of that default l ink . The re are two types of defaul t l inks w i t h except ions; their g raph ica l structures and t rans la t ions are shown in F igure 4.2. B A{x) : B{x) A -iC^x) A - A -^Cn{x) B(x) B A(x) : ^ B{x) A - . C i ( i ) A ... A - . C n ( x ) ^B(x) 1\ 4 ^ Cx ... C7„ Figure 4.2 — L i n k s w i t h except ions. W e i l lus t ra te w i t h an example f rom [Fah lman et al 1981]. Mo l l uscs are no rma l l y shel l-bearers. Cepha lopods must be Mo l l uscs but no rma l l y are not shel l-bearers. N a u t i l i mus t be Cepha lopods and must be shel l-bearers. O u r ne twork representat ion of these facts is g iven in F igu re 4.3. - 5 9 -She l l -bearer M o l l u s c C e p h a l o p o d N a u t i l u s Figure 4-$ — N e t w o r k representat ion of ou r knowledge about Mo l l uscs . T h e corresponding defaul t logic representat ion is: D = W = M(x) : Sb{x) A ~>C{x) C(x) : -ngfc(z) A -^(x) \ Sb{x) ' ^Sb[x) J ' (x). C{x) D M[x), (x). N(x) z> C{x), (x). N{x) D Sb{x) ) G i v e n a par t i cu la r Nau t i l us , th is theory has a unique extension 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 Shel l -bearer . A C e p h a l o p o d not k n o w n to be a Nau t i l us w i l l t u rn out to be a M o l -lusc w i t h no shel l . It is ins t ruc t ive to compare our network representat ions w i t h those of N E T L [ F a h l m a n et al 1981]. A basic difference is that i n N E T L there are no str ict l inks; a l l I S - A and I S N ' T - A l inks are po tent ia l l y cancel lab le and hence are defaul ts. Moreover , F a h l m a n et al a l low exp l i c i t except ion ( • U N C A N C E L ) l inks on ly for I S N ' T - A ( * C A N C E L ) l inks . If we restr ict the graph of F igure 4.3 to N E T L - l i k e l inks , we get F igu re 4.4(a), wh i ch is essential ly the graph g iven by F a h l m a n . a) She l l -bearer . b) Shel l -bearer . C e p h a l o p o d Nau t i l u s M o l l u s c Cepha lopod Nau t i l us M o l l u s c Figure 4-4 — N E T L - l i k e network representat ions of our knowledge about Mo l l uscs . T h e network i n F igure 4.4(a) corresponds to the defaults: - 60 -M[x) : Sb(x) C(x) : M(x) N{x) : C(x) ' Sb{x) ' M{x) ' C{x) ' C{x) : -,Sb{x) A ^N{x) N{x) : Sb{x) -.56 (z) ' Sb(x) A s before, a g iven Nau t i l u s w i l l also be a Cepha lopod , a Mo l l usc , and a Shel l -bearer . A Cepha lo -pod not known to be a Nau t i l us , however, gives rise to two extensions, corresponding to an ambiva lence about whether or not it has a shel l . W h i l e counter - in tu i t ive , th is mere ly indicates that an except ion to shel l -bear ing, namely being a Cepha lopod , has not been exp l i c i t l y represented i n the network. T h e amb igu i t y can be resolved by mak ing the except ion exp l ic i t , as in F igu re 4.3. O n the other hand , representat ions w h i c h do not permi t except ion l inks to po in t to I S - A l inks cannot make this except ion exp l ic i t in the g raph ica l representat ion. O t h e r vers ions of N E T L (and many other inher i tance reasoners) do not a l low exp l ic i t excep-t ion l inks at a l l . If on ly default I S - A and I S N ' T - A l inks are a l lowed, the representat ion of the Nau t i l us example becomes that of F igu re 4.4(b), w h i c h corresponds to the defaults: !M(x) : Sb(x) C{x) : M(x) N(x) : C(x) ' Sb(x) ' M(x) ' C(x) ' ( C{x) : -,Sb{x) N(x) : Sb{x) ^Sb{x) ' Sb{x) In such theories, there is a fur ther amb igu i t y about whether a Nau t i l us is a Shel l -bearer . H o w then do such systems conjecture that a Cepha lopod is not a Shel l -bearer, w i thou t also conjectur ing that i t is a Shel l -bearer? S u c h ambigui t ies are t yp ica l l y resolved by means of an inference procedure w h i c h prefers shortest paths. In terpreted in terms of default logic, this " sho r -test pa th heu r i s t i c " is in tended to favour one extension of the default theory. T h u s , in the net-works of F igu re 4.4, the paths f rom C e p h a l o p o d to -•Shel l -bearer are shorter than those to She l l -bearer so that the former w i n . Un fo r tuna te ly , th is heur ist ic is not sufficient to replace the exc luded except ion type in a l l cases. Re i te r a n d Cr iseuo lb [1983]; and E the r i ng ton [1982] show that it can lead to conclusions wh ich are un in tu i t i ve or even i n v a l i d — i.e., not in any extension. F a h l m a n et al [1981] and T o u r e t z k y [1981, personal commun ica t ion ; 1982] have also observed that shortest pa th a lgor i thms can lead to anomalous conclusions. T h e y describe a t tempts to restr ict the fo rm of networks to exc lude structures wh ich admi t such problems. One effect of these res-t r ic t ions appears to be to permi t on ly networks whose corresponding defaul t theories have unique extensions. A n inference a lgo r i thm for network structures is correct on ly i f it can be shown to derive conclusions a l l of w h i c h l ie w i t h i n a single extension of the under ly ing default theory. T h i s c r i -ter ion rules out shortest pa th inference for unrest r ic ted networks. T h i s is unfor tunate, since shor-test pa th inference has been popu lar for i ts re la t ive efficiency and ease of imp lementa t ion . O n a more posi t ive note, any network const ruc ted using the seven l ink- types g iven above corresponds to a network default theory. B y insist ing that any network const ructed must correspond to an ordered theory , the coherence of a network knowledge representat ion system can be assured. F o r such systems, the procedure g iven i n chapter 3 is a correct and a lways converg ing inference a lgo r i t hm. - 61 -It turns out that orderedness can be assured w i thou t reference to the fu l l comp lex i t y of the mechan ism descr ibed i n chapter 3. It is easy to see that any acyc l ic network gives rise to an ordered theory. T h e same is true i f on ly the subgraph consist ing of a l l I S - A l inks and exp l ic i t except ions thereto has no cycles invo lv ing at least one except ion l ink , or if there are no exp l ic i t except ions to I S - A l inks . Theorem 4.1 A n y network i n w h i c h the subgraph of IS -A l inks and exp l ic i t except ions thereto is acy-c l ic corresponds to an ordered theory. | Corollary 4.2 A n y acyc l i c network corresponds to an ordered theory. I Corollary 4.3 A n y network w i t h no exp l ic i t except ions to IS -A l inks corresponds to an ordered theory. | Corollary 4.4 T h e networks ment ioned in theorem 4.1 and corol lar ies 4.2 and 4.3 are coherent. | In add i t i on to po in t ing out the inadequacies of shortest pa th inferencing and to p rov id ing sufficient cond i t ions for coherence and a correct inference mechan ism, the f o r m a ! reconst ruc t ion of inher i tance we have presented clarif ies some of the outs tanding prob lems in network inference. O n e of these, how to per form inferences i n para l le l , is considered in the next sect ion. 4.1. Parallel Network Inference Algorithms T h e compu ta t i ona l comp lex i t y of inher i tance problems, combined w i t h some encouraging examples, has sparked interest in the poss ib i l i ty of de termin ing inher i tance i n para l le l . F a h l m a n [1979] has proposed a mass ive ly para l le l mach ine archi tecture, N E T L . T h i s arch i tecture assigns one processor to each predicate i n the knowledge base. " I n f e renc ing " is per formed by nodes pass-ing " m a r k e r s " to ad jacent nodes i n response to their own state and that of the i r immed ia te neigh-bours. F a h l m a n suggests that such archi tectures cou ld achieve logar i thmic speed improvements over t rad i t i ona l ser ia l machines. T h e fo rma l i za t ion of inher i tance networks as default theories suggests, however, that there m igh t be severe l im i ta t ions to this approach. F o r example , correct inference requires that a l l con-clusions share a c o m m o n extension. F o r networks w i t h more than one extens ion, in ter-extension interference effects must be prevented. T h i s seems impossib le for a one pass para l le l a lgor i thm w i t h pure ly l oca l commun i ca t i on , especia l ly in v iew of the inadequacies of the shortest pa th heur ist ic . - 62 -E v e n in knowledge bases w i th unique extensions, structures requ i r ing an a rb i t ra r i l y large rad ius of c o m m u n i c a t i o n can be created. F o r example [Ether ington 1982], the defaul t theories corresponding to the networks in F igure 4.5 each have unique extensions. A network inference a lgor i thm must reach F before propagat ing th rough B i n the first network and conversely i n the second. T h e sal ient d is t inc t ions between the two networks are not loca l ; hence they cannot be u t i l i zed to guide a pure ly loca l inference mechan ism to the correct choices. S i m i l a r networks can be const ructed w h i c h defeat marker-passing a lgor i thms w i t h any fixed rad ius. Figure 4-5 — Problems for local inheritance algorithms. This has prompted Touretzky [1981, personal communication; 1984a] to characterize a res-tricted 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 exten-sion. 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 res-trictions. Provided the network in question corresponds to an ordered theory, a form of limited paral-lelism can be achieved without sacrificing correctness. The 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. The "nodes asserted at level 0" are those in Th( W). These correspond to the nodes for - 6 3 -w h i c h the network is " a c t i v a t e d " . T h e result after step k is an ex tens ion . 2 There are two caveats associated w i t h this procedure: If bo th posi t ive and negat ive markers reach a node i n the same step, one must be chosen. E i t he r choice w i l l lead to an extension; we do not consider o ther rami f ica t ions of such choices here. Second, the a lgor i thm assumes that a l l str ict l inks propagate ins tantaneously . If th is is not the case, each step i n the a lgor i thm must be fo l -lowed by p ropagat ion a long str ict l inks , resolv ing conf l icts as above. No te that conf l icts are a lways resolved by chang ing assignments at the current leve l . P r o v i d e d that the inv io lab i l i t y of str ict l inks is ma in ta ined , that default l inks are act ive on ly i f the i r prerequis i tes are asserted and their just i f icat ions have not been denied, and that no node and i ts negat ion are asserted together (confl ict resolut ion), any reasonable propagat ion a lgor i thm (para l le l o r otherwise) m a y be used at each s tep . 3 T o i l lus t ra te the const ruc t ion, we app ly i t to the moderately complex network of F igure 4 . 6 . R a t h e r than restr ic t ourselves to a par t i cu la r para l le l propagat ion a lgor i thm at each step, we present a table showing a l l possibi l i t ies. A Figure 4-6— A mu l t i - l eve l inher i tance graph. T h e cor responding default theory, s impl i f ied to the propos i t iona l case and " a c t i v a t e d " for A , is: W = {A, ( A o f l ) , ( A O C)) ( A.^D A : - i F B: D C: F B : E E :-^H \ -.£> ' -,F ' D ' F ' E ' -itf 2 This construction is that used in the proof of Theorem 3.3, where it is shown to yield an extension. 3 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 propaga-tion is immaterial. - 6 4 -E: G f\ -<D G: H E: If\ ->F G ' H ' I I: A ~-H -<J T h e defaul ts above have been grouped accord ing to the level to w h i c h their consequents are assigned (see T a b l e 4.1). T a b l e 4.2 shows the possibi l i t ies at each step; a l ternat ives are L e v e l L i t e ra l s 1 A , B , C , D , n D , E , F , - . F , ->H 2 G , H , I 3 T a b l e 4.1 — Leve ls of l i terals. shown i n separate co lumns , w i t h ma jo r rows corresponding to steps i n the a lgor i thm. S tep 1 A , B , C , E , - . H D , F D, - F - . D , F - . D , - , F S tep 2 I G I, G Step 3 T a b l e 4.2 — Poss ib le outcomes using different propagat ion schemes. T h u s the a lgo r i t hm can , depending on the nature of the para l le l marker propagat ion procedure, find: E0 = Th( W U {A, B, C, E, D, F}) Ex = Th( W U {A, B, C, E, ->H, D, -,F, I, ->J}) E2 = Th\ W U {A, B, C, E, ->H, ->D, F, G}) E3 = Th( W U {A, B, C, E, ->H, -<D, /, G, ->J}) a l l of w h i c h are extensions. S ign i f i cant ly , no choice of para l le l marker-pass ing procedure w i l l enable the a lgo r i t hm to find the theory 's other two extensions: EA = Th( W U {A, B, C, E, H, -.£>, F, G}) E5 = Th( W U {A, B, C, E, H, ->F, G, /}) because ->H is at l eve l 1 and so can (and must) be inferred at step 1. H, being at leve l 2, is thus p rec luded before i t can be inferred. W e have not yet character ized the biases w h i c h th is i nab i l i t y to find a l l extensions wou ld induce in a reasoner. A n o t h e r po ten t ia l p rob lem w i t h th is approach stems f rom the fact that many network infer-ence systems " p r e f e r " one l ink - type over another (e.g., negat ion m a y overr ide assert ion). B y b reak ing the ne twork in to sub-networks w h i c h are processed in tu rn , the ab i l i t y to g loba l l y assert these preferences m a y be lost. W e have three responses to this. F i r s t l y , i f network s t ructure is rest r ic ted, i n the m a n n e r suggested b y T o u r e t z k y [1981, personal communica t ion ] , so that resul t -ing theories have un ique extensions, the above a lgor i thm produces the same results as any correct - 6 5 -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 dis-cuss 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 semi-normal defaults is inappropriate for two reasons: Firstly, adding new information to the knowledge-base requires modification of the defaults already in the knowledge-base. These become increasingly complex as the knowledge-base grows. Secondly, the translation of a link depends on other links in the network. 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. 4 Cott re 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. - 6 6 -T o our knowledge, the first a lgo r i t hm capable of correct ly reasoning w i t h an inher i tance net-work in para l le l was that presented in the preceding sect ion (see also [Ether ington 1983]). Because of the par t i t i on ing of the network, the a lgor i thm is incapable of finding some extensions of some defaul t theor ies; i t is not complete. T h i s a lgor i thm is correct; a l l of i ts conclus ions lie w i t h i n a single extens ion. However , it does not necessari ly produce the preferred extens ion, based on the in tu i t i ve semant ics for inher i tance networks. T o u r e t z k y [1984a] has developed a more sophist icated a lgor i thm, based on the " in fe ren t ia l d is tance" topology. T h i s in ferent ia l d is tance a lgor i thm is app l icab le to networks w i thou t exp l ic i t except ion l i nks , and is correct , in the sense that a l l of i ts conclusions l ie w i t h i n a single extension. Fu r the rmore , the in ferent ia l d istance concept is based on the pr inc ip le that ambiguous inher i tance shou ld be, when possible, resolved by appeal ing to the subclass/superc lass re la t ion w h i c h forms the basis of inher i tance. Inferent ia l d is tance is somewhere between the " b r a v e " and " c a u t i o u s " ends of the spect rum of non-monoton ic reasoning systems. Essent ia l l y , i f an i nd i v i dua l cou ld inher i t p roper ty P by v i r -tue of the fact that she I S - A B, and proper ty ->P because she I S - A C, then the amb igu i t y is resolved as fo l lows: If C I S - A B and not vice versa, inher i t ~<P, otherwise, if B I S - A C and not vice versa, inher i t P; otherwise, inher i t nei ther. Concep tua l l y , the inferent ia l d is tance a lgor i thm e l iminates those extensions w h i c h do not satisfy the h ierarch ica l nature of the representat ion, then draws those conclus ions wh ich ho ld in a l l of the remain ing extensions. T h i s approach captures the semant ic in tu i t ion (propert ies associated w i t h subclasses shou ld overr ide those associated w i t h superclasses) w h i c h is the fundamenta l raison d'etre for inher i tance representat ions. It also avoids the p i t fa l ls of incorrect behav iour wh ich curse shortest-path infer-ence a lgor i thms, as ev idenced by Theo rem 4.5. T h e o r e m 4 . 5 In the absence of " no - conc lus i on " l inks, a l l of the ground facts re turned by T o u r e t z k y ' s in ferent ia l d is tance a lgor i thm l ie w i t h i n a single extension of the default theory wh i ch corresponds to the inher i tance graph in quest ion. | T o i l lus t ra te the in ferent ia l d istance a lgor i thm, consider the network f r om F igu re 4.4(b). Because Nautilus is a subclass of Cephalopod, w h i c h is a subclass of Mollusc, in ferent ia l d istance gives the desired results: Nautili are Shell-Bearers, whi le Cephalopods not k n o w n to be Nautili are not. In the ne twork of F igu re 4.7, ne i ther Republican nor Quaker is a subclass of the other. T h u s in ferent ia l d is tance sanct ions no conclus ions about whether Nixon is a Pacifist. - 67 -. Pac i f i s t Q u a k e r . '. Repub l i can Figure J^.l— A genuinely ambiguous inher i tance graph. T h e o r e m 4.5 on ly begins to explore the connect ions between T o u r e t z k y ' s work and that repor ted i n chapters 3 and 4 of th is thesis (and in [Reiter 1980a]). W e have shown tha t g round facts re tu rned by in ferent ia l d istance - e.g., " C l y d e is an e lephant " , or " C l y d e is not grey" -belong to a c o m m o n extension of the corresponding default theory. Inferent ia l d is tance also sanc-t ions no rmat i ve conclusions, such as "A lb i no -e l ephan t s are [ typical ly] herb ivores" . W e have not exp lored the re la t ionsh ip such statements inferred under inferent ia l d istance bear to the under ly-ing defaul t theory. T o u r e t z k y also a l lows wha t he cal ls " no -conc lus ion " l inks. These l inks a l low inher i tance to be b locked w i t hou t exp l ic i t cancel la t ion. Defau l t logic has no analogue for the no-conc lus ion l ink, and we have exc luded them f rom cons iderat ion here. It appears that i t wou ld be s t ra ight forward to add a s im i l a r capac i t y to default logic, assuming that such l inks ac tua l l y prove useful . T h e proof of theorem 4.5 suggests that i ts genera l izat ion to networks w i t h no-conclus ion l i nks vis-a-vis such an ex tended defaul t logic w o u l d present no problems. T o u r e t z k y [1984a] provides a deta i led exp lorat ion of the propert ies of in ferent ia l d istance inher i tance reasoning, inc lud ing a const ruct ive mechan ism for determin ing the 'g rounded expan-sions' (analogous to extensions) of a network. M a n y of his results bear a superf ic ia l s im i la r i t y i n fo rm and proof to those in [Reiter 1980a] and in chapter 3 of this thesis. H i s proofs re ly on pa r t i a l acyc l i c i t y cond i t ions w h i c h seem s im i la r to the orderedness condi t ions we descr ibe. W e have speculated (as has Tou re t zky ) that the results i n [Touretzky 1984a] and those con ta ined herein m a y prove to be closely re la ted. F i n a l l y , T o u r e t z k y [1984a, 1985] explores the appl icat ions of in ferent ia l d istance to " i nhe r i t -able re la t ions" , c i t i ng examples such as C i t i z e n s d is l i ke crooks. E lec ted crooks are crooks. G u l l i b l e c i t izens don ' t d is l ike elected crooks. In th is examp le , c i t izens general ly d is l ike elected crooks, but F r e d , the gu l l ib le c i t i zen , doesn' t dis-l ike D i c k , the e lected crook. A complete t reatment of the re la t ion between T o u r e t z k y ' s work and defaul t log ic shou ld t ry to ex tend the correspondence presented here to inc lude T o u r e t z k y ' s in ferent ia l d is tance t reatment of inher i tab le re lat ions. T o u r e t z k y shows that , i n general , para l le l marker-pass ing a lgor i thms cannot der ive the con-c lus ions sanc t ioned b y the in ferent ia l d istance a lgor i thm. He also shows that an a rb i t ra ry net-work can be " c o n d i t i o n e d " , by add ing log ica l ly - redundant l inks, i n such a w a y tha t a para l le l marker -pass ing a lgo r i t hm can re turn correct results. Un fo r tuna te ly , this cond i t ion ing , w h i c h must - 6 8 -be per formed each t ime the network is modi f ied, is expensive (Toure tzky [1984a] gives a po lynomia l - t ime a lgo r i t hm w h i c h adds O^N2) l inks i n the worst case) and is apparen t l y not amen-able to para l le l marker -pass ing imp lementa t ion [Touretzky 1982, personal commun ica t i on ; 1983]. W e conc lude that , for cond i t ioned networks, there are correct ( in the sense we have described) para l le l , marker -pass ing a lgor i thms for determin ing inher i tance i n the presence of except ions. S u c h a lgor i thms can be v iewed as fast inference a lgor i thms for reasoning w i t h the t rac tab le class of defaul t theories w h i c h correspond to condi t ioned networks. C H A P T E R 5 Predicate Circumscription In th is chapter we focus on predicate c i rcumscr ip t ion , as presented in [ M c C a r t h y 1980]. O u r ob ject ive is to establ ish var ious results concerning the consistency of this fo rma l i sm, and to descr ibe some l im i ta t i ons of i ts ab i l i t y to conjecture new in format ion. O n e such l im i t a t i on is that pred icate c i r cumscr ip t i on cannot account for the s tandard k inds of default reasoning. A n o t h e r l im i t a t i on relates to equa l i t y ; predicate c i rcumscr ip t ion y ie lds no new ground facts about the equa l i ty predicate for a large class of first-order theories. T h i s has impor tan t consequences for the so-cal led " u n i q u e n a m e s " a n d " d o m a i n c losu re " assumpt ions. 5 . 1 . F o r m a l P r e l i m i n a r i e s Pred ica te c i r cumscr ip t i on was discussed i n deta i l in chapter 2. W e repeat some of the techn ica l deta i ls here for convenience. The semant ic in tu i t ion under ly ing predicate c i rcumscr ip -t ion is that c losed-wor ld reasoning about one or more predicates of a theory corresponds to t ru th in a l l models of the theory w h i c h are m i n i m a l in those predicates. Spec i f ica l ly , let T[Pi,...,P^ be a first-order theory , some (but not necessari ly all) of whose predicates are P = {P1,.../Pre}. A mode l M of T is a "P-submodel of a mode l M ' of T iff the extension of each P,- i n M is a subset of i ts extens ion i n M * , and M a n d M * are otherwise ident ica 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 den t i ca l to M . F o r finite theories, T(Pi,...,P^t M c C a r t h y [1980] proposes rea l iz ing predicate c i rcumscr ip -t ion syn tac t i ca l l y by add ing the fo l lowing a x i o m schema to T: Here are pred icate var iab les w i t h the same ari t ies as P ^ - . ^ P n , respect ive ly . !!($!,...,3> n) is the sentence ob ta ined by con jo in ing the sentences of T, then rep lac ing every occurrence of P i , . . v P „ i n T by $i,...,$„ respect ive ly . T h e above schema is ca l led the (joint) circumscription schema of Pu...,Pn in T. L e t CLOSUREP(T) - the closure of T with respect to P = { P l 7 . . . , P J -denote the theory consis t ing of T together w i t h the above a x i o m schema. M c C a r t h y fo rma l l y identi f ies reasoning about T under the c losed-wor ld assumpt ion w i t h respect to the predicates P w i t h first-order deduct ions f r om the theory CLOSURE-p{T). M c C a r t h y [1980] shows that any instance of the schema resul t ing f r om c i r cumscr ib ing a s in -gle pred icate P i n a sentence 1\P) is true in a l l { P } - m i n i m a l models of T. T h i s general izes - 6 9 -- 70 -di rect ly to the jo in t c i r cumscr ip t i on of mu l t ip le predicates; we omi t the proof of th is. W e use this genera l iza t ion extens ive ly i n the proofs of the results of this chapter . Because predicate cir-cumscr ip t i on is app l icab le on ly to finitely ax iomat izab le theories, we w i l l restr ict our a t ten t ion to such theories. 5 . 2 . O n t h e C o n s i s t e n c y 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 T h e m i n i m a l mode l semant ics of predicate c i rcumscr ip t ion guarantees that CLOSUREp[ T) is consistent whenever T has P - m i n i m a l models. T h i s suggests that cer ta in consistent first-order theories l ack ing m i n i m a l models m a y have inconsistent closures. Indeed, th is can happen, 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 c i r c u m s c r i p t i o n Cons ide r the fo l lowing 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 any mode l of T, the extension of NJ conta ins a sequence of elements isomorph ic to the na tu ra l numbers . A n { N } - s u b m o d e l can a lways be constructed by delet ing a finite i n i t i a l segment of th is sequence. Hence every mode l of T has a proper { N } - s u b m o d e l , so T has no { N } - m i n i m a l models . C i r c u m s c r i b i n g N i n th is theory, a n d let t ing $z be [Nz A 3y- z = succ(y^\ Ny] y ie lds Vz. N i D 3 y- [Ny l\ x = succ(y)\ w h i c h cont rad ic ts the first a x i o m . fl: In v i ew of th is example , it is na tu ra l to seek classes of first-order theories for w h i c h pred i -cate c i r cumscr ip t i on does not in t roduce inconsistencies. The "we l l - founded ' ' theories fo rm such a class. W e say that a first-order theory is well-founded iff each of i ts models has a P - m i n i m a l sub-mode l for every finite set of predicates P . A n y consistent wel l - founded theory obv ious ly has at least one P - m i n i m a l mode l . S ince every instance of the c i rcumscr ip t ion schema of P i n a theory T is t rue in a l l P - m i n i m a l models of T, we have: T h e o r e m 5.1 If T is a consistent wel l - founded theory, then CLOSURE-^ T) is consistent for any set P of predicates of T. I.e., predicate c i r cumscr ip t ion preserves consistency for wel l - founded theories. | W h i c h theor ies are wel l - founded? W e know of no complete syntac t ic charac te r i za t ion , but a pa r t i a l answer comes f rom a genera l izat ion of a result on un iversa l theories due to B o s s u and S iege l [1985]. A first-order theory is universal iff the prenex no rma l fo rm of each of its formulae - 71 -conta ins no ex is ten t ia l quant i f iers. T h e o r e m 5.2 U n i v e r s a l theories are wel l - founded. | In v i e w of T h e o r e m 5.1, we know that predicate c i rcumscr ip t ion preserves consistency for un iversa l theories: C o r o l l a r y 5.3 If T is a consistent un iversa l theory, then CLOSUREp(T) is consistent for a n y set P of predicates of T. | Not i ce that the class of un iversa l theories includes the H o r n theories, w h i c h have a t t rac ted considerable a t ten t ion f rom the P R O L O G , A l , and Database communi t ies . L i f sch i t z [1985b] has general ized theorem 5.2 to inc lude the class of " a l m o s t un ive rsa l " theories. A theory is almost universal in P iff i t has the fo rm Vx t A, where A does not con ta in pos i t ive occurrences of P € P w i t h i n the scope of quanti f iers. A l m o s t un iversa l theories inc lude un iversa l theories as we l l as the " sepa rab le " theories of L i fsch i tz [1985a] (see § 2.1.5.2). 5.3. W e i l - F o u n d e d T h e o r i e s a n d 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 h e proper ty of wel l - foundedness, taken together w i t h the "soundness" of predicate c i r-cumsc r ip t i on w i t h respect to the set of m i n i m a l models al lows us to character ize the power of predicate c i r cumsc r ip t i on . T h i s leads to some ra ther surpr is ing results. In this sect ion we describe some l im i ta t i ons of predicate c i rcumscr ip t ion w i th respect to wel l - founded theories. The first such resul t is that predicate c i r cumscr ip t ion y ie lds no new posi t ive ground instances of any of the predicates being c i rcumscr ibed . T h e o r e m 5.4 Suppose tha t T is a wel l - founded theory, P € P is an n-ary predicate, and ctl,...t'3k are n-tuples of g round terms. T h e n CLOSUREDT)|-rai v - v P^k *=» T \— P&i V . - V Pctk . | O n ref lect ion, th is is not too surpr is ing, since c i rcumscr ip t ion is in tended to m i n i m i z e the exten-sions of those predicates being c i rcumscr ibed . N e w posi t ive instances of such predicates should not arise f r om th is m in im i za t i on . - 72 -A more interest ing - even s tar t l ing - result is that no new ground instances, pos i t ive or negat ive, of unc i rcumscr ibed predicates can be der ived by predicate c i rcumscr ip t ion . T h e o r e m 5 .5 Suppose tha t T is a wel l - founded theory, P £ P is an n-ary predicate, and <£v...,ak are n-tuples of g round terms. T h e n (i) CLOSUREp(T) \-P3iy...y Petk T |— Paj V. . .V Pak , and (ii) CLOSURE^ T) \- -PoTi V. . .V - . P a f c T \- - P o ^ V. . .V ^ Pak . I In summary , Theo rems 5.4 and 5.5 te l l us that the on ly new ground l i tera ls that can be con-jec tured by pred icate c i r cumscr ip t ion of wel l - founded theories are negat ive instances of one of the predicates being c i rcumscr ibed ; A n unfor tunate consequence of this result is that the usua l k inds of defaul t reasoning cannot be rea l ized by predicate c i rcumscr ip t ion . T o see why , consider the s tandard A l example concern ing whether b i rds fly, g iven that " b y defau l t " b i rds fly. T h e re levant facts m a y be represented i n var ious ways, two of wh i ch fo l low: 1) In this representat ion, a l l of the except ions to flight are l is ted exp l i c i t l y in the a x i o m sanct ion-ing the conc lus ion that b i rds can fly. Vz. Bird{x) A -*Penguin(x) A -<Ostrich(x) A -*Dead(x) A ••• 3 Can-Fly(x) In add i t i on , there are var ious I S - A ax ioms, as we l l as mu tua l exc lus ion ax ioms: Vz. Canary(x) Z5 Birdix) Vz. Penguin(x) p 5tra(z) Vz. -if Canary(x) A Penguin(x\\ Vz. -i(Pen</tttn(z) A Ostrich(x)) 2) In th is representat ion, due to M c C a r t h y [19861, a new predicate, ab, s tand ing for " a b n o r m a l " , is i n t roduced . One then states that " n o r m a l " b i rds can fly: Vz. Bird(x) A --a6(z) D Can-Fly(x) T h e a b n o r m a l b i rds are l is ted: Vz. Penguin(x) D ab(x) Vz. Ostrich(x) D ab[x) F i n a l l y , one inc ludes the I S - A and m u t u a l exc lus ion ax ioms as in (1) above. B o t h representat ions (1) and (2) are un iversa l , and hence wel l - founded, theories. Therefore, if Bird(Tweety) is g iven , Theorems 5.4 and 5.5(i) te l l us that the defaul t assumpt ion Can-Fly( Tweety) cannot be conjectured by predicate c i rcumscr ip t ion . C a r e f u l readers of [ M c C a r t h y 1980] might find Theorems 5.4 and 5.5 inconsistent w i t h the resul ts i n Sec t ion 7 of that paper. In the b locks-wor ld example presented there to i l lust rate pred i -cate c i r cumsc r ip t i on , the ground instance on(A,C,result{move(A,C),SQJ) can be der ived by c i r -cumscr ib ing a dif ferent predicate, Xz.prevents(z,move(A,C),s0). T h i s appears to v io la te T h e o r e m 5.5(i). T h i s d iscrepancy stems f rom the fact that in formulat ing the c i r cumscr ip t i on schema for - 73 -this example, McCarthy 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 . E q u a l i t y 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 . T h e U n i q u e - N a m e s A s s u m p t i o n When told that Tom, Dick and Harry are friends, one naturally assumes that 'Tom', 'Dick' and 'Harry' denote distinct individuals: Tom 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. A l l pairs of individuals not specified as identical are assumed to be different. 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 cir-cumscription. For example, if all we know is that Opus is a Penguin, we can circumscriptively conjecture Vx. Penguin(x) = x = Opus. We cannot use this to deduce -<Penguin(Tweety), how-ever, unless we know Opus Tweety. How then can we formalize reasoning under the unique-names assumption? The 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: - 7 4 -Vz. x = x Vzy. x = y D y = x Vxyz. x — y/\y=z'Dx=z Vx v...,x myi,...,y„. z i = yi A ... A i „ = y„ A P fo , . . . , * , , ) D P(yy.--,y^, for each n-ary predicate s y m b o l P of T1. Vzi,...,z n )yi,... /y„. zi = yi A ... A z„ = y n D / (zi , . . . , in) = Avi.>"->yn)> f ° r e a c h w-ary funct ion symbo l / o f T. W h e n T is finite, i t is therefore possible to c i rcumscr ibe the equal i ty predicate, since the resul t ing schema is finite. T h e next result informs us that doing so y ie lds no th ing new. T h e o r e m 5 .6 (Rei ter) Le t T be a first-order theory con ta in ing ax ioms wh ich define the equa l i ty predicate, =. T h e n T \- CLOSURE[=){T). I In v i ew of th is resul t , one migh t a t tempt to capture the unique names assumpt ion by jo in t l y c i r cumscr ib ing severa l predicates of the theory, not just the equal i ty predicate. W e do not know whether there are any theories for w h i c h this might work, but it cannot succeed for wel l - founded theories. N o new ground equal i t ies or inequal i t ies can be der ived by c i rcumscr ib ing a we l l -founded theory, regardless of the predicates c i rcumscr ibed. T h e o r e m 5.7 Suppose that T is a wel l - founded theory conta in ing ax ioms w h i c h define the equa l i t y p red i -cate; ai,...,ak ; 01,...,0k are g round terms, a n d P is a set of some of the predicates of T. T h e n it Jfc (i) CLOSUREP(T) | - ( V a,- = 0,) < = • T \- ( V a,- = 0.) , and £=1 i=l (ii) CLOSURE^ T) \- ( Vat £ 0,) <=> T (— £ 0,). I C o r o l l a r y 5 .8 Suppose that T is a wel l - founded theory conta in ing ax ioms w h i c h define the equa l i t y predicate; P is an n-ary predicate; and a 1 , . . v a f o 01,...,0ka.re g round terms. T h e n CLOSURED T) \- ^ P{au...,ak) => T \- ( V a,: £ 0.) or T \f-P(0u-,0k) • I 1=1 R e t u r n i n g to the " P e n g u i n " example above, we see that predicate c i r cumscr ip t i on cannot conjecture -<Penguin( Tweety) unless it is known that Opus j= Tweety, otherwise we cou ld der ive Opus j= Tweety f rom CLOSURE({Penguin(Opus)}), cont rad ic t ing T h e o r e m 5.7. - 75 -T h i s last res t r ic t ion is somewhat puzz l ing . T h e model- theory fixes the d o m a i n and the in terpretat ions of constants and func t ion symbols when determin ing m i n i m a l models. G i v e n the soundness of predicate c i r cumscr ip t ion w i t h respect to th is model - theory, it is easy to see w h y ident i ty is not in f luenced by the set of m i n i m a l models. If the equal i ty predicate is in terpreted as a congruence re la t ion , ra ther than as ident i ty (i.e., if non-normal models are a l lowed, where pairs of d is t inct d o m a i n elements are permi t ted to be in the extension of '= ' ) , the s i tua t ion is less clear. Essen t ia l l y , i t can be shown that , for any pa i r of terms for wh i ch one might hope to c i r cumscr ip -t i ve ly conjecture ( in)equal i ty , there are m i n i m a l models wh i ch support e i ther side of the issue. So m u c h for the semant ic exp lana t ion . There rema in two quest ions. W h a t feature of the c i r -cumscr ip t i on schema gives rise to this anoma ly? W h a t does this te l l us about c i r cumscr ip t ion? A par t ia l answer to the first quest ion is that L e i b n i z ' pr inc ip le of subs t i tu t i v i t y - equals are every-where in tersubst i tu t ib le preserving t ru th - makes a stronger statement about equa l i t y than the c i r cumscr ip t i on schema. T h e second quest ion remains unanswered. In a recent paper, M c C a r t h y [1986] proposes a c i rcumscr ip t i ve approach to the unique-names assumpt ion by in t roduc ing two equa l i t y predicates. O n e of these is the s tandard equal i ty predicate, but rest r ic ted to arguments w h i c h are names of objects. T h e other equa l i ty predicate, e(x,y), means that the names x a n d y denote the same object, e is ax iomat i zed as an equivalence re la t ion w h i c h does not, however, satisfy the fu l l pr inc ip le of subst i tu t ion , i n contrast to " n o r m a l " equa l i ty . T h i s fa i lure of fu l l subs t i tu t i v i t y for the predicate e prevents our Theorems 5.6 and 5.7 f rom app l y i ng to e. B e n j a m i n Groso f (personal communica t ion) has independent ly proposed a s im i la r approach to the unique-names assumpt ion . He has also observed that our T h e o r e m 5.6 appl ies to M c C a r t h y ' s [1986] more general no t ion of c i rcumscr ip t ion . 5 . 4 . 2 . 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 T h e domain-c losure assumpt ion is the assumpt ion that , i n a g iven first-order theory T, the universe of discourse is rest r ic ted to the smal lest set wh i ch conta ins those ind iv idua ls ment ioned in T, and w h i c h is closed under the app l i ca t ion of those funct ions ment ioned in T. D o m a i n cir-cumscr ip t ion [ M c C a r t h y 1977, 1980] is a proposed formal iza t ion of th is assumpt ion . M c C a r t h y [1980] suggests that d o m a i n c i rcumscr ip t ion might be reduced to predicate c i r cumscr ip t ion . T h i s is i n fact false, as shown by Theorems 5.9 and 5.10. T h e s implest set t ing i n wh i ch the domain-c losure assumpt ion can arise is for a theory w i t h a finite H e r b r a n d Un iverse {c 1 , . . . ,c n } . In th is case we migh t want to conjecture the domain-closure axiom for th is theory: V z . z = cx V . . . V z = cn. S u c h an ax iom is impor tan t for the theory of first-order databases [Rei ter 1980b]. N o such a x i o m can arise f rom predicate c i r cumscr ip t ion for we l l -founded theories. - 7 6 -Theorem 5.9 Suppose tha t T is a wel l - founded theory; tv...,tn are ground terms; and P is a set of some of the pred icate symbo ls of T. T h e n CLOSUREP(T) |— Vx. x = tx V . . . V x = tn <=> T (— Vz . x= tx V . . . V x - <„ . | Theorem 5.10 If T is a we l l - founded theory and T has a mode l w i t h some doma in , D, then so does CLOSURE^!). I 5.4.3. Some Misconceptions There are a n u m b e r of c o m m o n misconcept ions about the use of predicate c i r cumscr ip t ion , w h i c h we discuss br ie f ly , below. It has been proposed that a rb i t ra ry formulae cou ld be c i rcumscr ibed using predicate c i r -cumsc r ip t i on by inc lud ing a new predicate let ter and a def in i t ion dec lar ing it to be equiva lent to the expression to be c i rcumscr ibed . T h i s w i l l not work, in general. Theorem 5.11 If T (— Vxt Px= $x * fo r some expression <5z* not invo lv ing predicate letters f rom P , then T\~ CLOSUREST). | T h i s resul t seems to be re lated to Doy le ' s [1984] comments on imp l i c i t def inab i l i ty . S ince the theory a l ready conta ins a definition for P, c i r cumscr ip t ion cannot fur ther const ra in P. A s it is general ly undec idab le whether T |—Vzt Px= $ 5 * for a par t i cu la r $ (let alone a l l <£), i t fo l lows tha t one cannot decide w h i c h predicates to c i rcumscr ibe. Corollary 5.12 It is genera l ly undec idab le whether CLOSURE^ T) is stronger t han T. | It is w ide ly (and correct ly) be l ieved that CLOSUREp(Qx. Px}) }= 3/z. Px (i.e., there is a un ique P). The re appears to be some misunders tand ing about how this is ach ieved, however. A f t e r some exper imen ta t ion , the idea of sko lemiza t ion comes to m i n d a n d , indeed, CLOSUREp(Pa) \= 3'z- Px (ac tua l ly Vz . Px = z = a). Sko lemiza t ion , however, can change the - 77 -set of m i n i m a l models of a theory (and hence the results of c i rcumscr ip t ion) . T o see this, not ice that in E x a m p l e 5 .1 , T has no m i n i m a l models, but the sko lemized form of T is un iversa l and hence wel l - founded. There has been a tendency to believe that the sko lemized fo rm of a theory T is equ iva lent to T, w h i c h is false. In fact, sko lemizat ion preserves satisfiability, not de r i vab i l i t y ; the existence of models , not the set of models. T h e ac tua l c i rcumscr ip t i ve der i va t ion of 3-'z- Px f rom 3 x. Px i nvo lves the subs t i tu t ion of a binary predicate for <&x, viz x = u, where « is a var iab le d is t inct f rom x. T h e skeleton of a correct c i rcumscr ip t i ve der iva t ion i n a na tu ra l deduc t ion system of 3 fx. Px f rom 3 x. Px fo l lows: 1 [(3z. $z) A (Vz. $z D Px)} D (Vz. Px D $z)] CL0SURE{P){3x.Px) 2 j(3z. z = u) A (Vz. z = u D Px)} Z> (Vz. Px D z = u)] 1, [z = u/$z] 3 Vu . [(3z. z = u) A (Vz. z = uz D Pz)] 15 (Vz. Pz O z = «)] 2, uniwcrsa/ generalization 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)] 3, universal instantiation 7 Pa D (Vz. P n i = a) 6, tautology 8 Vz . Pz D z = a 5,7, tautology 9 Pa A (Vz. Pz D x= a) 5,8, tautology 10 3j/- Pj/ A (Vz. Pz D z = y) 4,5,9, existential generalization 11 3 'z- Ps 10, definition There have been imp l i c i t [ M c C a r t h y 1980] and expl ic i t [Genesereth and N i l sson 1987] suggestions that the way a round some of the l im i ta t ions of predicate c i r cumscr ip t ion might be to c i rcumscr ibe on ly " r e l e v a n t " por t ions of the theory. T h e idea is that , by weakening T ( $ ) - hence e l im ina t ing some of the condi t ions that O must satisfy - perhaps some more useful results w i l l ob ta in . O b v i o u s l y , one must be carefu l ; c i rcumscr ib ing P in Pa, leav ing out Pa w i l l produce Vz. -<Pz. T h i s , be ing inconsistent w i t h Pa, is perhaps too useful. One idea is to d is t inguish, amongst the posi t ive l i tera ls i n each clause of the theory, one wh ich the clause is sa id to be " a b o u t " . T h e n on ly those clauses " a b o u t " P are taken in to account i n forming T(4>). T h i s may indeed a l low posi t ive facts to be der ived. F o r example, consider Vz. Bird(x) A ->Penguin(x) O Flies(x) Bird(Tweety), Penguin(Opus), Opus £ Tweety Vz. Penguin(x) D Bird(x) If the first a x i o m is taken to be about Flies, then we get $Opus A [Vz. $z D Pz] D [Vz. P z D $z] when we c i rcumscr ibe ( in this fashion) Penguin in T. F r o m th is we c a n der ive Vz. Penguin[x) = x = Opus and Flies(Tweety)\ There are two drawbacks w i t h this approach, - 7 8 -however. T h e first is that its semant ics are unknown. T h e y are not those of predicate c i r-cumscr ip t i on , a n d there is no known model - theory or "soundness" result corresponding to that for predicate c i r cumsc r ip t i on . T h u s i t is not c lear wha t this approach computes. M o r e ser iously, consistency is not necessar i ly preserved; the first a x i o m of T is also " a b o u t " Penguins, in the sense that T* = (T Li {->Flies( Tweety)}) |—Penguin(Tweety). T a k i n g the above approach to c i r -cumscr ib ing T* w i l l resul t i n an inconsis tency. 5 . 5 . W h a t t o C i r c u m s c r i b e ? O n e obv ious p rob lem w i t h using c i r cumscr ip t ion i n a g iven set t ing is knowing just what to c i rcumscr ibe. Some of ou r results prov ide clues i n this d i rect ion. (Coro l la ry 5.12 shows that clues are the best that can be hoped for, in general.) T h e o r e m 5.5 tells us that if we w ish to use predi -cate c i r cumscr ip t i on to conjecture - iP (a) i n some wel l - founded theory then we must inc lude P among the predicates being c i rcumscr ibed . Theorems 5.4 and 5.5 te l l us that predicate c i r -cumscr ip t i on w i l l not do at a l l i f we wish to conjecture P{ct), as is the case for most forms of defaul t reasoning, so tha t we must appea l to some other mechan ism, such as M c C a r t h y ' s more general f o r m of c i r cumscr ip t i on , discussed i n the next chapter . C H A P T E R 6 Generalizations of Circumscription 6 . 1 . F o r m 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 [1986] has recent ly fo rmu la ted a general izat ion of predicate c i rcumscr ip t ion , ca l led f o rmu la c i r cumscr ip t i on . T h i s genera l izat ion provides for the m i n i m i z a t i o n of arb i t rary first-order expressions ra ther than s imple predicates. It also provides for the t reatment of desig-na ted predicates as var iab les of the m i n i m i z a t i o n . In this vers ion of c i r cumscr ip t ion some of the l im i ta t ions of T h e o r e m 5.5 no longer app ly . Thus , as some of M c C a r t h y ' s examples show, it is possible to c i rcumscr ibe a predicate P , t reat ing another predicate Q as var iab le , and der ive new posi t ive and negat ive ground instances of Q. In par t icu lar , M c C a r t h y ' s new fo rma l i sm appears adequate for the t reatment of some forms of defaul t reasoning, as his examples show. M a n y of the l im i ta t ions of predicate c i rcumscr ip t ion stem f rom the fact that on ly those predicates being m i n i m i z e d are a l lowed to va ry . F o r m u l a c i rcumscr ip t ion retains many of the a t t rac t ive features of i ts predecessor, w i thout some of i ts l imi ta t ions. M c C a r t h y ' s def in i t ion of the fo rmu la c i r cumsc r i p t i on of E(P,x) i n the theory T\P) takes the fo rm of the second-order ax iom, (22), T(P) A V * . T ( * ) A [Vxt £($,x) z> E(P,x)] D [Vxt E(P,x) D E{&,x)\ (22) where E(P,x) is any wel l - formed expression whose free i nd i v i dua l var iab les are among x*= Xi,...,xh and i n w h i c h some of the predicate var iab les P = { P i , . . . , P , J occur free; E(3>,x) is the result of rep lac ing each free occurrence of the predicate letters, P,-, in E(P,x) w i t h predicate va r i -ables, of the same ar i ty . N o t everyone is conv inced of the need for second-order logic for c i rcumscr ip t ion [Perl is and M i n k e r 1986]. A first-order schema vers ion of fo rmu la c i rcumscr ip t ion , (23), is obta ined by delet-ing the second-order quant i f ier , V$. T{P) A I t * ) A [Vxt £(*,x) D E[P,x)\ D [Vxt E[P,x) D £(*,x)] (23) W e w i l l somet imes wr i te CLOSURE(T; P; E[P,xj) for ei ther a x i o m (22) or schema (23), ind icat -ing m i n i m i z a t i o n of the expression E[P,x), w i t h the predicates P t reated as var iab le , i n the theory, T. M c C a r t h y presented on ly a syntac t ic character izat ion of fo rmu la c i rcumscr ip t ion . M o t i v a t e d b y a bel ief i n the impor tance of semant ic character izat ions for reasoning systems, and by the s t r i k i ng consequences of exp lor ing the semant ics of predicate c i r cumscr ip t ion , we explored the poss ib i l i t y that an appropr ia te genera l iza t ion of the m in ima l -mode l semant ics of predicate - 79 -- 8 0 -c i r cumscr ip t i on w o u l d character ize fo rmu la c i r cumscr ip t i on . 1 T h i s led us to a fo rm of the general-ized m i n i m a l - m o d e l semant ics wh i ch has since been used in the exp l i ca t ion of a va r ie ty of c losed-wor ld reasoning formal isms (see §2.1). T h e precise detai ls are g iven below. D e f i n i t i o n : M < ^ p - ^ A f ' Le t T"(P) be a finitely ax iomat i zed (first- or second-order) theory, some (but not neces-sar i ly al l) of whose predicates are those in P ; let E(P,x) be a fo rmu la whose free var iab les are among ~x = xv...,xn, and i n wh ich some of the predicate var iab les P = { j - i , . . . , .? „ } occur free; and let M , M' be models of T. W e say M i s an E(P,x)-submodel of A f ' (wri t-ten A f < np^M') iff (i) |Af| = \M'\ , (ii) If t is a te rm, then \ t\M= \ , (ii i) I f Q £ P is a predicate letter of T, then \ Q\M= \Q\iJ , and (iv) \E(P,x)\MQ I ^ P . x ) ^ . I D e f i n i t i o n : i ? ( P , z ) - M i n i m a l M o d e l A mode l , Af, of T is E(~P,xj-minimal iff T has no model , M*, such that M' < ^ P ^ M a n d T h a t th is is the correct semant ics is suggested by Theorems 6.1 and 6.2. Theo rem 6.2 is appl icab le on ly to the f i rst -order-schema vers ion of fo rmu la c i rcumscr ip t ion ; T h e o r e m 6.1 appl ies bo th to that and to second-order f o rmu la c i rcumscr ip t ion . 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 sat isf ied by every ^ P . z J - m i n i m a l mode 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 (Per l is and M inke r ) If a l l models of T have finite extensions for each P G P (modu lo equa l i t y ) , then M satisfies every instance of CLOSURE(T) P ; E(P,xj) on ly if A f is an £ ( P , i ) - m i n i m a l mode l of T. | 1 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. - 81 -Per l i s and M i n k e r [1986] ac tua l ly prove a s l ight ly stronger resul t , app l icab le i f a l l models for CLOSURE(T; P ; E(P,x)) have finite extensions for each P S P . O f course, no general complete-ness result cou ld be fo r thcoming. The re is a unique (up to isomorphism) m i n i m a l m o d e l for the s tandard ax ioma t i za t i on of the na tu ra l numbers , but there is no recurs ive first-order ax ioma t i za -t ion w h i c h un ique ly character izes this mode l . If c i rcumscr ip t ion were complete, it cou ld be used to conjecture such a first-order ax iomat i za t i on . It is wor thwh i le de termin ing w h i c h of the differences between predicate c i r cumscr ip t i on and fo rmu la c i r cumscr ip t i on are rea l ly necessary. A s M c C a r t h y has suggested, the m i n i m i z a t i o n of a rb i t ra ry expressions is not. Theorem 6.3 T h e ab i l i t y to m i n i m i z e arb i t ra ry expressions, 2?(P,x), instead of s imple sets of p red i -cates, is an inessent ia l ex tens ion, p rov ided predicates other than those being m i n i m i z e d are a l lowed to va ry . | T h e o r e m 6.3 tel ls us that i t suffices to c i rcumscr ibe predicates. T o see this, observe that one can s imp ly ex tend the language w i t h a new predicate symbo l , \p and add the a x i o m : V x . rj>x= £ ( P , x ) to the theory. C i r c u m s c r i b i n g ^>5Tin the extended theory w i th P U {ip} var iab le results i n a con-servative extension (no new theorems over the or ig ina l language are der ivable) of the c i rcumscr ip -t ion of E{P,x) in the or ig ina l theory. 6.2. Generalized Circumscription M c C a r t h y ' s fo rmu la c i rcumscr ip t ion has late ly been general ized by L i fsch i tz [1984], explo i t -ing pre-orders, as discussed i n §2.1.5.2. L i f sch i t z ' general ized fo rm is: TpC) A V X ' . T ( X ' ) A ( X ' < j{X) z> ( X < * X ' ) (24) where < R denotes the pre-order on tuples of (predicate, funct ion, and ind iv idua l ) var iables induced by a ref lex ive, t rans i t ive re la t ion , R. W e ca l l th is generalized circumscription, and wr i te CLOSURE(T; X ; R) for (24) or the corresponding first-order schema. T h i s fo rmu la t ion al lows for a rb i t ra ry order ing re la t ions to dr ive the m in im iza t i on , and provides for the denotat ions of terms (constant a n d func t ion letters) to be affected by the m in im i za t i on process. T h e ex tended m i n i m a l - m o d e l semant ics ou t l ined above is amenab le to this fur ther general i -za t ion . T h e most s igni f icant change f rom the forms we have seen to th is point is that the denota-t ions of some constant and func t ion terms m a y change between a mode l and i ts submodels. The appropr ia te def in i t ions are: - 8 2 -D e f i n i t i o n : M < (X,R)M' Le t r(P) be a f in i te ly ax iomat i zed (first- or second-order) theory, whose predicate, func-t ion and constant letters inc lude (but need not be l im i ted to) those i n X ; let R be a b inary re la t ion o n tuples of type X ; let < R be the pre-order induced by R; and let M , M' be models of T. T h e n M i s an (X.,R)-submodel of M' (wr i t ten M < ^R)M') iff (i) | M | = \M'\ , (i i) If t is a t e r m and t £ X , then \t\M = [tl^1 , (i i i) If Q X is a predicate let ter of T, then | Q | A / = \Q-\M1 > a n < i (iv) < | X | M , | X | ^ > eR. I D e f i n i t i o n : ( X , i 2 ) - M i n i m a l M o d e l A mode l , M , of T is (X.,R)-minimal iff T has no mode l , M ' , such that M * < (x,iJ)A^ a n < i - ( M < ( X , * ) M ' ) . I W e have shown that general ized c i r cumscr ip t ion is sound vis-a-vis the set of m i n i m a l models specif ied by th is mode l theory. T h e o r e m 6 .4 — S o u n d n e s s CLOSURE(T; X ; R) is satisf ied by every ( X , i ? ) - m i n i m a l model of T. | W e do not know whether there is an analogue of Theo rem 6.2 ( f in i tary completeness) for general-ized c i r cumscr ip t i on . T h e prov is ion for var iab le terms leads to some surpr is ing results. These inc lude new pos i -t ive equa l i ty s ta tements, and the p rovab i l i t y of new posi t ive or negat ive ground facts i n p red i -cates not i nc luded among those specif ied as var iab le . P r o p o s i t i o n 6 .5 If terms are a l l owed to va ry , then new ground equal i ty statements m a y resul t f r om general ized c i r cumscr ip t ion . | P r o p o s i t i o n 6 .6 If terms are a l lowed to va ry , then new ground facts i nvo lv ing predicates Q (£ X m a y resul t f r o m CLOSURE^ T\ X ; R). | - 8 3 -E x a m p l e 6.1 Cons ide r the theory T= {Pa, Pb, Qb }. CLOSURE[T; {P, a}; {P}) is PaAPb/\Qb/\ V $ . V u . [$tt A A ( V z . $ z D Pz)] D ( V z . P z D $ z ) Ins tan t ia t ion w i t h [z = 6 /$z ] and [6/u] gives V z . P z 3 z = 6, f rom w h i c h we can infer a = 6 a n d hence Q a . | 6 . 3 . W e l l - F o u n d e d T h e o r i e s A s w i t h a l l of the forms of m i n i m a l - m o d e l semant ics we have discussed i n th is thesis, that for genera l ized c i r cumscr ip t i on prov ides for cer ta in elements to differ between a mode l and i ts submodels wh i le others r e m a i n fixed. Despi te Theorems 6.2-6.4, i t is not necessar i ly c lear that the syntac t ic man ipu la t ions of general ized (or formula) c i rcumscr ip t ion respect the in tent expressed by this semant ic charac ter iza t ion . It is conceivable that a l l models ref lect ing a par t i cu la r conf igurat ion of supposedly fixed at t r ibutes might have no m i n i m a l submodels. T h e semant ics then fai ls to guarantee that c i r cumscr ip t i on w i l l not affect these supposedly " i n v i o l a b l e " facets. It is na tu ra l to quest ion whether there is any proper ty analogous to the wel l - foundedness proper ty we discussed for predicate c i rcumscr ip t ion , w h i c h wou ld address this concern. In fact, as we sha l l see, there is such a not ion. Le t us redefine the te rm "we l l - f ounded" as fol lows: D e f i n i t i o n — W e l l - F o u n d e d n e s s T h e theory, T, is well-founded with respect to ( X , i i ) iff every mode l of T has an ( X , R)-minimal submode l . | T h i s de f in i t ion is s l igh t ly weaker t han that g iven in chapter 5, where we requ i red tha t every mode l of T have a P - m i n i m a l submode l for every finite tuple of predicates, P . T h i s weaker def in i t ion, re la t i v i zed to ( X , P ) , is sufficient for dec id ing whether a par t i cu la r c i r cumscr ip t i on is we l l -behaved. T h e more di rect genera l izat ion of the def in i t ion of chapter 5 is so strong that it excludes a l l theories. P r o p o s i t i o n 6 .7 (L i fsch i tz ) U n i v e r s a l theor ies are not necessari ly wel l - founded i f constants are a l lowed to va ry . | - 8 4 -E x a m p l e 6 .2 (L i fsch i tz ) T h e na tu ra l -number example of E x a m p l e 5.1, w i t h the ex is tent ia l ly specif ied i n d i v i d u a l rep laced by the constant ' 0 ' : NO A Vz. N z D succ(x) £ 0 V z . N z D Nsucc(z) Vzy. succ(z) = succ(y) D z = y is not we l l - founded w i t h respect to m in im iza t i on of N w i th { N , 0} var iab le . S ince the denota t ion of 0 is a l lowed to change f rom mode l to submodel , the inf in i te chains of models presented in E x a m p l e 5.1 serve to show that th is theory has no m i n i m a l models. | P r o p o s i t i o n 6.8 No c lass of theories is wel l - founded w i t h respect to a l l pre-orders. | E x a m p l e 6 .3 Cons ide r the theory w i t h no proper ax ioms, and m in im ize the express ion E(P,x) = Px A [Vz. -iPz] A @ z - Px A ->Psx\. Cons ider a mode l in wh i ch P is in terpret -ed b y the na tu ra l numbers, and s by the successor funct ion. C l e a r l y any non-empty i n i -t i a l subset of the na tu ra l numbers produces a proper submodel , bu t the mode l w i t h the emp ty in terpre ta t ion for P makes E t rue everywhere. | P ropos i t i on 6.8 a n d E x a m p l e 6.3 can best be understood in terms of Theo rem 6.3. M i n i m i z a t i o n of E(P,x) i n T is equiva lent to m in im i za t i on of yjx, w i t h {rp, P} var iab le , i n T' = | Vz . ybx = JPZ A [Vz. --Pz] A 0 i . Pz A --Paz] j w h i c h does not be long to any of the known classes of wel l - founded theories (because P occurs pos i t i ve ly w i t h i n the scope of ex is ten t ia l quant i f iers). In some sense, a l low ing a rb i t ra ry pre-orders enables one to " i m p o r t " a rb i t ra ry ax ioms in to the theory. W i t h these examples i n m i n d , we w i l l restr ict our a t tent ion i n the sequel to the case of s im-ple m i n i m i z a t i o n of some of the predicates of X . In other words, we w i l l consider a genera l izat ion of jo in t pred icate c i r cumscr ip t ion , in w h i c h other predicates and terms m a y be a l lowed to vary . W e w i l l wr i te < (x,p) f ° r * n e pre-order determined by the jo in t m i n i m i z a t i o n of each of the predi -cates in P , a l low ing the predicates and terms of X to vary . ( X is assumed to con ta in a l l of the predicate symbo ls of P . ) T h e quest ion remains, " A r e there any theories wh i ch are we l l - founded?" Fo r tuna te l y , the answer is " Y e s " . (Th i s result has been proved independent ly (using ra ther different techniques) by L i f sch i t z [1985].) - 85 -T h e o r e m 6 . 9 If T is a un iversa l theory, and X , P are finite tuples of predicate letters, then T is wel l - founded w i t h respect to < (x,P) • B T h e existence of well-founded theories p roved most distressing in the context of predicate c i r cumscr ip t ion . W h a t are the repercussions of T h e o r e m 6.9 for generalized c i rcumscr ip t ion? C e r t a i n l y , they are less pessimistic. Genera l ized c i rcumscr ip t ion affords m u c h greater cont ro l over w h i c h aspects of models must r ema in fixed when const ruct ing submodels. T h i s means that gen-eral ized c i r cumsc r ip t i on is not dr iven , w i l l y - n i l l y , to avo id conclusions w h i c h lead to the der iva-t i o n of new posi t ive in format ion . T h u s , for well-founded theories, general ized c i r cumsc r ip t ion al lows useful conclusions to be d rawn wi thou t sacrificing a clear semantic i n t u i t i o n of exact ly wha t is open to conjecture. A l s o on the posi t ive front, we have C o r o l l a r y 6.10: C o r o l l a r y 6 . 1 0 If T is consistent a n d wel l - founded w i t h respect to ( X , P ) , then CLOSURE(T; X ; P ) is consistent. | It is na tu ra l to quest ion the extent to w h i c h the negative results of chapter 5 a p p l y to gen-era l ized c i r cumsc r ip t i on . It is clear that , i n the case where only the m i n i m i z e d predicates are a l lowed to v a r y , that a l l the results i n chapter 5 continue to hold, since i n this case generalized c i r cumsc r ip t ion reduces to predicate c i rcumscr ip t ion . Fur thermore , Theo rem 5.4 and an appropr i -ate vers ion of T h e o r e m 5.5 continue to hold , even w i t h variable predicates. T h e o r e m 6 .11 If T is wel l - founded w i t h respect to ( X , P ) ; P S P is an n-ary predicate; X a set of predicate letters; a n d a 1 / . . v a ) k are n-tuples of ground terms; then CLOSURE(T; X ; P ) f- Pa\ V . . . V Pak <=• T (- Pa\ V . . . V Pak . | T h e o r e m 6 . 1 2 If T is wel l - founded w i t h respect to ( X , P ) ; X is a set of predicate letters; P ^ P U X is an n-ary predicate; a n d a1)...,ak are n-tuples of ground terms; then (i) CLOSURE^ T; X ; P ) f - Pc7t V . . . V Pak<=> T (— Pa x V . . . V ttk , and (ii) GLOSURE( T; X ; P ) | - -iPo^ V . . . V ^ ttk <=> T (- -Po^ V . . . V --P&?t. | - 86 -T h e fact that the model - theory out l ined in §6.2 for general ized c i r cumscr ip t ion (even w i t h var iab le terms) restr ic ts the submode l re la t ionship to models w i t h ident ica l domains suggests tha t general ized c i r cumscr ip t i on (and a fortiori f o rmu la c i rcumscr ip t ion) cannot be used to conjecture doma in closure ax ioms. F o r wel l - founded theories, this is the case. Theorem 6.13 If T is we l l - founded for (P, .R) and T has a mode l w i t h doma in , D, then so does CLOSURE(T(P);P;R). | T h u s nei ther genera l ized c i r cumscr ip t ion w i thou t var iab le terms nor fo rmu la c i r cumscr ip t i on sub-sumes d o m a i n c i r cumscr ip t i on . E q u a l i t y appears to rema in prob lemat ic i f on ly predicates are var iab le , but we have not pro-ven an analogue of T h e o r e m 5.7. T h e o r e m 5.6 cont inues to app ly even if terms are a l lowed to va ry . Theorem 6.14 If T is a first-order theory conta in ing ax ioms w h i c h define the equa l i ty predicate, = , then T f - CLOSURE( T ,X , {= } ) . | It appears that unique names ax ioms are der ivable (for theories w i t h finite domains) g iven var iab le terms, however [L i fschi tz 1984]. Un fo r tuna te ly , we have seen that var iab le terms can be prob lemat ic . T h e genera l fo rmu la t ion of c losed-wor ld reasoning about equal i ty us ing general ized c i r cumscr ip t i on w i t h var iab le terms remains an open quest ion. A l s o open axe the quest ions of analogues of Theorems 5.4 and 5.5 vis a vis a rb i t ra ry pre-orders a n d / o r var iab le terms. Because of the fai lure of wel l- foundedness for these forms of c i r -cumscr ip t i on , the tools we have used in this chapter and i n chapter 5 do not app l y to these more general problems. P ropos i t i on 6.6 suggests that such analogues may not be for thcoming. C H A P T E R 7 Domain Circumscription In chap te r 2, we discussed the mo t i va t i on for and one rea l i za t ion of doma in c i rcumscr ip t ion . In th is chapter , we invest igate the fo rma l i sm more thoroughly . D o m a i n c i r cumscr ip t i on [ M c C a r t h y 1977, 1980; Dav i s 1980] is in tended to be a syntac t ic rea l i za t ion of the model- theoret ic domain-c losure assumpt ion . It prov ides a mechan ism for con-jec tur ing domain-c losure ax ioms, e l im ina t ing the need to exp l i c i t l y state them. T o c i rcumscr ibe the d o m a i n of a sentence, A, the schema: is added to A. Axiom($) is the con junct ion of $ a for each constant s y m b o l a and VXl...xn. [ $ z x A—A 3>zJ D <&fxl...xnfor each n-ary func t ion symbo l / . A® is the resul t of rewr i t ing A, rep lac ing each un iversa l or ex is tent ia l quant i f ier, 'Vz.' or 'Ejz.', in A w i t h 'Vx.4>z D ' or ' 3 z . $ z A ' , respect ive ly . 7.1. A Revised Domain Circumscription Axiom Schema A s was no ted i n §2.1.5.3, the appropr ia te model- theoret ic charac te r iza t ion for doma in -closure invo lves rest r ic t ion of models to progressively smal ler domains, preserv ing agreement over c o m m o n terms. T h i s not ion of submode l corresponds rough ly to the s tandard not ion of " subs t ruc -tu re" . It is s l igh t ly stronger, however, in the sense that substructures are not requi red to be models of the theory i n quest ion. D a v i s [1980] shows that every instance of (25) is true in a l l m i n i m a l models of the or ig ina l sentence A. T h i s result is correct for most theories. However , inconsistency results when cir-cumscr ib ing un iversa l theories (theories whose prenex no rma l forms con ta in no leading ex is tent ia l quant i f iers) w i t h no constant symbols . F o r example, consider the re la t iona l theory: Because there are no constant o r func t ion symbols , Axiom($>) is empty , so the d o m a i n cir-cumsc r i p t i on schema for A is: Axiom(<&) A A * D V x. <J>(z) (25) A = { Vz. Px }. M e r c e r [1984, personal communica t ion ] has noted that subst i tu t ing ->Px for $ z gives: - 87 -- 8 8 -V i . PIJ D V i . ->PX w h i c h is c lear ly inconsistent w i t h A. T h e root of th is p rob lem is that , for such theories, $ can be chosen to be un iversa l ly false. M o d e l s of f i rst-order theories must have at least one doma in element, so the conjecture that every th ing is a $ (and hence there is nothing) is inconsistent. H a v i n g isolated the p rob lem, we have deve loped a s imple , easi ly mo t i va ted so lu t ion. S ince models must have non-empty domains , those $ ' s w h i c h are iden t ica l l y false must be exc luded. T o achieve this, the conjunct ^x. <&i is added to the lef t -hand-side of the c i r cumscr ip t ion schema (25), g i v ing : 3i- $ i A Axiom{$) A A* D V i . $ (x ) (26) D a v i s ' proof is easi ly corrected and amended to app ly to this revised schema. Schemas (25) and (26) are equ iva lent in a l l but the prob lemat ic cases out l ined above. If A conta ins a constant s y m -bo l , a , then $ a occurs o n the left of (25), and th is entai ls 9z. $ x . S im i l a r l y , if A has any leading ex is tent ia l quant i f iers, then ^x. $ z a l ready occurs i n (25). In those cases where ^x. $ i is not enta i led by the lef t -hand-side of (25), (25) results i n inconsistency. T h e rev ised schema m a y s t i l l take a consistent theory w i t h no m i n i m a l models to an inconsistent c i r cumscr ip t i on (for an example , see [Dav is [1980]), but so long as A has a m i n i m a l mode l , (26) preserves consistency. Theorem 7.1 — Soundness E v e r y instance of schema (26) is true i n every m i n i m a l mode l of the o r ig ina l theory. | 7.2. Some Properties of Domain Circumscription In th is sect ion we consider some propert ies of doma in c i rcumscr ip t ion . W e examine their consequences w i t h respect to using doma in c i rcumscr ip t ion to formal ize the domain-c losure assumpt ion . T o bet ter i l lus t ra te the propert ies of doma in c i rcumscr ip t ion , we refer to the fo l low-ing example. Example 7.1 Le t T = {Pa,Pc, Qb,Qc}. T has the fo l lowing m i n i m a l models. ( W e use the corresponding boldface let ter for the in terpretat ions of constant terms, and a , 8, and 7 represent the equiva lence classes {a, c}, {b, c}, and {a, b, c}, respect ively.) M i : \M,\ = {a, b, c} 1 1 ^ = {a. c} - 8 9 -| = | « i = { ( « , * ) , ( b ,b ) , (c,c)} Mi | M 2 | = { a , b } l<?k= {b , « } H M , = { ( A > A ) > ( B > B ) > ( C > C )> ( A > C ) . ( C > A ) } A 4 : | M 3 | = { a , 0} \P\MS = { a , 0} l<?k= {/?} H M , = {(a>»)> ( b ,b ) , (c ,c ) , (b ,c ) , (c,b)} M 4 : | M 4 | = {7} \Q\M,= {I} I=|M 4 = { (a ,a ) , ( b ,b ) , (c ,c ) , ( a ,b ) , (b ,a ) , (a,c) , (c ,a) , (b ,c ) , (c,b)} | Severa l impor tan t features are evident i n the above example. F i r s t , every mode l of T has one of Mt — M 4 as a m i n i m a l submode l . A s w i t h other forms of c i r cumscr ip t i on and their corresponding not ions of m in ima l i t y , i t is interest ing to know whether there is a class of theories each of whose models has a m i n i m a l submode l ( i .e., wel l - founded theories). It is for such theories that d o m a i n c i r cumscr ip t i on corresponds most closely w i t h one's in tu i t ions. In the case of doma in c i r cumscr ip t i on , the ma thema t i ca l logic l i terature provides a sufficient cond i t ion (c. / . [Barwise 1977, p 62]). P r o p o s i t i o n 7 .2 (tyo5-Tarski Theorem) U n i v e r s a l theories (possibly w i t h funct ion symbols) are wel l - founded for d o m a i n cir-cumscr ip t i on . I It is also c lear that theories w i t h on ly finite models are wel l - founded. Second, because the doma in c i rcumscr ip t ion schema is satisf ied by every m i n i m a l mode l , d o m a i n c i r cumsc r ip t i on does not produce any new ground- term equal i t ies or inequal i t ies, for we l l -founded theories. (The same l im i ta t i on also appl ies to predicate and fo rmu la c i rcumscr ip t ion. ) - 9 0 -T h e o r e m 7.3 If T is a wel l - founded theory wh i ch conta ins ax ioms wh ich define the equa l i t y predicate, = , and ay...,cx„ By...,8n&re g round terms, then (i) T\-(Vai= 8,)*=* DC(T)[-(\/ai= B,) t=l E=l (ii) T |— ( V a,- £ B,) <=> DC(T) \- ( V a , £ /?,) I «=i t=i T h e au tomat i c generat ion of a l l possible ground te rm inequal i t ies to capture the un ique-names assumpt ion [Rei ter 1980b] remains a thorny issue in knowledge representat ion. T h i r d , the amb igu i t y of the usua l s tatement of the domain-c losure assumpt ion is revealed. O n l y M4 has the m i n i m u m number of i nd iv idua ls necessary to satisfy T (i.e., 1), yet each of Mx — M 4 has on ly i nd iv idua ls named (and hence requi red to exist) by T. D o m a i n c i r cumscr ip t i on cap-tures a weak sense of the domain-c losure assumpt ion wh ich does not decide between these in terpreta t ions. Based on c o m m o n appl ica t ions of the domain-c losure assumpt ion ( t yp ica l l y in con junc t ion w i t h some fo rm of un ique-names assumpt ion) , this weak sense appears to be the pre-ferred sense. W h i l e new ground equa l i ty statements are not general ly fo r thcoming, the resul ts of doma in c i r cumscr ip t i on do in teract w i t h the equal i ty theory i n interest ing ways. T h e c i r cumscr ip t i on of T i n E x a m p l e 7.1 enta i ls a — b A 6 = c D •Hxiy. x = y, for example. T h e c i r cumscr ip t i on of {3x. Px, 3z- Qx} entai ls 3x. Px f\ Qx O EJzVy. x = y. Such formulae seem to precisely capture the difference between the var ious m i n i m a l models of the or ig ina l theory. In fact, a completeness result for d o m a i n c i rcumscr ip t ion can be obta ined. T h i s result guarantees that, for theories w i t h on ly finite models (among others), the set of m i n i m a l models of the or ig ina l theory const i tutes exac t ly the set of models of the c i rcumscr ibed theory. Such a precise charac ter iza t ion is very encouraging. T h e proof of th is result is analogous to Per l i s and M i n k e r ' s [1986] finitary complete-ness proof for predicate and fo rmu la c i rcumscr ip t ion . 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 ax iomat izab le theory, and every mode l of T is finite, then on ly the m i n i m a l models of T satisfy every instance of schema (26) for T. fl In the statement of Theo rem 7.4, the requi rement that a l l of T°s models be finite is stronger t han necessary. The theorem holds even if on ly the models wh i ch sat isfy schema (26) are finite. C o r o l l a r y 7.5 If T is a finitely ax iomat i zab le theory, and every mode l of T U schema (26) is finite, then on ly the m i n i m a l models of T satisfy every instance of schema (26) for T. | - 91 -7 . 3 . R e l a t e d F o r m a l i s m s M c C a r t h y [1980] c la ims that doma in c i rcumscr ip t ion is a specia l case of predicate c i r -cumscr ip t i on , i n tha t the d o m a i n c i rcumscr ip t ion schema for a theory, A, can be der i ved by predi -cate c i r cumsc r ip t i on of a theory, A', w h i c h is a conservat ive extension of A. In v iew of th is, i t m igh t appear that interest in d o m a i n c i r cumscr ip t ion is point less. A p a r t f rom the fact that doma in c i r cumscr ip t i on is a more di rect and somewhat s impler approach to domain-c losure, and that the mode l theory of doma in c i r cumscr ip t ion perhaps better captures our in tu i t ions about the conjectures i nvo l ved , there is another reason to reject this argument for abandonment . M c C a r t h y ' s demons t ra t ion of th is subsumpt ion actua l ly rests on a st rengthened fo rm of predicate c i r cumscr ip t i on w h i c h a l lows ax ioms of the o r ig ina l theory to be ignored dur ing the c i rcumscr ip -t i on process. A s we no ted i n chapter 5, this fo rm of c i rcumscr ip t ion does not a lways preserve con-s is tency, even 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 cumscr ip t i on cannot, i n general , y i e l d the d o m a i n c i r cumscr ip t i on schema. In fact, this is for tunate, since the fo rm of doma in c i r cumscr ip t i on M c C a r t h y was t ry ing to emulate in t roduced inconsistencies in to some theories w i t h m i n i m a l models . O u r rev ised fo rm of doma in c i rcumscr ip t ion , wh i ch preserves consistency for m i n i m a l l y modelab le theories, is s t i l l not obta inable using predicate c i rcumscr ip t ion . In chapter 5, we showed that predicate c i r cumscr ip t ion is too weak to conjecture domain-c losure ax ioms . Since doma in c i r cumscr ip t i on can conjecture such ax ioms, it fol lows that it is not subsumed by i ts predicate cous in . In chapter 6, we showed that nei ther fo rmu la c i r cumscr ip t i on nor general ized c i r cumscr ip t i on w i thou t var iab le terms subsumes doma in c i rcumscr ip t ion , i n general . O u r seman-t ic charac te r iza t ion suggests that it is un l i ke ly that any fo rm of general ized c i r cumscr ip t i on can conjecture domain-c losure ax ioms. It appears, therefore, that doma in c i r cumscr ip t ion cont inues to fill a n iche among the var ious mechanisms for c losed-wor ld reasoning. C H A P T E R 8 Connections Between Default Logic and Circumscription In chapter 3, we observed that the model-set semant ics for default logic bears a superf ic ia l resemblance to the m i n i m a l - m o d e l semant ics of the var ious forms of c i r cumscr ip t i on . Chap te rs 5 and 6 considered the feas ib i l i ty of do ing defaul t reasoning using c i r cumscr ip t ion . W e now con-sider the re la t ionships between defaul t logic and c i rcumscr ip t ion in more de ta i l . T h e na tu ra l quest ion is whether ei ther fo rm subsumes the other. Is there a d i rect correspon-dence between defaul t theories and c i r cumscr ip t ion , o r vice versa? Proposition 8.1 Defau l t logic can reach conclusions w h i c h cannot be obta ined by genera l ized c i r cumscr ip t i on w i thou t var iab le terms. | Example 8.1 chapter 6, we showed tha t general ized c i r cumscr ip t ion w i thout var iab le terms cannot con-jecture new inequal i t ies. | T h e converse of propos i t ion 8.1 is apparent ly false. A s s u m i n g that CLOSURE^T; X ; R) is consistent, the theory obv ious ly produces the requ i red results. Perhaps th is is not what one has i n m i n d w h e n one asks i f defaul t log ic can capture c i r cumscr ip t ion , however! W e w i l l re turn to this quest ion in la ter sec-t ions. T h e defaul t theory has a unique extension, conta in ing a J= b. In - 9 2 -- 9 3 -8 . 1 . " T r a n s l a t i o n " f r o m D e f a u l t L o g i c t o C i r c u m s c r i p t i o n In v iew of p ropos i t ion 8.1, the t i t le of this sect ion might seem paradox ica l . There has been some work on pa r t i a l t ranslat ions, however. G roso f [1984] presents two equiva lent t rans la t ion schemes for n o r m a l defaul t theories, one i nvo l v i ng 'ab' predicates (discussed i n §2.2.2), the other i nvo lv ing m i n i m i z i n g arb i t ra ry expressions. W e discuss the former. T h e t rans la t ion scheme carr ies the first-order ax ioms, W, over unchanged. F o r each closed ai: Pi n o r m a l defaul t , — - — , the a x i o m a,- A —A' 3 °H*) m added. T h e n ab is c i r cumscr ibed in the Pi resul t ing theory , v a r y i n g ab and each predicate wh i ch occurs i n any of the P-s. G roso f observes that th is " t r a n s l a t i o n " ac tua l l y differs f rom default logic in a number of respects. F i rs t , the equa l i t y predicate is not affected by the c i rcumscr ip t i ve theory. Grosof proposes to exclude defaults about equa l i t y to remedy th is, but th is is insuff icient. A n y defaul t w h i c h affects equal i ty w i l l not behave " c o r r e c t l y " in the c i rcumscr ip t i ve theory. A fur ther difference is tha t the cir-cumscr ip t i ve theory inher i ts c i rcumscr ip t ion 's " c a u t i o u s " nature. T h e mu l t i p l i c i t y o f extensions of a defaul t theory are ref lected i n d is junct ive statements i n the t rans la ted theory . F i n a l l y , G r o s o f s t rans la t ion of the n o r m a l defaul t a ® ac tua l l y more closely corresponds to the default ' a n ' s u i c e the t rans la t ion a l lows the conjecture of -<a f rom ->P, someth ing Groso f appears not a D p to have not iced. E v e n a l lowing for these discrepancies, Groso f presents no more t han in tu i t i ve arguments and examples in support of the correctness of the t rans lat ion scheme. Im ie l i nsk i [1985] takes the comp lementa ry tack of def in ing a t rans la t ion scheme to be ade-quate if the theory and its t rans la t ion produce precisely the same conclusions, and fur thermore the t rans la t ion scheme is " m o d u l a r " . M o d u l a r i t y requires that the t rans la t ion of the defaults a n d first-order facts must be independent . Im ie l i nsk i v iews the t rans la t ion of a set of defaults to consist of a co l lect ion of f i rst-order facts and a pre-order re la t ion . B o t h of these must be determined f rom the defaults a lone, w i thout reference to the specif ic facts at hand . T h i s is a desirable property , since one does not wish to have to recompute one's representat ion of knowledge ( in add i t ion to the necessary ad justments to the set of one's conjectures) every t ime a new fact is learned. G i v e n these st r ic tures, Im ie l insk i is able to prove that even no rma l defaults are not modu-la r ly t rans la tab le to genera l ized c i rcumscr ip t ion . The re are some defaults w h i c h do have modu la r t ranslat ions, however . These are the semi -no rma l defaults w i thout prerequisi tes (e.g., ——j^—)-These resul ts h igh l ight the necessity of the fundamenta l d is t inc t ion between the model-set-res t r ic t ion semant ics of defaul t logic (see chapter 3) and the m in ima l -mode l semant ics of cir-cumscr ip t i on . T h e prerequis i tes of the defaul ts are requi red to be provable . T h i s is a g lobal character is t ic o f the set of models. T h e submode l re la t ion, however, is on ly able to consider pairs of models. Prerequis i te- f ree defaul ts fit n ice ly in to c i rcumscr ip t ion precisely because they are prerequisi te-free. There are no (global) p rovab i l i t y requirements, on ly consistency requi rements. Cons is tency can be de termined by the existence of a single mode l , so can be loca l ly de termined. - 9 4 -The re remains the quest ion of whether the requirement of ident ica l sets of theorems is too s t rong. Im ie l insk i ' s theorem, prov ing that no rma l default theories are not modu la r l y t ranslatable, A : B rests on the fact that any modu la r t rans la t ion of the default — ' - — , where the sets {A, B} and B {A, ~<B} are both consistent, w i l l necessari ly y ie ld A D B as a theorem (assuming W \f—>B). W h i l e th is m a y be true, i f an extension conta ins A or B, it w i l l also con ta in A Z> B. It appears that the offending imp l i ca t i on is offensive only in those cases where it cannot be used to deduce any th ing " u s e f u l " . M o r e conv inc ing ly , we have noted that default logic is a " b r a v e " reasoner wh i le c i r cumscr ip t i on is " c a u t i o u s " . It seems reasonable to expect tha t a c i rcumscr ip t i ve t rans la-t ion of defaul t log ic wou ld reflect this caut ious nature, perhaps re turn ing those facts true i n all extensions. F i n a l l y , c i rcumscr ip t i ve conjectures app ly to a l l i nd iv idua ls , whereas those resul t ing f r om open defaul ts app l y on ly to ind iv idua ls w i t h names in the language. It m igh t be reasonable to expect that c i rcumscr ip t i ve versions of default theories w i t h open defaults w o u l d therefore prove stronger conjectures (at least for theories w i thou t doma in closure ax ioms) . These considerat ions suggest that Imie l insk i 's results might be taken as a "wo rs t case" scenar io, leav ing open the poss ib i l i ty of acceptable t rans lat ion schemes for defaul ts w i t h prere-quisi tes, g iven a weaker no t ion of " a c c e p t a b l e " . W e do not fur ther consider th is poss ib i l i ty here. 8.2. Translations from Circumscription to Default Logic T h e other side of the co in we have been examin ing is whether default logic can be used to per form c i r cumsc r ip t i on ( in any but the t r i v ia l sense ment ioned at the beginning of th is chapter) . T h e prev ious sect ion ou t l i ned a number of the very different capabi l i t ies of the two formal isms: brave vs caut ious, effects on equal i ty , g loba l (provabi l i ty ) vs loca l (consistency) compar isons in the model - theory (proof- theory) , and statements about " u n n a m e d " ind iv idua ls . In a l l but the last of these categories, default logic came out on the stronger end. T h i s suggests that the search for a direct imp lemen ta t i on of c i r cumscr ip t ion in default logic might be more successful that the con-verse a t tempt . T h e answer to this is, " Y e s , and no . " . There is one facet of general ized c i r -cumsc r i p t i on w h i c h is comple te ly absent f rom default logic. T h a t is the ab i l i t y to specify wh ich predicates are to be a l lowed to va ry dur ing the c i rcumscr ip t ion process. In default logic, there is no w a y to restr ic t the repercussions of the defaults to some par t i cu la r set of predicates ( and /o r i nd iv idua ls ) . T h u s we have Theo rem 8.2. Theorem 8.2 If T |— V i . I = &i V . . . V x — ctn and T |— a,- j= ctj, for i j for g round terms ctlt...,an ; and X inc ludes a l l of the predicates of L; then those formulae true in every ex tens ion of • - 9 5 -C o r o l l a r y 8 .3 If E is a n extens ion of A , then every mode l of E is an (X , { .P } ) -m in ima l mode l of T. | C o r o l l a r y 8 .4 If M i s an ( X , { P } ) - m i n i m a l mode l of T, then M i s a mode l for some extens ion of A . | C o r o l l a r y 8 .5 A captures the brave c i r cumscr ip t ion of P i n T w i t h every predicate var iab le . | No t i ce tha t T h e o r e m 8.2 requires that T have unique-name ax ioms as we l l as doma in -closure ax ioms. If we d rop the requi rement for un ique-name ax ioms, then the defaul t theory becomes stronger t han the c i rcumscr ip t i ve theory, in the sense that C o r o l l a r y 8.3 cont inues to ho ld but T h e o r e m 8.2 and Co ro l l a r y 8.4 do not. W e have not yet de termined whether these results general ize to the jo in t m i n i m i z a t i o n of several predicates. Because of the l im i ta t i on of open defaults to named ind iv idua ls , none of the results general ize to theories w i thou t doma in -closure ax ioms. P r o p o s i t i o n 8 .6 If T does not enta i l a domain-c losure ax iom, and T \/- Vz. ->Px, then every extens ion for A has models w h i c h are not ( X , {P } ) -m in ima l . | E v e n more pessimist ic is the result that fixed predicates preclude such a s t ra ight forward t rans la-t ion of c i r cumscr ip t i on to defaul t logic, even for c losed-domain , un ique-name 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 a,- £ ay, for t j= j and yet no comb ina t i on of the extensions of A = character izes the ( X , { P } ) - m i n i m a l models of T. | precisely 1 - 9 6 -W e exper imented w i t h an extended vers ion of default logic wh i ch a l lowed for the speci f icat ion of " f i x e d " predicates. A l t h o u g h we were able to show that the results in [Rei ter 1980a, chapters 2 and 3] ho ld for this logic, and - for finite theories - the obv ious genera l izat ion of the model-set res t r ic t ion semant ics of chapter 3 appl ies, we abandoned this approach when i t p roved incapab le of y ie ld ing an analogue for Theo rem 8.2 i n the presence of fixed predicates. (The best that cou ld be guaranteed was that those ground l i terals i n P conta ined in a l l extensions were t rue i n a l l m i n i m a l models. T h i s is s igni f icant ly weaker — suff ic ient ly so tha t we doubt that the (abundant) ex t ra mach ine ry requi red is wor thwhi le . E x a m p l e 8.2 Le t T be { V z . i = « V i = i , o/= 1, - . P a A ~^Pb Z> Qa} and let Q be fixed. T h e P - m i n i m a l models of T are ( loosely represented): { Pa, - . P 6 , ^Qa} { - .Pa , Pb, -.<?a} { - . P a , - . P 6 , Qa} There are no ground l i terals in P true in every P - m i n i m a l mode l . However , CLOSURE{T; {P}; {P}) \- ( 3 z . P z = Qa) A ( - P a V - P 6 ) . In other words, one can c i rcumscr ip t i ve ly conjecture that there is exact ly one P i f Qa, and none otherwise. | Ge l f ond and P r z y m u s i n s k a [1985] prove the weak result a l luded to above for the i r vers ion of M i n k e r ' s genera l ized c losed-wor ld assumpt ion, wh i ch al lows fixed predicates. Ge l f ond , P r z y m u -s inska, and P r z y m u s i n s k i [1985] prove a m u c h stronger result for thei r extended c losed-wor ld assumpt ion . P r o p o s i t i o n 8.8 (Ge l fond , P r z y m u s i n s k a , and P rzymus insk i ) A s t ruc ture , M, is a mode l for ECWA(T) iff i t is a m i n i m a l mode l for T. | A t first g lance this m igh t suggest that there shou ld be some analogous result for some defaul t theory. It appears that the E C W A ac tua l l y achieves th is power by the subterfuge discussed near the beg inn ing of th is chapter , by add ing every instance of the c i rcumscr ip t ion schema. T h i s is cer ta in ly the case i n the absence of var iab le predicates. - 9 7 -P r o p o s i t i o n 8 .9 If there are no var iab le predicates (Z={ }), then ECWA(T) adds to T every instance of the c i r cumscr ip t i on schema. | It seems tha t any general ized t rans la t ion f rom c i rcumscr ip t ion to default logic (for finite theories) - i f such a th ing exists, short of add ing defaults for each instance of the c i r cumscr ip t ion schema - requires more power than the c losed-wor ld default prov ides. T h e existence of an appropr ia te t rans la t ion remains open. C H A P T E R 9 Open Problems Don ' t confront me w i t h m y fai l ings ... I have not forgot ten them. - Jackson Browne Th roughou t the thesis, a catalogue of open problems has been compi led . R a t h e r than recap-i tu late this l ist of specif ic problems, th is chapter addresses a broader, ph i losoph ica l perspect ive. W e consider a general research programme, instead of a l i tany of isolated potholes in need of filling. A l t h o u g h there has been considerable ac t i v i t y in the area of non-monoton ic reasoning, a long w i t h some remarkab le successes, very l i t t le a t tent ion has been focussed on the dynamics of non-mono ton ic i t y . A s th is promises to be a par t i cu la r l y f ru i t fu l avenue of invest igat ion, this chapter addresses two aspects of th is p rob lem: how new in format ion is ass imi la ted in to a theory i nvo l v ing assumpt ions, and how non-monoton ic inference rules are acqui red and employed. These two areas are in t ima te ly re la ted. A major goal for future research shou ld be to develop a un i fy ing f ramework w h i c h makes their in terre lat ionships more apparent . T h i s point of v iew m a y be expected to prov ide new insights in to bo th non-monoton ic reasoning and updates. Fu r the rmore , m u c h of the work that has been done t reat ing these prob lems i n iso la t ion can , hope-fu l ly , be re in terpreted to advantage f rom th is more general s tandpoint . 9.1. Principles of Non-Monotonic Reasoning T h e impor tan t issue of non-monoton ic i ty w h i c h remains unaddressed is not p r ima r i l y how conclus ions are ob ta ined g iven some facts a n d some non-monotonic inference rules. Ra the r , the quest ion is h o w non-monoton ic rules are fo rmula ted, determined to be app l icab le , and app l ied. T h i s quest ion can be i l lus t ra ted by consider ing the c i rcumscr ip t ive examples of §2.1.5.2. G i v e n a representat ion of the facts about the " w o r l d " , cer ta in predicates must be c i r cumscr ibed , other predicates speci f ied as var iab le , appropr ia te subst i tu t ions discovered, and then the requi red con-jectures are ob ta ined . A s m u c h of the p rob lem lies i n these " a n c i l l a r y " tasks of dec id ing what and how to c i rcumscr ibe as i n the c losed-wor ld reasoning ach ieved by ac tua l l y per forming the c i r -cumsc r ip t i on . T o date , most of the work i n non-monoton ic reasoning ( inc lud ing this thesis) has focussed more on deve lop ing mechanisms for per forming cer ta in specia l ized reasoning tasks t han - 98 -- 9 9 -on under l y ing pr inc ip les or even an understanding of when and how to employ the mechanisms once they are deve loped. T h e cen t ra l quest ion is: can we d iscover ways to make non-monoton ic reasoning au tomat i c a n d / o r goal -d i rected? I.e., are there features of par t i cu la r problems w h i c h can guide the comple-t ion of an incomple te knowledge-base, w i thout external in tervent ion, to solve those problems? A first a p p r o x i m a t i o n to a theory of non-monoton ic theory const ruct ion was out l ined by Re i te r [1978a]. He exp la ined non-monoton ic reasoning i n terms of the c losed-wor ld assumpt ion. Re i te r ' s idea was that reasoners might assume thei r knowledge about re levant aspects of the s i tua t ion to be complete. C losed -wo r l d reasoning sanct ions exact ly those conclusions true in a wor ld com-pletely characterized by wha t is known. Such a c lear, s imple , un i fo rm charac ter iza t ion of non-monotonic reasoning appeals to int ros-pect ive in tu i t ions about the s imp l i c i t y and naturalness of commonsense reasoning. Un fo r tuna te ly , it p roved s imp l is t i c as we l l as s imple. N o t every knowledge state un ique ly character izes a state of the wo r l d . A s s u m i n g the rea l wo r ld is that wo r l d character ized by what is known is a dub ious step w h e n no wor ld is so character ized! Research since 1978 has focussed on mechan isms wh ich avo id the shor tcomings of the naive in terpretat ions of the C W A . L i t t l e effort has been di rected to finding a cor responding in tu i t i ve exp l i ca t ion of the under ly ing pr inc ip les. T h e m i n i m a l - m o d e l semant ics w h i c h we have discussed in one fo rm or another th roughout this thesis does not qual i fy as the in tu i t i ve exp l i ca t ion we seek, for two reasons. T h e first - a n d perhaps less compe l l i ng - is that not a l l theories have m i n i m a l models, and it is undecidable whether a pa r t i cu la r theory has a m i n i m a l model . C e r t a i n theories - qui te unexpected ly - tu rn out not to have m i n i m a l models. F o r example , we have shown that 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 models . T h i s is because any mode l has a cha in of i V s isomorphic to the na tu ra l numbers, N . B u t th is cha in has a subcha in , also isomorphic to N , wh ich satisfies the ax ioms. Hence every m o d e l has a proper submode l , and there are no m i n i m a l models. B u t , since every mode l conta ins a segment isomorphic to N , and since there are models exact ly isomorphic to N , surely commonsense d ic tates that N is an acceptable m i n i m a l model? M i n i m u m - m o d e l semant ics force the m i n i m i z a t i o n process to go beyond the bounds of commonsense in this case. M o r e te l l ing ly , m i n i m a l - m o d e l semant ics enter the picture after m u c h of the non-monoton ic reasoning process is comple te . O n l y after it has been decided what expression is to be m i m i m i z e d , and the connect ions between the m i n i m i z e d expression and the rest of the wor ld have been deter-m ined so tha t va r iab le predicates can be chosen, can the semant ic charac ter iza t ion te l l us what wor ld(s) the non-monoton ic theory character izes. T h e semant ics sheds no l ight on these other d imensions of the commonsense reasoning process. Hence, it is not the charac ter iza t ion we seek. W h a t ev idence is there that there is any under ly ing pr inc ip le? M i g h t not the d i f f icu l ty in finding such a p r inc ip le s tem, i n par t , f rom its non-existence? O f course, the on ly guarantee that the pr inc ip le we seek exists w i l l be i ts demonst ra t ion . There is evidence w h i c h suggests that some sort of un i f o rm rules m igh t underl ie commonsense reasoning. O n e ind ica t ion is the existence of - 100 -approx ima t ions w h i c h fill the role of the sought-af ter rule i n l im i ted cases. T h e C W A is one such ru le. O the rs inc lude m i n i m a l - m o d e l semant ics (for theories w i th m i n i m a l models), the model-set-res t r i c t ion semant ics for default log ic , and the in ferent ia l distance concept in semant ic network reasoning systems. A final example is " O c c a m ' s R a z o r " , a hypothes is- rank ing ru le w h i c h suggests tha t the s implest exp lana t ion for any phenomenon is the best. O f course, there m a y be no un i fo rm under ly ing pr inc ip les. So be it. T h a t is, i n itself, in terest ing. Bes ides, i f humans use no un i fo rm procedures at a l l , we can s t i l l hope to uncover heur is t ics w h i c h can he lp guide the task of commonsense reasoning. F o r example, even a way to au tomat i ca l l y de termine, for some class of theories, wh ich expressions to c i rcumscr ibe a n d / o r w h i c h predicates to v a r y based on the goal at hand and the current knowledge state w o u l d be a s igni f icant con t r i bu t i on . 9.2. Update T h e prob lems of upda t ing theories w i t h in fo rmat ion inconsistent w i t h their cur rent state are obv ious ly prob lems of non-monoton ic inference: such new facts must force the re t rac t ion of p rev i -ously accepted facts if consistency is to be preserved. A second major open p rob lem is to develop a v iew of updates w h i c h integrates them' w i t h other forms of non-monoton ic reasoning. Instead of b l i nd add i t i on a n d de le t ion - wh i ch obv ious ly w i l l not work - or the pro l i fera t ion of a l ternate theories - w h i c h increases uncer ta in ty — it seems appropr ia te to v iew updates as new in fo rmat ion w h i c h leads to the reasoned assert ion or re t rac t ion of facts. T h e exact fo rm that th is research might take is unclear. T h e f inal result w i l l l i ke ly be heav i l y in f luenced by work i n five areas: 1) Re levance Log i c (Anderson and Be lnap 1975]: i n Relevance Log ic , cont rad ic t ions do not au toma t i ca l l y lead to chaos. T h e repercussions of the var ious facts in an inconsistent theory c a n be exp lo red w i thout in t roduc ing " a r t i f a c t s " of the inconsistency. T h i s seems l ike an idea l env i ronment for invest igat ing the effects of cont rary updates. 2) Coun te r fac tua ls and H y p o t h e t i c a l [Rescher 1964, 1976; Lewis 1973]: These branches of phi loso-phy dea l w i t h what wou ld be true in a wor ld wh i ch differs f rom the real wo r ld i n that (at least) cer ta in speci f ied facts ho ld . T h e update prob lem can easi ly be construed in these terms. O n e m igh t therefore expect th is work to shed l ight on updat ing . 3) Change- reco rd ing , correct ing, a n d knowledge-adding updates: W i l k i n s [1983] and K e l l e r [& W i l k i n s 1984a, b] d is t ingu ish different k inds of updates depending on whether the update expresses a change in the state of the wor ld , an error in the database, or s imp ly new knowledge. In a database w i t h incomplete in fo rmat ion , an update can be expected to have different semant ics depending on to w h i c h of these categories it belongs. 4) Non -mono ton i c reasoning systems appear to prov ide useful theoret ica l tools for examin ing the repercussions of updates. Upda tes cont rary to wha t was inferred by defaul t can be made to au toma t i ca l l y exc lude these offending defaults after the update. Re i te r [1980a] has considered updates to defaul t theories i n l im i ted c i rcumstances. He shows that cer ta in classes of updates are knowledge-conserv ing; they do not force the reject ion of any conclusions. 5) Be l i e f R e v i s i o n Sys tems: T h e assumpt ion-based approach to bel ief rev is ion [Mar t ins 1983, de K l e e r 1984] p rov ides an a t t rac t ive book-keeping system for deal ing w i t h s t ra ight forward reper-cussions of chang ing sets of assumpt ions. Re i te r and Grosof [1985, personal communicat ions] - 101 -have each worked on fo rmal iz ing these systems in default logic. Non -mono ton i c reasoning and update are in t imate ly connected: non-monoton ic reasoning is non-monoton ic precisely because of i ts behav iour when confronted by updates. In fact, i t is possi-ble to v iew wha t we have been ca l l ing non-monoton ic reasoning as a monoton ic , v a l i d , fo rm of inference. A n y update w h i c h forces assumpt ions to be ret racted can be const rued as cont rary to the o r ig ina l knowledge-base (i.e., assumpt ions are v iewed as enta i led by the knowledge-base under a mod i f ied en ta i lment re la t ion [Israel 1980, N u t t e r 1983].) U n d e r this v iew, non-monoton ic i t y becomes s t r i c t l y a p rob lem of deal ing w i t h cont rary updates. T h e p rob lem of updates is also impor tan t w i t h i n the context of non-monoton ic reasoning. G i v e n a sys tem for d raw ing non-monoton ic inferences, one is faced w i t h the p rob lem of adapt ing to new in fo rmat ion . E v e n updates w h i c h do not represent a change in the state of the wo r l d are p rob lemat i c w h e n non-monoton ic i t y is i nvo lved . T h e obvious p rob lem is that cont rary in forma-t ion m a y have been prev ious ly inferred by default . In such cases, the conf l ict can perhaps be detected. T h e defaul t inference can then s imp ly be revoked (if the system remembers its default genesis) or va r ious consistency restorat ion techniques can be app l ied to reject some set of " o f f e n d i n g " beliefs. T h e update p rob lem i n non-monoton ic theories is compounded by the fact that inferences m a y have been based on the absence of wha t is now being asserted. In such c i rcumstances, there m a y be no inconsistencies to s ignal the necessity of bel ief rev is ion. Unless the assumpt ions under-l y ing facts i n the knowledge-base can be examined for compatab i l i t y w i t h updates in the same w a y that the facts themselves are, no th ing can prevent the knowledge-base f rom being " c a t a -p u l t e d " in to sel f -support ing - but otherwise unjust i fed - bel ief sets. F o r example, the default theory: leads to the beliefs P and R. Un less care is taken, bel ief in R m a y suppor t bel ief in P after Q is asserted, even though R was or ig ina l ly inferred because of a lack of bel ief i n Q. W o r k on t ru th -main tenance systems [Doyle 1979; D o y l e and L o n d o n 1980] has shed some l ight on these prob-lems. In a re la ted ve in , there are issues of how knowledge representat ion languages should be designed to address these issues. W o r k on bo th database theory and non-monoton ic i t y has tended to dea l w i t h tenseless languages, v iew ing the knowledge-base as a snap-shot of some state-of-affairs. U p d a t e is seen as an a tomic process of t ransforming f rom one snap-shot to the next, w i t h the state of the knowledge-base defined on ly before and after - not dur ing - the update . O t h e r work in A l has embraced t ime - e i ther reservedly, by adopt ing " s i t u a t i o n s " o r " s t a t e s " and " f l u e n t s " w h i c h t rans form the wor ld f rom one state to another [ M c C a r t h y & Hayes 1969; Moo re 1979], o r who lehear ted ly , by adopt ing a fu l l -b lown tempora l logic [McDermot t 1981; A l l e n 1984], or somewhere in between. Perhaps the best w a y to deal w i t h non-monoton ic i ty is monoton ica l l y , by represent ing the state of an agent 's beliefs at a par t i cu la r t ime. C H A P T E R 10 Conclusions I don ' t unders tand it. I don ' t even unders tand the people who unders tand it. — Queen J u l i a n a of T h e Nether lands 1 0 . 1 . D e f a u l t L o g i c a n d I n h e r i t a n c e W e presented a correspondence between default theories and inher i tance networks w i t h except ions, analogous to that ou t l ined by Hayes [1977] between f irst-order theories and except ion-free inher i tance networks. T h i s correspondence a l lowed us to specify m i n i m u m correct-ness c r i te r ia for any inher i tance-determin ing a lgor i thm, ident i fy ing the not ion of correct inference w i t h that of de r i vab i l i t y w i t h i n a single extension of the corresponding default theory. These c r i -te r ia show tha t proposed para l le l marker -pass ing implementat ions of inher i tance networks w i t h except ions are not feasible for genera l theories. Cor rec t behav iour w o u l d require tha t severe (and di f f icul t to define) const ra in ts be p laced on the structure of the inher i tance ne tworks they cou ld represent a n d reason w i t h . G i v e n a no t ion of correct inference, i t became possible to quest ion whether inher i tance net-works w i t h except ions are a lways coherent, i n the sense of a lways represent ing a reasonable set of beliefs. Inher i tance graphs are t yp i ca l l y acyc l i c . W e showed that acyc l ic networks are coherent and , i n fact , tha t weaker c r i te r ia are suff icient to ensure coherence. T h i s led to a genera l iza t ion of the no t ion of acyc l i c i t y w h i c h can be app l ied to default theories, cal led "o rderedness" . T h e ordered theories const i tu te a na tu ra l class of theories a l l of wh ich have at least one extens ion. W e p rov ided an inference a lgo r i thm for ordered inher i tance networks w i t h except ions w h i c h is prov-ab l y correct w i t h respect to th is concept of der ivab i l i t y . O u r fo rmu la t ion suggests that i t m a y not be possible to correct ly real ize mass ive ly para l le l marker -pass ing hardware of the k i n d env isaged by N E T L wh ich is appl icable to a rb i t ra ry inher i -tance graphs. It appears that the best that can be achieved for such networks is a rest r ic ted, quas i -para l le l inference a lgor i thm. W e have sketched such an a lgor i thm, but have shown that not every set of conclus ions just i f ied by the network is accessible to it. It remains to be seen whether the l im i ta t i ons imposed by the a lgo r i thm are acceptable. Fo r tuna te ly , these pessimist ic observa-t ions do no t prec lude para l le l archi tectures for su i tab ly restr icted networks. W e have shown that T o u r e t z k y ' s in fe rent ia l d is tance a lgo r i t hm produces correct conclusions. T o u r e t z k y shows how to restr ic t a network so that pa ra l l e l marker-pass ing produces the same conclusions as the in ferent ia l - 1 0 2 -- 103 -distance a lgo r i thm. W e conclude that , for such restr icted networks, para l le l marker -pass ing is correct. W e have shown default logic to be a useful too l for fo rma l iz ing the reasoning processes invo lved i n A l systems. S u c h a speci f icat ion provides a method for eva luat ing correctness a n d a met r ic by w h i c h var ious approaches can be measured and compared. A defaul t logic speci f icat ion of a system can prov ide bo th a more complete v isua l i za t ion of how the system performs and a guarantee that that performance is coherent. T o faci l i ta te such appl icat ions, we have presented a number of resul ts on defaul t logic. These inc lude a semant ics for a rb i t ra ry s ingle- just i f icat ion defaul t theories, a charac te r iza t ion of a large class of theories for w h i c h coherent reasoning is a lways possible (i.e., theories w h i c h a lways have at least one extension), and a to ta l l y correct inference a lgo r i t hm for a subclass of these theories. It m ight be — and has been — argued that a dec larat ive fo rma l i sm such as defaul t logic is inadequate for the tasks of knowledge representat ion and reasoning. W h i l e we c lear ly disagree w i t h this pos i t ion , we expect defaul t logic to be useful even to "p rocedu ra l i s t s " . E v e n i f some sys tem were fundamen ta l l y more t han the s u m of i ts declarat ive content, defaul t log ic cou ld be used to formal ize tha t dec larat ive content . T h e non-declarat ive " c o n t r o l " in fo rmat ion cou ld then be t reated as an inference a lgo r i thm for the resul t ing default theory. T h e correctness of the sys tem w o u l d be de te rmined by whether this inference a lgor i thm was correct w i t h respect to the proof theory of defaul t logic. Defau l ts , i n one fo rm or another, are ext remely common in A l . Re i te r [1978b, 1980a] discusses a w ide va r ie ty of c o m m o n s i tuat ions to w h i c h they can be app l ied, inc lud ing severa l A l knowledge representat ion schemes. M a n y of these m a y be amenable to analys is us ing an approach s im i la r to that w h i c h we have used for inher i tance networks. If some are not, two possibi l i t ies arise: the features not so amenable m a y prove incorrect or inessent ial , or they m a y point out shor tcomings of defaul t logic. E i t h e r resul t wou ld raise interest ing quest ions. 10.2. Predicate Circumscription A l t h o u g h a mode l - theory for predicate c i rcumscr ip t ion has been ava i lab le since 1980; together w i t h an at tendant soundness resul t , very l i t t le was known about the strengths a n d weaknesses of predicate c i rcumscr ip t ion un t i l recent ly . W e explored the constra ints imposed by c i rcumscr ip t ion 's model - theory and were surpr ised to find them very r i g id indeed. P rev ious expectat ions for predicate c i r cumscr ip t ion had been very h igh; examples in the l i terature had pushed the techn ique beyond the safety of i ts semant ic just i f icat ions, and this fact had gone unno-t i ced. P red ica te c i r cumscr ip t i on (and fo rmu la c i rcumscr ip t ion) can lead to inconsistent conjectures when app l i ed to theories w i thou t m i n i m a l models. In retrospect, th is is not surpr is ing, but it does not appear to have occur red to anyone un t i l we d iscovered an example . T h i s is perhaps a t t r ibu t -able to the schemat ic nature of predicate c i rcumscr ip t ion . N o t every subst i tu t ion produces incon-s istency, so unless an inconsistent subs t i tu t ion is d iscovered, c i rcumscr ip t ion of theories w i thout - 104 -m i n i m a l models m a y appear s imp ly ineffectual. T h e existence of theories w i t h inconsistent c i r -cumscr ip t ions suggests that one must be carefu l to c i rcumscr ibe only those theories w i t h m i n i m a l models. A l a s , i t is undec idab le w h i c h theories have m i n i m a l models. W e have character ized a class of theories, w h i c h we ca l l well-founded, wh i ch a lways have m i n i m a l models. W e then exp lored the propert ies of predicate c i r cumscr ip t ion vis-a-vis these wel l - founded theories. W e discovered that the semant ic character iza t ion of predicate c i rcumscr ip -t ion - so i n tu i t i ve l y appeal ing on the surface - r ig id ly constra ined the effectiveness of c i rcumscr ip -t ion i n con jec tur ing new ground facts. T h e only g round facts wh i ch predicate c i r cumscr ip t ion can conjecture are negat ive instances of one of the predicates being c i rcumscr ibed - and then on ly insofar as such conjectures prov ide no new in format ion about the extensions of non-c i rcumscr ibed predicates. Fu r the rmore , the equa l i t y predicate is somehow resistant to predicate (and formula) c i r cumscr ip t i on . 10.3. Generalizations of Circumscription T h e success of our model- theoret ic invest igat ions in to predicate c i rcumscr ip t ion (pessimist ic though the resul ts were) suggested that a s im i la r exp lorat ion of the var ious general ized forms of c i r cumsc r ip t i on m igh t also prove wor thwh i le . M c C a r t h y [1986] d i d not prov ide a model - theory for f o rmu la c i r cumscr ip t i on , however. T h e first task for this invest igat ion, thus, was to develop a model - theory . T h e model - theory presented is a general izat ion of that of predicate c i rcumscr ip -t ion , w i t h appropr ia te changes to accomodate the in t roduct ion of var iable predicates. T h e m i n i m i z a t i o n of expressions, rather than predicates, also forces modi f icat ions to the def ini t ions of submode l and m i n i m a l mode l . T h e soundness (and, for cer ta in classes of theories, completeness) of f o rmu la c i r cumscr ip t i on w i t h respect to this model- theory has been proven. U n i v e r s a l theories a lways have m i n i m a l models regardless of the predicates va r i ed or m i n i m -ized. F o r these theories, the consistency of general ized c i rcumscr ip t ion is assured. In fact, the proof shows that every mode l of a un iversa l theory has at least one m i n i m a l submode l . A s a coro l la ry of th is, general ized c i rcumscr ip t ion of un iversa l theories does not affect the extensions of any predicates not designated as var iab le . F o r such theories, the repercussions of c i rcumscr ip t ion do not ex tend beyond those predicates exp l i c i t l y ind ica ted as l iable to change. L i fsch i tz [1984, 1985a,b] has developed extensions to c i rcumscr ip t ion a l lowing constants a n d funct ions to be t reated as var iab les dur ing the m in im iza t i on process! a n d a l low ing a rb i t ra ry pre-orders to be speci f ied; m i n i m i z a t i o n proceeds accord ing to this pre-order. Su i tab le modi f icat ions to the genera l ized c i rcumscr ip t ion model - theory , wh ich accommodate these extensions, were presented. L i f s c h i t z ' innovat ions were shown to be sound w i th respect to this mode l theory. W e examined the effects of some of these formula t ions on the existence of m i n i m a l models, on con-s is tency, a n d o n the types of conjectures wh ich c a n be obta ined. - 105 -10.4. Domain Circumscription M c C a r t h y [1980] c la ims that d o m a i n c i rcumscr ip t ion is a specia l case of predicate c i r -cumscr ip t i on . W e showed that the demonst ra t ion ac tua l ly rests on a st rengthened fo rm of pred i -cate c i r cumsc r ip t i on w h i c h does not a lways preserve consistency, even for theories w i t h m i n i m a l models. W e showed that none of predicate c i rcumscr ip t ion , f o rmu la c i r cumscr ip t ion , or general-ized c i r cumsc r ip t i on w i thout var iab le terms supercedes doma in c i rcumscr ip t ion , i n general . W e conjectured tha t even var iab le terms are un l i ke ly to suffice to make general ized c i r cumscr ip t ion subsume d o m a i n c i rcumscr ip t ion . In fact , the d o m a i n c i r cumscr ip t i on schema presented by M c C a r t h y [1980] and Dav i s [1980] is also too s t rong. C e r t a i n theories w i t h m i n i m a l models turn out to have inconsistent doma in c ir-cumscr ip t ions . A f t e r iso lat ing the p rob lem, we out l ined a s t ra ight forward cor rec t ion w h i c h preserves the appea l ing semant ic charac ter iza t ion presented by D a v i s [1980], and proved its correctness. W e have also no ted the amb igu i t y of the do main-c losure assumpt ion, as i t is usua l ly stated. W e argue that the most c o m m o n d isambigua t ion agrees w i t h the results ob ta ined f rom doma in c i r cumscr ip t i on . A l s o , we conjectured that the completeness of d o m a i n c i r cumscr ip t i on for cer ta in classes of theories m igh t be provable. 10.5. Relations Between Circumscription and Default Logic W e have considered the re la t ionsh ip between default logic and c i rcumscr ip t ion . W e showed that , in some cases, the c losed-wor ld defaul t coincides w i t h c i rcumscr ip t ion ; that , in a par t i cu -la r l y useless way , defaul t logic subsumes c i rcumscr ip t ion ; and that default logic is capable of affect ing the equa l i t y theory wh i le predicate, fo rmula , and doma in c i rcumscr ip t ion are not. W e showed tha t the in t roduc t ion of fixed predicates and appl ica t ions to open domains each prov ide c i r cumscr ip t i on w i t h capab i l i t ies not avai lable using s imple c losed-wor ld defaul t theories. F i n a l l y , we used semant ic compar isons to h ighl ight a number of the essent ia l differences between the two approaches. T h i s a l lowed us to suggest that some of the work o n t rans lat ions between the two formal isms m a y not have not iced the essent ial character is t ics w h i c h should be carefu l ly cons idered i n de termin ing adequacy condi t ions for t ranslat ions. References A l l e n , J . F . [1984], " T o w a r d s a G e n e r a l T h e o r y of A c t i o n and T i m e " , Artificial Intelligence 28 (2), N o r t h - H o l l a n d , J u l y , 1984, pp 123-154. Ande rson , A . R . , a n d B e l n a p , N . D . [1975], Entailment: The Logic of Relevance and Necessity, Pr ince ton Un i ve r s i t y Press , P r i n c e t o n , N J , 1975. Barw ise , J o n (ed) [1977], Handbook of Mathematical Logic, N o r t h - H o l l a n d , N e w Y o r k , 1977. Ba rw ise , J o n , a n d P e r r y , J o h n [1983], Situations and Attitudes, M I T Press, C a m b r i d g e , M A , 1983. B e t h , E . W . [1953], " O n P a d o a ' s me thod in the theory of def in i t ions" , Indag. Math 15, 1953, pp 330-339. Bob row , D . G . and W i n o g r a d , T . [1977], " A n Ove rv i ew of K R L - 0 , a Know ledge Representa t ion L a n g u a g e " , Cognitive Science 1(1). Bossu , G . a n d Siege l , P . ]1985], " S a t u r a t i o n , Non -Mono ton i c Reason ing, and the C l o s e d - W o r l d A s s u m p t i o n " , Artificial Intelligence 25(l), No r t h -Ho l l and , pp 13-63, 1985. B r a c h m a n , R. [1982], " W h a t ' I S - A ' Is and Isn ' t " , Proc. Canadian Soc. for Computational Studies of Intelligence-82, Saska toon , Sask. , M a y 17-19, pp 212-220. C l a r k , K . L . [1978], " N e g a t i o n as F a i l u r e " , in Logic and Databases, H . Ga l l a i r e and J . M i n k e r (eds.), P l e n u m Press , N e w Y o r k . Co t t r e l l , G . W . [1985]. " P a r a l l e l i s m in Inher i tance Hierarch ies w i t h E x c e p t i o n s " , Proc. Ninth International Joint Conference on Artificial Intelligence, Los Angeles, C A , A u g . 18-23, 1985, pp 194-202. D a v i s , M . [1980], " T h e M a t h e m a t i c s of N o n - M o n o t o n i c Reason ing " , Artificial Intelligence IS, N o r t h - H o l l a n d , pp 73-80. de K lee r , J . [1984], " C h o i c e s W i t h o u t B a c k t r a c k i n g " , Proc. Amer. Assoc. for Artificial Intelligence-84, A u s t i n , T X , pp 79-85. Doy le , J . [1979], A Truth Maintenance System, A l M e m o 521, M I T A r t i f i c i a l Intel l igence L a b o r a -tory, C a m b r i d g e , M a s s . Doy le , J . [1982a], " S o m e theories of reasoned assumpt ions: A n essay in ra t iona l psycho logy" , T e c h n i c a l Repo r t , C a r n e g i e - M e l l o n Un ive rs i t y . Doy le , J . [1982b], The Ins and Outs of Reason Maintenance, T e c h n i c a l Repor t , Ca rneg ie -Me l l on Un i ve rs i t y , 1982. Doy le , J . [1984], " C i r c u m s c r i p t i o n and Imp l ic i t De f inab i l i t y " , P rep r in t , Amer. Assoc. for Artificial Intelligence Workshop on Non-Monotonic Reasoning, New P a l t z , N Y , O c t o b e r 1984. Doy le , J . , and L o n d o n , P . [1980], A Selected Descriptor-Indexed Bibliography to the Literature on Belief Revision, A l M e m o 568, M I T , C a m b r i d g e , M a s s . - 106 -- 107 -Ethe r i ng ton , D . W . [1982], Finite Default Theories, M . S c . Thes is , Dept . C o m p u t e r Sc ience, Un i ve r -s i ty of B r i t i s h C o l u m b i a . E the r ing ton , D . W . [1983], Formalizing Non-Monotonic Reasoning Systems, T e c h n i c a l Repor t 83-1, Depar tmen t of C o m p u t e r Science, Un ive rs i t y of B r i t i s h C o l u m b i a , 1983. ( T o appear in Artificial Intelligence). Ethe r i ng ton , D . W . a n d Merce r , R . E . [1986], " D o m a i n C i r cumsc r i p t i on R e v i s i t e d " , Proc. Canadian Society for Computational Studies of Intelligence-86, Mon t rea l , M a y 1986. E the r i ng ton , D . W . , Merce r , R . E . , and Rei ter , R. [1985], " O n the adequacy of predicate c i r -cumscr ip t i on for c losed-wor ld reason ing" , Computational Intelligence 1(1), Feb rua ry 1985. E the r i ng ton , D . W . , and Re i te r , R . [1983], " O n Inher i tance Hierarchies W i t h E x c e p t i o n s " , Proc. American Assoc. for Artificial Intelligence-88, Wash ing ton , D . C . , Augus t 24-26, pp 104-108. F a g i n , R. , U l l m a n , J . D . , and V a r d i , M . Y . [1983], " O n the Semant ics of Updates i n Da tabases " , Proc. of the Second ACM SIGACT-SIGMOD Symp. on Principles of Database Systems, A t l a n t a , G A , pp 352-365. F a h l m a n , S . E . [1979], NETL: A System for Representing and Using Real- World Knowledge, M I T Press , C a m b r i d g e , M a s s . F a h l m a n , S . E . [1982], " T h r e e F lavo rs of P a r a l l e l i s m " , Proc. Canadian Soc. for Computational Stu-dies of Intelligence-88, Saskatoon, Sask. , M a y 17-19, pp 230-235. F a h l m a n , S . E . , T o u r e t z k y , D .S . , and v a n Roggen, W . [1981], " C a n c e l l a t i o n in a P a r a l l e l Semant i c N e t w o r k " , Proc. Seventh International Joint Conference on Artificial Intelligence, Vancouve r , B . C . , A u g . 24-28, p p 257-263. Ga l l a i r e , H . and M i n k e r , J . (eds) [1978], Logic and Data Bases, P l e n u m Press, 1978. Ge l f ond , M . , a n d P r z y m u s i n s k a , H . [1985], Negation as Failure: Careful Closure Procedure, Un ive r s i t y of T e x a s at E l Paso , unpub l i shed draft, 1985. Genesereth , M . and N i l sson , N . [1987], draft of chapter ent i t led " N o n m o n o t o n i c Reason ing " , T o appear i n Fundamentals of Artificial Intelligence, M o r g a n K a u f m a n n Pub l i shers , Los A l t o s , C A , fo r thcoming 1987. Ge l f ond , M . , P r z y m u s i n s k a , H . , and P r z y m u s i n s k i , T . [1985], The Extended Closed-World Assumption and its Relation to Parallel Circumscription, Un ive rs i t y of T e x a s at E l Paso , unpub l i shed draft , 1985. G o o d m a n , Ne lson [1955], Fact, Fiction, and Forecast, Harvard ' Un ive rs i t y Press, C a m b r i d g e , M A , 1955. G r i c e , H . P . [1975], " L o g i c a n d C o n v e r s a t i o n " , in Syntax and Semantics, Vol S: Speech Acts, P . C o l e and J . L . M o r g a n (eds.), A c a d e m i c Press. Grosof , B . [1984], " D e f a u l t Reason ing as C i r c u m s c r i p t i o n " , T e c h n i c a l Repor t , S tan fo rd Un ive r -s i ty , S tan fo rd , C A . Hayes , P . J . [1973], " T h e F r a m e P r o b l e m and Re la ted Prob lems in A r t i f i c i a l In te l l igence" , i n Artificial and Human Thinking, A . E l i t h o r n and D . Jones (eds.), Jossey-Bass Inc., San F r a n -cisco. Hayes , P . J . [1977], " I n Defense of L o g i c " , Proc. Fifth International Joint Conference on Artificial Intelligence. C a m b r i d g e , M a s s . , pp 559-565. Hew i t t , C . [1972], Description and Theoretical Analysis (Using Schemata) of PLANNER: A Language for Proving Theorems and Manipulating Models in a Robot, A l M e m o 251, M I T P r o -ject M A C , C a m b r i d g e , Mass . Hughes, G . E . and Cresswe l , M . J . [1972], An Introduction to Modal Logic, M e t h u e n and C o . L t d . , - 108 -L o n d o n . Im ie l insk i , T . [1985], " R e s u l t s on T rans la t i ng Defaul ts to C i r c u m s c r i p t i o n " , Proc. Ninth Interna-tional Joint Conference on Artificial Intelligence, Los Angeles, C A , A u g . 18-23, 1985, pp 114-120. Israel , D . J . [1980], " W h a t ' s wrong w i t h N o n - M o n o t o n i c L o g i c " , Proc. Amer. Assoc. for Artificial Intelligence-80, 1980. K a p l a n , S . J . and D a v i d s o n , J . [1981], " In te rp re t ing na tu ra l language database upda tes " , Proc. 19th Ann. Meeting of the Assoc. for Computational Linguistics, S tan ford , 1981, pp 139-142. K e l l e r , A . M . and W i l k i n s , M . W . [1984a], " A p p r o a c h e s for U p d a t i n g Databases w i t h Incomplete In fo rmat ion and N u l l s " , Proc. IEEE Computer Data Engineering Conference, A p r i l 1984, Los Angeles . Ke l l e r , A . M . and W i l k i n s , M . W . [1984b], " O n the Use of an E x t e n d e d Re la t i ona l M o d e l to H a n -dle C h a n g i n g Incomplete In fo rmat ion " , IEEE Transactions on Software Engineering, 1984. K r a m o s i l , I. [1975], " A No te on D e d u c t i o n Ru les w i t h Negat ive P rem ises " , Proc. Fourth Interna-tional Joint Conference on Artificial Intelligence. Levesque, H . J . [1982], A Formal Treatment of Incomplete Knowledge, T e c h n i c a l Repor t 3, F a i r -ch i l d Labo ra to r y for A r t i f i c i a l Intel l igence Research, M e n l o P a r k , C A . Levesque, H . J . [1984], " F o u n d a t i o n s of a F u n c t i o n a l A p p r o a c h to Know ledge Rep resen ta t i on " , Artificial Intelligence 28(2), Ju l y , 1984, N o r t h - H o l l a n d , pp 155-212. Lewis i D. [1973], Counterfactuals, H a r v a r d Un ive rs i t y Press, 1973. L i f sch i t z , V . [1984], " S o m e Resul ts on C i r c u m s c r i p t i o n " , T e c h n i c a l Repor t , S tan fo rd Un ive rs i t y , S tan ford , C A . L i f sch i tz , V . [1985a], " 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 and Separable F o r m u l a s " , T e c h n i c a l Repor t , S tan fo rd U n i v e r s i t y , S tan ford , C A . L i f sch i t z , V . [1985b], " O n the Sat is f iab i l i t y of C i r c u m s c r i p t i o n " , Techn i ca l Repor t , S tanford Un i ve rs i t y , S tan ford , C A . L i n s k y , L . (ed) [1971], Reference and Modality, O x f o r d Un ive rs i t y Press, London , 1971. L i p s k y , W . J r . [1979], " O n Semant ic Issues Connec ted w i t h Incomplete In format ion Da tabases " , ACM Transactions on Database Systems 4 (8), Sept. 1979, pp 262-296. tyukaszewicz, W . [1984], " N o n - m o n o t o n i e logic for default theor ies" , P r o c . Sixth European Confer-ence on Artificial Intelligence, 1985, pp 165-193. J iukaszewicz, W . [1985], " T w o results on default log ic " , Proc. Ninth International Joint Confer-ence on Artificial Intelligence, Los Angeles , C A , A u g . 18-23, 1985, pp 459-461. M a r t i n s , J . [1983], Re asoning in Multiple Belief Spaces, P h D thesis, S U N Y at Buf fa lo , C o m p u t e r Science T e c h n i c a l Repor t 203. M c A l l e s t e r , D . A . [1978], A Three-Valued Truth Mainte nance System, A l M e m o 473, M I T , C a m -br idge, M a s s . M c A l l e s t e r , D . A . [1980], An Outlook On Truth Maintenance, A l M e m o 551, M I T , C a m b r i d g e , M a s s . M c C a r t h y , J . [1977], " E p i s t e m o l o g i c a l P rob lems of A r t i f i c i a l In te l l igence" , Proc. Fifth Interna-tional Joint Conference on Artificial Intelligence. 1977, pp 1038-1044. M c C a r t h y , J . [1980], " C i r c u m s c r i p t i o n - a F o r m of Non -Mono ton i c R e a s o n i n g " , Artificial Intelli-gence 18(1,2), N o r t h - H o l l a n d , 1980, pp 27-39. - 109 -M c C a r t h y , J . [1986], " A p p l i c a t i o n s of C i r c u m s c r i p t i o n to Fo rma l i z i ng Commonsense K n o w l e d g e " , Artificial Intelligence 28, 1986, pp 89-116. M c C a r t h y , J . and Hayes , P . J . [1969], " S o m e Ph i l osoph ica l P rob lems f rom the S tandpo in t of A r t i f i c i a l In te l l igence" , i n Machine Intelligence 4i B . Me l t ze r and D . M i c h i e (eds.), E d i n b u r g h U n i v e r s i t y Press , E d i n b u r g h . M c D e r m o t t , D . [1981], " A T e m p o r a l Log i c for Reasoning A b o u t Processes and P l a n s " , Research Repor t 196, C o m p u t e r Science Depar tment , Y a l e Un ive rs i t y , N e w H a v e n , C T , 1981, (also in Cognitive Science 6(2), 1982). M c D e r m o t t , D . [1982], " N o n - M o n o t o n i c Log ic II", J.ACM29(1). M c D e r m o t t , D . , and Doy le , J . [1980], " N o n - M o n o t o n i c Log ic I", Artificial Intelligence 1S(1,2), ( A p r i l 1980), N o r t h - H o l l a n d , p p 41-72. Mende l son , E . [1964], Introduction to Mathematical Logic, V a n Nos t rand Re inho ld , N e w Y o r k . M i n k e r , J . [1982], " O n Indefinite Databases and the C l o s e d - W o r l d A s s u m p t i o n " , Proc. Sixth Conf. on Automated Deduction, N e w Y o r k , 7-9 June, 1982, Spr inger -Ver lag , N Y . M i n k e r , J . a n d Pe r l i s , D . [1983], On the Semantics of Circumscription, T e c h n i c a l Repo r t , U n i v e r -s i ty of M a r y l a n d . M i n k e r , J . a n d Pe r l i s , D . [1984a], Circumscription: Finitary Completeness Results, Unpub l i shed Dra f t , U n i v e r s i t y of M a r y l a n d . M i n k e r , J . a n d Pe r l i s , D . [1984b], " A p p l i c a t i o n s of P ro tec ted C i r c u m s c r i p t i o n " , Lecture Notes in Computer Science 170: Proc. of the Seventh Conference on Automated Deduction, pp 414-425, Spr inger . M i n s k y , M . [1975], " A F r a m e w o r k for Represent ing K n o w l e d g e " , i n The Psychology of Computer Vision, P . W i n s t o n (ed.), M c G r a w - H i l l , N e w Y o r k . M o o r e , R. [1983a], " S e m a n t i c a l considerat ions on non-monoton ic l og ic " , Proc. Eighth Interna-tional Joint Conference on Artificial Intelligence. Ka r l s ruhe , W e s t G e r m a n y , pp 272-279. Moore , R. [1983b], Semantical considerations on non-monotonic logic, S R I T e c h n i c a l Note 284, M e n l o P a r k , C A . M o o r e , R . C . [1984], " P o s s i b l e - W o r l d Semant ics for Au toep is temic L o g i c " , T e c h n i c a l Repor t , S R I , M e n l o P a r k , C A . N u t t e r , J . T . [1983], Default reasoning in Al systems, M S c Thes is , S U N Y at Buf fa lo , C o m p u t e r Science T e c h n i c a l Repor t 204. Per l i s , D . and M i n k e r , J . [1986], "Comp le teness Resul ts for C i r c u m s c r i p t i o n " , Artificial Intelli-gence 28, 1986, pp 29-42. P r z y m u s i n s k i , T . [1985], Minimal-Model Resolution and Query Answering in Circumscriptive and Closed- World Theories, Un ive rs i t y of T e x a s at E l Paso , unpubl ished draft , 1985. Q u i l l i a n , M . R . [1968], " S e m a n t i c M e m o r y " , in M . M i n s k y (ed), Semantic Information Processing, M I T Press , C a m b r i d g e , Mass . Re i te r , R. [1978a], " O n C l o s e d - W o r l d D a t a B a s e s " , i n [Gal la i re and M i n k e r 78], 55-76. Re i te r , R . [1978b], " O n Reason ing by D e f a u l t " , Proc. Second Symposium on Theoretical Issues in Natural Language Processing, U r b a n a , I l l inois, 1978, 210-218. Re i te r , R . [1980a ] , " A L o g i c for De fau l t R e a s o n i n g " , Artificial Intelligence IS, ( A p r i l 1980), N o r t h - H o l l a n d , pp 81-132. Re i te r , R . [1980b], " E q u a l i t y and D o m a i n C losure i n F i r s t -O rde r D a t a B a s e s " , J ACM 27(2), - 110 -1980, 235-249. Rei ter , R. [1982], " C i r c u m s c r i p t i o n Impl ies Pred ica te C o m p l e t i o n (Somet imes) " , Proc. Amer. Assoc. for Artificial Intelligence-82, 1982, 418-420. Rei ter , R. [1984], " T o w a r d s a L o g i c a l Recons t ruc t ion of Re la t i ona l Database T h e o r y " , in M . L . B rod ie , J . M y l o p o u l o u s , and J . W . Schmid t (eds): On Conceptual Modelling, Sp r inger -Ver lag , N e w Y o r k , p p 191-233. Re i te r , R . , and Cr i scuo lo , G . [1983], " S o m e Representa t iona l Issues in Defau l t R e a s o n i n g " , Int. J. Computers and Mathematics 9 (l), (Spec ia l Issue on C o m p u t a t i o n a l L ingu is t i cs ) , 1983, pp 1-13. Rescher , N . [1964], Hypothetical Reasoning, N o r t h - H o l l a n d , A m s t e r d a m , 1964. Rescher, N . [1976], Plausible Infe rence, V a n G o r c u m , Assen , T h e Nether lands, 1976. Rousse l , P . [1975], PROLOG, Manuel de Reference et d'Utilisation, G r o u p d ' Inte l l igence Ar t i f i c ie l le , U . E . R . de Marse i l l e , F rance . Sandewa l l , E . [1972], " A n A p p r o a c h to the F r a m e P r o b l e m and Its Imp lemen ta t i on " , i n Machine Intelligence 7, B . M e l t z e r and D . M i c h i e (eds.), E d i n b u r g h Un i ve rs i t y Press, E d i n b u r g h . Schuber t , L . K . [1976], " E x t e n d i n g the Express ive P o w e r of Semant i c N e t w o r k s " , Artificial Intelligence 7(2), N o r t h - H o l l a n d , pp 163-198. Shepherdson, J . C . [1984], " N e g a t i o n as Fa i l u re : A Compar i son of C l a r k ' s C o m p l e t e d D a t a Base and Re i te r ' s C l o s e d - W o r l d A s s u m p t i o n " , Journal of Logic Programming 1(1), June 1984, 51-79. S ta lnaker , R. J1980], " A note on non-monoton ic l og i c " , Unpub l i shed note, Dep t of Ph i losophy , C o r n e l l Un i ve rs i t y . T o u r e t z k y , D . S . [1982], " E x c e p t i o n s i n an Inher i tance H i e r a r c h y " , Unpub l i shed Manusc r i p t , Depar tmen t of C o m p u t e r Science, Ca rneg ie -Me l l on Un ivers i ty . T o u r e t z k y , D . S . [1983], " M u l t i p l e Inher i tance and E x c e p t i o n s " , Unpub l i shed M a n u s c r i p t , Depar t -ment of C o m p u t e r Science, Ca rneg ie -Me l l on Un ive rs i t y . T o u r e t z k y , D . S . [1984aJ, The Mathematics of Inheritance Systems, P h D Thesis , Dep t of C o m p u t e r Sc ience, C a r n e g i e - M e l l o n Un i ve rs i t y . T o u r e t z k y , D . [1984b], " I m p l i c i t order ing of defaults in inher i tance sys tems" , Proc. Amer. Assoc. for Artificial Intelligences^-T o u r e t z k y , D. [1985], " Inher i tab le relat ions: a log ica l extension to inher i tance h ierarch ies" , Proc. Theoretical Approaches to Natural Language Understanding, Ha l i fax , 28-30 M a y 1985, pp 55-60. W i l k i n s , M . W . [1983], " P a r t i a l In format ion and A l te rna t i ve S e t s " , Techn i ca l Repor t , S tanford Un ive rs i t y , S tan fo rd , C A . W i n o g r a d , T . [1980], " E x t e n d e d Inference Modes i n R e a s o n i n g " , Artificial Intelligence 1S(1,2), ( A p r i l 1980), N o r t h - H o l l a n d , p p 5-26. W o o d s , W . A . [1975], " W h a t ' s In A L i n k ? " , Representation and Understanding, A c a d e m i c Press, pp 35-82. A P P E N D I X A Proofs of Theorems B a c k g r o u n d I n f o r m a t i o n The re are a few def in i t ions and results due to Re i te r [1980a] on w h i c h we draw freely i n the fo l lowing proofs. W e reproduce t hem here for the reader's convenience. 1) T h e o r e m 0.1 [Reiter 1980a, T h e o r e m 2.1] oo E is an extens ion for A = (D , W ) i f and only i f E = U E ; , where i=0 E 0 = W , and for i >0 E i + 1 = Th(E j ) U {w |; SLlL e D, a e E t , and ->B £ E } 1 w 2) T h e Generating/ Defaults for E w i t h respect to A are defined as: G D ( E , A ) = {.2-L£ € D I a € E , -.0 £ E} CO 3) If D is a set of defaul ts, then CONSEQUENTS (D) is def ined, as one wou ld expect , as: CONSEQUENTS ID) = {w I e D} w 4) ; T h e o r e m 0 .2 [Reiter 1980a, Theo rem 2.5] If E is an extens ion for A = (D, W ) , then E = T h ( W U CONSEQUENTS(GD(E,A))). 5) T h e o r e m 0.3 [Reiter 1980a, Co ro l l a r y 2.2] If E is an extens ion for A = (D , W ) , then E is consistent if a n d on ly if W is. In the proofs of resul ts f rom chapters 3 and 4, we w i l l usual ly assume that formulae are in c lausa l f o rm: i.e., expressed as a con junct ion of d is junct ions of l i terals. W e define the funct ions CLA USES (•) a n d LITERALS (•) as fol lows: If P = V .. . V BUm) A .. . A ( A ^ V .. . V B^nJ then CLAUSES(6) = {(Ai V ... V BUm) | 1 < i < m } LITERALS (B) = {A,j I 1 < i < m, 1 < j < m i } A b u s i n g the no ta t i on somewhat we somet imes use CLAUSES (T), where T is a set of formulae, to 1 Note the explicit reference to E in the definition of E i + 1 . - I l l -- 112 -refer to U CLAUSES(7). 7 6 T W e w i l l define other no ta t ion as i t is required. Definition: Satisfiability, admissibility, and applicability L e t X be a set of models; T a set of formulae; a, 0, and w formulae, and 5 = — a defaul t . T h e n i) a is X-satisfiable (X-valid) iff ] x £ X . x [= « ( V x € X . x j= a ) ii) r is X-admissible (X permits T) iff V 7 € l \ 3 X € X . x ^ = 7 i i i ) 6 is X-applicable iff a is X v a l i d and /? is X-sat is f iab le . | Definition: Result of a default Le t X , T, and 6 be as above. T h e n the result of S in (X , T) is: f ( X , T) i f 6 is not X -app l i cab l e and T is X-admiss ib le , 6{X, F ) = { ( ( X - { N I N }= -.w}), ( r U { £ } ) ) if £ is X -app l i cab le and T is X -admiss ib le , and I otherwise. | Definition: Result of a sequence of defaults Le t X and T be as above, and let <S(> be a sequence of defaults. T h e n « * i > ( X , r ) = ( n X ; , U TO where (X0=X; T0 = T; and \ ( x i + 1 , r i + 1 ) = ^ ( x , r j , i > o . I Definition: Stability L e t Y be a non-empty set of models , T a set of formulae, and A = (D , W ) a defaul t theory. T h e n ( Y , T) is stable for A iff (1) ( Y , T) = <$ i> (X , { }) for X = { M I M j= W } , and some C D , (2) V * € D . 6(Y, T) = ( Y , T ) , a n d - 113 -(3) T is F-admiss ib le . | T h e o r e m 3.1 — S o u n d n e s s If E is an extens ion for A , then there is some set T such that ( { M | M j= E}, T) is stable for A . P r o o f Def ine G D = j<5 = " : 8 G D | a € E, - . ; 9 ^ E J . F r o m theorem 0.2, we have E = T h ( W U G D ) . There are 2 cases: G D = { }: T h e n E = T h ( W ) . C l e a r l y < > ( X , { }) = (Y , { }). Cons ider 6 = SLlL G D. If a is Y - v a l i d w and 8 is Y-sa t is f iab le , then E )— a, E \f—<B, so S G G D , w h i c h is a con t rad ic t ion . Hence 5 (X , { }) = (Y , { }) . C l e a r l y { } is Y -admiss ib le . Hence (Y , { }) is stable w i t h respect to A . G D jfe { }: Le t {6 l v . . } be any order ing of G D . Def ine S'i by S'i = 6)t where j is the smal lest integer such that 5j is <S'o...6'i_i>{X,{ Inapplicable, and Sj £ {<^ 'o)---j^ 'i-i}j where 0 < i < n. It c a n easi ly be seen that th is is wel l -def ined, and uses a l l of <S{>. Obv ious l y , if T = J U S T I F I C A T I O N S ^ ' ; } ) , then S G D impl ies 6{Y,T) = ( Y , r ) . It remains to show that <6\>(X,{ }) = ( Y , r ) . It is easi ly p roved that <S'o...S'i>(X,{ }) = ( X ^ i ) , where X ; is the set of a l l models for T h ( W U {w'o,...,u'i}). - where the w ' i ' s are the consequents of the respect ive 6' i 's -and r , = J U S T I F I C A T I O N S ( { 5 ' o , . . . , 5 ' i } ) . Hence < 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 , J U S T I F I C A T I O N S ( G D ) ) . 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 -admiss ib le . Hence ( Y , r ) is stable for 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) is s tab le for A then Y is the set of models for some extension of A . (I.e., {w | V y G Y . y |= w} is an extension for A . ) - 114 -Proof Since ( Y , r ) is stable, ( Y , r ) = < ^ > ( X , { }) where X = { M | M \= W } and {6J C D . W i t h o u t loss of genera l i ty , let <^> be inf in i te. (If finite, repl icate 5„). Def ine (X^I?;) as fol lows: f X o . r o ) = (X , { }), and for i > 0, ( X i + 1 , r i + 1 ) = ^ ( X ^ ) . T h e n Y = D X ; , and V = U a : 0 f S ince ( Y , T ) is stable, for any defaul t , S = — € E D, either S is not Y -app l i cab le , or w is Y -u> v a l i d and B S T. In e i ther event, T is Y -admiss ib le . a i : A Assume 6{ = . L e t F ; be the set of X j - va l i d formulae. W e show that F 0 = T h ( W ) and that if a ; e F ; , a n d -.ft £ Fi» then F i + 1 = T h ( F i U {w ;}). Otherwise F i + 1 = F ; . T h i s is t r i v i a l for Fo- Assume i t is t rue for F;, and consider F ; + 1 . S ince V is Y - a d m i s s i b l e , each T ; is Xpadm iss i b l e . If a ; S F;, then a ; is X r v a l i d . If ->A ^ F;, then A is Xj-sat is f iable. Hence X i + i = X ; - { N | N \= ^w;}, and F i + 1 = T h ( F i U ( w j ) . Otherwise X i + 1 = X ; , so F i + 1 = F ; . L e t E = U F ; . C l e a r l y Y = { M | M \= E } . It remains to show that E is an extens ion for A . i=l Define E 0 = W , and E i + 1 = T h ( E i ) U {w | e D . a e E ^ ^ E } . W e show E = U E ; . w i=0 U E i C E = U F ; : C l e a r l y E 0 C F 0 C E . Assume E ; C E , and consider w € E i + 1 . T r i v i a l l y , if w e T h ( E j ) , w e E . a • B Otherw ise , there is a defaul t , 5 = — € D, such that a € E ; and ->0 & E . Since a € E ; , and E i C E , a is Y - v a l i d . S im i l a r l y , 0 is Y-sat is f iab le . B y ( f ) , w is Y - v a l i d , so u 6 E . C O 0 0 E = U F; C U E ; : i=0 i=0 0 0 0 0 0 0 C l e a r l y F 0 = T h ( W ) C E x C U E ; . Assume F ; C U E ; and consider F i + 1 . S ince U E ; is i=0 i=0 fc=0 0 0 0 0 closed a n d F 5 C U & „ it suffices to show tha t a-, € F ; , and ->0i F-„ whence w ; € U E; . i=o i=o If a ; S F j , a n d —<0 F j then S{ is X p a p p l i c a b l e . S ince (Y,r ) is stable, T is Y -admiss ib le . B u t 0 0 {A} £ C T , so E \/~ -i0„ SO --A £ E . a{ £ F ; C U E i , so a ; € E j , for some j . i=0 T h u s E is an ex tens ion for A , b y T h e o r e m 0.1. - 115 -QED T h e o r e m 3.2 L e m m a 3.3.1 If E ' ( i > 0) is an extens ion for the default theory A ; = (D;, E ' _ 1 ) and E _ 1 = W , then the fo l low ing are equ iva lent : (1) a S E 1 (2) E'f-ot (3) ( W U U CONSEQUENTS(GD(Er, A r ) ) ) j - a r=0 P r o o f (1) a € E ' < = • E ! f - a T h i s fol lows f rom the fact that E 1 is an extension and thus log ica l ly c losed. (2) E 1 f— a <=> ( W U U CONSEQUENTS(GT>(ET, A r ) ) ) f - a r=0 If E is an extens ion for A , then by Theo rem 0.2 we know that E = T h ( W U CONSEQUENTS(GD(E, A ) ) ) . Hence E ' = T h ( E w U CONSEQUENTS(GD(E', A J ) ) = ^ ( T h f E ^ U CONSEQUENTS(GDfE1"1, A M ) ) ) U CONSEQUENTS(GD(Ei, A ; ) ) ) = T h ( T h . . . ( W U CONSEQUENTS(GD(E°, A 0 ) )> U ... U CONSEQUENTS(GD(E\ A J ) ) Since T h ( T h ( A ) U B ) = T h ( A U B ) , E 1 = T h ( W U U CONSEQUENTS(GD(Er, A r ) ) ) . r=0 F r o m th is, the result fol lows by the def in i t ion of T h . QED L e m m a 3.3.1 - 116 -D e f i n i t i o n 3 . 3 . 2 : « : and «C Le t A = ( D , W ) be a c losed, semi -norma l defaul t theory. W i t h o u t loss of general i ty , assume a l l fo rmulae are i n c lausa l fo rm. T h e par t i a l relat ions, <C and <C , on Literals X Literals, are defined as fo l lows: (1) If a € W then a = ( a x V ... V a j , for some n > 1 . F o r a l l a ; , ctj G {cti,...,<*„}, i f a ; j= ctj , let ->aj <C a j . (Since: ( « ! V . . . V a j = [(- .aj A... A - a j - i A -<Qrj+1 A-A -x* „ ) 3 « j ]) (2) If £ G D then £ = —'—— - . L e t ait ... aT , p\, ... 0S, and flt ... 7 t be the l i tera ls of the P c lausa l forms of a, 0, and 7, respect ive ly . T h e n (i) If a ; G {au...,at} and 0; G {/?!,...,&} let a, ^  0} . (ii) If T, G {7i,-.7t}, Pj 6 {&>->&} a n d 7 i £ {&, . . . ,&} let - , 7 i « 0-} . (i i i) A l s o , 0 = 0i A ••• A An j for some m > 1. F o r each i < m, A = (A,i V ... V 0-^ , where m; > 1 . T h u s if Aj , A,k G { ^ , 1 , - , ^ ^ } and Aj £ A,k I<* -Aj « ftk . (3) T h e expected t rans i t i v i t y re lat ionships ho ld for <SC and <C . JT.e., (i) If a « y9 and 0 « 7 then a ^ 7. (ii) II a « 0 and 0 1 then a <3C 7. (ii i) If a <C 0 and £ « 7 or a ^ and /? <SC 7 then a <C 7. | D e f i n i t i o n 3 . 3 . 3 : O r d e r e d n e s s A semi -no rma l defaul t theory is said to be ordered iff there is no l i te ra l , cx, such that 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 losed, semi -no rma l default theory, A = (D , W ) , define the Universe of A , U ( A ) , as fol lows: U ( A ) = {a I a G Literals and [ 3f. [ (a V £) G CLAUSES(W U CONSEQUENTS(D))] or [ ( - .a V £) G CLAUSES^ U COWStfQHENrSfD))])} U { a ; I 3 a , A7. G D and a t G LITERALS (a) } U {-7i I 3 a , A T a ' ^ A 7 G D and 7 i G LITERALS (7) } - 1 1 7 -Observe tha t £ m a y be the n u l l c lause. | D e f i n i t i o i i 3 . 3 . 5 : / : U ( A ) (-• N F o r a c losed, ordered, semi -no rma l default theory, A = (D , W ) , we define the func t ion / : U ( A ) |-» N , as fo l lows: If a,0 G U ( A ) and a « 0 then I (a) < I (0). If a <K 0 then I (0) > I {oc) + l. If 0 G U ( A ) and for no a € U ( A ) is ( a <SC 0) or ( a « 0) then / (0) = 0. If n G N , 0 G U ( A ) , and f (/9) > n then 3 « G U ( A ) . ( a « p") and / (a) = n. S ince A is ordered, / is we l l def ined. Observe that / is a to ta l func t ion on U ( A ) w h i c h assigns a na tu ra l number to each l i te ra l in U ( A ) . 1(a) m a y be thought of as the length of the longest cha in of sem i -no rma l defaul ts w h i c h cou ld figure i n an inference of a. | D e f i n i t i o i i 3 . 3 . 6 : / M A X . ' M I N If 0 is a c losed fo rmu la , and the c lausa l fo rm of 0 is V ... V 0hm) A ... A (0mi V ... V then define /MAX(0 = M A X ( / (AJ) W $ = M I N ( / (/Jy)) . | L e m m a 3 .3 .7 If A = (D , W ) is an ordered, c losed, semi -norma l default theory, then there is a par t i t i on , {Dj} , for D induced by : V f 6 D J = a - PJ\1 a n d 1^(0) = i iff s e D ; . P Proof C l e a r l y LITERALS (CONSEQUENTS ({5 G D}) ) C U ( A ) , and / is to ta l on U ( A ) . - 1 1 8 -Therefore: 1) Vtf G D. Vi. Vj. (S G D{ A 6 G Dj) impl ies i = j . 2) V < 5 G D . 3 1 ( 5 e D 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 Do, then 5 is a n o r m a l defaul t . P r o o f If 5 = Q : *'A 7 e D q t h e n / m m ( ^ > / m a x ( ^ ) > o . P 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 and D j j= { }, there is at least one non-norma l (i.e., semi-normal) defaul t i n D ; . P r o o f If D ; conta ins on ly no rma l defaul ts, then the m in ima l i t y of / guarantees that lhm(CONSEQUENTS(D^) < i, wh i ch is a cont rad ic t ion . Q E D C o r o l l a r y 3 . 3 . 9 L e m m a 3 . 3 . 1 0 If T is consistent , i f / M A X H 3 ) < J. a n d i f / (7) is def ined for a l l 7 € LITERALS (T), then there is a l i near reso lu t ion re fu ta t ion of 8 f r om V i f and only if there is a l inear resolut ion re fu ta t ion of 8 f rom where * C T and rp G * iff / M I N ( ^ ) < J-P r o o f - 119 -(-0 T h e proof is by const ruc t ion of such a refutat ion. Since T is consistent , i f there is a re fu ta t ion of B f rom T, there is a re fu ta t ion w i t h top clause i n C L A U S E S (8). I.e., R o ^ C 0 A A K-k+l = D and R 0 e CLAUSES (B), C 0 € V. W e proceed by i nduc t ion on the steps i n the re fu ta t ion, b a s e Assume 8 is i n c lausa l fo rm, i.e., 8 = Br A ... A Bn and A = A,i V ... V A^ , for i = l,...,n . B y hypothesis, / (;->Ar) < J - W i t h o u t loss of general i ty, assume that R 0 . = A = A,i v ••• V A, n i l , that C 0 = C Q ! V ... V C0mo , and that C Q ^ resolves on A,i t o produce R x . T h u s C Q ! = ->Ai,i so / ( C 0 i l ) < j a n d / MIN ( CO ) < j • It fo l lows that C 0 € * . S ince for i > l , - .Co, ; « C Q I , / (-.Co,;) < / (Co, i ) < j . Thus , if R x = R M V ... V R l t then V s . / ( - ,R 1 < 8 ) < j . s t e p Assume tha t R ; . = R{1 V ... V R ^ , that V s . / ( - . R ^ J < j, and that V r < i . C R e * or C r £ { R ^ . - . R ^ J . Cons ider the resolut ion of R[ w i t h C ; . C ; = C ^ V ... V CUm. . W i t h o u t loss of genera l i ty , assume C ^ x =- iR^ x . Hence f ( 0 ^ ) = / ( - . R ^ J < j and so /MINCCO < j • So C ; e * or Q e {Ro,. . . ,Ri} . F o r r > l , / ( - C J < / ( C M ) < j . T h u s V s . / ( - R i + i , s ) < j . B y i nduc t ion , for every clause, C ; , in the re fu ta t ion of 8, C ; S or C ; is a descendent of ^ U {Al-T h u s , there is a l inear resolut ion refutat ion of 8 f rom "9. (H T r i v i a l : S ince $ C f , the re fu ta t ion f rom iff serves as a refutat ion f rom V. 120 -Q E D l e m m a 3 . 3 . 1 0 T h e o r e m 3 .3 — C o h e r e n c e If A = ( D , W ) is an ordered, semi -norma l default theory, then A has an extens ion. P r o o f If W is inconsis tent , then A has the t r i v i a l extension, L. Hence assume W is consistent. W e proceed by const ruc t ing an extension, E for A . F i rs t , let {D;} be a par t i t i on of D i nduced by / , as descr ibed i n L e m m a 3.3.7. R e c a l l that by Co ro l l a r y 3.3.8, i f 8 € D 0 then 8 is a no rma l defaul t , and tha t by Co ro l l a r y 3.3.9, for i > 0, D; must conta in at least one semi -norma l default , say 8 — , and / M A X ( - ' 7 ) < ' M T N ( ^ ) -W e now const ruct an extension for A . Le t A 0 = (Do, W ) . S ince A 0 is a n o r m a l default theory and W is consistent, A 0 has a consistent extens ion, say E° . F o r i > 0, const ruc t A ; as fo l lows: D ^ i ^ l \ 2-^- e D; V SLlIJSSL € DS , - . 7 £ E i _ 1 } A l = ( D ^ E - 1 ) Where E 1 _ 1 is an extens ion for A w . S ince each A ; is a no rma l default theory, each A ; has at least oo one ex tens ion, E \ L e t E = U E ' . S ince W is consistent, so is E°, by T h e o r e m 0.3. S ince E 1 is an i=0 extens ion for ( D / , E^1), E ' is consistent i f E 1 _ 1 is, and E 1 - 1 C E 1 . B y induc t ion E is consistent . W e oo now show tha t E is an extension for A . B y Theo rem 0.1, i t is sufficient to show that E = U F ; , i=0 - 1 2 1 -where F 0 = W , and for i >0 F i + 1 = T h ( F j ) U {w | SLLL e D , a G F ; , and -i/3 <£ E } . w oo (1) W e first show that 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 that F i + 1 C E . F i + 1 = T h ( F j U {0 | SLLiASL g D , a 6 F j , (-•/? V -vy) $ E } i) S ince F ; C E and E is log ica l ly c losed, T h ( F i ) C E . ii) Cons ide r 0 G {0 \ a ^ A 1 G D , a G F ; , {-.0 V S ince a G F j , a G E , and hence a G E 1 for some j S ince (-.0 V - . 7 ) £ E , -.7 £ E ' " 1 , so G D / . B u t - , /? £ E , so -. /? £ E* Therefore, since E j is an extension for A j = ( D / , E j *)) and a G E j , 0 G E j . Therefore j3 6 E . O O B y i nduc t i on , U F £ C E . 0 0 (2) F i n a l l y , we show that E C U F ; . i=0 A ) Cons ide r w G E° . E ° is an extens ion for A 0 , so by Theo rem 0.1 E ° = U G ; , where i=0 G 0 = W , and for i>0 a : w w G i + 1 = Th(G0 U {w I G D 0 , a G G ; , and - w £ E0 } . O O It therefore suffices to show that U Gx C U F t 1=0 i=0 0 0 a) G 0 = W = F 0 C U P , . i=0 00 b) A s s u m e G ; C U F; , and consider w G G i + 1 . i=0 G i + 1 = Th(G0 U {w j - 2 - L i i G D 0 , a G G ; , -w $ E 0 } 0 0 0 0 i) If w € T h ( G ; ) then w G U F ; by hypothesis since U F ; is log ica l ly c losed. i=0 i=0 i i) O therw ise w G {w I a W G D 0 , a G G : , ->UJ f£ E°>. w B u t : 1) If w G G i + 1 and E ° = U G j then u G E ° C E . i=0 Since E is consistent, ->w E . 00 2) If a G G j then a G U F j by hypothesis, so a G F k for some k. i=0 - 122 -3 ) D 0 C D O O T h u s w g F k + 1 C U F ; . i=0 oo oo B y induc t ion , U G ; C U F , . i=o i=0 B ) Assume E T 1 C U F , , a n d show E j C U F-,. i=0 i=0 oo Cons ide r w g E j . E J is an extension for A j = ( D / , E j _ 1 ) , so E J = U G s , where i=0 G 0 = H T 1 , and for i > 0 G i + 1 = Th (G j ) U {w | SLUL € D / , a € G ; , and $ E^} . w oo a) B y hypothesis , G 0 = E i _ 1 C U F ; . i=0 oo b) A s s u m e G ; C U F ; and consider w g G i + 1 . i=0 oo oo i) I f w g T h ( G i ) then w g U F ; by hypothesis since U F j is log ica l ly c losed. i=0 i=0 i i) O therw ise | — ' • — g D j ' , a g G ; , and ->u £ E*} . CO oo Since a g G j , we know that a g E J and a g U F j . A l so , if w g G i + 1 then w g E 1 i=o so w 6 E . Therefore -icu ^ E , since E is consistent. If 6 = g Dj , then ei ther g D or d 7 - ! - ! - L € D . Thus there w w to are two cases: a • co °° °° a) E i t h e r — : — g D, a g U F ; , and -iw £ E and hence w g U F ; , CO i=0 i=0 oo b) O r Q : " A 7 g D, a g U F ; , and £ E . w i=o oo Clea r l y , i f (-.7 V -.w) £ E then w g U F ; . i=0 Since co g E , i t can be shown that (-17 V ->w) g E iff -17 g E . W e show that -17 ^ E . C l e a r l y lyucdrl) < ' M I N M = j - Assume -.7 g E . T h e n 3r>j . (-.7 g E r ) . B y L e m m a 3 . 3 . 1 , ( W U U CONSEQUENTS ( G D ( E ; , A J ) ) j - -7. i=0 T h u s there is a l inear resolut ion refutat ion of 7 f rom T = ( W U U CONSEQUENTS(GD(E\ A O ) ) . i=0 Observe that i f 6 g G D ( E ' , A ; ) then 6 g D j ' and ' M I N (CONSEQ UENTS (6)) = i . B y L e m m a 3 . 3 . 1 0 , the existence of a refuta-t ion of 7 f rom T, g iven / M A X ( - , 7 ) < J> impl ies that there is a re fu ta t ion f rom so - 1 2 3 -$ C T such that ip € W *-* /MIN (^) < J- T h u s there is a re fu ta t ion f rom # = ( W U U CONSEQUENTS(GD(E\ A ; ) ) ) . i=0 Hence * j — -.7 a n d , by L e m m a 3.3.1, * | .7 iff E i _ 1 ] .7. B u t if 6 e D / then -.7 £ E i _ 1 and so E 1 " 1 - 7 since E J 1 is log ica l ly c losed. Hence we ob ta in a con t rad ic t ion by assuming that -17 € E , so -17 £ E . 00 T h u s (-.7 V ->w) £ E and so w e U F ; . j=o 00 00 00 W e see that G i + 1 C U F ( , a n d by induc t ion U G ; C U F ; . i=0 i=0 i=0 Therefore E j C U F ; . i=0 00 B y i nduc t i on , E C U F s . i=0 oo Together , (1) and (2) show that E = U F j , so E is an extension for A . i=0 Q E D T h e o r e m 3 .3 Before present ing the proof of T h e o r e m 3.4, we repeat the def in i t ion of the procedure to gen-erate extensions g iven earl ier. Superscr ip ts have been added wh ich serve only as reference points i n the proofs. T h e y do not effect the computa t ion . - 124 -H 0 - W ; j ^ O ; r e p e a t j « - j + 1; h 0 - W ; G D 0 - { }; i « - 0; r e p e a t V> _ { !L±A. G D | (n; j _ a)t ( h i ^ ( H H I/- -,/J) }; i f - 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 / U {8}; h ^ i « - h / U { C O N S E Q U E N T ^ } } ; e n d i f ; i « - i + 1; u n t i l n u l l p j . ! - G D ^ ) ; Hj = h k u n t i l Hj = H j ^ L e m m a 3.4.1 If A is a finite defaul t theory, then the a lgo r i thm can fa i l to converge on ly if one of the app rox ima t ions is repeated. I.e., for some j and some k > j+1 , Hj = H k . P r o o f If A is finite, there are on ly a finite number of different combinat ions possible. T h u s there are on ly a finite number of d is t inc t H^s w h i c h can be constructed. If H j = H j + 1 , the a lgo r i thm con-verges. Q E D L e m m a 3.4.1 L e m m a 3.4.2 If A is a finite, semi -no rma l defaul t theory, and W is consistent, then ^ \- 0 - ^ \f- ->p. P r o o f A s s u m e H ; ]— 0, —>p. L e t r, s be the smal lest integers such that h*)— P, hg11 >p. Assume r < s, - 125 -so \f- -iff. B y hypothes is , h,1 j — 0, ->0. N o w h 8 ' = h ^ U {w}, where a : w A ^ e D , a e hU, H j - i y- V - , 7 ) , and h ^ \f- V -,7) . B u t i f hs* j — A -./9, then ( h ^ U {w}) )— 0, ->0 so h ^ ) .w and hence h ^ j — (-<u V -.7), wh i ch is a con t rad ic t ion . T h e proof is s im i la r i f s < r. Q E D L e m m a 3 .4 .2 D e f i n i t i o n 3 . 4 . 3 : N e t w o r k D e f a u l t T h e o r y A defaul t theory , A = (D , W), is a network theory i f i t satisfies the fo l lowing condi t ions: (1) W conta ins only : a) L i te ra l s (t.e., A t o m i c formulae or thei r negat ions), or b) D is junc ts of the fo rm ( a V 0) where a and 0 are l i tera ls . (2) D conta ins on ly n o r m a l and semi -norma l defaults of the form: a:0 a : 0 A 7i A ... A 7n —^— or 7. 0 0 where a, 0, and 7; are l i terals. | L e m m a 3 .4 .4 If A is a finite, ordered, network defaul t theory, if W is consistent, and if 0 is a l i te ra l , then f - 0 -+ H; \j-P r o o f 00 A s s u m e Hj_i j — 0, and consider Hj = U h}. Assume Hj )— —>0. T h e proof proceeds by induc t ion . i=o b a s e h 0 = W. S ince H ^ j / - c lear ly W \f- -i£ Therefore h 0 \f- ->0. s t e p Assume hj 1 \f- ->0 and h ^ ] <0. hf+1 = h} U {w}, where " : i A w e D , W \- a, hi V- ( - 7 V - w ) , and H H \f- ( -7 V - w ) . - 126 -C l e a r l y , co j=- ->B or else H ^ | <co. Note that : i) H j conta ins on ly d is junct ions of two l i terals. i i) hJ = W U CONSEQUENTS {GD}) i i i) GD{ Q D iv) CONSEQUENTS(GD>) C Literals. Cons ide r a l inear reso lu t ion re fu ta t ion of B (i.e., a proof of ->B) f rom h ^ i , w i t h top clause 8. W e cont inue b y i nduc t i on on the structure of this refutat ion. B / C 0 D . A b a s e co G Literals and co ^B so C 0 £ co. C lea r l y , C 0 £ 8. T h u s C 0 € h s j . If C 0 6 W - W , then C 0 € Literals. B u t then C 0 = ~<B w h i c h leads to the cont rad ic t ion that h} )—->B. T h u s C 0 € W . C l e a r l y C 0 ^ Literals, as above. Hence C 0 = ( _ , / 3 V ^ ) , w i t h £ g Literals. T h u s R i = £ £ D. step Assume : i) w £ { C 0 C ^ } ii) { C » , . , C J C W i i i ) {Rx R n } C Literals. Le t R n = fj € Literals. If C „ = w then w = - . I J so W U {w} \— ->B but W C H H and H j ^ |— ft so H j ^ j — ft -ift wh i ch cont rad ic ts L e m m a 3.4.2. C l e a r l y r] j= ->B, so C n /= 6, or else W |— -1/9 w h i c h is false. T h u s C n g W . C lea r l y C n ^ Literals, as above, hence C n = (-"7 V A) w i t h A e Literals. Therefore R n + 1 = A j= D-So: i) u £ { C 0 , . . . , C n } ii) { C 0 , . . . , C n } C W ii i) { R i Rn+i} ^ Literals . B y i nduc t ion , there is no such resolut ion refutat ion and the requi red result is p roved . Q E D L e m m a 3 .4 .4 - 127 -L e m m a 3 .4 .5 If A is a f ini te, ordered, network defaul t theory, and {o^ , . . . ,a „ } C Literals , then H ; \— (o^ V ... V a j i f and on ly if W f— ( a 1 V ... V c t j or H;)— ctj , for some j . P r o o f (<-) T r i v i a l . (—•) A s s u m e false, and consider a l inear resolut ion proof of ( « ! V ... V a j (i.e., a refuta t ion of (->cti A ••• A ""O n^)) f r om H j , w i t h top clause R 0 € {->ai,...,->an}. Ro C 0 Q W e know that C 0 £ H i U {-ia1)...,->an}, and that , for i>0, C ; € H i or C ; e {Rj | j< i } or C ; € {-ia1,...,-ian}. W e proceed by induc t ion . b a s e W i t h o u t loss of general i ty , assume R 0 = - I O ^ . C lea r l y a j ^  {-^a1,...,-'aI}, or else W {j— (ax V ... V a j , so C 0 {-!«!,...,,-ia,,}. C lea r l y C 0 £ aj, or else H j f— a x wh i ch cont rad ic ts our assumpt ion . Hence C 0 = (a t V 7) € W , for some 7 € Literals, a n d so R x = 7 f=- Q. s t e p Assume a) { R 0 R „ } C Literals b) { C 0 C ^ } C W . L e t R n = rj e Literals. If C n = ->fj e {-•a1,...,->an} then W j — (a^ V ... V a j w h i c h cont rad ic ts our hypothesis . I f C B=->i)eHiU { R 0 ) . . . , R n } then H ; j— a x wh i ch also cont rad ic ts the hypothesis . Hence C n = (->rj V £) € W , w i t h £ €E Literals and R n + 1 = £ ^ = Q. B y i nduc t i on , there is no such resolut ion refutat ion, and the lemma is proved. Q E D L e m m a 3 .4 .5 - 128 -L e m m a 3 .4 .6 If A is a finite, ordered, network default theory, and a G Literals, then Hi }— a if and on ly i f W \- a or 33 £ L i te ra ls . I (0) < I (a), j ? 6 H ; , and W ) - ( / 3 3 a ) . Proof (<-) T r i v i a l . (—•) A s s u m e false and consider a l inear resolut ion proof of a (i.e., a re fu ta t ion of -io:) f rom H i , w i t h top clause ->a. W e proceed by induc t ion . base C l e a r l y CQJ= a or else a G H ; and / (a) < / (a) and W \— (a 3 a) wh i ch cont rad ic ts the hypothesis. Hence C 0 = (a V 7) G W , for 7 G Literals. B y def in i t ion, / (-17) < / (a ) . R x = 7 £ • . C l e a r l y W |— (-.7 3 a ) . s t e p A s s u m e : a) { C Q J . - J C ^ } C W b) {Ro,...,R„} C Literals c) 1 ( ^ R J < 1(a) d) W j - ( - . R n D a ) L e t R„ = r?. If C N = ->tj G H ; then H ; {-a, -itj G H;, W j — (-.77 3 a ) , and / (-"^J = / (—Rn) — ' (<*) w h i c h cont rad ic ts our assumpt ion. If C N = ->rj = —*a then W J— a w h i c h is also a con t rad ic t ion . Hence C N = (->r] V £) G W , w i t h £ G Literals, R n + i = £ Q , and / ( - i R ^ ) = / (-.£) < / (-.77) = / ( - R J < 1(a). B y modus ponens, W f - (-.£ 3 a ) . T h u s there is no such re fu ta t ion , and the result is p roved. Q E D L e m m a 3 .4 .6 - 129 -Lemma 3.4.7 If A is a f ini te, ordered, network default theory, and a G L i te ra ls , a £ H , , a ^ Hj , and a G H k for i < k < j , then 30 e L i te ra ls . (/ (0) < I (a)) and 0 6 U H ; A H r . i<r<j Proof Let j be the least j > k such that a £ H j . Def ine D a = {8 G D | 8= 1 " A "* A - A } a oo GDi = U G D r ' n D a r=0 C l e a r l y GDi""1 j= { } and GD£ = { }. Cons ider 8 G GDJ"1. Since 5 £ GD£ three cases are possi-ble: 1 ) Hj_j | — (—Wx V ... V n u j . B y L e m m a 3 . 4 . 5 , there is an w r , say w, such that H j_ x j iw. B y L e m m a 3 . 4 . 6 . there is a 0 G H j_ x such that 1(0) < I (w) and W f - (0 D w). B u t then I (0) < I (a). C lea r l y 0 £ Hj_ 2 , so 0 is the required l i tera l . 2 ) H j j — (-"Wi V ... V -<u^. T h e argument for case 1 appl ies. 3 ) Hj \f~ 7 . B y recurs ive ly app l y ing the foregoing arguments to 7 , we can construct a set of 7 r ' s w h i c h were i n H j_ x and are not in Hj . T h e first of these to go into H j_ x mus t also go in to H j , unless H j ^ U Hj conta ins a 0 <SC 7 r a w h i c h was not i n H ; . QED Lemma 3.4.7 Lemma 3.4.8 If A is a finite, ordered, network default theory, and a G L i te ra ls , a G H ; , a 6 Hj , and a ^ H k for i < k < j , then e i ther 1 ) 30 G L i t e ra l s . (/ (0) < I (a)) and 0 G U H ; A H r , or i<r<j 2 ) 30 G L i te ra ls . (/ (0) < I (a)) a n d 0 G H j and 0 £ H ; . Proof Le t k be the least k > i such that a ^ H k . L e t j be the least j > k such that a G H j . - 130 -Cons ide r 6 = 7 : a A ^ G G D * . C l e a r l y G D i jfe { }, and <S 0 G D * . Cases: 1) H k ) 'ft H j \J— ->0. T h i s gives the first of the required condi t ions, by L e m m a s 3.4.5 a n d 3.4.6. 2) H k _ i ) — -i0, H j (/- -<0. T h e argument for case 1 appl ies. 3) H k \f- % H j j — 7 . B y L e m m a 3.4.6, 3 7 l « a . 7 l G H j , 7 l £ H k . Cases : a) 7 X ^ H ; . T h i s is the second of the requi red cond i t ions . b) 7 ! G H j . Repea t i ng the above arguments for 7 ^ y ie lds a (possibly cyc l ic) cha in of 7 r ' s such that 7 r G H k _ ! , 7 r £ H k . Cons ide r the first 7 r to go into H k _ i . It mus t also go into H k , w h i c h is a con t rad ic t ion . QED Lemma 3.4.8 Theorem 3.4 — Convergence T h e procedure presented above a lways converges when appl ied to a finite, ordered, network default theory. Proof B y L e m m a 3.4.1, non-convergence impl ies there is a cyc le. I.e., for some i and some j > i , H ; = Hj and H ; j= H i + 1 . Choose a e U ( H i A H j such that a G Literals and for every 0 G U ( H i A H J , ->(/ (0) < I (a)). »<k<j i<k<j T h u s a is the " l e a s t " l i t e ra l to change state between H i and Hj . There are two cases: (1) If a ^ H ; a n d a G H k then, by L e m m a 3.4.7, 30 G U ( H i A H k ) . I (0) < I (a), so a is not i<k<j the least such a, w h i c h is a cont rad ic t ion . (2) If a G H i and a £ H k then, by L e m m a 3.4.8, ei ther a) 30 G U ( H i A H J . / (0) < I (a) i<k<j so a is not the least such a, w h i c h is a cont rad ic t ion , or b) 30. 0G H j <md0(E H j w h i c h imp l ies that H i j=- Hj w h i c h is also a cont rad ic t ion. Therefore, there is no cyc le , and so the procedure converges. - 1 3 1 -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 procedure g iven above a lways converges immed ia te ly when app l ied to a finite, n o r m a l defaul t theory A = (D , W ) - i.e., T h ( H i ) is an extension. P r o o f E t h e r i n g t o n [1982] shows that = H 2 i f and on ly i f T h f i y is an extension for A . If W is incon-sistent, then Th (H ] ) = L w h i c h is an extension for A . Hence assume W is consistent. T o show oo tha t T h ( H i ) is an extens ion for A , we invoke Theo rem 0.1 and show tha t T h ( H i ) = U E j , where i=0 E 0 = W E , + 1 = T h ( E 0 U {w | £±2- € D , a € E i , ^ T h ( H i ) } . O O O O a) W e first show tha t U E ; C T h ( H i ) . R e c a l l that Ej_ = U h^. i=0 i=0 b a s e C l e a r l y E 0 = W = h^ Q T h ( H 1 ) . s t e p . Assume E ; C T h ( H 1 ) and consider w S E i + 1 . i) If w e T h ( E i ) then w € T h ( H i ) , by hypothesis and closure. ii) Otherwise w € {w | " = " 6 D , a £ E ; , -w (£ T h f f i J } . Therefore Uiy~-<w. Hence w H 0 \f—>OJ since H 0 = W C H i . A l s o , a € E ; , so a € T h ( H 1 ) , by hypothesis. It fol lows by [E ther ing ton 1982, L e m m a 3.3] that H i ]— w. Hence E i + X C T h ( H i ) . O O b) F i n a l l y , we show tha t T h ( H i ) C U E r . r= l oo oo Since U E r is log ica l ly c losed, i t suffices to show that H x C U E r . r = l r=l b a s e C l e a r l y h^ = W = E 0 C U E r . s t e p oo Assume tha t h ; 1 C U E r , and consider hj+i . - 1 3 2 -= h; 1 U {w}, for some u G CONSEQUENTS (D; 1 ) . oo oo Since h j 1 C U E r by hypothesis, we need on ly show that w € U E r . i = l r=l Since u e CONSEQUENTS (Bi1), for some 6 = g D, a <5 V, a; H 0 — i w , and h * j / - ->w. oo B y hypothes is , s ince a 6 t ' , a € U E r , so a € E j for some j . r=l Since w € h ; ! ^ C Ku i t fol lows by L e m m a 3.4.2 that Hx ]/—>w. oo B u t then by def in i t ion of E j + 1 , w 6 E j + 1 C U E r . =i C o m b i n i n g (a) and (b), we have the desired resul t . Q E D T h e o r e m 3 .5 - 1 3 3 -T h e o r e m 4 .1 A n y ne twork i n w h i c h the subgraph of I S - A l inks a n d except ions thereto is acyc l i c corresponds to an ordered theory. P r o o f T h e l i nks cor responding to a D —>B, a : -JB a n d a: -<B A ~>li A - A ->7i give rise to a ^ -<B and 7,- <K —>B. The re are no l inks w h i c h make a t rans i t ion f rom negat ive to posi t ive or negative to negat ive, so such l inks cannot par t ic ipate i n any cycle leading to w « u for any w. W h a t remains are I S - A l inks and except ions thereto. Q E D T h e o r e m 4 .1 T h e o r e m 4 . 5 In the absence of no-conclus ion l inks , a l l ground facts returned by Tou re t zky ' s in ferent ia l d is tance a l go r i t hm l ie w i t h i n a single extension of the default theory corresponding to the inher i tance ne twork i n quest ion. W e prove that a l l the ground facts i n a n y " g r o u n d e d expans ion" of the network l ie w i t h i n a single extens ion. F r o m th is the result fo l lows. A s a no ta t iona l shortcut , we w i l l use dtP to s tand for +P or -P (or, occas iona l l y , for P or ~<P). T h e in tended meaning should be clear f rom context . Le t F be a ne twork i n T o u r e t z k y ' s sense. L e t $ be a grounded expansion for F . Define facts($) = { < + a , ± P > € C(4>) [ a is an i n d i v i d u a l token} , and facts'{$) - {Pa | < + a , + P > e facts($)} U {->Pa | <+a,-P> S facts{$)}. If <+a,±P> S facts($) then for some Pv...,Pn, we have <+a,+Pi,...,+Pn ,±P> G by def in i t ion. Hence , by [Toure tzky 1984a, theorem 2.3], <+a,+Pi> <+a,+Pn>, <+a,±P> e facts($). T h u s P x a Pna, ±Pa € facts'($). Fu r thermore , < + P , - , + P , + 1 > € $ for t = l , . . . , n - l , and <+Pa±P> e by [Toure tzky 1984a, theorem 2.3]. Hence they are a l l i n Pfl : Pi+1x P^c : ±Px T by T o u r e t z k y 1984a, theorem 2.2 . Hence — and € D. Pi+iX ±Px W e c l a i m tha t facts'($) is inconsistent iff W is. B y def in i t ion, W — {±Ra \ <+a,±R> ET, where a is an i n d i v i d u a l token} . Therefore, W is inconsistent iff <-ra,+R>, <+a,-R> G T, for some a a n d T. P r o o f - 134 -T h e right-to-left d i rec t ion of the c l a i m is t r i v ia l . F o r the left-to-r ight d i rec t ion, assume that facts'($) is inconsistent . T h e n Ra, -*Rct G facts'($) so <+a,+R>, <+a,-R> G facts($), so ay = <+a , j / 1 , . . . ,yy ,+R> and cr 2 = <+a,xll...,xk ,-R> G So $ cont rad ic ts and cr2 , and <& is inconsistent . Hence T is inconsistent , by [Toure tzky 1984a, theorem 2.8]. Fu r the rmore , nei ther o~i nor cr 2 is inher i tab le i n so bo th are i n T, since $ is a grounded expansion of T. B u t then /= k = 0, so < + a , + P > a n d < + a , - P > G I\ Hence, P a , — P a G W, so W inconsistent . N o w i f facts'(<&) inconsistent , W is inconsistent , so A has a unique extension, Th(L) 3 facts'(<&). In the sequel, we assume facts'(<&) consistent. W e show that E' = Th(facts'($)) is an extension for A ' = ( I > ' , W), where , f P,<x : ±Pi+1a . \ D = < <+a,+Pi,...,dcPk> G 1 < t<A:>. T h e n , by the semi -monoton ic i t y of n o r m a l defaul t theories, there w i l l be an extension, ED E' for A , since D' C CLOSED-DEFAULTS(A) [Rei ter 1980a, theorem 3.2]. O O A s usua l , we show that E = U Et. £=0 oo E' D U E{: Cons ide r w = ±Ra G E0= W = {±Ra \ <+a,±R> G T}. T h e n < + a , ± P > G F C <£, so < + a , ± P > G /acfcs(<3>), so ±Ra G facts'($). F o r the induc t i ve step, assume Ej C E', and consider tii G If tii G Th(Et), then tii G i ? ' . Otherwise, Pfic : Pi+ia. . . w G {Pi+ict | 5 = G D , P , a G Eh and - P i + i a ^ £ }. Since 8 G D , we have P ^ - i " < + a , + P 1 , . . . , ± P J t > G for some k > Hence < + a , + P , > , < + a , ± P , + 1 > G <5, since $ is a grounded expans ion . So P , a , ± P , + 1 a G /acis'(<5). oo C U E{: Cons ide r ±Ra G facts'($). T h e n <+a,+Rll...,+Rp±R> G $ . B y [Toure tzky 1984a, theorem 2.3], < + a , + P 1 > < + a , + P y > , <+a,±R> G facts(§), so Rxa Rfc, ±Ra G facts'($). If < + a , + i 2 i > G $ , then < + a , + i ? ! > G F , by [Touretzky C O 1984a, theorems 2.3, 2.2], so Rya G W C U E,. F O F the induct ive step, assume t=0 oo °° Rka : Ru+ia . Rycx R/fic G U E{ , for k<j. W e show that P ^ a G U E{. N o w 5 = — — G D . i=0 £=0 Rk+la oo Since P * a G U Ej, P ^ a G i?,-, for some t. Since <+a,+Ru...,+Rk+x> G t=0 < + a , P f c + 1 > G C ( $ ) so < + a , P A H . 1 > G / ac t s ($ ) , so R^a G facts'($). B y the consis tency of T, oo E' \f—'P^a, so Pjfe+ia G ^ - i . S o P ^ a G U for 1 < k < n, by induc t ion . S i m i l a r l y for fc=o ±Ra. oo T h u s E' = U 2?,. So £ ' is an extension for A ' , by Theo rem 0.1. «=o - 135 -QED Theorem 4.5 - 136 -T h e proof of T h e o r e m 5.1 fol lows immed ia te l y f rom M c C a r t h y ' s proof of the soundness of predicate c i r cumsc r ip t i on and the def in i t ion of wel l- foundedness. T h e o r e m 5 .2 U n i v e r s a l theories are wel l - founded. P r o o f T h e proof is i den t i ca l to tha t of P rope r t y 1.3.2 i n [Bossu and Seige l 1985]. T h e def in i t ion of sub-mode l used there is less restr ic t ive than that used here, but this does not al ter the fo rm 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 we l l - founded theory, ai,...,ak are n-tuples of ground terms, and P s P , is an n-ary pred icate , then CLOSUREP(T) \r- PSj. V . . . V P3k <==> T \- PS* V . . .V Pak . P r o o f T h e r ight- to- lef t d i rec t i on is immed ia te . W e prove the contraposi t ive of the left- to-r ight d i rec-k k t ion. Assume tha t CLOSUREp(T) ( r - V P a , - and T \f- V P a , - . T h e n T has a mode l , Af, in w h i c h P a , - is false, for al l 1 t = l,...,k. S ince T is wel l - founded, there is a P - m i n i m a l submode l , A f ' , of Af: Fu r the rmore , since the c i r cumscr ip t i on is true i n al l 1 P - m i n i m a l submodels, P a , - is true i n A f ' , for some 1 < i < k. B u t then A f is not a P - s u b m o d e l of Af; and this cont rad ic ts the fact that A f ' is a P - m i n i m a l submode l of M Therefore CLOSUREP{T) \f- Pal V . . . V Pak . 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 we l l - founded theory, al,...,ak are n-tuples of ground terms, and P ^ P is an n-ary predicate, then (0 CLOSURE^T) (— P a ! V . . . V PSk T\-PS1 V . . . V P a * , and - 137 -(ii) CLOSUREp{T) ( - - .Pc? ! V...V -,PSk <==> T \- -^PS^ V...V -*P3k . P r o o f (i) T h e right-to-left d i rec t ion is immedia te . W e prove the contraposi t ive of the left-to-right k di rec t ion . A s s u m e T \f- V P a , - . T h e n there is a model , Af, for T i n w h i c h Pc?,- is false, for a l l *=1,...,k. S ince T is well-founded, there is a P - m i n i m a l submodel , A f ' , of M. B y the defini t ion of submodel , the in te rpre ta t ion of P remains the same i n A f and A f ' , since P ^ P . Hence Pa , - is false i n A f ' , for a l l t=l,. . . ,fc . Since the c i r cumsc r ip t ion 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 proof for (ii) is s imi lar . Q E D T h e o r e m 5.5 In the proofs of Theorems 5.6 and 5.7 we use the fol lowing no ta t iona l conventions: 1. SCHEMA(T,T?) is the c i r cumsc r ip t ion schema result ing from c i rcumscr ib ing the predicates of P i n T. 2. CLOSURE{y(7) = T. (The closure of T w i t h respect to the empty set of predicates is defined to be T itself.) o 3. If A f is a model , A is true i n Af. (The empty conjunct ion is vacuously true i n a l l models.) T h e o r e m 5.6 (Rei ter) If T is an a rb i t ra ry , finitely-axiomatized theory conta in ing axioms wh ich define the equa l i ty predicate, = , then T |— CLOSURE^=.}(T). P r o o f Cons ider the schema resul t ing from c i rcumscr ib ing ' = ' 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 that \— ( V z . #zz) D ( V z y . I = J D * z y ) for any predicate letter, \P. Fur thermore , V z . ^ z z is one of the conjuncts of T ( $ ) i n SCHEMA(T,{'=*}) since V z . z = z must be an a x i o m of any theory w i t h equa l i ty . T h u s i f any instance of T ( $ ) is true i n a mode l of T, so is the corresponding instance of V z y . z = y D $ z y . Hence, every instance of SCHEMA(T,{'='}) is true i n every m o d e l of T, so T |— CLOSURE^iT). Q E D T h e o r e m 5.6 - 138 -T h e o r e m 5.7 If T is a we l l - founded theory con ta in ing ax ioms wh ich define the equa l i ty predicate; and a, ~$ are tuples of ground terms; then (i) CLOSURE^T) ( - 5? = <=> T\-S = ^, and (ii) CLOSURE^T)\-c? £0* T \-~a j=0*. P r o o f (i) T h i s is a coro l la ry of Theorems 5.4 a n d 5.5(i). (ii) T h e r ight- to- lef t d i rec t ion is immed ia te . T o prove the left-to-r ight d i rec t ion , we consider the compos i t ion of P . If does not occur in P , the result fol lows d i rec t ly f rom T h e o r e m 5.5(i i). If P = {=} , the result fo l lows f rom T h e o r e m 5.6. F i n a l l y , consider P = P ' U {'—'), for an a rb i -t ra ry set of predicates P ' = {Pi,...,P^ not i nc lud ing equal i ty . B y T h e o r e m 5.5(H), CLOSUREp'(T) | - S j t ^ < r = > T\-cl£~$ W e show that CLOSUREp(T) \-aj=~$ <==> T\-ct j=~$ . W e have SCHEMA( T" ,P ' ) 3t*i , - ,*J A (.A (V*. 3 P^)j n D A (Vx*. P? D <J>,x) SCHEMA(T,P) = J 2 X * i » — A (,A(Vz. A (Vxy. ¥xy D x = y)j D ^ (Vx. PflZ) $p) A Vxy. x = y D *xy Assume CLOSUREp( T) |— a £ /2> and T \f-~a j=~$. It fol lows that CLOSUREpi( T) \f- a £ % A n y mode l of T i n wh i ch every instance of SCHEMA(T,P) is true is also a mode l for CLOSUREp(T). Hence a j= ~$ is true i n that mode l . Fur thermore , there is some mode l of T in w h i c h every instance of SCHEMA(T,P') is true and a J= 0* is false. W e show that in every mode l of T i n w h i c h every instance of SCHEMA(T,P') is true, every instance of SCHEMA(T,P) is also t rue. F i r s t observe that |— (Vx. ^xx) 3 (Vxy. x = y D tyxy) for any predicate letter, Fu r the rmore , Vx. *xx is one of the conjuncts of T(<& 1 , . . . ,$ n , * ) i n SCHEMA(T,P). T h u s i f any instance of T[$i,...,$n jty) is t rue i n a mode l of T , so is the corresponding instance of Vxy. x = y D Vxy. L e t A f b e a mode l of T where every instance of SCHEMA(T,P') is true. C o n -sider an ins tance, 7, of SCHEMA(T,P), w i t h the predicates and subst i tu ted for and \P , respect ive ly . The re are two cases: n 1) A (Vx*. P&Z) $,'x) is t rue i n M. B y the observat ion above, ei ther r($ x ' , . . . ,$ „ ' , * ' ) is false i n M o r Vxy. x = y D *'xy is true. In e i ther case, lis true in M. n 2) A (Vx. P jX D $,'x) is false i n M B u t then T ( $ 1 ' , . . . , $ n ' ) is false or - 1 3 9 -A (Vx*. $,'x*D Pjiz) is false, since every instance of SCHEMA(T,P') is true in M. In t=i the la t ter case / is also true in M. In the former case, if [ r ( $ i ' , , A Vxy. ty'xy Z> x = y] is false i n M , then / is true. Otherwise , by the observa t ion above, Vxy. x = j D ^ 'xy is true and, hence, so is Vxy. x — y = \P 'xy. B u t T ( $ 1 ' , . . . , $ I t ' , 1 I r ' ) is * n e result of subst i tu t ing # ' for some of the occurrences of ' = ' in T ( $ ! ' , . . . , $ „ ' ) , so T ( 4 > i ' , . . . , $ „ ' , * ' ) is false, because ' , . . . , $ „ ' ) is, and th is is a con-t rad ic t i on . T h u s , for every m o d e l of T, i f SCHEMA(T,T') is true, so is SCHEMA(T,P). B u t then a j= p* is t rue in every mode l of CLOSUREp>[T). Hence CLOSUREpi(T) |— a /= p\ w h i c h is a cont rad ic-t ion , since T \/-a £p*. W e conc lude that CLOSUREp(T) \f-ct j=~$. Q E D T h e o r e m 5 .7 C o r o l l a r y 5 .8 If T is a we l l - founded theory conta in ing ax ioms wh ich define the equa l i ty predicate, P is an n-ary pred icate , and a is an n-tuple of g round terms, then CLOSUREp( T) |— -*Pa impl ies T (— a i= ~j$ for a l l g round n-tuples such that T (— P/3. P r o o f Otherw ise CLOSUREp(T) |— a f /3 and T \f-~a j=~$ wh i ch contradic ts Theo rem 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 we l l - founded theory; a j a „ are ground terms; and P is a set of some of the predicate symbo ls of T\ then CLOSUREp{T) \- Vx. x = a x V...V x = «„<==>• T\-Vx.x=al V...V x = a „ . P r o o f T h e r ight- to- lef t d i rec t ion is immed ia te . F o r the left-to-r ight d i rec t ion , assume that T\f- Vx.x = a x V...V x = a „ . T h e n T has a mode l w h i c h falsifies Vx.x = ax V...V x = an . S ince T is we l l - founded, th is mode l has a P - m i n i m a l s u b m o d e l B u t Vx. x = ct^ V...V x = an is false i n this submode l , because the extension of the equa l i ty predicate in this submode l mus t be a subset of i ts extens ion i n the o r ig ina l mode l . S ince the c i rcumscr ip t ion is true i n a l l m i n i m a l models, - 1 4 0 -CLOSUREp(T) \/- Vi.i= Q lV...Vi=a n. Q E D T h e o r e m 5 .9 T h e o r e m 5 . 1 0 If T is a wel l - founded theory, and T has a mode l w i t h some doma in , D, then so does CLOSURE^ T). P r o o f M c C a r t h y [1980] shows that CLOSUREp(T) is true in a l l m i n i m a l models. S ince T is we l l -founded, every m o d e l has a m i n i m a l submode l . B y the def in i t ion of submodel , the d o m a i n of a m i n i m a l submode l of M is the same as that of M. Q E D T h e o r e m 5 . 1 0 T h e o r e m 5 .11 If T |— Vx*. Px = 3>af for some expression $5*, not invo lv ing predicate letters f rom P , then T \- CLOSUREP(T)'. P r o o f Tfi!), on the le f t -hand side of the c i rcumscr ip t ion schema, inc ludes Vx*. ^ 5 * = <I>z*. B u t any choice of mode l , M, a n d predicate, ^ , w h i c h satisfies the L H S clear ly a l ready satisfies the R H S , Vx. F x * D ^x*, since every mode l of T satisfies Vx*. $ 5 * = P i * . Q E D T h e o r e m 5 .11 - 1 4 1 -D e f i n i 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 T h e circumscription of the formula E(P,x) in the theory T, w i t h the predicates P t reated as var iab le , is g iven by: T ( P ) A V $ . T ( $ ) A | W . E{&,x) D E{P,x)} D [ V ? . E{P,x) 3 E{$,x)} D e f i n i t i o n : M < ^-^M1 Le t T ( P ) be a finitely-axiomatized (first- or second-order) theory, some (but not neces-sar i ly al l) of whose predicates are those in P ; let E[P,x) be a fo rmu la whose free var iab les are among x *= xv...,xn, and i n w h i c h some of the predicate var iab les P = { P 1 , . . . , P n } occur free; a n d let M , M ' be models of T. W e say M i s an E(P,x^-submodel of M ' (wri t-ten M < ^ p ^ M ' ) iff (i) | M | = | M ' | , (ii) If t is a t e rm, then |<|JI/= I'IA/ I (ii i) If Q P is a predicate let ter of T, then |<2|A/= \Q\IJ , and (iv) \E(P,?)\MC | £ ( P , x * j | ^ . I D e f i n i t i o n : 2 ? ( P , 2 ^ - M i n i m a l M o d e l A mode l , M , of T i s £ ( P , i ) - m m t ' m a / iff T h a s no model , M1, such that M ' < ^ P ^ M and - ( M < ^ M ' ) . I 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 sat isf ied by every £ { P , z > m i n i m a l mode l of T. P r o o f T h e proof fol lows M c C a r t h y ' s [1980] proof of the soundness of predicate c i rcumscr ip t ion . C o n -sider a m i n i m a l mode l , M , and an ins tan t ia t ion , w i t h some predicate, of the schema (or second-order ax iom) w h i c h makes the le f t -hand side true and the R H S false. T h e n by the second conjunct of the L H S , | £ ( P , I ) | J ^ C [ £ ( $ , 1 ^ 1 ^ B u t then a proper submodel , M ' , cou ld be con-s t ructed b y le t t ing P agree w i t h B u t th is cont rad ic ts the fact that M i s m i n i m a l . Q E D T h e o r e m 6.1 - 142 -T h e o r e m 6 .3 T h e ab i l i t y to m i n i m i z e arb i t ra ry expressions, E(P,x), instead of s imple sets of p red i -cates, is an inessent ia l extens ion, p rov ided predicates other than those being m in im i zed are a l lowed to va ry . P r o o f W e show that the theory, T, can be extended by adding a new predicate symbo l , St, and the def in i t ion Vx*. ^ 5 * = E[P,x), and that c i rcumscr ib ing \P in the extended theory, T', w i t h P va r i -able is equiva lent to c i r cumscr ib ing E{P,~x) in the o r ig ina l theory. I.e., that T A A [Vi*. £ ( * , 2 ) 3 -E(P,x}]j 3 [Vx*. E ( P , x j D £($,xj] (27) and T' A [ T(*,rj>) A [Vx. rjix = £($,?)] A [Vx. tfx[ D [Vx*. tfx D V i ] (28) are equiva lent over the language of T. T o see that (27) enta i ls (28), let M be a mode l w h i c h satisfies (27). Since (27) does not ment ion we can in terpret * as we choose. Therefore, let | ¥ | M = | £ ( P ) | M . C l e a r l y , M J = (28). C o n -versely, let M sat isfy (28), and let be a tuple of predicate var iab les sat isfying the L H S of (28). C lea r l y , 7*' f— T, and T'($) \— T ( $ ) . B y subst i tu t ion of equivalents, we get the rest of (27), so M\= (27). Q E D T h e o r e m 6.3 D e f i n i t i o n : 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 Le t X be a tup le of predicate, funct ion, a n d / o r constant symbols, and let R be a b inary re la t ion on tuples of type X . T h e generalized circumscription of X in the theory, T, accord ing to the pre-order, < R , i nduced by R is g iven by: r ( x ) A V x ' . r ( x ' ) A ( x ' < * x ) D ( x < j pc ' ) D e f i n i t i o n : M < (x, i?)M' L e t T ( P ) be a finitely ax iomat i zed (first- or second-order) theory, whose predicate, func-t ion a n d constant letters inc lude (but need not be l im i ted to) those i n X ; let IE be a b inary re la t ion o n tuples of type X ; let < R be the pre-order induced by R; and let M , M' be models of T. T h e n M i s an (X,iZ)-submodel of M ' (wr i t ten M < (x,j?)M') iff (i) | M | = | W | , (ii) If t is a t e rm and t X , then |t|ji/ = \t\fj , - 1 4 3 -(iii) If Q X is a predicate letter of T, then |Q|A/= \ Q-\M1 » A N D (iv) <|XU, |X| V > e i 2 . I D e f i n i t i o n : ( X , i ? ) - M i n i m a l M o d e l A mode l , M , of T is (X.,R)-minimal iff T has no model , A f ' , such that Af* < ( x , i j ) M a n d - . (A f < (X,J?)M'). | T h e o r e m 6 .4 — S o u n d n e s s CLOSURE(T; X ; i?) is satisf ied by every (X , J? ) -m in ima l model of T. T h e proof is s im i l a r to that of T h e o r e m 6.1, except that the interpretat ions of each of the var iab le terms mus t also be set. I D e f i n i t i o n : W e l l - F o u n d e d n e s s T h e theory, T, is well-founded with respect to ( X , i i ) iff every mode l of T has an ( X , i 2 ) - m i n i m a l submode l . | T h e o r e m 6 . 9 If T is a un iversa l theory, and X , P are finite tuples of predicate letters, then T is wel l - founded w i t h respect to < (x,p) • P r o o f W e show tha t any cha in of submodels of a mode l of T has a lower bound among the submodels of that mode l . It fo l lows b y Zorn ' s l e m m a that every mode l has a m i n i m a l submodel . Le t Mo,... be a cha in of models of T , ordered under the submodel re la t ion. If the cha in is finite, i t has a lower bound , hence assume i t is inf in i te. Le t {dj,...} be the elements of |A^>|. E x t e n d the language of T, L, to L' by add ing a new constant s y m b o l , d{, for each d , - . Le t T' = T U {Pi | for a l l i, Afj-1= Pet} U { - P c f | for some i, MifcPl). - 144 -Assume T' is inconsistent . T h e n , by compactness, so is a finite subset. B u t then some M,- must set each Pet i n this finite set accord ing ly , so M,- \f= T, wh i ch is a cont rad ic t ion , since the cha in {Mi} is ordered. Hence T' is consistent, so T' has a model , M'. N o w we can a d d the d iagrams (over a l l g round terms of L') of the equal i ty predicate and a l l fixed predicates f r om MQ to T' to get T". B y the above argument, T" must be consistent. Hence there is an M " such that M" j= T". B y v i r tue of the fact that M" satisfies the d iag ram of the equa l i t y pred icate f r om MQ , we can isomorph ica l l y embed the domain of M0 in to M". (Because T" conta ins the d iagrams of the equa l i t y predicate over a l l g round terms of L ' , it is clear that the resu l t ing subst ruc ture is closed under and preserves the funct ions.) F i n a l l y , since T" 3 T, M" j= T. Since T is a un iversa l theory, the rest r ic t ion, M, of M" to | M 0 | is a mode l of T. C lea r l y M< (x,p)M,- > f ° r a l l * i so M i s the lower bound we require. Q E D T h e o r e m 6.9 T h e o r e m 6.11 If T is we l l - founded w i t h respect to ( X , P ) ; P € P is an n-ary predicate; X is a set of predicate let ters; and a\,...,ak are n-tuples of ground terms; then CLOSURE{T; X ; P ) f - Pa^ V . . . V Pb?k^=> T f— Pa1 V . . .V Pak . I T h e o r e m 6.12 If T is we l l - founded w i t h respect to ( X , P ) ; X is a set of predicate letters; P j ^ P U X is an n-ary pred icate ; and ay...,ak are n-tuples of ground terms; then (i) CLOSURE{ T; X ; P ) | - Pd? ! V . . . V PoT f c T \— Paty V . . . V P3K , and (ii) CLOSURE(T; X ; P ) (- - P a \ V . . . V - P o ? * T |— - P c ^ V . . .V - P a f c . I T h e o r e m 6.13 If T is we l l - founded for ( P , P ) a n d T has a mode l w i t h doma in D, then so does CLOSURE( r(P);P;P). I T h e o r e m 6.14 If T is a first-order theory con ta in ing ax ioms w h i c h define the equa l i ty predicate, = , then T \- CLOSURE( T ; X ; { =}). | - 145 -T h e proofs of Theorems 6.11, 6.12, 6.13, and 6.14 are essential ly a lphabet ic var iants of those of Theorems 5.4, 5.5, 5.10, a n d 5.6, respect ively. W e do not repeat t hem here. - 1 4 6 -T h e o r e m 7.1 — S o u n d n e s s E v e r y instance of the rev ised d o m a i n c i rcumscr ip t ion schema for a theory, T, is true in a l l m i n i m a l mode ls of T. P r o o f T h e proof is i den t i ca l to that presented i n [Davis 1980, p75], except that, in the proof of the l emma, the rev ised schema guarantees that D0 is non-empty and hence N is wel l -def ined. 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 wel l - founded theory w h i c h conta ins ax ioms wh ich define the equa l i ty predicate, = , and ai,...,an, are ground terms, then (i) T |— ( V a,- = ft) DC( T) [— ( V a,- = ft) «=i t=i (ii) T ( - ( V a , - £ ft) < = • DC(T)\- (Va{ £ ft) P r o o f E v e r y mode l of a wel l - founded theory has a m i n i m a l submodel . L e t M ' be a m o d e l of T. Le t M< M' be a m i n i m a l submode l of M?. T h u s M a n d M! agree on a l l g round terms, and M i s the res t r ic t ion of M ' to a smal ler doma in . B u t then c lear ly they must have the same set of ground ( in)equal i t ies, since new equal i t ies i m p l y that M is not a restr ic t ion of M ' , and new inequal i t ies i m p l y that | M | does not con ta in the in terpreta t ion of some of the ground terms (since M i s a res-t r i c t ion of M * ) , wh i ch 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 ax iomat izab le theory, a n d every mode l of T is finite, then on ly the m i n i m a l models of T satisfy every instance of the doma in c i rcumscr ip t ion schema for T, DC(T). P r o o f Assume every m o d e l of T is finite. Cons ide r some non -m in ima l model , M . W e assume that every instance of DC( T) is t rue in M and ar r ive at a cont rad ic t ion . M is finite, w i t h m elements i n i ts d o m a i n . Since M is not m i n i m a l , there is a submode l , N < M, - 1 4 7 -w i t h n < m d o m a i n elements. Le t <&x be x = xx V . . . V x = xn where the x/s are var iables . W e can instant iate these x,'s i n M to be the n elements w h i c h survive the submodel ing to N. C l e a r l y 3x. $ x is true, as is AXIOM($). A* must be true, as follows: Cons ide r an a rb i t ra ry expression, \Px. V x . tyx Z) [ V x . $ X D \PX ] , a n d the existent ials given by T must be satisfied i n N (since TV is a model ) . Fur the rmore , $ is true for a l l of \N\. T h u s [3x. G T w i l l mean that 3x. $ z A * z w i l l be true i n M. B u t since n < m, V x . $ x is c lear ly 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 f ini tely ax iomat izab le theory, a n d every model of D C ( T ) is finite, then only the m i n i m a l models of T satisfy every instance of DC[T). P r o o f DC(T) is true 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 models. DC(T) false i n a l l infinite models, so on ly finite n o n - m i n i m a l models r emain to be e l imina ted . E v e r y finite mode l has a m i n i m a l submodel (there can ' t be an infinite cha in of proper submodels) . T h e argument for T h e o r e m 7.4 serves to rule 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 - 148 -Theorem 8.2 If T |— V z . x = a x V...V x = a „ and T |— a,• j= ay , for i j= j for ground terms cty...,an ; a n d X inc ludes a l l of the predicates of L; then those formulae true in every extens ion of : - P x ' A = iPz are precisely those enta i led by CLOSURE(T\ X ; { P » . Proof L e m m a 8.2.1 shows that every mode l for any extension of A is ( X , { P } ) - m i n i m a l . L e m m a 8.2.2 shows that every ( X , { P } ) - m i n i m a l mode l of T is a mode l for some extension of A . F r o m these the resul t fo l lows. QED Theorem 8.2 L e m m a 8.2.1 If T |— V z . z = cti V...V x = an for g round terms a x , . . . , a n ; and X inc ludes a l l of the ' : - . P z ' predicates of L; then any mode l of any extension of A -m i n i m a l mode l for T. -.Pz is an ( X , { P » -Proof A n y mode l , M for an extens ion, E, for A has d o m a i n |A4| = U { J a i l ^ . Assume that M i s not m i n i m a l . T h e n there is a n M ' < M W i t h o u t loss of general i ty , assume \P\M= 0<k < n}, and \P\M = {av...,aT \ 0 < r<k). (k>0 or there is no M ' < M ) N o w , g iven the existence of M ' , i t is c lear that E \j- Pak so —Pajt must be in E, so M \f= E, wh i ch is a con t rad ic t ion . Hence, M i s m i n i m i a l . QED Lemma 8.2.1 Lemma 8.2.2 If T |— V z . x = oti V...V x = an and T \— a,- j= ay , for t =^ / for g round terms a ^ . - . ^ a ^ and X inc ludes a l l of the predicates of L; then any ( X , { P } ) - m i n i m a l mode l for T is a i . ^ P z mode l of some extens ion of A = - 1 4 9 -P r o o f W e construct the extens ion, E, f rom the m i n i m a l mode l , M . C lea r l y M\= T. If M \= - . P a , - , put : - . P a , — in GD(E,A). Obv ious l y , TU CONSEQUENTS(GD(E,A)) then entai ls P a , for each a , - . P a , -such that Pa}<£ CONSEQUENTS(GD(E,A)). (Otherwise M i s not m in ima l ) . T h e existence of M guarantees that E \f- Pa , - for the a,-'s wh ich make up GD. T h u s E= Th{TU CONSEQUENTS(GD[E,A)) is an extension for A . C l e a r l y M j= E. Q E D 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 en ta i l a domain-c losure a x i o m , and T J / - V z . ->Px, then every extens ion for A has models w h i c h are not ( X , {P } ) -m in ima l . T h e proof of th is propos i t ion l ies i n the observat ion that one can a lways sei - Pa for some doma in element a w h i c h does not cor respond to any term- i n the language. Since T does not enta i l a doma in closure a x i o m , a mode l w i th such a n element w i l l a lways exist . | T h e o r e m 8 .7 There are theories, T, such that T f— V z . z = ax V . . . V for i j= j and yet no comb ina t ion of the extensions of A character izes the ( X , { P } ) - m i n i m a l models of T. JL = an and T' i— a{J= a3-, precisely T h e proof of th is theorem fol lows f rom 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 var iab le predicates (Z = { }), then ECWA(T) adds to T every instance of the c i r cumsc r ip t i on schema. T h e proof of th is fo l lows d i rec t ly f r om of the th i rd coro l lary to Ge l fond , P r symus inska , and P rsymus insk i ' s [1985] theorem 1. | APPENDIX B Dictionary of Symbols Symbol Definition G Set membersh ip ^ Set non-membersh ip U Set un ion n Set in tersect ion { } T h e emp ty set Set difference: $ - r = { a | a £ * and a (£ T} A S y m m e t r i c set difference: I A T = ( f - T ) U ( f -h ¥= 3 A V 3 V F i rs t -o rder p rovab i l i t y F i rs t -o rder non-p rovab i l i t y L o g i c a l enta i lment L o g i c a l non-enta i lment L o g i c a l imp l i ca t i on L o g i c a l negat ion L o g i c a l and L o g i c a l or L o g i c a l equivalence E x i s t e n t i a l quant i f ier Un i ve r sa l quant i f ier P reced ing quant i f ier 's scope extends over 1st enclosing fo rmu la . iff, D Th T h e nu l l clause C o n t r a d i c t i o n L o g i c a l c losure operator " I t fo l lows tha t " or " Imp l i es " If and on ly if <sC S t rong precedence re la t ion on Literals x Literals < i W e a k precedence re la t ion o n Literals x Literals )-» F u n c t i o n mapp ing L T h e first-order language (i.e., a l l wel l - formed formulae) N T h e set of a l l N a t u r a l numbers Literals T h e set of a l l a tomic formulae and their negat ions | M a r k s end of def in i t ion, example, or theorem - 150 -A P P E N D I X C Useful Logical Definitions C l a u s e - A clause is a finite d is junc t ion of l i terals. C l o s e d F o r u m u l a - A fo rmu la is closed iff i t conta ins no free var iab les. G r o u n d - A n express ion ( l i teral , te rm, or formula) is ground iff i t conta ins no var iab les. H e r b r a n d U n i v e r s e - If T is a un ive rsa l theory, then the Herbrand Universe of T is H(T) = {r[h,...,t^ | f1 is an n-ary funct ion- let ter of T, and ty...,tne H(T)}. (Th is is we l l -def ined because the O-ary funct ion- let ters (or constants) provide the base for the recursion.) H e r b r a n d B a s e - If T is a un iversa l theory, then the Herbrand Base of T is H(T) = {P"(tu...,Q | F1is an n^ary predicate- let ter of T, and tu...,tne H[T)}. H e r b r a n d I n t e r p r e t a t i o n - If T is a un iversa l theory, then a Herbrand Interpretation, I, of T is a subset of Ts H e r b r a n d base, H(T). Those a tomic formulae Pn(t1,...,t^ G I are in terpreted as true in I, a l l others are in terpreted as false. H e r b r a n d M o d e l - If T is a un iversa l theory, then a Herbrand Model of T is a Herb rand in terpre ta t ion of T w h i c h satisfies every fo rmu la i n T, accord ing to the usua l def in i t ion of sat is fact ion b y an in terpre ta t ion . H o r n - A set of clauses, T, is Horn iff every clause i n T contains at most one posi t ive l i tera l . L i t e r a l - A literal is an a tomic fo rmu la or the negat ion of an a tomic formula . S k o l e m i z e d f o r m - T h e Skolemized form of a theory is the theory obta ined by conver t ing to p renex-norma l fo rm then progressively, f rom the r ight-most quanti f ier, rep lac ing each ex is tent ia l ly quant i f ied var iab le by a new func t ion-symbol tak ing as arguments each of the var iab les cap tu red by quant i f iers occur r ing fur ther to the left. T h e process of obta in ing the sko lemized fo rm of a theory is ca l led skolemization. - 151 -

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0051930/manifest

Comment

Related Items