is caused by a latet utreated frm f syphilis, althugh the prbability that latet utreated syphilis leads t paresis is ly 25%. Nte that the directialiti

Transcription

1 Tempral Reasig with i-abducti Secd Draft Marc Deecker Kristf Va Belleghem Departmet f Cmputer Sciece, K.U.Leuve, Celestijelaa 200A, B-3001 Heverlee, Belgium. marcd@cs.kuleuve.ac.be Abstract Abducti ca be deed as reasig frm bservatis t causes. I the ctext f dyamic systems ad tempral dmais, a imprtat part f the backgrud kwledge csists f causal ifrmati. The secti shws hw i the ctext f evet calculus, dieret reasig prblems i a brad class f tempral reasig dmais ca be mapped t abductive reasig prblems. The dmais csidered may ctai dieret frms f ucertaity, such as ucertaity the evets, the iitial state ad eects f determiistic actis. The prblems csidered iclude predicti, ambiguus predicti, pstdicti, ambiguus pstdicti ad plaig prblems. We csider als applicatis f itegratis f abducti ad cstrait prgrammig fr reasig i ctiuus chage applicatis ad ressurce plaig. 1 Itrducti Abducti has bee prpsed as a reasig paradigm i AI fr fault diagsis [4], atural laguage uderstadig [4], default reasig [14], [34]. I the ctext f lgic prgrammig, abductive prcedures have bee used fr plaig [13], [43], [31, 2], kwledge assimilati ad belief revisi [21], [1], database updatig [20]. [11] shwed the rle f a abductive system fr frms f reasig, dieret frm plaig, i the ctext f tempral dmais with ucertaity. The term abducti was itrduced by the lgicia ad philspher C.S. Pierce ( ) [33] wh deed it as the prcess f frmig a hypthesis that explais give bserved phemea [35] [43]. Ofte Abducti is deed as \iferece t the best explaati" where best refers t the fact that the geerated hypthesis is subjected t extra quality cditis such as (a frm f) miimality r maximality criteri. There are dieret views what a explaati is. Oe view is that a frmula explais a bservati i it lgically etails this bservati. A mre crrect view is that a explaati gives a cause fr the bservati [17]. Fr example, the street is wet may lgically etail that it has raied but is t a cause fr it ad it wuld be uatural t dee the rst as a abductive explaati fr the secd. Ather mre illustrative example is cited frm [37]: the disease paresis 1

2 is caused by a latet utreated frm f syphilis, althugh the prbability that latet utreated syphilis leads t paresis is ly 25%. Nte that the directialities f lgical etailmet ad causality here are ppsite: syphilis is the cause f paresis but des t etail it, while paresis etails syphilis but des t cause it. Yet a dctr ca explai paresis by the hypthesis f syphilis while paresis cat accut fr a explaati fr syphilis. The term abducti has bee used t cver hypthetical reasig i a rage f dieret settigs, frm huma scietic discvery i philsphical treatmets f huma cgiti t frmally deed reasig priciples i frmal ad cmputatial lgic. I a frmal lgic, abducti is fte deed as fllws. Give a lgical thery T represetig the expert kwledge ad a frmula Q represetig a bservati the prblem dmai, a abductive sluti is a frmula E such that E is satisable 1 w.r.t. T ad it hlds that 2 T j= E! Q I geeral, E may be subjected t further restrictis: the afremetied miimality criteria, but mre imprtatly criteria the frm f the explaati frmula. This frmal deiti implemets the lgical etailmet view abductive explaatis. Hwever, i may applicatis f abducti i AI, the thery T describes explicit causality ifrmati. This is tably the case i mdel-based diagsis ad i tempral reasig, where theries describe eects f actis. By restrictig the explaati frmulas t the predicates describig primitive causes i the dmai, a explaati frmula which etails a bservati gives a cause fr the bservati. Hece, fr this class f theries, the lgical etailmet view implemets the causality view abductive iferece. Abducti is a frm f hypthetical reasig. Makig hyptheses makes ly sese whe there is ucertaity, that is whe T des t etirely x the state f aairs f the dmai f discurse. Abducti is a versatile ad ifrmative way f reasig icmplete kwledge ad ucertaity, kwledge which des t fully describe the state f aairs i the wrld. I the presece f icmplete ifrmati, deducti is the reasig paradigm t determie whether a statemet is true i all pssible states f aairs; abducti returs pssible states f aairs i which the bservati wuld be true r wuld be caused. Hece, abducti is strgly related t mdel geerati ad satisability checkig: it is a reemet f these frms f reasig. By deiti, the existece f a abductive aswer prves the satisability f the bservati. But abducti returs mre ifrmative aswers, i the sese that it describes e, r i geeral a class f pssible states f aairs i which the bservati is valid. I the ctext f tempral reasig, Eshghi [13] was the rst t use abducti. He used abducti t slve plaig prblems i the Evet Calculus [23]. This apprach was further explred by Shaaha [43], Missiae et al. [32, 2], [11] ad [18]. Plaig i the evet calculus ca be see as a variat f reasig frm bservatis t causes. Here, the bservati crrespds t the desired al state. The eect rules describig eects f actis prvide the causality ifrmati. The 1 If E ctais free variables, (E) shuld be satisable w.r.t. T. 2 Or, mre geeral, if Q ad E ctai free variables: T j= 8(E! Q). 2

3 causes are the actis t be perfrmed t trasfrm the give iitial state it a al gal state. I Evet Calculus, predicates describe the ccurreces f actis ad their rder (evet = ccurrece f a acti). A abductive explaati fr a gal represetig the al state is expressed i terms f these primitive predicates ad prvides a pla (r pssibly a set f plas) t reach the iteded al state. I [11], this apprach was further reed ad exteded by shwig hw abducti culd be used als fr ther frms f reasig tha plaig, icludig (ambiguus) pstdicti ad ambiguus predicti. This paper als claried the rle f ttal versus partial rder, ad shwed hw t implemet a crrect partial rder plaer by extedig the abductive slver with a cstrait slver CLP(LO) fr the thery f ttal rder (r liear rder). This chapter aims at presetig the abve research results i a simple ad ui- ed ctext. Oe part f the secti is devted t represetig dieret frms f ucertaity i the ctext f evet calculus ad shwig hw abducti ca be used t slve dieret srts f tasks i such represetatis. The tasks that will be csidered are (ambiguus) predicti, (ambiguus) pstdicti ad plaig prblems. We will csider ucertaity the fllwig levels: - the iitial state, - the rder f a kw set f evets, - the set f evets, - the eect f (idetermiate) evets A predicti prblem is e i which the state at a certai pit must be determied give ifrmati the past. A predicti prblem is ambiguus if the al state f the system cat be uiquely determied. A ambiguus predicti prblem arises whe the iitial state is ly partially kw, r whe kwledge abut the sequece f actis previus t the state t be predicted is t r ly partially available, r whe sme f these actis have a determiistic eect. I a pstdicti prblem, the prblem is t ifer sme ifrmati abut the iitial state r the evets usig cmplete r partial ifrmati the state f aairs at later stages. A pstdicti prblem is ambiguus if the iitial state is t uiquely determied by the al state. I a plaig prblem, the set f evets is ukw ad must be derived t trasfrm a iitial state it a desired al state. I all these cases, we illustrate hw abductive reasig ca help t explre the space f pssible evlutis f the wrld. We csider als applicatis f itegratis f abducti ad cstrait prgrammig fr reasig i ctiuus chage applicatis ad ressurce plaig. The utlie f the chapter is as fllws. I secti 2 we mtivate the chice fr rst rder lgic as a represetati laguage. Secti 3 briey discusses hw t cmpute abducti. Secti 4 itrduces a simple variat f evet calculus, ad i several subsectis, dieret kids f ucertaity are itrduced ad dieret applicatis f abducti are shw. Secti 5 prpses a partial rder plaer based a itegrati f abducti ad a cstrait slver fr the thery f liear rder. Secti 6 csiders applicatis f a itegrati f CLP(R) ad abducti fr reasig ctiuus chage ad ressurce plaig. Secti 7 briey explres the limitatis f abductive reasig. 3

4 2 The lgic used: FOL + Clark Cmpleti = OLP-FOL We will use classical rst rder lgic (FOL) t represet tempral dmais. Fr a lg time, FOL was csidered t be usuitable fr tempral reasig. As Mc- Carthy ad Hayes pited ut i [27], the mai prblem i tempral reasig is the s-called frame prblem: the prblem f describig hw actis aect certai prperties ad what prperties are uaected by the actis. At the ed f the seveties, FOL was believed t be iapprpriate fr slvig the frame prblem due t its mticity [27]. These prblems have bee the mai mtivati fr mtic reasig [26, 28, 3]. Hwever, i the begiig f the 0-ties, several authrs prpsed slutis fr the frame prblem based Clark cmpleti, als called explaati clsure [42, 40]. The priciple is simple ad well-kw. Give a set f implicatis: 8X i :p(t i ) i that we thik f as a exhaustive eumerati f the cases i which p is true. The cmpleted deiti f this predicate is the frmula: 8X:p(X) $ (X 1 :X = t 1 ^ 1) _ :: _ (X :X = t ^ ) A variat f cmpleti is used i Reiters situati calculus [40], curretly e f the best explred tempral reasig frmalisms. Als tempral reasig appraches i lgic prgrammig as i [43, 11, 41] ca be uderstd as classical lgic appraches usig cmpleti. Cmpleti plays a crucial rle i the theries that we will csider, bth the declarative level ad the reasig level. The lgic theries csidered here essetially csist f cmpleted deitis ad ther rst rder lgic axims. Cmpleted deitis will be writte as sets f implicatis r rules, i ucmpleted frm, as i: 8 >< >: p(t 1 ) 1 :: p(t ) Smetimes, whe a deiti csists f grud atms, we will write als: >= >; p 4 = p(t 1 ); ::; p(t ) We call such a set f rules a deiti. A thery csistig f (cmpleted) deitis ad FOL axims will be deted as i: >< >: p(f(x)) 8Z:q(X; Z) :: p(b)). 8X:p(X)! Z:q(X; Z) >= >;

5 Uless explicitly metied, we always iclude the Clark Equality Thery (CET) [5] r the uique ames axims [38]. Hece, we assume that tw dieret terms represet dieret bjects. We assume the reader t be familiar with sytax ad mdel sematics f classical lgic. Sme detatial cvetis: variables start with a capital; cstats ad fuctrs with a small letter; free variables i a rule r a axim are assumed t be uiversally quatied. Predicates which have a cmpleted deiti, will be called deed, therwise, they are called pe. S, i a FOL thery withut cmpleted deitis, all predicates are pe. Ofte sme further sytactical restrictis will be applied. Dee a rmal clause p(t) F as e i which F is a cjucti f literals, i.e. f atms q(s) r egated atms :q(s). As fte, the cjucti symbl is deted by the cmma. A rmal deiti is a set f rmal clauses with the same predicate i the head. A rmal axim is a deial f the frm l 1 ; ::; l i which l i are psitive r egative literals; its lgical meaig is give by the frmula 8(:l 1 _::_:l ). A rmal thery csists f rmal deitis (e deiti per deed predicate) ad rmal axims. Imprtat is that every deiti ad FOL axim ca be trasfrmed i a equivalet rmal e usig a simple trasfrmati, the Llyd-Tpr trasfrmati [25]. By the detatial cveti f represetig a deiti as a set f rules withut explicit cmpleti, rmal theries sytactically ad sematically crrespd t Abductive Lgic Prgrams r Ope Lgic Prgrams [8] 3 uder the 2-valued cmpleti sematics f [6]. As a csequece f this, abductive prcedures desiged i the ctext f ALP ca serve as special purpse abductive reasers fr FOL but tued t deitis. 3 Abducti fr FOL theries with deitis The abducti that will be used here is tued t the presece f cmpleted deitis; we will refer t it as i-abducti. Give a thery T ctaiig deitis ad FOL axims ad a bservati Q, i-abducti geerates a explaati frmula fr Q csistig ly f pe predicates such that T j=! Q ad is csistet with T. Essetially the cmputati f this ca be thught f as a prcess f repeatedly substitutig deed atms i Q by their deiti (ad pssibly drppig disjucts frm the deiti) util a explaati frmula i terms f the pe predicates ca be derived which etails the bservati Q. I case T ctais FOL axims, the FOL axims are reduced simultaeusly with the query such that the resultig explaati frmula als etails the FOL axims. This frm f abducti related t cmpleted deitis was rst extesively described i [6]. It shws strg crrespdece with gal regressi [40], a reasig techique fr situati calculus based rewritig usig cmpleted deitis. Thugh i-abducti implemets the etailmet view abducti (see secti 3 These tw terms refer t dieret kwledge theretic iterpretatis f sytactically the same frmalism. Whereas ALP is deed as the study f abductive reasig i lgic prgrams, OLP- FOL is deed as a lgic t express deitis ad axims, ad as a sub-frmalism f FOL with cmpleted deitis. See [8]. 5

6 1), it will geerate causes fr bservatis whe the set f deitis is desiged apprpriately. Ideed, the desig f the deitis may have subtle, extra-lgical iuece the abductive reasig. Csider the fllwig example. We represet the fact that streets are wet i it rais, ad it rais i there are saturated cluds. Each f these tw simple equivaleces ca be deted as deitis i tw dieret directis. Fr example, this ifrmati ca be represeted as the fllwig theries. Bth csist f tw deitis: h streets wet rai ; rai saturated cluds i but als as: h rai streets wet ; saturated cluds rai i Bth theries are lgically equivalet; evertheless, i bth cases i-abducti will geerate dieret aswers fr the same queries. Fr example, the bservati streets wet will be explaied by saturated cluds i the rst thery, but by itself as a primitive fact i the secd thery. Satisfactry causal abductive explaatis will ly be geerated usig theries with deitis where the directi f are lied up with the causal arrw. The abve example shws that extra-lgical aspects may be ivlved i the desig f deitis. The directiality f the deitis determies the reducti ad rewritig prcess. By desigig the deitis alg the arrw f causality, i-abducti will implemet the causality view abducti, althugh its frmal characterisati crrespds t the lgical etailmet view f abducti. Crrect use f i-abducti impses a methdlgical requiremet: that rules i the deiti fllw the directi f causality. Ather example shws the disticti betwee deiti rules ad lgical implicatis. We represet that e is walkig implies that e is alive; t be br causes that e is alive. Obviusly, the rst implicati is t a causal rule, while the secd e is. Csider the fllwig thery: " ( alive br ) # alive walkig Give this thery, tw i-abducti explaatis fr alive are br ad walkig. Oly the rst e is a causal explaati; the secd e is t. This leads t a secd methdlgical requiremet: -causal implicatis shuld t be added tgether with causal rules i e deiti. A crrect represetati is: h alive br ; alive walkig I this example, the sluti geerated by i-abducti fr alive is br; fr walkig it is walkig ^ br. These are atural ad iteded aswers. Ideed, what the implicati represets is that alive is a ecessary precditi fr walkig; the deiti expresses that t be br is the ly cause fr beig alive. Hece, t be i 6

7 br is a ecessary (but t suciet) precditi fr beig walkig 4 We discuss sme restrictis f i-abducti. First, te that s far we assumed a set f causal rules t be exhaustive. Oly if a set f rules prvides a exhaustive eumerati f the causes, this set f rules ca be crrectly iterpreted as a deiti. Assume that fr a certai bservable p, ly a icmplete set f causes represeted by a set f rules p 1; :::; p is kw. Because this set is icmplete ad there may be ther causes fr p, the cmpleti f this set is icrrect. T abductively explai p, we wat explaatis usig each f these rules but als thers i which p is caused by sme ukw cause. The latter sluti will t be btaied if the set f kw causes is iterpreted as a deiti. There is a simple techique t exted i-abducti i case f icmplete kwledge causal eects. Oe pssibility is that e itrduces a ew symbl, e.g. pe p, adds the rule p pe p t the rule set f p ad adds the FOL axim pe p! : 1 ^ :: ^ : t the thery. pe p ca be thught f as the sub-predicate f p caused by the ukw causes f p. This srt f traslati was rigially metied i [1]. Iabducti will the prduce aswers usig the kw causes, but will als geerate aswers i terms f the ukw causes. Secd, aswers geerated by i-abducti lgically etail the explaied bservati. Recall the syphilis example f secti 1: causal explaatis d t ecessarily etail the bservati. I secti 4.3, we will see examples with a similar avur, ivlvig actis with determiistic eects. Als this srt f causal explaati ca be easily implemeted with i-abducti. We illustrate it with the example f the itrducti. Syphilis pssibly causes paresis ad it is the ly cause. We culd thik f this situati as that paresis is caused by syphilis i cmbiati with sme ther uticeable primitive cause. Fr this residual part f the cause, we itrduce a ew predicate, here simply bad luck. With this ew ccept i mid, the fllwig deiti is a crrect represetati, beyig the methdlgical requiremet fr represetig causal rules usig deitis: paresis utreated syphilis; bad luck I the area f Abductive Lgic Prgrammig, algrithms have bee desiged which cmpute i-abducti fr cmpleted deitis r fr sets f rules uder 4 Nte that i this example, there seems t be a cict betwee the causality view ad the lgical etailmet view abducti. I the secd view, the hypthesis walkig is a crrect explaati fr alive, while clearly it is t a cause fr it. I-abducti is csistet with the causality view ad will ly geerate the explaati br. Thugh i-abducti des t geerate the explaati walkig, it is still csistet with the lgical etailmet view i the weaker sese that it geerates a lgically mre geeral sluti. Ideed, br is lgically mre geeral tha walkig because the thery etails walkig! br; the set f pssible states f aairs i which walkig is true is a subset f the set f states f aairs i which br is true. 7

8 strger sematics such as stable ad well-fuded sematics. Fr a verview f these abductive algrithms, we refer t [10]. The mst direct implemetati f iabducti is the algrithm f [6]; it is based rewritig a frmula by substitutig the righthadside f their cmpleted deiti fr deed atms util a frmula is btaied i which ly pe predicates ccur. There are several prblems which makes this algrithm usuitable fr may abductive cmputatis. Oe is that it is ly applicable t -recursive (sets f) deitis; ather e is that this algrithm des t prvide itegrated csistecy checkig f the geerated aswer frmula. Imprved implemetatis f i-abducti are fud i SLDNFA [, 10] ad the i-prcedure [15]. Bth algrithms ca be see as extesis f the SLDNFalgrithm [24] which prvides the uderlyig prcedural sematics fr mst curret Prlg systems. We fcus SLDNFA; belw, we describe the aswers geerated by SLDNFA ad its crrectess results. I secti 3.1, we give a brief verview f the algrithm. The abductive aswers that will be csidered here have a particular simple frm. Give is a OLP-FOL thery T csistig f deitis D ad FOL axims T, ad a query Q t be explaied. Deiti 3.1 A grud abductive aswer is a pair f a set f grud atmic deitis fr all pe predicates, pssibly ctaiig sklem cstats, ad a substituti such that: D [ j= 8((Q)), D [ j= T ad D [ is csistet. Nte that the existece f a grud abductive aswer prves the csistecy f (Q). I may cases, the pe predicates capture the essetial, primitive features f the prblem dmai. These ccepts are the features i terms f which the thers ca be deed. As a csequece, the set, which gives a exhaustive eumerati f all primitive pe predicates, ca be csidered as a simple descripti f a sceari i which the bservati wuld be true. Cmputatis f SLDNFA r f the i-prcedure retur pssibly cmplex explaati frmulas 5 i a rmal frm, ut f which a aswer i the frm f a grud atmic aswer ca be straightfrwardly extracted. The crrectess therem states a slightly weaker result tha required i deiti 3.1: i geeral it cat be prve that D [ is csistet w.r.t. 2-valued sematics; hwever, D [ is csistet w.r.t. t a 3-valued cmpleti sematics. Icsistecy f (sets f) deitis is due t egative cyclic depedecies. A bvius example is the deiti p :p. Frm a theretical pit f view, abductive reasers used fr reasig i 2-valued lgics shuld perfrm csistecy checkig f the deitis. Whereas i-abducti thrugh rewritig usig deitis ly accesses ad expads deitis relevat fr the explaadum, csistecy checkig f a thery icludig may deitis requires that als irrelevat 5 These frmulas satisfy the prperty that D j= 8(! Q) ad D j= ( ) w.r.t. 3-valued cmpleti sematics. 8

9 deitis are prcessed. This ca be very cstly. Frtuately, this geeral csistecy checkig is uecessary i may cases. Ideed, fr a brad class f deitis, csistecy is kw t hld 6. Fr example, this is the case with hierarchical ad acyclic rule sets [1]. Als the deitis used i the tempral theries csidered i the fllwig sectis, have the csistecy prperty. The fllwig deiti frmalises the csistecy prperty. Deiti 3.2 Give is a thery D csistig f deitis, J a class f iterpretatis f the fucti symbls ad the pe predicates. T is i-deitial w.r.t. J i fr each J 2 J, there exists a uique mdel M f D that cicides with J the fucti ad pe symbls. Therem 3.1 Let D be a acyclic set f deitis [1], J the class f Herbrad iterpretatis f the fucti symbls ad the pe predicates. D eff is i-deitial w.r.t. J. This therem is prve i [1] 7. Therem 3.2 Let T = D [ T be a thery, D a set f deitis which is ideitial w.r.t. t a class J f iterpretatis f pe ad fucti symbls. Let (; ) be a SLDNFA-aswer geerated fr a query Q. If there exists a mdel f amg J the (; ) is a crrect grud abductive aswer fr Q w.r.t. T. Prf The crrectess therem f SLDNFA states that 8 : D [ j= 8((Q)); D [ j= T. It suces t prve that D [ is csistet. But this is trivial, sice there is a mdel f amg J ad this mdel ca be exteded t a uique mdel f D, sice D is i-deitial w.r.t. J. 2 Whereas the rle f abducti is t search fr e r fr a class f pssible state f aairs f the prblem dmai which satisfy a certai prperty, the rle f deducti is shw that all pssible states f aairs satisfy a give prperty. A imprtat prperty f SLDNFA ad i-prcedure is that they have the duality prperty. Give a thery T ad a query Q t be explaied, they satisfy the fllwig prperty: Deiti 3.3 (Duality prperty) If failure ccurs i ite time the it hlds that T j= 8(:Q). 6 I certai applicatis f lgic prgrammig (fte uder stable sematics), egative cyclic depedecies are explicitly explited t represet itegrity cstraits. Fr such applicatis, reasers are eeded that d perfrm csistecy checkig f the deitis. 7 [1] prves that the 2-valued cmpleti f a acyclic lgic prgram has a uique Herbrad mdel. 8 I [10], these tw results are prve fr 3-valued sematics. Hwever, because a 2-valued mdel f the cmpleti is als a mdel i 3-valued cmpleti, these results hld als fr 2-valued cmpleti.

10 This duality prperty is at the same time a cmpleteess result fr i-abducti. The duality prperty is imprtat: it implies that these algrithms ca be used t ly fr abducti but als fr deducti tued t i-deitis. If the abductive reaser fails itely the query :Q, the this is a prf fr Q. I the applicatis belw, this duality prperty will be explited fr therem-prvig. Nte that we view these abductive prcedures as special purpse reasers t reas FOL theries with cmpleted deitis. S, we avid all epistemlgical prblems ccerig the rle f LP ad ALP i kwledge represetati, the ature f egati as failure ad mre f these. 3.1 A algrithm fr i-abducti The SLDNFA prcedure is a abductive prcedure fr rmal theries 10. We will call the cjuctis i PG a psitive gal, a rmal axim i N G a egative gal. Bth psitive ad egative gals may have -pssibly shared- free variables. SLDNFA als maitais a stre f abduced pe atms. The algrithm tries t reduce gals i PG t the empty gal ad tries t build a itely failed tree fr the gals i N G. Iitially, N G ctais all rmal FOL axims, ad PG ctais the iitial query. At each step i the cmputati, e gal ad a literal i it is selected ad a crrespdig cmputati step is perfrmed. Belw we sketch the steps: Whe a pe atm A is selected i a psitive gal A ^ Q, A is stred i the set ad Q is substituted fr A ^ Q i PG. Whe a deed atm A is selected i a psitive query A ^ Q, the e f the rules H B deig the predicate f A is selected, the mst geeral uier f A ad H is cmputed, ad A ^ Q is replaced by (B ^ Q) i PG. Als, because may bid free variables, is applied all frmulas ivlved i PG; N G ad. Whe a egative literal :A is selected i a psitive gal :A^Q, the latter gal is replaced by Q i PG ad A added t N G. Aalgusly, whe a egative literal :A is selected 11 i a egative gal 8X: A; Q, the the cmputati prceeds determiistically by either deletig the egative gal ad addig A t PG, r substitutig 8X:Q fr the egative gal 8X: A; Q i N G. Assume a deed atm A is selected i a egative gal 8X: A; Q. I that case, all reslvets f 8X: A; Q ad all rules H B f the deiti f A are cmputed ad are added t N G. Hwever, i these resluti steps, the free variables f the egative gal e had ad the uiversal variables f the egative gal ad the variables f the rules the ther had must be Thugh deducti i FOL is semi-decidable, SLDNFA ad the i-prcedure are t cmplete fr deducti. 10 Recall that these csist f rmal axims ad e deiti per deed predicate csistig f rmal rules. 11 :A may be selected ly whe A ctais uiversally bud variables. Otherwise, the cmputati termiates i errr. This errr state is called uderig egati. 10

11 treated dieretly. We illustrate this with a simple example. Csider the deiti: p(f(g(z); V )) q(z; V ) ad the executi f the query :p(f(x; a)), where X is a free variables. Belw, the selected atm at each step is uderlied. Oly the mdied sets PG; N G ad at each step are give. Iitially N G ad are empty. PG = f:p(f(x; a))g ; N G = fg ; = fg Switch t N G PG = fg ; N G = f p(f(x; a))g Negative resluti T slve the egative gal p(f(x; a)), the terms f(x; a) ad f(g(z); V ) must be uied. Nte that if we make the default assumpti that 8Z: X = g(z), the the uicati fails ad therefre :p(f(x; a)) succeeds. S, this assumpti 8Z: X = g(z) yields a sluti. But i geeral, X may appear i ther gals; t succeed these gals, it may be ecessary t uify X with ther terms at a later stage. Assume that due t sme uicati, X is assiged a term g(t). I that case, we must retract the default assumpti ad ivestigate the ew egative gal q(t; a). Otherwise, if all ther gals have bee slved, we ca cclude the SLDNFA-refutati as a whle by returig 8Z:X 6= g(z) as a cstrait the geerated sluti. As we will shw, addig these cstraits explicitly may be avided by substitutig a ew sklem cstat fr the variable X. SLDNFA btais this behaviur as fllws. First the uicati algrithm is executed the equality f(x; a)) = f(g(z); V ), prducig fv = a; X = g(z)g. The part with uiversally quatied variables fv = ag is applied as i rmal resluti. The part with the free variables fx = g(z)g which ctais the egati f the default assumpti, is added as a residual atm t the reslvet ad the resultig reslvet 8Z: X = g(z); q(z; a) is added t N G. The selecti f the etire gal ca be delayed as lg as value is assiged t X. Whe such a assigmet ccurs ad fr example the term g(t) is assiged t X, the the gal g(t) = g(z); q(z; a) reduces t the egative gal q(t; a) which the eeds further ivestigati. Otherwise, further refutati is eeded. Fially csider the case that a pe atm A is selected i a egative gal 8X: A; Q. We must cmpute the failure tree btaied by reslvig A with all abduced atms i. The mai prblem is that the al may t be ttally kw whe the gal is selected. We illustrate the prblem with a example. Csider the prgram with pe predicate r: q r(x); :p(x) p(x) r(b) Belw, a SLDNFA refutati fr the query r(a) ^ :q is give. PG = fr(a) ^ :qg Abducti PG = f:qg ; = fr(a)g Switch t N G 11

12 PG = fg ; N G = f qg Negative resluti N G = f r(x); :p(x)g Selecti f abducible atm If r was a deed predicate the at this pit we shuld reslve the selected gal with each clause f the deiti f r. Istead, we are cmputig a deiti fr r i. Therefre, the atm r(x) must be reslved with all facts already abduced r t be abduced abut r. The prblem w is that the set fr(a)g is icmplete: ideed, it is easy t see that the resluti f the gal with r(a) will ultimately lead t the abducti f r(b). Hece, the failure tree cat be cmputed cmpletely at this pit f the cmputati. SLDNFA iterleaves the cmputati f this failure tree with the cstructi f. This ca be implemeted by strig the tuple ((8X: A; Q) ; D) where D is the set f abduced atms which have already bee reslved with A. Belw, the set f these tuples is deted N AG. We illustrate this strategy the example. At the curret pit i the cmputati, N AG is empty ad the ly abduced fact that ca be reslved with the selected gal is r(a). The tuple ((8X: r(x); :p(x)); fr(a)g) is saved i N AG ad the reslvet :p(a) is added t N G: N G = f :p(a)g ; N AG = f((8x: r(x); :p(x)); fr(a)g)g Switch t PG PG = fp(a)g, N G = fg Psitive resluti PG = fr(b)g Abducti PG = fg ; = fr(a); r(b)g N AG gal selected Due t the abducti f r(b), ather brach startig frm the gal i N AG has t be explred: N G = f :p(b)g ; N AG = f((8x: r(x); :p(x)); fr(a); r(b)g)g Switch t PG PG = fp(b)g ; N G = fg Psitive resluti PG = fr(b)g Abducti PG = fg At this pit, a sluti is btaied: all psitive gals are reduced t the empty gal, the set f egative gals is empty ad with respect t, a cmplete failure tree has bee cstructed fr the egative gal i N AG. I geeral, the cmputati may ed whe the set PG is empty, each egative gal i N G ctais a irreducible equality atm X = t with X a free variable, ad fr each tuple ((8X: A; Q); D) i N AG, D ctais all abduced atms f that uify with A. A grud abductive aswer ca be straightfrwardly derived frm such a aswer, by substitutig all free variables by sklem cstats, ad mappig t a set f deitis fr all pe predicates. 12

13 4 A liear time calculus Kwalski ad Sergt prpsed the rigial evet calculus (EC) [23] as a frmalism fr reasig abut evets with durati, abut prperties iitiated ad termiated by these evets ad maximal time perids durig which these prperties hld 12. Mst subsequet develpmets f the EC used a simplied variat f the rigial EC based time pits istead f time perids. This simplied evet calculus EC was applied t prblems such as database updates [22], plaig [13, 2], explaati ad hypthetical reasig [43, 36], mdelig tempral databases [47], air trac maagemet [46], prtcl specicati [12]. I this paper, we will use the Evet Calculus as deed i [45]. I this evet calculus, the tlgical primitive is the time pit rather tha the evet. The basic predicates f the laguage f the calculus are listed belw. The laguage icludes srts fr time pits, uets, actis ad fr ther dmai depedet bjects: happes(a; t): a acti a ccurs at time t. t 1 < t 2 : time pit t 1 precedes time pit t 2. hlds(p; t): the uet p hlds at time t. clipped(e; p; t): the uet p is termiated durig the iterval ]e; t[. clipped(p; t): the uet p is termiated befre t. pss(a; t): the acti precditis f acti a hld at time t. iitially(p): p is true iitially. iitiates(a; p; t): a acti a at time t is a cause fr the uet p t becme true 13. termiates(a; p; t): a acti a at time t is a cause fr p t becme false 14. icmpatible(a 1 ; a 2 ; t): actis a 1 ; a 2 cat ccur simultaeusly at time t. Deiti 4.1 A state frmula i time variable T is ay frmula i which T is the ly variable f srt time ad each ccurrece f T i is free ad ccurs i a atm hlds(p; T ) with p a uet term. The EC theries csidered here csist f the fllwig parts: 12 The rigial evet calculus icluded rules e F which derived the existece f a evet e previus t sme bserved fact F caused by e. Such rules d t match the causality arrw. As a csequece, abductive reasig i the frm described here is quite useless because it wuld explai certai evets i terms f facts caused by them. 13 Nte that p may be already true at time t. 14 p may be false at time t. 13

14 T per : this is the law f iertia ad csists f the fllwig deiti fr the predicate hlds 15 8 >< >: hlds(p; T ) hlds(p; T ) iitially(p ) ^ :T 1 ; A: happes(a; T 1 ) ^ T 1 < T ^ termiates(a; P; T 1 ) happes(a; E) ^ E < T ^ iitiates(a; P; E)^ :T 1 ; A 1 : happes(a 1 ; T 1 ) ^ E < T 1 < T ^ termiates(a 1 ; P; T 1 ) >= >; T T O : a thery expressig that < is a strict liear r ttal rder the time pits. The axims express atisymmetry, trasitivity ad liearity: T 0 < T 1 ; T 1 < T 0 T 0 < T 2 T 0 < T 1 ^ T 1 < T 2 T 0 < T 1 _ T 0 = T 1 _ T 1 < T 0 The rigial EC ad may prpsals usig variats f the EC require implicitly r explicitly that time is a partial rder but the liearity requiremet is abset. I secti 4.2, we argue that a umber f amalies reprted i [11] ad i [2] i the ctext f plaig i Evet Calculus are slved by addig this thery 16. D eff : this thery csists f deitis fr iitiates ad termiates. They csists f eect rules f the frm: iitiates(a; p; T ) termiates(a; p; T ) where is a state frmula i T, p a uet term, a a acti term. ; p ad a may share variables. These rules describe iitiatig ad termiatig eects f actis. As usual i tempral reasig, we assume that these rules exhaustively describe the eects f the actis the uets; uder this assumpti, the cmpleti f the rule sets f the predicates iitiates ad termiates hld. 15 Trasfrmig these rules t rmal frm usig the Llyd-Tpr trasfrmati wuld itrduce tw ew predicates clipped=2 ad clipped=3 : 8 < : hlds(p; T ) iitially(p ) ^ :clipped(p; T ) hlds(p; T ) happes(a; E) ^ E < T ^ iitiates(a; P; E)^ :clipped(e; P; T ) clipped(e; P; T ) happes(a; T 1) ^ E < T 1 < T ^ termiates(a; P; T 1) clipped(p; T ) happes(a; T 1) ^ T 1 < T ^ termiates(a; P; T 1) = ; 16 Nte that the axims f T T O are satised i appraches such as [45] i which time is ismrphic with the atural umbers r real umbers. As a csequece, i these appraches, the amalies discussed i secti 4.2 d t appear. 14

15 T pre : the acti precditi thery csists f the acti precditi axim A pre which expresses that pss(a; T ) is a ecessary precditi fr a acti A t happe at time T : happes(a; T )! pss(a; T ) ad a deiti D pss f the predicate pss csistig f rules f the frm: pss(a; T ) with a a acti term, a state frmula i T. a ad may share free variables. We assume that this set f rules exhaustively eumerates the situatis i which a acti may ccur. T state : the state cstrait thery csists f axims f the frm: 8T: [T ] where [T ] is a state frmula i T. They express that the prperty is satised at each time pit. T cc : the ccurrecy thery csists f the axim A cc : happes(a 1 ; T ); happes(a 2 ; T ); icmpatible(a 1 ; A 2 ; T ) ad a deiti D icmpatible csistig f rules icmpatible(a 1 ; A 2 ; T ) where is a state frmula i T. Uless stated therwise, we exclude ccurret actis etirely fr simplicity reass by deig icmpatible as fllws: icmpatible(a 1 ; A 2 ; T ) :A 1 = A 2 T ar : the thery describig the arrative. This thery is a pssibly icmplete descripti f the iitial state, f a umber f evets (acti ccurreces) ad their rder ad f a umber f ther user deed predicates. T ar des t ctai predicates hlds; clipped; iitiates; termiates; pss; icmpatible. This thery may csists f deitis ad f FOL axims, depedig whether cmplete kwledge is available r t. We illustrate the dmai depedet axims i the case f the Turkey Shtig prblem [16]: iitiates(lad; laded; E) ( termiates(sht; alive; E) hlds(alive; E) ) termiates(sht; laded; E) 8 >< >: pss(lad; T ) pss(wait; T ) pss(sht; T ) >= >; 15

16 I the case f the YTS, cmplete kwledge is available iitial state ad evets. T ar csists f three deitis 17 : iitially(alive) happes 4 = happes(lad; t 0 ); happes(wait; t 1 ); happes(sht; t 2 ) < 4 = t 0 < t 1 < t 2 < t 3 All predicates have a deiti; the resultig EC is a executable lgic prgram (uder cmpleti sematics). The thery etails :hlds(alive; t 3 ); this ca be veried by ruig Prlg the query :hlds(alive; t 3 ); the query succeeds. The fllwig therem is imprtat. It allws t prve the csistecy f a brad class f EC's ad the csistecy f abductive slutis geerated by SLDNFA ad i-prcedure w.r.t. 2 valued sematics. Therem 4.1 Let T be ay evet calculus. Let J be the class f iterpretatis I that satisfy T T O ad i which happes is iterpreted by a ite set f atms. The set f deitis fd hlds ; D iitiates ; D termiates ; D pss ; D icmpatible g i T is ideitial w.r.t. the class J. This therem was prve i [45]. I the fllwig subsectis, we shw hw i-abducti is a pwerful istrumet t explre dieret frms f ucertaity. 4.1 Ucertaity the iitial state A well-kw bechmark example with ucertaity the iitial state is the Murder Mystery [2]: iitially the turkey is alive; there is a shtig fllwed by a waitig evet; the the turkey is dead. The prblem assciated with the Murder Mystery sceari is the pstdicti prblem f iferrig that iitially, the gu must have bee laded. Baker used this prblem t shw a prblem with chrlgical miimisati. I this prblem there is full kwledge the evets ad their rder but there is icmplete ifrmati the iitial situati. I ur represetati, iitially is the ly pe predicate. The dmai idepedet ifrmati ad the geeral dmai kwledge with the D eff ; P pss ; D cc are as i the YTS sluti. The deitis f happes, ad <, ad the FOL axims are: happes 4 = happes(shtig; t 1 ); happes(waitig; t 2 ) < 4 = t 1 < t 2 < t 3 iitially(alive) ^ :hlds(alive; t 3 ) Here a sequece t 1 < :: < t, detes the trasitive clsure ft i < t j j1 i < j g. 18 Trasfrmig the FOL axim t rmal frm yields: :iitially(alive) hlds(alive; t 3 ) 16

17 Nte that here the FOL cstrait :hlds(alive; t 3 ) is phrased i terms f a deed predicate ad idirectly cstrais the state f the pe predicate iitially. The abve represetati is crrect ad prvides a crrect sluti t the pstdicti prblem. SLDNFA yields e abductive sluti fr the query true 1 ; the geerated aswer is: iitially 4 = iitially(alive); iitially(laded) Nte that this aswer des t prve that the gu is iitially laded; it ly asserts that it is pssible that the gu is iitially laded. Hwever, the strger cclusi hlds that it is ecessary the gu is laded iitially. SLDNFA ca prve this. The algrithm fails itely the query :iitially(laded). Frm the duality prperty, it fllws that this EC etails iitially(laded). 4.2 Ucertaity the rder f evets Oe way i which ur evet calculus diers frm ther variats is that it ctais the thery f liear/ttal rder as fudametal axim 20. I [23], it is argued that e f the advatages f evet calculus ver situati calculus is that the time precedece f evets ca be a partial rder represetig a icmpletely kw rder f evets. Hwever, as argued i [11], such a represetati leads t icrrect results whe acti eects are iterferig. A example illustrates the prblem: iitially the light is ; at tw dieret times t 1, t 2, a light switch is ipped; the rder f t 1, t 2 is ukw. The predicti prblem t be slved is t ifer that the light is at the al state t 3. We specify the dmai as fllws: iitiates(switch; ; T ) termiates(switch; ; T ) pss(switch; T ) :hlds(; T ) hlds(; T ) happes 4 = happes(t 1 ; switch); happes(t 2 ; switch) < 4 = t 1 < t 3 ; t 2 < t 3 iitially 4 =fg Nte that the deiti f < etails :t 1 < t 2 ^ :t 2 < t 1 ; mrever by the Clark Equality Thery, t 1 6= t 2. As a csequece, the liearity axim f T T O is vilated. The abve thery augmeted with the persistece thery T per ctais deitis fr all predicates ad crrespds t a lgic prgram. This thery etails hlds(; t 3 ); Prlg succeeds the query hlds(; t 3 ). Hece the thery fails t slve the predicti prblem. 1 Nte that it wuld als be atural t drp :hlds(alive; t 3 ) frm the thery ad t give it as a bservati t be explaied as iput t the abductive prcedure. This makes dierece whatsever. 20 As metied earlier, ay apprach i which time is iterpreted by the atural umbers r the real umbers uder stadard rder implicitly ctais the liearity axim. 17

18 The cause fr this amaly is the fact that < is a -liear partial rder, i particular that it etails :t 1 < t 2 ^ :t 2 < t 1 ^ (:t 1 = t 2 ). Because f this fact, the itervals ]? 1; t 1 [= ftjt < t 1 g ad ]t 1 ; t 3 [= ftjt 1 < t < t 3 g are empty. Because is false iitially ad ]? 1; t 1 [ is empty, is false at t 1 ad hece switch iitiates at t 1 ; because ]t 1 ; t 3 [ is prvably empty, there is termiati f durig this iterval, hece hlds(; t 3 ) ca be derived. Represetig a partially kw rder f evets by a prvably -liear partial rder is i geeral t a crrect strategy t represet icmplete kwledge the rder f evets 21. A mre crrect represetati f the arrative is btaied by drppig the deiti f < ad icludig the FOL axim t 1 < t 3 ^ t 2 < t 3 ad T T O. This way, the ucertaity the rder f time pits is represeted by the fact that the resultig thery has mdels i which t 1 < t 2 ad thers i which t 2 < t 1. Hwever, i all mdels, < is a ttal rder ad a liearisati f the set f kw rder atms. Deiti 4.2 A liearisati f a partial rder < a dmai D is ay ttal rder < t D which icludes the partial rder. I.e. if t < t 0 the t < t t 0. ad Abductive reasig the query :hlds(; t 3 ) yields tw slutis: < 4 = t 1 < t 2 < t 3 < 4 = t 2 < t 1 < t 3 The existece f these slutis ly prves that it is pssible that the light is at t 3. Hwever, the thery augmeted with T T O etails that it is ecesseary that the light is at t 3. SLDNFA fails itely the query hlds(; t 3 ). By the duality prperty, it fllws that this EC etails :hlds(; t 3 ). Observe here that i each liearisati f the partial rder ft 2 < t 3 ; t 1 < t 3 g, it hlds that hlds(; t 3 ) is false, while i the partial rder itself, hlds(; t 3 ) is true. 4.3 Ukw eects ad determiism Ather way i which ucertaity ca arise i tempral dmais is by the presece f actis with determiistic eects: the utcme f such a acti is t uiquely determied by the circumstaces ad the way i which it is executed. The state f the wrld at all times f a dmai ivlvig determiistic actis is depedet the iitial state, the ccurrig actis ad their rder, ad ifrmati determiig fr each determiistic acti ccurrece which f the pssible utcmes it has. Typically the latter ifrmati is ukw, therwise we are t talkig abut determiism. As befre, abducti ca fte be used t ll sme gaps i the described kwledge. First f all, we eed a sytax t represet determiistic actis. We assume that each acti has a kw set f pssible utcmes, e f which will be realised 21 This prblem with partial rder time ccurs als i the ctext f plaig. The prblem with the light switch sceari is similar t the amalies reprted i [30, 2] the plaig appraches i [13], [43], [30]. I secti 4.2, we cme back t this issue. 18

19 wheever the acti ccurs. This ca be represeted at a high level by a rule f the frm: A causes E 1 j : : : j E if i which the E i are the pssible utcmes f acti A if it is executed whe hlds. The symbl j detes a disjucti. As a example, csider the Russia turkey shtig prblem, which ivlves a determiistic acti f spiig the chamber f the gu. If the gu is laded, this acti may r may t ulad it: spi causes :laded j true if laded where true detes just the empty eect. Nte the crrespdece betwee this prblem ad the syphilis-pssibly-causesparesis prblem f secti 1. The techique t represet this determiistic causati is the e prpsed i secti 3. A traslati f determiistic rules f the abve type t the preseted calculus eeds t d a way arud the disjuctis, as the calculus des t allw fr disjuctive rules 22. Such a traslati ca be achieved by itrducig \degree f freedm" predicates. These are pe predicates f which the truth r falsehd determie the utcme f each determiistic acti ccurrece. As such, they represet exactly the ifrmati usually missig i a determiistic dmai. As a example, the abve rule represets e determiistic acti with tw pssible utcmes (amg which the empty eect). Hece, there is e degree f freedm ad ly e predicate eeds t be itrduced. If this predicate is true at a certai time, the executi f the acti at that time has the eect f uladig the gu. If it is false, the acti has eect ad the gu remais laded. Adptig the turkey's pit f view, we call this predicate gd luck. It is parameterised with a time pit, t idicate that the acti may have dieret utcmes at dieret times. The traslati f the determiistic eect rule w becmes termiates(spi; laded; T ) hlds(laded; T ); gd luck(t ): I the ther case, i.e. whe hlds(laded; T ); :gd luck(t ) is true, the acti has its secd pssible utcme. Hwever, this is eect at all. Fr this reas there is secd rule i this case. Degree f freedm predicates explicitly represet the missig kwledge which wuld determie the utcme f determiistic actis. As a result, abducti ca be used t derive ifrmati these predicates ad hece the way determiistic actis tur ut. A simple example i the turkey wrld is the fllwig pstdicti prblem: we have a sequece f actis f ladig, spiig, ad shtig, ad we bserve that after the shtig the turkey is still alive. Des this tell us aythig abut the utcme f spiig? Frmally, we get the fllwig deitis: happes 4 = happes(lad; t 1 ); happes(spi; t 2 ); happes(sht; t 3 ) 22 Mrever the iteded disjucti is t really stadard disjucti at the level f the calculus, but a precise discussi f this issue is -tpic. Suce it t say that the traslati we prvide iterprets the disjucti i the right way. 1

20 < 4 = t 1 < t 2 < t 3 < t 4 ad ifrmati the iitial state. The bservati hlds(alive; t 4 ) ca ly be explaied if hlds(laded; t 3 ) is false ad iitially(alive) true. Sice t 1 iitiates laded ad the acti spi is the ly ther acti betwee t 1 ad t 3, spi must have termiated laded agai, ad s gd luck(t 2 ) is abduced. I additi, als iitially(alive) must be abduced t justify the bservati hlds(alive; t 4 ). Ather bechmark example which ca be iterpreted as a prblem ivlvig ukw eects f actis is the stle car example [2]. Smee parks his car at a rather usafe lcati, leaves it there fr tw ights, ad cmes back the third day t d his car ge. I e represetati, the tw ights ca be represeted by tw wait actis. I this frmulati f the prblem, there is cmplete kwledge the iitial state, the evets ad their rder. Nrmally the wait acti has eect. Hwever, give the bservati that the car is stle i the al state ad assumig that we have full kwledge f the evets that ccur, a iteded cclusi is that e f the wait evets had a (abrmal) eect f termiatig parked. A prblem f this srt ivlves default reasig. The desired mtic prperty f the prblem represetati is that withut the bservati that parked is false i the al state, it etails that parked is true i this state; with the bservati icluded i the thery, it shuld etail that e f the wait eects has termiated parked. By FOL's mticity, it fllws that a default f this srt cat be represeted i FOL. Fr example, the fllwig EC: iitially 4 =fg iitiates(park; parked; T ) termiates 4 =fg pss(wait; T ) happes 4 = happes(park; t 1 ); happes(wait; t 2 ); happes(wait; t 3 ) < 4 = t 1 < t 2 < t 3 < t 4 etails hlds(parked; t 4 ) ad hece is icsistet with :hlds(parked; t 4 ). Nte that this EC dees all predicates. The query hlds(parked; t 4 ) ca be prve by Prlg. Oe way t vercme this prblem is t t take the clsure f the eect rules fr iitiates ad termiates. The resultig thery is csistet with hlds(parked; t 4 ) but des t etail it. A abductive sluti fr this query is: termiates 4 =fg iitiates 4 = iitiates(park; parked; t i ) j 1 i 4 20

21 The thery is als csistet with :hlds(parked; t 4 ). A abductive sluti fr this query is: termiates(wait; park; t 2 ) iitiates 4 = iitiates(park; parked; t i ) j 1 i 4 A aalgus sluti exists with termiates(wait; park; t 3 ). SLDNFA fails the query: :hlds(parked; t 4 ) ^ :termiates(wait; parked; t 2 ) ^ termiates(wait; parked; t 3 ) By the duality prperty, this etails that the thery etails the egati f this query, beig: :hlds(parked; t 4 )! termiates(wait; parked; t 2 ) _ termiates(wait; parked; t 3 ) A alterative sluti based determiistic evets culd be the fllwig: we itrduce tw types f \waitig" actis wait ight ad wait day, assumig that durig ay ight, cars ca get stle (parked is termiated). As such, wait ight is a determiistic acti with tw pssible utcmes. Eect rules are w iitiates(park; parked; T ) termiates(wait ight; parked; T ) hlds(parked; T ); bad luck(t ) where bad luck(t ) is a pe predicate. The abve sceari is w represeted as happes 4 = 8 >< >: happes(park; t 1 ); happes(wait ight; t 2 ); happes(wait day; t 3 ); happes(wait ight; t 4 ) < 4 =ft 1 < t 2 < t 3 < t 4 < t 5 g The bservati :hlds(parked; t 5 ) ca w be explaied i ly tw ways: by abducig either bad luck(t 2 ) r bad luck(t 4 ). I ther wrds, the car is deitely stle, but it is t certai durig which ight this has happeed. The thery augmeted with the bservati etails: ad als: termiates(wait ight; parked; t 2 ) _ termiates(wait ight; parked; t 4 ) bad luck(t 2 ) _ bad luck(t 4 ) SLDNFA prves these frmulas by failig their egati. >= >; 21

22 4.4 Cmbiig ucertaity evets ad iitial state I [41], the fllwig simple example was used t demstrate sme errrs with existig versis f evet calculus. Csider the fllwig sceari:: at time t 1 Bb gives a bk at Mary; at time t 2 > t 1, Jh gives the bk at Tm. The cmplexity f this simple sceari lies i the fact that it cmbies state cstraits ad acti precditis with ucertaity iitial state ad evets. The deitis f iitiates, termiates ad pss are: iitiates(give(x; O; Y ); has(y; O); T ) termiates(give(x; O; Y ); has(x; O); T ) pss(give(x; O; Y ); T ) hlds(has(x; O); T ) ^ X 6= Y I this prblem, there is als the state cstrait that i each state a bject has at mst e wer: 8T:hlds(has(X; O); T ) ^ hlds(has(y; O); T )! X = Y The ifrmati the predicates iitially, <, happes describig the arrative, is icmplete. The EC based the cmpleti f the set f kw iitially atms ad kw happes atms etails :hlds(has(bb; bk); t 1 ) ad :hlds(has(jh; bk); t 2 ), hece is icsistet with the acti precditis f the kw give actis. A crrect represetati is give by the FOL axims: happes(give(bb; bk; mary); t 1 ) happes(give(jh; bk; tm); t 2 ) t 1 < t 2 < t 3 A abductive sluti with miimal cardiality fr the query true is give by: happes 4 = 8 >< >: happes(give(bb; bk; mary); t 1 ); happes(give(mary; bk; jh); t 0 ); happes(give(jh; bk; tm); t 2 ) iitially 4 = iitially(has(bb; bk)) >= >; The thery etails 23 : < 4 =ft 1 < t 0 < t 2 < t 3 g hlds(has(bb; bk); t 1 ) hlds(has(mary; bk); t 2 ) X; T:happes(give(X; bk; jh); T ) ^ t 1 < T < t 2 X; T:happes(give(mary; bk; X); T ) ^ t 1 < T < t 2 23 The curret implemetati f SLDNFA cat prve these therems ad lps, due t ctrl prblems. 22

23 A iteded cclusi f this thery culd be als that Tm has the bk at t 3. Oe culd argue that sice ifrmati is available that suggests r implies that Tm has give the bk away after receivig it, a huma expert will fte ifer by default that such a acti has t happeed. The curret thery des t etail the absece f this acti, ad hece des t etail that Tm has the bk at t 3. Hwever, as Ple pited ut i [34], default reasig f this kid ca be perfrmed by abductive reasig. Ideed, bserve that i the abductive slutis which have miimal cardiality r are miimal w.r.t. set iclusi, a acti happes(give(tm; bk; x); t 00 ) with t 2 < t 00 < t 3 will be abset. 5 A cstrait slver fr T T O I the ctext f tempral reasig, abducti was itrduced rigially fr plaig i the Evet Calculus [13]. A plaig prblem is frmulated as a EC describig eects f actis, acti precditis, the iitial state ad by a gal describig the desired gal state. Subject f the search is a sequece f actis that trasfrms the iitial it the gal state. The predicates describig this sequece are happes ad <; they are subject f the plaig search; a frtiri these predicates are pe predicates i the EC describig the plaig dmai. As reprted i [2], early abductive plaig systems ([13], [43], [32]) smetimes geerated erreus partial plas, i which tw iterferig actis (such as ippig a switch) are assumed t be urelated i time. While these erreus plas lgically etail the desired al state, liearisatis f these plas did t i geeral. As a csequece, real executi f the cmputed pla prduces a erreus al state. I priciple, this prblem is slved by icludig the axim f ttal rder i the EC. Usig this techique, a abductive prcedure will geerate ly liear plas. Ufrtuately, liear plaig is very ieciet. I geeral, i a pla may evets d t iterfere with each ther. A liear plaer is ieciet because it geerates a expetial umber f permutatis f these idepedet evets. I geeral, it wuld be desirable t get partial plas which satisfy the crrectess criteri give i [30]: the gal state must be prvable frm each liearisati f the pla. Such a partial pla leaves pe the rder f idepedet evets. Cmpared t a partial plaer, the umber f plas cmputed by a liear plaer is expetial i the umber f idepedet pairs f evets. T slve this prblem, [11] prpsed a extesi SLDNFA-LO with a simple cstrait slver fr <, checkig the satisability f the abduced < atms agaist the thery f ttal rder. This slver is based a simple idea. Prcedurally, it cmputes the trasitive clsure f the abduced < facts. Whe a atm t 1 < t 2 is abduced, it is checked whether t 1 ad t 2 are dieret ad whether the symmetric atm t 2 < t 1 des t already ccur i the precedece relatiship. If e f these situatis happes, the abducti step fails. Otherwise, the atm is abduced ad the trasitive clsure f the exteded relati is cmputed. I additi t this, whe SLDNFA-LO selects a atm t 1 < t 2 i a egative gal, it makes this atm false by abducig the symmetric atm t 2 < t 1. Whe later t 1 < t 2 ccurs i a egative gal, the abducti f t 1 < t 2 fails. As a result, a derivati ever depeds the absece f tw symmetric precedece facts. 23