In: Proceedings of the Fourteenth International Joint Conference on Articial Intelligence (IJCAI-95), C. Mellish

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1 I: rceedigs f the Furteeth Iteratial Jit Cferece Articial Itelligece (IJCAI-95), C. Mellish ed, Mrga Kaufma, I press. Sematics ad Cmplexity f Abducti frm Default Theries (Exteded abstract ) Thmas Eiter, Gerg Gttlb Christia Dppler Lab fr Expert Systems Techical Uiversity f Viea aiglgasse 16, A-1040 Wie, Austria (eiterjgttlb)@dbai.tuwie.ac.at Abstract Sice lgical kwledge represetati is cmmly based classical frmalisms like default lgic, autepistemic lgic, r circumscripti, it is ecessary t perfrm abductive reasig frm theries f classical lgics. I this paper, we ivestigate hw abducti ca be perfrmed frm theries i default lgic. Dieret mdes f abducti are plausible, based credulus ad skeptical default reasig they appear useful fr dieret applicatis such as diagsis ad plaig. Mrever, we aalyze the cmplexity f the mai abductive reasig tasks. They are itractable i the geeral case we als preset kw classes f default theries fr which abducti is tractable. 1 Itrducti Abductive reasig has bee recgized as a imprtat priciple f cmm-sese reasig havig fruitful applicatis i a umber f areas such diverse as mdelbased diagsis le, 1989], speech recgiti Hbbs et al., 1988], maiteace f database views Kakas ad Macarella, 1990], ad visi Chariak ad McDermtt, 1985]. Util w, maily abducti frm theries f classical lgic has bee studied. Hwever, lgical kwledge represetati is cmmly based classical frmalisms like default lgic, autepistemic lgic, r circumscripti. Thus, i such situatis it is ecessary t perfrm abductive reasig frm theries (i.e. kwledge bases) f classical lgics. Sice default lgic is widely prpsed fr kwledge represetati, it is imprtat t ivestigate hw abducti ca be perfrmed frm theries hw Di i default lgic. We ifrmally pursue this a example. Example 1 Csider the fllwig set f default rules, which represet kwledge abut Bill's skiig habits: : :skiig Bill weeked : :swig : :swig :skiig Bill skiig Bill :swig A mre elabrate versi icludig prfs is available request t the authrs. y Wrk carried ut while visitig the Christia Dppler Lab. Nicla Lee y Istitut per la Sistemistica e l'ifrmatica C.N.R. c/ DEIS { Uiversity f Calabria 8706 Rede, Italy ik@si.deis.uical.it The defaults ituitively state the fllwig: (i) Bill is usually t ut fr skiig (ii) Bill is ut fr skiig weekeds, if we ca assume that it is t swig (iii) usually it is t swig. Fr the certai kwledge W = f weeked g (ecdig that it is Saturday r Suday), the default thery T = hw Di has e extesi which ctais :swig ad skiig Bill. Suppse w we bserve that Bill is t ut fr skiig (which is icsistet with the extesi). Abducti meas t d a explaati fr this bservati, that is, t idetify a set f facts, chse frm a set f hyptheses, whse presece i the thery at had wuld etail the bservati :skiig Bill, i.e., cause that :skiig Bill is i the extesi. We d such a explaati by adptig the hypthesis swig. Ideed, if we add swig t W, the default thery T 0 = hf weeked swig g Di has a sigle extesi, which ctais :skiig Bill. We say that swig is abduced frm the bservati :skiig Bill, r that it is a abductive explaati f :skiig Bill. Observe that the descripti f the abve situati requires the specicati f sme default prperties that ca t be represeted prperly i classical lgic. I geeral, as ppsed t the example, a default thery may have several r eve extesis. Fr deductive etailmet, this gives rise t credulus etailmet, uder which is etailed frm a default thery T (deted T `c ) i belgs t at least e extesi f T, ad t skeptical etailmet, uder which fllws frm T (T `s ) i belgs t all extesis f T. Accrdigly, tw variats f abducti frm default theries arise: credulus abducti, where etailmet f a bservati is based `c, ad skeptical abducti, which is based `s. I practice, the user will chse credulus r skeptical abducti the basis f the particular applicati dmai. We argue that credulus abducti is well suited fr diagsis, while skeptical abducti is adequate fr plaig. (Cf. le, 1989] ad Eshghi, 1988 Ng ad Mey, 1991] fr abducti i lgic-based diagsis ad plaig & pla recgiti, respectively.) I fact, csider a system represeted by a default thery hw Di. If it receives sme iput, reected by addig a set A f facts t W, the each extesi f hw A Di is a pssible evluti f the system, i.e., each extesi represets a pssible reacti f the system t A.

2 Abductive diagsis csists, lsely speakig, i derivig frm a bserved system state (characterized by the truth f a set F f facts), a suitable iput A which caused this evluti (cf. le, 1989]). Nw, sice each extesi f hw A Di is a pssible evluti f the system with iput A, we ca assert that A is a pssible iput that caused F if hw A Di `c F. Thus, diagstic prblems ca be aturally represeted by abductive prblems with credulus etailmet. Example Assume there are tw sky rutes, rv1 ad rv, betwee Rme ad Viea, ad three sky rutes mv1, mv, ad mv betwee Mila ad Viea. Rute mv1 itersects rute rv1, ad mv itersects rv. O rmal speed ad ight cditis, tw plaes frm Mila ad Rme t Viea will cllide if the plae frm Mila takes 0 miutes after the plae frm Rme ad they y itersectig rutes. This kwledge abut pssible cllisis is represeted i simplied frm by the fllwig set D f prpsitial defaults: mv1 ^ rv1 ^ m 0mi later : cllisi cllisi mv ^ rv ^ m 0mi later : cllisi cllisi : :rv :rv : :cllisi :cllisi : :rv1 :rv1 : :mv1 :mv1 : :mv :mv : :mv :mv Nw, yu are ifrmed that plaes yig frm Mila ad Rme t Viea cllided. A diagsis fr the cllisi ca be btaied by abducig a explaati fr the bservati cllisi frm the thery T = h Di. I this case, we wat t kw pssible ight schedules that ca have caused the cllisi. I ther wrds, we are lkig fr schedules S such that cllisi is i sme extesi f the thery T 0 = hs Di (T 0 `c cllisi). Credulus abducti crrectly ideties such explaatis. Fr istace, it is easy t recgize that bth E1 = fmv1 rv1 m 0mi laterg ad E = fmv rv m 0mi laterg are credulus explaatis fr cllisi. Suppse w we wat that the system evlves it a certai state (described by a set F f facts), ad we have t determie the \right" iput that efrces this state f the system (plaig). I this case it is t suciet t chse a iput A such that F is true i sme pssible evluti f the system rather, we lk fr a iput A such that F is true i all pssible evlutis, as we wat be sure that the system reacts i that particular way. I ther wrds, we lk fr A such that hw A Di `s F. Hece, plaig activities ca be represeted by abductive prblems with skeptical etailmet. Example We kw that a plae frm Rme t Viea left at 7.50 (r 7:50), but we d t kw which rute. We have t schedule the ight f a plae frm Mila t Viea, where take is pssible at 8.10 (m 8:10) ad at 8.0 (m 8:0). The cllisi-free schedules ca be btaied by dig a abductive explaati ut f the hyptheses m 8:10, m 8:0, mv1, mv, mv fr the bservati :cllisi frm the thery T = hw D1i, W = f r 7:50 rv1 _ rv m 8:10 _ m 8:0 mv1 _ mv _ mv r 7:50 ^ m 8:10 m 0mi later g : :m 8:10 D1 = D :m 8:10 : :m 8:0 :m 8:0 As we ca t risk a cllisi, we wat that every pssible evluti f the system is cllisi-free. Thus, we have t lk fr skeptical explaatis f :cllisi. Fr istace, bth E = fm 8:0g ad E4 = fmvg are skeptical explaatis fr :cllisi that is, take at 8.0 r usig rute mv prevets a cllisi, where the rute i E ad the time i E4 ca be chse freely. The tw examples abve supprt the ituiti that credulus abducti is feasible fr diagsis, while skeptical abducti is well-suited fr plaig. O the ther had, Secti 4 shws that skeptical abducti has mst likely a higher cmplexity tha credulus abducti thus, frm the abve pit f view, plaig is mst likely harder tha diagsis. Fr space reass, we ly preset sme prf sketches. rfs f all results are give i the full paper. relimiaries ad Ntati We assume that the reader kws the basic ccepts f default lgic Reiter, 1980] (cf. als Marek ad Truszczyski, 199] fr a extesive study). We fcus prpsitial default theries T = hw Di ver a prpsitial laguage L (icludig? fr falsity), i.e. W is a subset f L ad D a set f defaults :1:::m, m 1 where, 1 : : : m, are frm L. The extesis f T, which are deductively clsed sets E L, are deed by a xpit equati i particular, L is a extesi f T (ad, i this case, uique) i W is t csistet. Recall that T is rmal i each default i D is rmal, i.e., f frm : a rmal T always has a extesi. Fr N-cmpleteess ad cmplexity thery, cf. Jhs, 1990]. The classes k ad k f the plymial hierarchy are deed as fllws: = 0 0 =, ad k = N k?1 k = c- k fr all k 11: I particular, N = 1 ad c-n = 1. The class D k, which is deed as the class f prblems that csist f the cjucti f tw (idepedet) prblems frm k ad k, respectively, is csidered t be further restricted i cmputatial pwer. Fr all k 1, clearly k D k k+1 bth iclusis are believed t be strict. May mtic reasig prblems are cmplete fr classes at the lwer ed f the plymial hierarchy Cadli ad Schaerf, 199 Nebel, 1994]. It is well-kw that decidig whether a prpsitial default thery has a extesi is - cmplete, ad that credulus ad skeptical reasig frm default theries are cmplete fr ad, respectively. This remais true if icsistet extesis are excluded ad, fr the latter prblems, if default theries are i additi rmal Gttlb, 199

3 Stillma, 199]. Cases f lwer cmplexity ad tractable fragmets were idetied i Kautz ad Selma, 1991 Stillma, 1990]. Frmalizig default abducti I this secti, we describe a basic frmal mdel fr abducti frm prpsitial default theries ad state the mai decisial reasig tasks fr abductive reasig. Our frmalizati f a abducti sceari is as fllws. Deiti 1 A prpsitial default abducti prblem (DA) is a quadruple hh M W Di where H is a set f prpsitial literals (called hyptheses, r abducibles), M is a set f prpsitial literals (bservatis, r maifestatis), ad hw Di is a prpsitial default thery. is rmal i each default i D is rmal. Nte that hyptheses ad maifestatis may be literals rather tha atms. Allwig literals as hyptheses is cmm i abducti, cf. Selma ad Levesque, 1990]. Hwever, this has eect the expressive pwer r cmplexity f the frmalism i geeral. Credulus ad skeptical explaatis are as fllws. Deiti Let = hh M W Di be a DA, ad let E H. The, E is a credulus explaati fr i (i) hw E Di `c M, ad (ii) hw E Di has a csistet extesi. Similarly, E is a skeptical explaati fr i (i) hw E Di `s M ad (ii) hw E Di has a csistet extesi. The existece f a csistet extesi fr hw E Di (i this case, all extesis are csistet) assures that the explaati E is csistet with the kwledge represeted i hw Di. This is aalgus t the csistecy criteri i abducti frm classical theries. It is cmm i abductive reasig t prue the set f all explaatis ad t fcus, guided by sme priciple f explaati preferece, a set f preferred explaatis. The mst imprtat such priciple is, fllwig Occam's priciple f parsimy, t prefer redudat explaatis, i.e., explaatis which d t ctai ay ther explaati prperly, cf. eg ad Reggia, 1990 Selma ad Levesque, 1990 Klige, 199]. We refer t such explaatis as miimal explaatis. I Example E = fm 8:0g ad E4 = fmvg are the miimal explaatis they represet the smallest partial schedules that ca be arbitrarily cmpleted t cllisi-free schedules, ad thus prvide the greatest exibility. I the sequel, we will write Exp() fr the set f explaatis fr the DA, abstractig frm the chse type f explaatis (credulus, skeptical, miimal credulus, r miimal skeptical). The fllwig prperties f a hypthesis i a DA are imprtat with respect t cmputig explaatis. Deiti Let = hh M W Di be a DA ad h H. The, h is relevat (resp. ecessary) fr i h E fr sme (resp. every) E Exp(). The ppsite f ecessity is als termed dispesability (cf. Jsephs et al., 1987]). I Example, m 0mi later is ecessary, while each hypthesis rv1 rv mv1 mv is relevat, but t ecessary. Mrever, i Example mv is relevat w.r.t. miimal (skeptical) explaatis, but t ecessary. Nte that i the same example rv1 is relevat uder arbitrary explaatis, but t relevat uder miimal explaatis. The mai decisial prblems i abductive reasig amut t the fllwig. Give a DA = hh M W Di, (Csistecy): des there exist a explaati fr? (Relevace): is a give hypthesis h H relevat fr, i.e., des h ctribute t sme explaati f? (Necessity): is a give hypthesis h H ecessary fr, i.e., is h ctaied i all explaatis f? Due t the fllwig simple fact, we shall t deal i ur aalysis explicitly with Necessity i the case f miimal explaatis. rpsiti 1 Let = hh M W Di be a DA ad let h H. The, h is ecessary fr uder miimal credulus (resp. skeptical) explaatis i h is ecessary fr uder credulus (resp. skeptical) explaatis. 4 Results The mai results the cmplexity f abducti frm geeral prpsitial default theries are summarized i Table 1. I ur aalysis, we pay particular atteti t rmal DAs, sice this class crrespds t the mst imprtat fragmet f default lgic. All hardess results i Table 1 have bee derived fr the case where the uderlyig default thery hw Di is rmal. Thus like deducti, abducti frm rmal default theries is as hard as abducti frm arbitrary default theries. We itrduce sme additial tati. Fr a set A f prpsitial atms, we dete by :A the set f:a j a Ag ad by A 0 the set f atms fa 0 j a Ag. 4.1 Arbitrary explaatis Our rst result shws that abducti frm default theries based credulus explaatis ca be ecietly reduced t deductive reasig frm prpsitial default theries. This is smewhat uexpected ad surprisig, sice i case f classical theries, abducti ca t be ecietly reduced t deducti. Give a DA = hh M W Di, we cstruct a default thery T = hw D i such that the credulus explaatis f crrespd t the extesis f T. Ideed, dee W = W fa h h j h Hg, j m M D = D ::m? :a h a h ::a h :a j h h H where fr each h H, a h is a ew prpsitial atm. The, we have: Therem 1 Let = hh M W Di be a DA. The, (i) if E is a credulus explaati fr, the there exists a csistet extesi E 0 f T such that E = fh H j a h E 0 g (ii) if E 0 is a csistet extesi f T, the E = fh H j a h E 0 g is a credulus explaati fr.

4 DA = hh M W Di arbitrary explaatis miimal explaatis rblem: credulus skeptical credulus skeptical Exp() 6= E Exp() E Exp() is miimal D D h H is relevat fr 4 h H is ecessary fr Table 1: Cmplexity results fr abducti frm prpsitial default theries Usig (i) ad (ii), the mai decisial abductive reasig tasks ca be ecietly trasfrmed t similar deductive reasig tasks i default lgic. Crllary 1 Let be a DA based credulus explaatis. The, (i) Csistecy, (ii) Relevace, ad (iii) Necessity are equivalet t (i') existece f a csistet extesi f T, (ii') membership f a h i sme csistet extesi f T, ad (iii') membership f a h i all extesis f T, respectively. By the results the cmplexity f prpsitial default lgic Gttlb, 199 Stillma, 199], it fllws that (i) ad (ii) are i ad that (iii) is i. We als btai matchig hardess by reductis frm deductive default reasig. Let T = hw Di be a rmal default thery such that W is csistet, ad a frmula. Let h q be ew prpsitial atms. The, the DA () h fqg W f qg Di has a credulus explaati i T `c h is relevat fr the DA () hfhg fqg W f qg Di i T `c ad h is ecessary fr the DA ( ) hfhg fqg W f _ h qg Di i T 6`c. Sice the reasig prblems fr T i (*), (**) are -hard ad the e i (***) is -hard Gttlb, 199], the hardess results fllw. It is iterestig t te that verifyig a credulus explaati is as hard as dig e. The frmer prblem ca be easily reduced t the latter mrever, is the ly pssible credulus explaati fr the DA (*). Thus, Therem Let = hh M W Di be a DA. Decidig if E H is a credulus explaati fr is - cmplete, with hardess hldig eve fr rmal. Nw csider abducti based skeptical reasig. It wuld be useful t have a reducti f abductive reasig t deductive reasig which ca be cmputed ecietly. Hwever, by usig skeptical reasig the abductive reasig tasks grw mre cmplex, by e level f the plymial hierarchy. This strgly suggests that such a eciet reducti is t pssible. We rst csider the prblem f recgizig skeptical slutis. Clearly, this reduces t decidig if a certai default thery has a csistet extesi (which is i ) ad if each extesi icludes all maifestatis ( ). Thus, the prblem is a lgical cjucti f a prblem i ad a prblem i, ad hece i the class D. Mrever, it is als hard fr this class. Therem Let = hh M W Di be a DA. Decidig if E H is a skeptical explaati fr is D - cmplete. Thus, as i the case f credulus explaatis, recgizig a skeptical explaati is at the secd level f the plymial hierarchy. Hwever, sice this prblems ivlves bth a ad a -hard subtask (as ppsed t ly a -hard e), dig a skeptical explaati resides at the third level. We sketch here the -hardess prf fr Csistecy by a trasfrmati f decidig if a quatied Blea frmula (QBF) = 9X8Y 9ZF is valid (cf. Jhs, 1990] fr a deiti f QBFs). Dee ::a :a :a a j a X Y :F F = hx f:x j x Xg ffg f f F g Di where f is a ew atm. The, has a skeptical explaati i is valid. Hw des this result cmpare t ther mtic lgics, i particular, which mtic lgic has similar cmplexity? We kw that Klige's mderately gruded autepistemic lgic Klige, 1988] ad several ther grud mtic mdal lgics have the same cmplexity Eiter ad Gttlb, 199 Dii et al., 1995] thus, we ca use a therem prver fr such lgics t perfrm abductive reasig frm default theries based skeptical explaatis. 4. Miimal explaatis As metied abve, e is usually iterested i miimal explaatis fr bservatis. The results i Eiter ad Gttlb, 1995] were that the cmplexity f abducti frm classical theries des t icrease if miimal explaatis are used istead f arbitrary explaatis. Hwever, this is t true i fr abducti frm default lgic. Here, checkig miimality f a explaati is a surce f cmplexity, which causes a icrease i cmplexity by e level f the plymial hierarchy. Csider rst credulus explaatis. Checkig miimality f a explaati E has cmplemetary cmplexity f checkig the explaati prperty. Ntice

5 that E is t miimal i fr sme h E, the DA he? fhg M W Di has a credulus sluti hece, it fllws that the prblem is i. O the ther had, recsider (***). Clearly, fhg is a credulus explaati mrever, it is miimal i h is ecessary fr. Thus, -hardess fllws. Nte that recgizig miimal credulus explaatis, which csists i checkig the sluti prperty ad testig miimality, is i D, ad als cmplete fr this class. Thus, this prblem ca be trasfrmed it recgiti f skeptical explaatis fr a certai DA ad vice versa. Due t the cmplexity f miimality checkig, prblem Relevace migrates t the ext level f the plymial hierarchy. Therem 4 Let be a DA based miimal credulus explaatis. The, prblem Relevace is - cmplete, with hardess hldig eve fr rmal. rf. (Sketch) Membership. A guess E fr a miimal credulus explaati fr such that h E ca be veried by tw calls t a racle. Hece, the prblem is i. Hardess. We utlie a reducti frm decidig validity f a QBF = 9X8Y 9ZF. Let s ad q be ew atms, ad dee ::s s^y :q :s q x:x 0 x 0 :x:x0 x 0 :s^:f :q q j x X ::y :y j y Y : Let = hx :X Y fsg X 0 fqg Di. The, e ca shw that s is relevat fr a miimal credulus explaati fr i is valid. Nw let us csider miimal skeptical explaatis. Testig miimality f a skeptical explaati is much mre ivlved tha f a credulus explaati. While the latter has rughly the same cmplexity as testig the explaati prperty, the frmer is harder by e level f the plymial hierarchy. Ituitively, this ca be explaied as fllws. Sice verifyig a credulus explaati E is i, it has a plymial-size \prf" which ca be checked with a N racle i plymial time. Thus, if we ask fr a smaller explaati E 0 E, we ca simultaeusly guess E 0 ad its prf, ad check the prf i plymial time with the N racle. Hwever, verifyig a skeptical explaati E is -hard, ad hece E des t have such a \prf". Here, vericati eeds the full pwer f a racle. Therem 5 Let = hh M W Di be a DA. Decidig if a skeptical explaati E fr is miimal is - cmplete, with hardess hldig eve fr rmal. rf. (Sketch) Membership. A guess fr a smaller skeptical explaati E 0 E ca be veried with tw calls t a racle, ad hece decidig the existece f such a E 0 is i. Csequetly, the prblem is i. Hardess. We describe here a reducti frm decidig whether a QBF = 8X9Y 8ZF is valid. Let s ad q be ew atms, ad dee ::s :s s^x:q q :y y :s::f ^q :F ^q ::y :y j y Y : ::x :x j x X Let = hx fsg fqg Di. Check that E = X fsg is a skeptical explaati fr. Mrever, E is miimal i is valid. Nte that recgizig miimal skeptical explaatis is i, sice the cmplexity f decidig miimality ( ) dmiates the cmplexity f the sluti prperty (\ly" ), ad is als cmplete fr this class. The cmplexity f decidig relevace f a hypthesis icreases by the same amut as testig miimality if skeptical explaatis are used istead f credulus explaatis. I fact, the prblem resides at the furth level f the plymial hierarchy. Therem 6 Let be a DA based miimal skeptical explaatis. The, prblem Relevace is 4 - cmplete, with hardess hldig eve fr rmal. rf. (Sketch) Membership. A guess fr a miimal skeptical explaati E fr such that h E ca be veried with e call t a racle. Hardess. We utlie a reducti frm decidig validity f a QBF = 9R8X9Y 8ZF, which is a extesi t the reducti i the prf f Therem 5. Let as there be s ad q ew atms, ad dee r D1 = D 0 :r^r 00 r^r 00 :r0 ::r^r 00 :r^r 00 j r R where D is the same set f defaults as i the prf f Therem 5. Dee = hh R 00 fqg D1i, where H = R 0 :R 0 X fsg. (Nte that if W wuld be empty, the wuld be idetical t the DA i the prf f Therem 5). It hlds that fr each subset R1 R, the set R1 0 :(R? R1) 0 X fsg is a skeptical explaati fr. Mrever, it ca be shw that s is relevat fr a miimal skeptical explaati fr i is valid. There is well-kw mtic lgic that has similar cmplexity, ad thus e ca t take advatage f therem prvers fr such lgics t perfrm skeptical abducti frm default theries. 4. Tractable cases Frm the practical side, the results frm abve are discuragig, sice abducti frm default theries has eve higher cmplexity tha deducti, i particular fr skeptical explaatis. The reasig tasks suer frm several itermigled surces f cmplexity, whse umber is (at least) the level at the plymial hierarchy. Fr example, Relevace fr = hh M W Di usig miimal skeptical explaatis (cmplete fr 4 ) suers frm the fllwig fur \rthgal" surces f cmplexity: (1.) classical deductive iferece (j=), (.) the umber f extesis f hw E Di, (.) the umber f cadidates E fr a skeptical explaati, ad (4.) the umber f pssible smaller explaatis, where each umber ca be expetial. Fr dealig with abducti frm default theries i practice, we have t d tractable cases r cases where gd algrithms fr hadlig hard prblems like GSAT Selma et al., 199] are applicable. A example f the latter case is credulus abducti frm default theries where all prpsitial frmulas

6 are frm a tractable fragmet f the prpsitial laguage, e.g. Hr frmulas r Krm frmulas (clauses with at mst tw literals). I such a case, classical iferece j= vaishes as surce f cmplexity. I particular, the -cmplete abductive reasig tasks fall back t N. Thus, we ca use e.g. GSAT Selma et al., 199], which prvides a gd heuristics fr slvig N-cmplete prblems, t slve the prblems quickly. Fr tractable cases f default abducti, all surces f cmplexity must be elimiated. I particular, the uderlyig default reasig tasks must be tractable. Kautz ad Selma Kautz ad Selma, 1991] ad Stillma Stillma, 1990] gave a very detailed picture f plymial vs. itractable cases f deductive default reasig. Fr the fllwig tw classes f default theries hw Di, they prved tractability f credulus iferece hw Di `c ` f a sigle literal `: Literal-Hr Kautz ad Selma, 1991]: W is a set f literals ad each default i D is Hr, i.e., f frm, where the a i 's are atms ad ` is a literal. a 1^^a k:` ` Krm-pf-rmal Stillma, 1990]: W is a set f Krm frmulas, ad each default i D is f frm :`1^^`k, `1^^`k where all `i's are literals. A atural geeralizati f the prf i Kautz ad Selma, 1991] yields the fllwig. Lemma 1 Let hw Di be a Literal-Hr default thery, ad let `1 : : : ` be literals. The, decidig hw Di `c `1 ^ ^ ` is plymial. Fr Krm-pf-rmal, such a geeralizati is t evidet as hw Di `c `1^ ^` is N-hard. Hwever, it is pssible fr a small cjucti. I what fllws, we call a set L f literals small i jlj c fr sme xed cstat c. Lemma Let hw Di be Krm-pf-rmal, ad let L = f`1 : : : `kg be a small set f literals. The, decidig hw Di `c `1 ^ ^ `k is plymial. Based these tractable cases f credulus default reasig, we btai tractable cases f credulus default abducti. Similar tractability results fr skeptical default abducti are ulikely, sice the uderlyig skeptical iferece hw Di `s ` is c-n-cmplete i bth cases (cf. Kautz ad Selma, 1991] fr Literal-Hr). Literal-Hr default theries I this case, the mai reasig tasks fr credulus abducti are tractable. Therem 7 Let = hh M W Di be a DA based credulus explaatis ad hw Di Literal-Hr. The, Csistecy, Relevace, ad Necessity are plymial. rf. (Sketch) Cstruct a Literal-Hr T 1 = hw bh:h h ::bh :b h :bh b h where D1i, where D1 = j h H each b h is a ew prpsitial atm. The, it ca be shw that has a explaati i W is csistet ad T 1 `c `1^ ^`k, where M = f`1 : : : `kg. By Lemma 1, this ca be decided i plymial time. Thus, Csistecy is plymial. Relevace ad Necessity ca be easily reduced t Csistecy resp. its cmplemet. Ntice that a plymial algrithm fr dig a credulus explaati (eve ctaiig a give hypthesis), ca be extracted frm the prf. Mrever, there is als a plymial algrithm fr dig a miimal credulus explaati. Ideed, a explaati E fr = hh M W Di is miimal i he? fhg M W Di has explaati fr each h E. Thus, fr as abve, e ca check i plytime whether E is miimal ad, if t, d a smaller explaati E1 E. By repeatig this test, we ca miimize E. Therem 8 Let = hh M W Di be a DA where hw Di is Literal-Hr. The, a miimal credulus explaati fr ca be fud i plymial time. Hwever, Relevace based miimal credulus explaatis fr DAs with Literal-Hr default theries ca be shw t be N-cmplete. Krm-pf-rmal default theries Fr this fragmet, we have the fllwig results. Therem 9 Let = hh M W Di be a DA based credulus explaatis such that M = f`1 : : : `kg is small ad hw Di is Krm-pf-rmal. The, Csistecy, Relevace, ad Necessity are plymial. rf. (Sketch) Cstruct a Krm-pf-rmal default thery T = hw Di, where W = fc h h j h Hg D :ch ch ::ch :ch j h H where each c h is a ew prpsitial atm. The, has a explaati i W is csistet ad T `c `1 ^ ^ `k, which are bth plymial. Csistecy f W ad T `c `1 ^ ^ `k ca be decided i plymial time (cf. Lemma ). Hece, Csistecy is plymial. Sice Relevace ad Necessity ca be easily reduced t Csistecy resp. its cmplemet, these prblems are als plymial. Agai, a plymial time algrithm fr dig a explaati ca be extracted frm the prf. Ufrtuately, Therem 9 ca t be geeralized t a arbitrary set M f literals. I fact, due t the N-hardess f hw Di `c `1 ^ ^ ` fr Krm-pf-rmal hw Di, the prblem is N-hard. Iterestigly, the umber f hyptheses i a miimal credulus explaati is buded by the umber f maifestatis. Ituitively, this is explaied by the fact that always a sigle hypthesis ca explai a maifestati. rpsiti Let E be ay miimal credulus explaati fr = hh M W Di where hw Di is Krm-pfrmal ad M = f`1 : : : `g. The, jej jm j. I particular, fr a sigle maifestati (M = f`g), the miimal explaatis csist f sigle hyptheses, if hyptheses are eeded fr a explaati. A csequece f this characterizati ad Lemma is that all miimal credulus explaatis fr a small set M ca be cmputed by exhaustive testig f all subsets E H with jej jm j i plymial time. Therem 10 Give a DA = hh M W Di where hw Di is Krm-pf-rmal ad M is small, all miimal credulus explaatis fr ca be cmputed i plymial time.

7 Csequetly, als Relevace fr miimal explaatis is plymial if M is small. 5 Cclusi ad further research We prpsed a basic mdel f abducti frm default theries, ad aalyzed its cmputatial cmplexity. Mrever, we have shw that credulus abducti frm the previusly kw classes f Literal-Hr ad Krmpf-rmal default theries is tractable. Besides idetifyig further tractable ad maageable cases f default abducti, the fllwig issues are curretly uder ivestigati. The size f a explaati (cf. eg ad Reggia, 1990]) r, mre geeral, its cst, give by the sum f the predeed csts f its hyptheses, ca be used fr further pruig miimal (i.e., redudat) explaatis. Results fr abducti frm classical theries Eiter ad Gttlb, 1995] suggest usig such explaatis, abducti frm default theries yields cmplete prblems fr the classes k ad k O(lg )] f the plymial hierarchy. Ather issue is default lgic with a uderlyig laguage richer tha a plai prpsitial e. A geeralizati f ur abducti mdel t a prpsitial laguage ver atms p(t 1 : : : t ) where the t i are variables r cstats, is straightfrward here, a istace f a abducti prblem reduces t the prpsitial abducti prblem btaied by replacig frmulas with all grud istaces. Sice the gruded prpsitial versi ca be expetially larger, this leads ituitively t a expetial icrease i cmplexity. Thus, abducti frm default theries i this grud laguage is expected t be cmplete fr the expetial aalgues f k, k etc. Ackwledgmet Nicla Lee has bee partially supprted by the EC-US prject "Deus ex Machia" ad by MURST40% prject "Sistemi frmali e strumeti per basi di dati evlute". Refereces Cadli ad Schaerf, 199] M. Cadli ad M. Schaerf. A Survey f Cmplexity Results fr N-mtic Lgics. Jural f Lgic rgrammig, 17:17{160, 199. Chariak ad McDermtt, 1985] E. Chariak ad. Mc Dermtt. Itrducti t Articial Itelligece Dii et al., 1995] F.M. Dii, D. Nardi, ad R. Rsati. Grud Nmtic Mdal Lgics fr Kwledge Represetati. I rc. Wrld Cgress fr AI (WOCFAI-95), Frthcmig. Eiter ad Gttlb, 199] Thmas Eiter ad Gerg Gttlb. Reasig with arsimius ad Mderately Gruded Expasis. Fudameta Ifrm., 17(1,):1{5, 199. Eiter ad Gttlb, 1995] Thmas Eiter ad Gerg Gttlb. The Cmplexity f Lgic-Based Abducti. Jural f the ACM, Jauary Abstract i rc. STACS-9, LNCS 665, pp. 70{79, 199. Eshghi, 1988] K. Eshghi. Abductive laig with Evet Calculus. I rc. 5th It'l Cf. ad Symp. Lgic rgrammig, pp. 56{ Gttlb, 199] Gerg Gttlb. Cmplexity Results fr Nmtic Lgics. Jural f Lgic ad Cmputati, ():97{45, Jue 199. Hbbs et al., 1988] J. R. Hbbs, M. E. Stickel,. Marti, ad D. Edwards. Iterpretati as Abducti. I rc. 6th Aual Meetig f the Assc. fr Cmputatial Liguistics, pp. 95{10, Bual (NY), Jhs, 1990] D. S. Jhs. A Catalg f Cmplexity Classes. I Hadbk f Theret. Cmp. Sc. A, Jsephs et al., 1987] J.R. Jsephs, B. Chadrasekara, Jr. J. W. Smith, ad M.C. Taer. A Mechaism fr Frmig Cmpsite Explaatry Hyptheses. IEEE Trasactis, TSMC-17:445{454, Kakas ad Macarella, 1990] A.C. Kakas ad. Macarella. Database Updates Thrugh Abducti. I rc. VLDB-90, pp. 650{661, Kautz ad Selma, 1991] H. Kautz ad B. Selma. Hard rblems fr Simple Default Lgics. Articial Itelligece, 49:4{79, Klige, 1988] K. Klige. O the Relatiship betwee Default ad Autepistemic Lgic. Articial Itelligece, 5:4{8, 1988, + 41:115, 1989/90. Klige, 199] K. Klige. Abducti versus clsure i causal theries. Art. Itell., 5:55{7, 199. Marek ad Truszczyski, 199] W. Marek ad M. Truszczyski. Nmtic Lgics. Spriger, 199. Nebel, 1994] B. Nebel. Articial Itelligece: A Cmputatial erspective. Nvember T appear i \Lgic ad Cmputati i AI", G. Brewka ed. Ng ad Mey, 1991] H. T. Ng ad R. J. Mey. A eciet rst-rder Hr clause abducti system based the ATMS. I rc. AAAI-91, pp. 494{499. eg ad Reggia, 1990] Y. eg ad J.A. Reggia. Abductive Iferece Mdels fr Diagstic rblem Slvig le, 1989] D. le. Nrmality ad Faults i Lgic Based Diagsis. I rc. IJCAI-89, pp. 104{110. Reiter, 1980] R. Reiter. A Lgic fr Default Reasig. Articial Itelligece, 1:81{1, Selma ad Levesque, 1990] Bart Selma ad Hectr J. Levesque. Abductive ad Default Reasig: A Cmputatial Cre. I rc. AAAI-90, pp. 4{ 48, July Selma et al., 199] B. Selma, H. Levesque, ad D. Mitchell. A New Methd fr Slvig Hard Satisability rblems. I rc. AAAI-9, pp. 440{446. Stillma, 1990] J. Stillma. It's Nt My Default: The Cmplexity f Membership rblems i Restricted rpsitial Default Lgic. I rc. AAAI-90, pp. 571{579, Stillma, 199] J. Stillma. The Cmplexity f rpsitial Default Lgic. I rc. AAAI-9, pp. 794{799.