Alternative Approaches to Default Logic. Fachgebiet Intellektik. Technische Hochschule Darmstadt. Alexanderstrae 10. W. Ken Jackson. Burnaby, B.C.

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1 Alterative Appraches t Default Lgic James P. Delgrade Schl f Cmputig Sciece Sim Fraser Uiversity Buraby, B.C. Caada V5A 1S6 Trste Schaub Fachgebiet Itellektik Techische Hchschule Darmstadt Alexaderstrae 10 D-6100 Darmstadt Germay W. Ke Jacks Schl f Cmputig Sciece Sim Fraser Uiversity Buraby, B.C. Caada V5A 1S6 August 1993 Abstract Reiter's default lgic has prve t be a edurig ad versatile apprach t mtic reasig. Subsequet wrk i default lgic has ccetrated i tw majr areas. First, mdicatis have bee develped t exted ad augmet the apprach. Secd, there has bee gig iterest i sematic fudatis fr default lgic. I this paper, a umber f variats f default lgic are develped t address dierig ituitis arisig frm the rigial ad subsequet frmulatis. First, we mdify the maer i which csistecy is used i the deiti f a default extesi. The idea is that a glbal rather tha lcal ti f csistecy is emplyed i the frmati f a default extesi. Secd, we argue that i sme situatis the requiremet f prvig the atecedet f a default is t strg. A secd variat f default lgic is develped where this requiremet is drpped; subsequetly these appraches are cmbied, leadig t a al variat. These variats the lead t default systems which cfrm t alterative ituitis regardig default reasig. Fr all f these appraches, a xedpit ad a pseud-iterative deiti are give; as well a sematic characterisati f these appraches is prvided. I the cmbied apprach we argue als that e ca w reas abut a set f defaults ad ca determie, fr example, if a particular default i a set is redudat. We shw the relati f this wrk t that f Lukaszewicz ad Brewka, ad t the Therist system. Curret address IRISA, Campus de Beaulieu, Rees cedex, Frace 1

2 1 Itrducti Reiter's default lgic [Reiter, 1980] is e f the best kw appraches t mtic reasig. I this apprach, dmai-specic rules f iferece r defaults are added t classical prpsitial r rst-rder lgic t capture patters f iferece f the frm \i the absece f ifrmati t the ctrary, cclude that ". A example f a default Bird(x) F ly(x) represetig the asserti \Birds y" is F ly(x). This rule is rughly iterpreted as \if smethig ca be iferred t be a bird, ad if that thig ca be csistetly assumed t y, the ifer that it des y". The meaig f a rule the rests tis f prvability ad csistecy with respect t a give set f beliefs. A set f beliefs sactied by a set f defaults, with respect t a iitial set f facts, is called a extesi f this set f facts. Hwever, as discussed i the ext secti, this frmalism lacks several imprtat prperties, icludig existece f extesis ad semi-mticity [Reiter, 1980], ad cumulativity [Makis, 1989]. I additi, dierig ituitis ccerig the rle f default rules lead t dierig piis ccerig ther prperties, mst tably that f \cmmitmet t assumptis" [Ple, 1989]. With regard t cmmitmet t assumptis, the geeral idea is that the ti f csistecy ca be emplyed i either a brad r \glbal" fashi, r i a mre limited r \lcal" fashi. Varius prpsals have bee put frward t address these issues ad diculties; as we shw hwever, e f these prpsals csider the full set f issues. Secti 3 describes a variat f default lgic, called cstraied default lgic. This variat essetially marries ad exteds the wrk fud i [Delgrade ad Jacks, 1991] ad [Schaub, 1991b]. I this variat, a default extesi is characterised by a set f frmulas (as befre) givig the set f default cclusis, ad a secd set f frmulas givig the \ctext f reasig", determied by the justicatis ad cclusis f applied defaults. Csistecy is treated i a \glbal" fashi, ad the justicatis f default rules are take as expressig uderlyig assumptis, rather tha a ti f pssibility. As a result, this variat cfrms with the ti f cmmitmet t assumptis. Furthermre, the prblems f semi-mticity ad existece f extesis are addressed, ad i a simpler framewrk tha ther prpsals. It may als be argued that i certai situatis default lgic is verly weak, i that desirable cclusis are t frthcmig. We argue that i such cases the requiremet that the atecedet be prvable is t strg, ad i Secti 4 preset a variat where defaults are replaced by prerequisite-free cuterparts. The missi f prerequisite cditis leads t a simpler frmulati, wherei e may reas by mdus tlles ad by cases. This variat is referred t as prerequisite-free default lgic. This secti als examies the system btaied by jiig cstraied default lgic with prerequisite-free default lgic. I this variat, prblems that arise i the rigial frmulati are fully addressed, icludig that f cumulativity, ad i a simpler system tha ther variats. We als shw hw e ca w reas abut a default i the metathery t determie, fr example, if it may ever be applicable i the frmati f a extesi. Fr each f these variats, xed-pit, iterative, ad sematic characterisatis are give ad shw t be equivalet. As well, i Secti 5, the relati t previus wrk by Lukaszewicz [1988] ad Brewka [1991a] is give; i additi we demstrate the relati f 2

3 ur mdicatis t the Therist system [Ple, 1988]. Hece the results reprted i this paper prvide liks amg these systems. The rst appedix ctais tw tables summarisig the systems discussed here. The rst table lists the ames ad abbreviatis f these systems, tgether with the sectis i which they are discussed. The secd table ctais a summary f prperties f the variats f default lgic. The secd appedix ctais the prfs f the results stated i the bdy f the paper. 2 Default Lgic 2.1 Itrducti t Default Lgic Default lgic (r DL) exteds classical rst-rder lgic by the additi f dmai-specic (~x) (~x) rules f iferece f the frm (~x). A rule may be ifrmally iterpreted as \If, fr sme set f istaces ~c, (~c) is prvable frm what is kw ad (~c) is csistet, the cclude by default that (~c)." (~x) is called the prerequisite; (~x) is the justicati; ad (~x) is the csequet. Fr cveiece, we dete the prerequisite f a default by Prereq(), its justicati by Justif () ad its csequet by Cseq(). These prjectis exted t sets f defaults i the bvius way. A rmal default is e where the justicati ad csequet are equivalet. A semi-rmal default is e where the csequet is a lgical csequece f the justicati. Almst all \aturally ccurrig" defaults are rmal [Etherigt, 1988], althugh semi-rmal defaults are required fr \iteractig" defaults [Reiter ad Criscul, 1981]. A default thery is a pair (D; W ) where D is a set f defaults ad W is a set f clsed rst-rder (r prpsitial) frmulas. 1 A clsed default thery is e where e f the frmulas i the defaults ctais a free variable. 2 Als, a default thery is said t be rmal r semi-rmal if it ctais ly rmal r semi-rmal defaults, respectively, The set f defaults is iteded t capture hypthetical r -strict ifereces. Hwever defaults dier markedly frm stadard rules f iferece i a aximatic specicati f a lgic. First, they are \dmai-specic" i that a rule makes referece t specic frmulas. Secd, these rules allw fr ifereces based t ly what ca be prve frm the facts, W (as give i a rule's prerequisite), but als frm what cat be prve (as give i the justicati). A set f defaults iduces e r mre extesis f the frmulas i W. A extesi ca be viewed as a maximal acceptable set f beliefs that e may hld abut the wrld W, suitably augmeted by the defaults D. Reiter species three prperties which shuld hld fr a extesi it shuld ctai the iitial set f facts W ; it shuld be deductively clsed; ad fr each default rule, if the prerequisite is i a extesi, but the egati f the justicati is t, the the csequet is i the extesi. If Th(S) is used t dete the deductive clsure f S, we btai 1 I what fllws, we simply say frmula istead f clsed rst-rder frmula. 2 Fr simplicity we deal ly with clsed default theries; pe theries are described i [Reiter, 1980]. Als we restrict urselves t default rules with ly a sigle justicati. I the systems we develp, multiple justicatis crrespd t their cjucti. 3

4 Deiti 2.1 [Reiter, 1980] Let (D; W ) be a default thery. Fr ay set S f frmulas, let (S) be the smallest set satisfyig the fllwig prperties 1. W (S), 2. (S) = Th( (S)), 3. If 2 D ad 2 (S) ad 62 S the 2 (S), the E is a extesi f the default thery (D; W ) i (E) = E. That is, E is a xed pit f. A pseud-iterative specicati f a extesi is als give Therem 2.1 [Reiter, 1980] Let (D; W ) be a default thery ad let E be a set f frmulas. Dee E 0 = W ad fr i 0 E i+1 = Th(E i ) [ 2 D; 2 E i; 62 E The E is a extesi fr (D; W ) i E = S 1 i=0 E i. The abve prcedure is t strictly iterative sice E appears i the specicati f E i+1. If we had fr example that Quakers are typically pacists while Republicas typically are t (i.e. D = Q(x) P (x) R(x) P (x) ; P (x) P (x) ) the, fr W = fq(sue); R(sue)g, there are tw extesis, e i which P (sue) is true ad e i which P (sue) is true. If D 0 = D [ Q(x) G(x) A(x) E(x) ; (say, Quakers are typically geerus ad adults are typically G(x) E(x) emplyed) the agai we btai tw extesis where P (sue) is true i e ad P (sue) is true i the ther, ad where additially G(sue) is true i bth, ad E(sue) is true i either. While these characterisatis f a default thery are clear ad ituitive, they etheless are essetially sytactic i ature. As a rst step i develpig sematic uderpiigs fr DL, [ Lukaszewicz, 1985] prvides a sematic characterisati f rmal default theries. The geeral idea is that every rmal default ca be regarded as a mappig frm sets f mdels it sets f mdels, such that the rage f the mappig is the subset f the dmai where the csequet is true. I [Etherigt, 1987; Etherigt, 1988] sematic characterisatis f geeral default theries are give, based a ti f preferece betwee sets f mdels. A default prefers a set f mdels 1 i which the csequet f the default hlds, ver a superset f mdels 2 where the prerequisite is true ad the justicati is csistet but the csequet is t ecessarily satised. If we let MOD(W ) dete the set f all mdels f W, the preferece relati is deed as fllws Deiti 2.2 [Etherigt, 1988] Let = be a default rule, a set f mdels ad 1 ; The relati crrespdig t,, is deed as fllws 1 2 i 4

5 1. fr every 2 2 we have j=, 2. there is 2 2 such that j=, 3. 1 = f 2 2 j j= g. This rder is easily exteded t a set f defaults D Deiti 2.3 [Etherigt, 1988] Let D be a set f default rules, a set f mdels ad 1 ; The relati crrespdig t D, D, is the trasitive clsure f the ui f the relatis fr every 2 D 1 D 2 i 1. there exists 2 D, such that 1 2 r else 2. there is a such that 1 3 ad 3 2. Fr rmal default theries, we eed ly csider the D -maximal elemets f 2 MOD(W ) [ Lukaszewicz, 1985]. Hwever fr geeral default theries, we eed t esure that the justicati f each default is csistet with the al extesi. Etherigt dees a prperty which he calls stability that esures this cditi Deiti 2.4 [Etherigt, 1988] Let (D; W ) be a default thery ad 2 2 MOD(W ). is stable fr (D; W ) i there is a set f default rules D 0 D such that 1. D 0 MOD(W ), ad 2. fr every 2 D 0, we have fr sme 2 that j=. Stable, maximal sets f mdels fr a default thery are shw t crrespd t extesis i a default thery, thus prvidig a aalgue t sudess ad cmpleteess results. 2.2 Prperties f Default Lgic There are a umber f prperties which are t preset i DL. These may be rughly divided it, rst, limitatis f the frmalism itself, ad, secd, prperties that arise due t dierig ituitis ccerig the ature f a default rule. This divisi is i sme sese a arbitrary e, ad sme prperties that may be prblematic t e pers r applicati may appear t be quite reasable t ather. Sice the examples illustratig these prperties are used thrughut the paper, they are develped i sme detail here. First, it is t always the case that a default thery has a extesi. Example 2.1 (Existece f Extesis) The default thery A A ; ; has extesi. 5

6 Nrmal default theries guaratee the existece f extesis, which is t the case fr semi-rmal default theries. The ext example illustrates the prperty f semi-mticity ad thus ccers icreasig the set f defaults i a thery. Ituitively, we wuld wat that if D 0 D fr tw sets f defaults, the if E 0 is a extesi f D 0 the there is a extesi E f D where E 0 E. While this ideed is the case fr rmal defaults [Reiter, 1980, p. 96], the ext example, adapted frm [ Lukaszewicz, 1988], illustrates that it is t s i geeral. Example 2.2 (Semi-mticity) The default thery The default thery A B^C B A B^C B ; fag has a uique extesi Th(fA; Bg). ; fag has a uique extesi Th(fA; Cg). ; A C C As [Reiter, 1980] discusses, semi-mticity has the imprtat practical csequece that it allws fr lcal prf prcedures that may discard sme f the defaults. Mrever, if we iterpret justicatis as supplyig tetative assumptis (r wrkig hyptheses) the the fact that we btai ly e extesi, Th(fA; Cg), i the secd part f the precedig example is arguably uituitive if we kw ly W = fag rigially, the ifrmally it seems that the rst default shuld be \applicable". The argumet might ru \Iitially I kw thig at all; hece B ^ C is csistet ad I ca cclude that B. Hwever, C is icsistet with my rigial assumpti f B ^ C, ad s I cat apply the secd default." Thus we btai a extesi Th(fA; Bg) by cmmittig t B ^ C. Similar reasig begiig with the secd default yields the secd extesi Th(fA; Cg). Uder this argumet, we wuld btai semi-mticity, sice addig a default wuld ly elarge r preserve existig extesis. Nte that this iterpretati f justicatis as cstitutig \wrkig hyptheses" is just that uderlyig the failure t cmmit t assumptis, discussed belw. Hwever if we d't treat justicatis as cstitutig wrkig hyptheses, but take them simply as csistecy cditis (as i the rigial frmulatis), the i ther cases this behaviur ca be see t be quite reasable. Csider a iterpretati f Example 2.2 expressig the ifrmati that, wherever I am, if a televisi picks up Chael 19 ad it is pssible t tur t the theatre prgramme ad the recepti is t pr, the I will tur t the theatre prgramme. That is, A is \The televisi receives chael 19"; B is \I tur t the theatre prgramme"; C is \The recepti is pr". Give the rst default thery I will tur t the prgramme by default. Hwever if we als A C kew that Chael 19 typically gets pr recepti (i.e. ), the arguably this default C verrides the previus, ad by default I d't attempt t watch the prgramme. Sice extesis are iteded t represet maximal csistet sets f beliefs, e wuld expect that distict extesis wuld be icsistet. I a rmal default thery this ideed is the case; hwever, it is t the case fr semi-rmal default theries. 6

7 Example 2.3 (Orthgality f Extesis) The default thery ; ; has tw extesis Th(fBg) ad Th(fAg). A^B B ; A^B A I Secti 3 we suggest that a mre apprpriate prperty is weak rthgality where, if a thery has distict extesis, the the extesis tgether with a \ctext f reasig", supplied by the set f justicatis f applied defaults, are icsistet. The ext example, dealig with cumulativity, is due t David Makis [1989]. The ituitive idea is that if a therem is added t the set f premises frm which the therem was derived, the the set f derivable frmulas shuld remai uchaged. Ufrtuately this is t the case fr DL Example 2.4 [Makis, 1989] (Cumulativity) The default thery A A_B A ; ; ; has e extesi Th(fAg). A A Csequetly the extesi ctais A _ B. The default thery A A_B A ; ; fa _ Bg has tw extesis A A Th(fAg) ad Th(fA; Bg). This example suggests that the failure f cumulativity is caused by chagig the way i which a default's prerequisite has bee derived. I the rst part f the example, the prerequisite A _ B is derived by default frm the applicati f the rule A A Here the secd default is iapplicable. I the secd part, A _ B cstitutes the wrld kwledge W, s that w we btai a secd extesi geerated by the secd default. Hwever, the failure f cumulativity ca als be caused by chagig the way i which a justicati f a default is refuted, as the ext example illustrates. Example 2.5 (Cumulativity) A B B ; A B B ; C The default thery C Csequetly the extesi ctais C. The default thery A B B ; A B B ; C C Th(fA; B; C g) ad Th(fA; B; C g) ; fa; B C g has e extesi Th(fA; B; C g) ; fa; C ; B C g has tw extesis I the case f the rst default thery, the third default is ly blcked wheever the rst default applies. This is because the csequet f the rst default A B B implies C, which ctradicts the justicati f the third default C C As a csequece, there is extesi geerated by the secd default. This chages whe the default cclusi C is added t the set f facts, as de i the secd default thery. Nw, we additially btai a secd extesi geerated by the secd default. Ather issue is cmmitmet t assumptis [Ple, 1989] Example 2.6 (Cmmitmet t Assumptis) The default thery B C ; B ; ; has e extesi Th(fC; Dg). D If we take justicatis as specifyig strict csistecy cditis (as Deiti 2.1 suggests) the this is e. Hwever, a alterative view takes justicatis as prvidig \implicit assumptis" r \wrkig hyptheses". Thus e might argue that the default 7

8 cclusi C ^ D is implausible, sice bth default rules are applicable eve thugh they have ctradictry justicatis. That is, the default cclusi C relies B beig assumed as the csistecy cditi, whereas D relies B. The implausibility arises frm the fact that B ad B cat jitly hld, ad s it wuld seem that C ad D shuld t jitly hld. This may have further eects, as the fllwig shws Example 2.6 (ct'd) The default thery B C ; B D ; C E ; D ; ; has e extesi Th(fC; Dg). F Thus, a family may decide t d e f tw thigs, g t the beach r t a mvie (C r D) the weeked, depedig whether it is suy (B) r t. Of the tw childre e des't like the beach, the ther des't like gig t mvies. S Chris will be happy (E) by default if they d't g t the beach, ad Leslie will be happy (F ) if they d't g t a mvie. We wuld expect t have three extesis, e i which fc; Fg are true; e i which fd; Eg are true; ad e i which fe; Fg are true. Thus i the rst case, assumig that it is suy, the family ges t the beach ad Leslie is happy; i the secd case the family ges t a mvie ad Chris is happy; i the last case, based the assumpti that the family ges t either the beach r t a mvie, bth childre are happy. Hwever i DL we cclude ly that fc; Dg, ad the family ges t bth the beach ad a mvie. The situati is eve wrse if we assert that gig t the beach ad gig t a mvie are mutually exclusive, ad s W = fc _ Dg. I this case i DL we have extesis. This pheme shws up mre subtly i the \brke arms" example Example 2.7 [Ple, 1989] (Cmmitmet t Assumptis) The default thery ; fb _ Dg has e extesi Th(fA; Cg). A^B A ; C^D C Sice e f B r D must be false, e f the default cditis, it wuld seem, cat hld, ad s bth default cclusis shuld t jitly hld. Csider, ccretely, where we assert that by default a pers's left arm is usable (A) uless it is brke (B); similarly a pers's right arm is usable (C) uless it is brke (D). The precedig thery the directs us t cclude that bth arms are usable, eve if e f them is kw t be brke (B _ D). The diculty, frm a techical pit f view, is that i Deiti 2.1 (ad i Therem 2.1) justicatis eed ly be idividually csistet with a extesi. Hwever the abve lie f argumet relies the view that justicatis i default rules are iteded t fucti as uderlyig assumptis. The rigial frmulati f DL thugh tk justicatis as straightfrward csistecy cditis. Csider the iterpretati f Example 2.6, where we wish t use the default rules t help us decide what t take a trip if it is csistet (i.e. pssible) that it will be ht (B) the take a T-shirt (C); if it is csistet that it will t be ht the take a sweater (D). S w if I kw thig abut the weather I take a T-shirt ad sweater. I this case ur iteded iterpretati f the justicatis i the rules is clser t \is pssible". Hece it makes sese t cclude by default fc; Dg, eve thugh the justicatis are jitly icsistet. The failure f DL t cmmit t assumptis the may be see t as a \bug", but rather as reectig a alterative ituiti ccerig default rules. I this paper, as 8

9 metied, we primarily explre alterative frmulatis i which we btai cmmitmet t assumptis. There are a umber f further prperties that e might expect f default ifereces that are t preset i DL. These, like cmmitmet t assumptis, are a matter f e's ituitis, but here the ituitis ccer the iteded iterpretati f a default rule as a whle, rather tha the justicati ale. First, DL prvides a ti aki t defeasible mdus pes, i that frm W = fag ad A B if B is csistet with A we ca ifer by B default that B. Hwever this des t exted t mdus tlles, r reasig by cases. Example 2.8 (Mdus Tlles) The default thery ; ffg has e extesi Th(fFg). B F F Hece, fr example, if birds y by default 3 ad we kw that a give idividual des t y, the we cat cclude by default that the idividual is t a bird. Example 2.9 (Reasig by Cases) The default thery A B B ; A B B ; ; has e extesi Th(;). While A _ A is a therem, either A r A ale are prvable, ad s either default ca be applied. This situati als arises where mre tha e class may have the same default prperty Example 2.9 (ct'd) The default thery Q P P ; V P ; fq _ V g has e extesi Th(fQ _ V g). P Thus if Quakers ad vegetarias are bth pacists by default, ad if smee is either Quaker r vegetaria, we still cclude thig abut pacism. These examples rely the implicit assumpti that a default rule shuld behave like classical implicati, except that it is defeasible. Such a assumpti als seems t uderly the wrk f [Reiter ad Criscul, 1981]. There, fr example, it is assumed that defaults f the frm A B ad B C B C shuld be trasitive uless explicitly blcked. Csequetly if W = fag we wuld cclude C by default, uless the trasitivity was explicitly blcked accrdig t the recipe give i that paper. If e accepts the argumet that defaults behave like defeasible implicatis the it makes sese that e be able t reas usig the \ctrapsitive" f a default, uless it is explicitly blcked. Hwever, agai, this argumet is t uiversally applicable. Fr example, i a diagstic prgram, we might wat t express that peple are rmally pers diabetic diabetic t diabetic, (i.e. ); clearly i this case we wuld t wat t emply the ctrapsitive t cclude that a diabetic was by default t a pers. I Secti 4 we address the issues raised by the last tw examples. Fially, while e ca draw default cclusis i DL e cat reas abut defaults. I classical lgic, fr example, if ad are true, the _ must als be 3 Fr simplicity we use a prpsitial \glss" i sme f the examples, rather tha the mre apprpriate B(x) F (x) F (x). 9

10 true. I DL, there is explicit cecti betwee, ad _, eve thugh we kw ituitively that the third default ca be applied wheever either f the rst tw ca. That is, i DL, i rder t determie whether a default is applicable (fr sme extesi) i a give default thery, we must eectively cmpute the extesis f the thery. I Secti 4.3 we address the issue f reasig abut defaults. I particular, we shw that, i the al variat f default lgic that we develp, it is pssible t deductively reas abut defaults i the metathery t determie, fr example, whether the applicability cditis f e default is subsumed by thers. 2.3 Related Wrk This secti itrduces subsequet wrk i default lgic. The majr appraches described here are discussed i detail i Secti 5, where they are cmpared t the preseted framewrk. Lukaszewicz [1988] presets a variat f DL that addresses the issues f semi-mticity ad existece f extesis. I this apprach, a tw-place xed-pit peratr is used i the deiti f a extesi. The rst argumet accumulates the csequets f applied defaults, while the secd accumulates the justicatis. Fr a default t be applicable, Lukaszewicz requires that a csequet be csistet with the csequets f applied defaults, ad als csistet with the idividual justicatis f the ther applied defaults. The apprach guaratees semi-mticity ad the existece f extesis. It des t address cumulativity r cmmitmet. These last tw prperties are csidered i [Brewka, 1991a], where a variat called cumulative default lgic is described. Tw mdicatis t the rigial frmulati are itrduced. First, the applicability cditi fr a default is stregtheed s that a justicati must be csistet with bth the set f csequets ad the set f justicatis f the applied defaults. The set f justicatis is kept track f by itrducig assertis, where a asserti is a rst-rder frmula (the asserted frmula) tgether with a set f reass (r supprt) fr believig the frmula. A extesi csists f a set f assertis, where the rst elemet f each asserti csists f either wrld kwledge r the csequet f a applied default; i the latter case, the secd elemet is the set f justicatis ad csequets f the ther defaults that eabled the default t be applied. This apprach yields a variat that makes explicit the ti that justicatis are t be emplyed as tetative assumptis. This variat is cumulative ad cmmits t assumptis. As well it is semi-mtic ad guaratees the existece f extesis. Brewka et al. [1991] idetify a diculty with this apprach called the \atig cclusis" prblem. The diculty essetially is that, sice extesis are lger sets f rst-rder frmulas but are sets f assertis, it is uclear (at best) hw the extesis ca be jied t yield cclusis cmm t every extesi. While [Brewka, 1991a] prvides a xed-pit ad pseud-iterative specicati f a extesi i cumulative default lgic, it des t prvide a sematic characterisati. This missi is addressed i [Schaub, 1991a] where a extesi f Etherigt's sematics is preseted, alg with a sematical appraisal f the apprach. [Delgrade ad Jacks, 1991] presets a series f sytactic variats f DL. Oe f these variats essetially cmbies the 10

11 appraches f Lukaszewicz ad Brewka, ad crrespds t the system described i [Schaub, 1991b]. I this apprach, justicatis f applied defaults must be csistet as a grup; extesis are as i the rigial frmulati ad s the atig cclusis prblem des t arise. This apprach is further elabrated i [Schaub, 1992b], where the relatiship t the Brewka ad Lukaszewicz variats are als ivestigated. Examples 2.8 ad 2.9 illustrate that there are arguably-useful situatis where default rules cat be applied. [Besard, 1989] describes a trasfrmati f rmal default rules which permits mdus tlles ad reasig by cases with default rules. [Delgrade ad Jacks, 1991] idepedetly prpses the same trasfrmati, but exteded t semirmal defaults. 3 Cstraied Default Lgic The \brke arms" example (Example 2.7) illustrates that, depedig e's ituitis, DL may prduce cclusis that are strger tha desired. The example suggests that the set f justicatis used i the specicati f a extesi shuld be csistet, rather tha each idividual justicati. I this secti we develp ad explre this ituiti. We itrduce the ti f a cstraied extesi ad call the resultig system cstraied default lgic r CDL. A cstraied extesi is cmpsed f tw sets f frmulas E ad C, where E C The extesi E ctais all frmulas which are assumed t be true ad the set f cstraits C csists f E ad the justicatis f all applied defaults. I this apprach, we regard the csistecy assumptis (i.e. the justicatis) as cstraits a give extesi. This is illustrated i Figure 1. The gure illustrates the atural set iclusi betwee the facts W, their deductive clsure Th(W ), the extesi E, ad its cstraits C. I this respect Th(W ) cstitutes a lwer bud whereas the cstraits C cstitute a upper bud fr ur set f beliefs give by E. W Th(W ) E C Figure 1 A cstraied extesi (E,C ) f a default thery (D; W ). Fr a default t apply i DL its prerequisite must be i E ad its justicati must be csistet with E. I CDL, hwever, the prerequisite must be i extesi E 11

12 whereas the csistecy f the justicati is checked with respect t the set f cstraits C. The cstraits ca be regarded as a ctext established by the premises give i W, the mtic therems (i.e. cclusis derived by meas f defaults), ad the uderlyig csistecy assumptis. I this sese, CDL aturally exteds the itrisic ctext-sesitive character f defaults by distiguishig betwee ur set f beliefs give i the extesi, ad the uderlyig cstraits that frm a ctext guidig ur beliefs. While this slightly cmplicates the deiti f a extesi, it als meas that rules ad extesis are w represeted uifrmly, i that bth csist f a csistecy cditi alg with cclusis based the csistecy cditis. Deiti 3.1 Let (D; W ) be a default thery. Fr ay set f frmulas T let (T ) be the pair f smallest sets f frmulas (S 0 ; T 0 ) such that 1. W S 0 T 0, 2. S 0 = Th(S 0 ) ad T 0 = Th(T 0 ), 3. Fr ay 2 D, if 2 S0 ad T [ fg [ fg 6`? the 2 S 0 ad ^ 2 T 0. A pair f sets f frmulas (E,C ) is a cstraied extesi f (D; W ) i (C) = (E,C ) The set f cstraits is geerated by accumulatig the justicatis frm the applied defaults alg with the cclusis. Thus, each justicati is jitly csistet with the extesi ad all ther justicatis f applied defaults. Cmpared with Deiti 2.1, the xed-pit cditi relies ly the cstraits (T ). Ituitively, this meas that ur ctext f reasig has t cicide with ur set f accumulated cstraits. A immediate csequece f this deiti is that semi-rmal ad geeral default theries are equivalet. This is see by tig that the deiti cicides fr defaults f the frm ^ ad. Fr the default thery A B C ; fag ; istead f a \at" extesi Th(fA; C g) as i DL, we btai i CDL a extesi that is embedded i a ctext, viz. the cstraied extesi (Th(fA; C g); Th(fA; B; Cg)) Fr the brke arms example (Example 2.7), we w btai tw cstraied extesis. Example 3.1 The default thery A^B A ; C C ^D ; fb _ Dg has tw cstraied extesis (Th(fA; B _ Dg); Th(fA ^ B; Dg)) ad (Th(fC ; B _ Dg); Th(fC ^ D; Bg)) We btai e cstraied extesi i which A is true ad the value f C is uspecied, ad ather i which C is true ad the value f A is uspecied. I the rst cstraied extesi, the cstraits csist f the justicati A ^ B frm the rst default ad D frm the wrld kwledge. I the secd cstraied extesi, the cstraits ctai the justicati C ^ D frm the secd default ad B frm the wrld kwledge. I Example 2.2 we w als btai tw cstraied extesis. This reects the fact that CDL is semi-mtic (cf. Therem 3.3 belw). Example 3.2 The default thery A B^C B ; A C C (Th(fA; Bg); Th(fA; B ^ Cg)) ad (Th(fA; Cg); Th(fA; Cg)) 12 ; fag has tw cstraied extesis

13 Thus, i a cstraied extesi (E; C), the extesi E represets what we believe abut the wrld whereas the cstraits C tell us what assumptis we have made i rder t adpt ur beliefs. Hece, a extesi is ur evisiig f hw thigs are, whereas the cstraits represet ur assumptis i drawig these cclusis. Similar t [Reiter, 1980], we are able t prvide a mre ituitive but still \pseuditerative" characterisati f cstraied extesis Therem 3.1 Let (D; W ) be a default thery ad let E; C be sets f frmulas. Dee E 0 = W ad C 0 = W ad fr i 0 E i+1 = Th(E i ) [ C i+1 = Th(C i ) [ ^ 2 D; 2 E i; C [ fg [ fg 6`? 2 D; 2 E i; C [ fg [ fg 6`? (E,C ) is a cstraied extesi f (D; W ) i (E,C ) = ( S 1 i=0 E i ; S 1 i=0 C i ) Whe cmputig a extesi, referece is made t the previus partial extesi E i whereas the csistecy is checked with respect t all cstraits. Thus, we btai that cstraied extesis are uiquely determied by their sets f cstraits, i that if (E; C) ad (E 0 ; C 0 ) are cstraied extesis where C = C 0, the E = E Prperties f Cstraied Default Lgic The precedig discussi demstrates that CDL cmmits t assumptis. Cstraied extesis are maximal, i.e. fr cstraied extesis (E,C ) ad (E 0 ; C 0 ) f (D; W ) we have that if E E 0 ad C C 0 the E = E 0 ad C = C 0 4 Hwever, the example demstratig the failure f cumulativity give fr DL carries ver t CDL as well, ad s the system is t cumulative. Als, the set f beliefs i a extesi ale is t ecessarily maximal, as the ext example illustrates [ Lukaszewicz, 1988]. BA ; B D (Th(fAg); Th(fA; Bg)) ad (Th(fA; Bg); Th(fA; B; Dg)) Example 3.3 The default thery ; fag has tw cstraied extesis S the actual extesi f the rst cstraied extesi is icluded i the extesi f the secd cstraied extesi. Ntably, CDL guaratees the existece f extesis ad pssesses the prperty f semi-mticity. Therem 3.2 (Existece f extesis) Every default thery has a cstraied extesi. Csider the default thery f Example 2.1, which illustrates that DL des t guaratee the existece f extesis 4 Cf. [Schaub, 1992a, Therem 4.3.4]. 13

14 Example 3.4 The default thery A ; ; has e cstraied extesi (Th(;); Th(;)) A The default A A is t applicable sice, accrdig t Deiti 3.1, its justicati ad its csequet must be csistet. Therem 3.3 (Semi-mticity) Let (D; W ) be a default thery ad D 0 a set f defaults such that D D 0 If (E,C ) is a cstraied extesi f (D; W ), the there is a cstraied extesi (E 0 ; C 0 ) f (D 0 ; W ) such that E E 0 ad C C 0 That is, CDL is mtic with respect t the defaults. Csider agai Example 2.2. Example 3.5 The default thery (Th(fA; Bg); Th(fA; B ^ Cg)). The default thery A B^C ; fag has e cstraied extesi B A B^C B ; A C C ; fag has tw cstraied extesis (Th(fA; Bg); Th(fA; B ^ Cg)) ad (Th(fA; Cg); Th(fA; Cg)). Semi-mticity ad cmpactess imply that cstraied extesis are cstructible i a truly iterative way by applyig e applicable default after ather. The, i the case f a ite umber f clsed defaults, the csistecy f each justicati ca be checked with respect t the previus partial set f cstraits iduced by the facts ad all previuslyapplied defaults. I the iite case, e has t emply sme srt f diagalizati methd. Ather prperty which hlds fr cstraied extesis, we refer t as weak rthgality. This is aalgus t the prperty f rthgality i DL, which hlds fr rmal default theries, ad states that distict extesis are mutually ctradictry. Therem 3.4 (Weak rthgality) Let (D; W ) be a default thery. If (E,C ) ad (E 0 ; C 0 ) are distict cstraied extesis f (D; W ), the C [ C 0 is icsistet. That is, give tw dieret cstraied extesis the cstraits f the extesis are mutually ctradictry. Nte that weak rthgality is iapplicable fr DL, sice meti is made f the set f cstraits used i cstructig a extesi. Example 3.6 The default thery B C ; C B ; ; has tw cstraied extesis (Th(fC g); Th(fC ; Bg)) ad (Th(fBg); Th(fB; C g)) The tw cstraied extesis result frm icmpatible sets f cstraits. Thus, ad i ctrast t stadard extesis which hide their uderlyig csistecy assumptis ad therefre lack trasparecy, cstraied extesis exhibit their csistecy assumptis. 3.2 Cstraied versus Stadard Default Lgic We discuss here the relatiship betwee DL ad CDL. I DL, the prperties discussed i the previus subsecti (fr example, semi-mticity ad existece f extesis) actually hld fr rmal default theries. Thus e view f CDL is f a variat f DL 14

15 that exteds the prperties fud i rmal default theries t geeral default theries. The tight relatiship betwee DL ad CDL i the case f rmal default theries is shw by the fact that the appraches cicide i this case. 5 Therem 3.5 Let (D; W ) be a rmal default thery ad E a set f frmulas. The, E is a stadard extesi f (D; W ) i (E; E) is a cstraied extesi f (D; W ). At rst glace, it seems that CDL is strictly weaker tha its classical cuterpart ad that every cstraied extesi is \subsumed" by a stadard e. T see that this is t the case csider agai the family-beach-mvie situati f Example 2.6. Example 3.7 The default thery BC ; D B ; C E ; D F ; ; has e stadard extesi Th(fC ; Dg); but three cstraied extesis 1. (Th(fC ; Fg); Th(fB; C ; D; Fg)); 2. (Th(fD; Eg); Th(fB; C ; D; Eg)); 3. (Th(fE; Fg); Th(fC ; D; E; Fg)) Hece, CDL is either strger r weaker tha its classical cuterpart. I particular, the ly stadard extesi is either a superset r a subset f the extesis btaied i CDL. We ca describe the relatiship betwee DL ad CDL by usig the justicatis f the geeratig defaults, that is, fr a stadard extesi E, C E = 2 D; 2 E; 62 E Therem 3.6 Let (D; W ) be a default thery ad let E be a stadard extesi f (D; W ). If E [ C E is csistet, the (E; Th(E [ C E )) is a cstraied extesi f (D; W ). Observe that the cverse f the abve therem des t hld sice DL des t guaratee the existece f extesis. Hwever, if the extesis cicide we have Therem 3.7 Let (D; W ) be a default thery. If (E,C ) is a cstraied extesi f (D; W ) ad E is a stadard extesi f (D; W ), the C Th(E [ C E ) Fially, it is iterestig t bserve hw the specicati f a prblem i CDL ca be represeted i DL. 6 The idea is t shift the ifrmati give by the cstraits C i a cstraied extesi (E,C ) t the justicatis f the defaults. The, each such justicati is supplied with a additial but xed csistecy cditi represetig the cstraits i C. Agai, this is de by usig the set f geeratig defaults 7 f a cstraied extesi GD((E; C); D) = 2 D 2 E; C [ fg [ fg 6`? 5 We refer t extesis i Reiter's apprach as stadard extesis, i rder t distiguish them frm thers. 6 We wuld like t thak the aymus referees fr brigig this relatiship t ur atteti. 7 See Deiti B.1 fr a frmal deiti f the set f geeratig defaults. 15

16 Therem 3.8 Let (D; W ) be a default thery ad E ad C sets f frmulas. Let ^C = V 2GD((E ;C);D) Cseq() ^ Justif () with ite GD((E; C); D) ad D 0 = ^^ ^C 2 D The, if (E,C ) is a cstraied extesi f (D; W ) the E is a stadard extesi f (D 0 ; W ) 3.3 The Fcussed Mdels Sematics I rder t characterise cstraied extesis sematically, we dee a preferece relati similar t that give i [Etherigt, 1988]. Istead f sets f mdels, we csider pairs (; ) f sets f mdels. These pairs admit mre structure; we refer t them as fcused mdels structures. The ituiti behid a fcused mdels structure is as fllws. If we view the justicatis f defaults as \wrkig assumptis", the the csistecy cditi f DL is lger adequate. A primary diculty that arises is -cmmitmet t assumptis (cf. Example 2.6 ad 2.7). Sematically, we als eed t csider thse mdels satisfyig ur implicit assumptis, give i the ttality f the justicatis f the applied defaults. Sice we d t require that the justicatis be valid, there may exist mdels that falsify them. Csequetly, we impse mre structure the sets f mdels uder csiderati, viewig the secd cmpet, which is a subset f, as ur fcused set f mdels. We illustrate the crrespdig structure f fcused mdels structures i Figure 2. The set f mdels represets the set f mdels i which the set f csequets f applied defaults is true, while additially icludes thse mdels where the justicatis are true. Mdels Fcused Mdels Figure 2 A fcused mdels structure (; ). I rder t illustrate this, csider agai the default thery A B C ; fag Etherigt [1988] characterises the stadard extesi Th(fA; C g) by meas f a \at" set f mdels = f j j= A ^ C g 16

17 Hece, fr example, if B ad C are lgically idepedet the there are as may mdels satisfyig ur \wrkig assumpti" B as there are mdels falsifyig it. The apprach take by the fcused mdels sematics yields a pair (; ) = (f j j= A ^ C g; f j j= A ^ B ^ C g) that crrespds t a structured set f mdels icludig a fcus which additially satises ur implicit assumptis. This is the set f mdels satisfyig A, C, ad B. I ther wrds, we admit mre structured sets f mdels by fcusig thse mdels that satisfy ur assumptis. Sematically, a default prefers a fcused mdels structure ( 1 ; 1 ) t ather ( 2 ; 2 ) if its prerequisite is valid i 2 ad the cjucti f its justicati ad csequet ^ is satisable i sme fcused mdel i 2, ad lastly if 1 ad 1 etail the csequet (i additi t the previus requiremets). Frmally, we achieve all this by deig a rder relatig the csistecy f the justicatis with their satisability i the fcused mdels. Deiti 3.2 Let = ad let be a set f mdels, ad ( 1; 1 ); ( 2 ; 2 ) The relati crrespdig t, ; is deed as fllws ( 1 ; 1 ) ( 2 ; 2 ) i 1. fr every 2 2 we have j=, 2. there is 2 2 such that j= ^, 3. 1 = f 2 2 j j= g; 4. 1 = f 2 2 j j= ^ g The iduced rder D is deed as the trasitive clsure f all rders such that 2 D Deiti 3.3 Let D be a set f defaults, a set f mdels, ad ( 1 ; 1 ); ( 2 ; 2 ) The relati crrespdig t D, D ; is the trasitive clsure f the ui f the relatis fr every 2 D ( 1 ; 1 ) D ( 2 ; 2 ) i 1. there exists 2 D such that ( 1 ; 1 ) ( 2 ; 2 ) r else 2. there is ( 3 ; 3 ) such that ( 1 ; 1 ) D ( 3 ; 3 ) ad ( 3 ; 3 ) D ( 2 ; 2 ) We will refer t the D -maximal sets abve (MOD(W ); MOD(W )) as the preferred fcused mdels structures fr a default thery (D; W ). Give a preferred fcused mdels structure (; ); a extesi is frmed by all frmulas that are valid i, whereas the fcused mdels express the cstraits surrudig the extesi. Accrdigly, we have the fllwig crrectess ad cmpleteess therem establishig the crrespdece betwee cstraied extesis ad preferred fcused mdels structure fr a default thery (D; W ). 17

18 @ R@ Therem 3.9 Let (D; W ) be a default thery. Let (; ) be a pair f sets f mdels ad E; C deductively clsed sets f frmulas such that = f j j= Eg ad = f j j= Cg The, (E,C ) is a cstraied extesi f (D; W ) i (; ) is a D -maximal elemet abve (MOD(W ); MOD(W )) As i [Etherigt, 1988], we btai a simpler sematical characterisati i the case f rmal default theries 8. The larger set f mdels cllapses t the fcussed mdels sice rmal defaults require their justicatis t be valid after they have bee shw t be satisable 9. Cmpared with [Etherigt, 1988], we have stregtheed the ti f csistecy i extesis, by requirig that all justicatis ad csequets be jitly satisable by the fcused mdels. I particular, we d t require a stability cditi (cf. Deiti 2.4). Techically, this is due t the fact that we are dealig with a semi-mtic default lgic. Frm the viewpit f the fcused mdels sematics, hwever, we esure the ctiued csistecy f the justicatis f applyig defaults by allwig ly thse defaults t be applied that are cmpatible with the established fcus. Figure 3 illustrates why we btai tw cstraied extesis i Example 3.1. Oce we have \applied" e f the defaults the ther default is lger applicable the fcus des t satisfy its justicati. Applyig e f the defaults des t just require the validity f its csequet; it als makes us fcus its uderlyig assumpti (i.e. its justicati) i rder t preserve its satisability. Fr example, addig A uder the assumpti that A ^ B is csistet (by applyig the default A^B A ) prhibits us frm assumig that C ^ D is csistet tgether with the kwledge that B _ D; ad vice versa. B _ D A^B A C ^D C A B _ D C B _ D B ^ D B ^ D Figure 3 Cmmitmet t assumptis i CDL. The abve example shws hw the fcused mdels structures sematically accuts fr 8 Recall Therem We will see i Secti 5.2 that the fcus plays a fudametal rle i the case f rmal assertial default theries i rder t capture sematically the ti f cumulativity. 18

19 cmmitmet t assumptis. I view f the fact that the fcused mdels sematics als captures Brewka's cumulative default lgic (cf. Secti 5.2), ad the fact that Brewka's variat cmmits t assumptis as well, we may regard the fcused mdels sematics as a geeral apprach t cmmitmet t assumptis i default lgics. I additi, the sematics supplies us with several isights it the prperties f CDL. The existece f fcused mdels structures ad s the existece f cstraied extesis, is guarateed sice Deiti 3.2 esures that ever becmes the empty set. The same deiti als takes care f semi-mticity, sice there has t exist a fcused mdel satisfyig the prerequisite, csequet, ad justicati f a added default befre it is applied. Weak rthgality is mirrred by the fact that there ever exists a fcused mdel which is cmm t tw dieret preferred sets f fcused mdels. We will see i Secti 5.2 that the fcused mdels sematics captures als the prperty f cumulativity. A alterative sematics fr CDL has bee prpsed i [Besard ad Schaub, 1993]. The basic idea is t emply Kripke structures such that a fcused mdels structure (; ) crrespds t a set M f Kripke structures, where is captured by the actual wrlds i M ad by the accessible wrlds i M. The, a \preferred" set f Kripke structures M characterises a cstraied extesi (E,C ) such that E = f -mdal j M j= g ad C = f -mdal j M j= 2g. 4 Prerequisite-Free Appraches This secti discusses variats f DL ad CDL, wherei defaults are replaced by prerequisitefree cuterparts. This trasfrmati allws fr reasig by cases ad default ifereces ivlvig the ctrapsitive. Fr DL, the resultat system diers little frmally frm the rigial system; the majr dierece is that e lger eeds t prve the atecedet. This variat is discussed i Subsecti 4.1. Surprisigly, the prerequisite-free variat f CDL has quite dieret frmal prperties frm regular CDL. First, the resultat system is cumulative. Secd, i a limited, but -trivial maer, e ca w reas abut a set f defaults t determie, fr example, whether the applicability cditi f e default is subsumed by thers. This variat is discussed i Subsectis 4.2 ad 4.3. Fr clarity, we assume thrughut this secti that rmal ad semi-rmal defaults cstitute distict classes f defaults. This is t say, fr istace, that a semi-rmal default is e where the csequet is a lgical csequece f the justicati but t vice versa. 4.1 Prerequisite-Free Default Lgic This subsecti deals with the traslati f rmal ad semi-rmal defaults it their prerequisite-free cuterparts i DL. We have ted that i default reasig it wuld be useful t be able t reas by cases r reas by mdus tlles, uless such prperties are explicitly blcked. A ratiale is that a default cditial shuld retai prperties f the classical cditial, uless explicitly blcked. The emphasis the shifts t the cditial itself, rather tha a rule ivlvig a prerequisite ad justicati fr a cclusi. 19

20 Csider rst rmal defaults. The default is read as \if is prvably true ad is csistet, the is true". The readig we prpse is \if it is csistet that the cclude that ". Thus is ccluded (whe csistet) regardless f the BF prvability f. \Birds y" the is expressed prpsitially as BF. T be precise, we trasfrm rmal defaults it prerequisite-free rmal defaults i regular DL i the fllwig way. 7 () The resultig subsystem f DL, amely prerequisite-free default lgic, is referred t as PfDL. As a rst example, suppse that we have that, implausibly, A's are rmally B's ad A's are rmally t B's Example 4.1 The default thery A B The default thery B ; A B B AB AB ; AB AB ; ; has e extesi Th(;). ; ; has e extesi Th(fAg). Hece i the traslated thery we btai a reducti result if A's are rmally B's ad A's are als rmally t B's, the by default we ca cclude that A. Csider ext what this traslati meas fr reasig by cases. Fr Example 2.9 we btai Example 4.2 The default thery The default thery Q P P ; V P P QP QP ; V P V P ; fq _ V g has e extesi Th(fQ _ V g). ; fq _ V g has e extesi Th(fQ _ V; Pg). Thus if Quakers are rmally pacists, ad vegetarias are rmally pacists, the if smee is either a Quaker r vegetaria the, sice either Q r V are prvable, i DL either rule is applicable ad we cclude thig abut P. Hwever, ituitively, we kw that e f Q r V must be true, ad that P fllws by default frm either Q r V ; s arguably it shuld fllw by default that P. This ideed is the case fr the traslated defaults. A similar argumet applies t the ctrapsitive. Give that birds rmally y, the if smethig is kw t t y the it seems reasable t cclude by default that this thig is t a bird. Agai, we ca accmplish this if the cditial itself is ccluded wheever csistet. Example 4.3 The default thery The default thery B F F BF BF ; ffg has e extesi Th(fFg). ; ffg has e extesi Th(fB; Fg). There are hwever times whe we wat t blck the ctrapsitive; this issue is addressed usig semi-rmal defaults. The primary use f semi-rmal defaults is t specify that e default takes precedece ver thers [Reiter ad Criscul, 1981]. Fr example, csider the statemets \uiversity studets are typically adults", \adults are typically emplyed", ad \studets are typically t emplyed". 20

21 S A A ; A E E ; S E ; fsg has tw extesis E Th(fA; E; Sg) ad Th(fA; E; Sg). Example 4.4 The default thery The rst extesi is btaied by applyig the rst ad last default; the secd by applyig the rst tw defaults. Ituitively we wat ly e extesi, where E is true the mre specic default that studets are t emplyed shuld take precedece ver the less specic default that adults are emplyed. Oe pssible x is t assert that adults are t rmally A S A^S E studets (viz. ) ad the replace the secd default by. This results i a S E rmal default thery, but requires that we be able t assume that adults are t rmally studets. I additi, it frces us t cclude that a adult is a -studet, uless we ca prve therwise; it may well be that we wuld wat t remai agstic abut the fact f a pers's studethd. If we are uwillig r uable t make such a assumpti, semi-rmal defaults appear t be required. Example 4.4 (ct'd) The default thery S A A ; A S^E E ; S E E ; fsg has e extesi Th(fA; S; Eg). The same csideratis apply fr PfDL. Sice PfDL extesis are geeralisatis f extesis i regular DL (see Therem 4.1 belw), the prblem f ctrllig iteractis is mre acute. Csider the precedig example, traslated it PfDL Example 4.4 (ct'd) The default thery SA SA ; AE AE ; SE SE ; fsg has three extesis Th(fA; S; Eg), Th(fA; S; Eg), ad Th(fA; S; Eg). The third extesi results frm applyig the third ad secd defaults. We wuld like t blck the trasitivities implicit i the secd ad third extesis. The apprach here is t exted that f [Reiter ad Criscul, 1981] t prerequisite-free defaults. There are tw pssible traslatis replace ^ with ()^ () ; r replace ^ with (^) The rst pssibility carries as a \glbal" csistecy cditi, while the secd uses as a \lcal" csistecy cditi, uder the assumpti f. Hwever the secd alterative fails t blck the third extesi i the example abve. I additi, this alterative cat be used t satisfactrily blck the ctrapsitive (see belw). Hece we adpt the rst alterative. That is, we trasfrm semi-rmal defaults it prerequisite-free rmal defaults i DL i the fllwig way. ^ 7 ()^ I the precedig example, the default \adults are typically emplyed" is t applicable fr studets ad s we have Example 4.4 (ct'd) The default thery SA SA ; (AE)^S (AE) ; SE ; fsg has e extesi Th(fA; S; Eg). SE 21

22 Blckig the ctrapsitive f a cditial,, is a special case f this where is. Fr example, the ctrapsitive f the default, \A's are typically B's" is blcked by writig the rule as (AB)^B. If we are give W = fbg the clearly the default is iapplicable AB ad s we cat cclude A B r, csequetly, A. Nte that the default (AB)^B AB is equivalet t B AB. This last frm was suggested i [Brewka, 1991b], ad s the abve traslati may be regarded as a geeralisati f Brewka's. The fllwig therem shws that this traslati gives a system i which mre cclusis are frthcmig tha i the stadard apprach. Therem 4.1 Let (D; W ) be a semi-rmal default thery ad let (D 0 ; W ) be the thery where (^) 2 D i ()^ 2 D 0. If E is a extesi f (D; W ) the there is a () extesi f (D 0 ; W ), E 0, such that E E 0. Hwever we may als get mre extesis frm (D 0 ; W ). Example 4.5 The default thery A B^C ; B A^D B A (AB)^C (AB) The default thery Th(fA; Bg), Th(fA; Bg). ; fag has e extesi Th(fA; Bg). ; fag has tw extesis ; (BA)^D (BA) Of curse, the same result hlds fr rmal default theries. As a example, csider the previus e, where the last cjuct i the justicatis f each default is drpped. At rst sight, the abve disticti betwee rmal ad semi-rmal seems t be redudat. Hwever, tice that the precedig traslatis are sytax-depedet. This is why we distiguish betwee rmal ad semi-rmal defaults. 10 As a example, csider the default A B^B This is clearly a rmal default. Hwever, e culd be tempted t B trasfrm this default accrdig t the recipe give fr semi-rmal defaults. Hwever, this yields a dieret result tha trasfrmig the default A B accrdig t the recipe give fr B rmal defaults, as ca be easily veried. 11 As regards geeral defaults, we fllw [Etherigt, 1988; Lukaszewicz, 1990] i arguig that it is reasable t replace geeral defaults by semi-rmal es by cjiig the csequet t the justicati. I all, this amuts t the fllwig traslati i the case f geeral defaults i Reiter's DL 7 ^ 7 ()^ Fially, let us summarize the diereces betwee Reiter's DL ad its fragmet PfDL. I geeral, prerequisite-free rmal default theries ejy the same advatages as stadard rmal default theries, ad geeral prerequisite-free default theries suer frm the same drawbacks as stadard default theries. Apart frm cumulativity, bth types f rmal default theries have e f the diculties discussed i Secti 2.2, while bth types f geeral default theries have all f these diculties. 10 Recall that accrdig t the deiti give at start f this secti, rmal ad semi-rmal defaults are mutually exclusive. 11 We thak e f the referees fr this example. 22