Extracting Duration Facts in Qualitative Simulation using Comparison Calculus

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1 Exracng Duraon Facs n Qualave Smulaon usng Comparson Calculus Tolga Könk 1 and A. C. Cem Say 2 1: konk@umch.edu Compuer Scence and Engneerng ATL., Unv. Mchgan, 1101 Beal Ave., Ann Arbor, MI, USA 2: say@boun.edu.r Deparmen of Compuer Engneerng, Boğazç Unversy Bebek 80815, İsanbul, Turkey Absrac In a prevous work, we descrbed he duraon conssency fler, whch could exrac nformaon abou he relave lenghs of he me nervals (duraon facs n he oupu of qualave smulaon and use he nconssences among hem o elmnae spurous behavors. In hs paper, we descrbe a new mehod for exracng duraon facs ha canno be obaned by our prevously descrbed mehod. The power of he duraon conssency fler s herefore ncreased. To prove he correcness of our approach, we have developed a new sgn algebra, he Exended Sgn Algebra SR1* and a new comparson formalsm; Comparson Calculus. Inroducon Qualave smulaon algorhms (Weld and Kleer 1990 symbolcally solve ses of ordnary dfferenal equaons, predcng a se of rajecory descrpons, such ha any acual soluon of he equaons n he npu se s guaraneed o mach one of hese predcons n he oupu. Along wh qualave descrpons of all possble behavors ha can be exhbed by sysems n he npu model, qualave smulaors may produce spurous predcons, behavor descrpons whch no such sysem would exhb. These spurous predcons lm he usefulness of qualave reasoners n applcaons lke desgn and dagnoss (Kupers Several echnques (Kupers 1994 (Say and Kuru 1993 (Say 1998 (Say 2001 for reducng he number of spurous behavors n he algorhms oupu have been developed. As a resul of hese sudes, new flerng mechansms have been proposed. The ncorporaon of some of hese flers o he algorhm requres addon of exra ems of nformaon o he npu se (lke sysem-specfc knowledge or resrcons on he space of possble relaonshps beween he sysem s varables (lke he second dervave sgn-equaly assumpon (Kupers In (Könk and Say 1998, we descrbed he duraon conssency fler, whch was able o exrac nformaon abou he relave lenghs of he me nervals (duraon facs n he oupu of qualave smulaon. Ths fler were usng smple echnques employng conceps lke symmery, perodcy, and comparson of he crcumsances durng mulple raversals of he same nerval o buld a ls of facs represenng he deduced nformaon abou relave duraons. These facs were fed o a smple lnear nequaly conssency checker, whch elmnaes proposed spurous behavors leadng o nconssen duraon daa. In hs paper, we propose a new mehod, ponwse comparson, for exracng duraon facs ha canno be obaned by our prevous mehod. The power of he duraon conssency fler s herefore ncreased. To prove he conssency of hs new mehod, we descrbe he relevan pars of he wo mahemacal formalsms ha we have developed, namely, he Exended Sgn Algebra SR1*, and Comparson Calculus. Due o space lmaons, some proofs wll be skpped whle some wll only be oulned. A more horough reamen can be found n (Könk v 0 a ( x dx v = d dv a = d da a = a = c d x: hegh of he ball (ref. frame: elevaor v: velocy of he ball (ref. frame: elevaor a: acceleraon of he ball(ref. frame: elevaor c: a posve consan Fgure 1. Upwards Thrown Ball n Elevaor wh Increasng Acceleraon The Idea Consder he sysem n Fgure 1 depcng an upwards hrown ball n an elevaor wh ncreasng upwards acceleraon. Fgure 2 shows he behavors of he poson x, velocy v and he acceleraon a of he ball wh respec o he reference frame of he elevaor 1. In hs problem, we wan o be able o exrac he duraon facs < auomacally. If hs model s par of a To keep he model smple, we have assumed a consan value for a, bu our echnque s equally applcable as long as a s decreasng durng he compared nervals.

2 bgger sysem ha produces conflcng duraon facs, we can elmnae spurous behavors. We show n (Könk 2000, ha he consrans we presened n (Könk and Say 1998 are no suffcenly srong o exrac hs duraon fac, alhough nuvely s no very dffcul o verfy. If we consder he behavor of wo magnary balls sarng a me 1 a he maxmum hegh, wh zero velocy and movng accordng o he graph n Fgure 2, one backwards n me and he oher forward, we observe ha he ball movng forward n me wll have hgher speed a each correspondng me pon, snce s acceleraon s greaer n magnude a each correspondng me pon. Consequenly, we can conclude ha he magnary ball n he forward drecon wll h he ground n a shorer me compared o he one n he backward drecon, herefore 2 1 < 1 0. In he res of he paper, we wll develop he mahemacal framework ha enables us o oban such resuls auomacally. x 1 x 0 x( I 1 I 2 v 0 v 2 v( a( v 1 I 1 I 2 a a 1 a 2 Fgure 2. Comparson A Change of Drecon The Exended Sgn Algebra SR1* Qualave reasonng algorhms make heavy use of sgns for represenng quanes. A bref overvew of he new sgn algebra SR1* ha we use n our echnques s presened here. A sgn s ofen defned as one of he followng four subses of he se of real numbers R: [ + ] = def (0, +, [ ] = def (, 0, [?] = def (, +, [0] = def {0}. Wllams (1991 has formalzed he Sgn Algebra S1 on he se S = {[ ], [0], [ + ], [?]}, wh he sgn addon and mulplcaon operaons. For example, we have [ + ] + [ ] = [?] and [ + ] [ ] = [ ] n S1. Here are some smple properes of S1 ha we are gong o use n hs paper: Proposon 1. Some Smple Properes of S1 For he sgns s 1, s 2, s3 S1. s1 s2 s1 = s2 f s2 [?]. s1 = s2 s3 s1 s2 = s3 f s2 [0],[?] Wllams SR1 (1991 s an exenson of S1 where real numbers are elemens of he doman n addon o sgns of S. For example, an expresson such as [ + ] + 3 = [ + ] s vald n SR1. Buldng on SR1, we have developed a new sgn/real hybrd algebra called SR1* whch proves useful n he forhcomng dscusson. The doman of SR1* s he I 1 I 2 se of all nonempy subses of R. The followng se operaors are well-defned on SR1*. Defnon 1. Se Operaors For A and B, wo nonempy subses of R,. A B a + b : a A, b B. + def B def = { } = { b : b B}. A B = def A + ( B v. A B def v. A def = { a b : a A, b B} = { a : a A} Lke SR1, SR1* conans all sgns of S and all real numbers 2 as elemens. Unlke SR1, SR1* conans arbrary subses of R as elemens, and he operaor semancs s slghly dfferen. [ + ] + ( 3 s mapped o [?] n SR1 whle s mapped o ( 3, n SR1*. More nformaon abou SR1* and s relaon wh SR1 can be found n (Könk Proposon 2. A Se Theorecal Propery of SR1* Gven A1, A2,..., A n, B1, B2,..., Bn SR1* such ha A1 B1, A2 B2,..., An Bn and Φ, a formula wren usng he bnary and unary operaors n Defnon 1, he followng relaonshp holds: Φ( A, A,..., A Φ ( B, B,..., B 1 2 n 1 2 The sgn absracon operaor [.] n SR1* s smlar o s counerpar n SR1. Gven A SR1*, [A] s he smalles sgn n S such ha A [ A ]. For example, he expressons [(0, + ] = [ + ], [{ 1, + 1}] = [?], and [5] = [ + ] are vald n SR1*. See (Könk 2000 for a formal defnon of he absracon operaor n SR1*. The followng are some properes of he absracon operaor of SR1*. Alhough Wllams (1991 has presened smlar saemens for SR1, hey deserve a separae proof n SR1*. Proposon 3. Absracon Properes n SR1* For any A, B SR1*,. [ A + B] [ A] + [ B]. [ A B] = [ A] [ B]. [ A] = [ A] v. [ A B] [ A] [ B]. A + B [ A] + [ B] snce A [ A], B [ B] (Prop. 2 [ A + B] [[ A] + [ B]] = [ A] + [ B] by frs absracng boh sdes hen usng [ A] + [ B] S. [ A B] [ A] [ B] s smlar o bu full equaly has a 2 As n SR1, he real numbers are consdered o be n SR1*, because all sngleon ses of real numbers are here, and he real numbers are somorphc o hem. For example 3+2=5 s consdered o be n SR1* snce {3}+{2} = {5} s n. n

3 long proof n (Könk usng, by nserng B = { 1} Theorem 1. Absracon of an SR1* Expresson Gven A1,..., An SR1* such ha A1 Φ ( A2,..., A n where Φ s a formula wren usng he operaors{,, + }, we ge: [ A1 ] Φ ([ A2 ],...,[ A n ] [ A1 ] [ Φ ( A2,..., A n ] Φ ([ A2 ],...,[ A n ] usng Prop. 3 repeaedly Defnon 2 Absracon of a Funcon Gven a funcon f, and F, he mage of f on a doman I, we call he sgn valued expresson [ F ] he mage absracon of f on he doman I and for a I we call [ f ( ], he pon absracon of f a. Theorem 2. Absracon of Funconal Relaons Gven n funcons f 1, f 2, f 3,, f n and her mages F 1, F 2, F 3,, F n on a doman I, f we have ( ( (, (,..., n ( f = Φ f f f I such ha Φ s a formula wren usng he operaors{,, + }, we ge: ( [ ] Φ [ ],[ ],...,[ ] F1 F2 F3 F n Usng F1 ( F2, F3,..., Fn Φ and Theorem 1. Proposon 4. Image Absracon of Sgn Mulplcaon of Funcons Gven hree funcons f, g, h and her mages F, G, H on I such ha f ( g ( h ( [ ] = [ ] [ ] I, we ge: [ F] = [ G] [ H ] f [ G] [?] [ H ] [?] Comparson Calculus Comparson Calculus s desgned o accommodae auomac reasonng based on comparsons of behavors of dynamc sysems. I s bul on op of SR1* Algebra descrbed n prevous secon. Gven a QSIM smulaon, we wll have algebrac comparson facs ha can be exraced drecly from a QSIM behavor. These facs wll be fed n he algebrac consrans ha we wll prove n hs secon o deduce new algebrac facs ha represen nformaon abou relave duraons of me nervals n he smulaon. These duraon facs, whch can be also represened as lnear nequales of me pons, can be checked for conssency usng a smple algorhm such as he one used n (Clarke and Zhao 1992 and f an nconssency s deeced, he behavor can be elmnaed. In he remander of hs secon, we presen a subse of Comparson Calculus ha enables one o prove duraon facs such as he one n he accelerang elevaor problem. See (Könk 2000 for a full presenaon of he formalsm, and s oher applcaons such as QSIM fler represenaon or Comparave Analyss. Defnon 3. Real Comparson arables For wo real numbers x 1 and x 2, we defne he real comparson varables x, Σ x, and x such ha:. x = def x 2 x 1. Σ x = def x2 + x1. x = def x2 x1 ( x : absolue value of x Defnon 4. Real Quanes ha Summarze Some Propery of a Funcon on an Inerval For wo connuous funcons f1 and f 2 on nervals I 1 = ( b1, e1 and I 2 = ( b2, e2, we defne he followng quanes (for j = 1, 2 :. bj.. v. v. j f = def lm j ( bj f f ej = def lm f j ( ej f j = def f ej f bj j = def ej bj f = def j ( : nal value of f j on I j : fnal value of f j on I j : change of f j on I j 1 ej j bj : duraon of he me nerval I j f d :average value of f j on I j f bj and f ej wll also be called he end pons of f j on I j The real quanes above wll be nsered no Defnon 3 o oban expressons ha are useful o make comparsons. For example for wo me nervals I 1 and I 2 and her duraons 1 and 2, we ge well-defned sgn valued expressons such as [ ] = [ 2 1 ]. Expressons lke hese allow us o represen comparson facs usng algebrac formulas. For example f we consder I 1 = ( 0, 1 and I 2 = ( 1, 2 n Fgure 2, he duraon fac, 2 1 < 1 0 we wan o exrac can be represened as [ ] = [ ]. Smlarly, f we consder he funcons x 1 and x 2 o be he resrcon of he funcon x on he nervals I 1 and I 2, usng he change varables x 1 and x 2 we oban he algebrac expresson [ x ] = [ x 2 x 1 ]. Ths expresson gves us he change-of-x comparson n wo nervals. For example n Fgure 2, we have [ ] = [0], meanng ha he change of x n he x nerval I 1 s equal n magnude o he change of x n I 2. Theorem 3. Real Comparson Consrans For wo real numbers x 1 and x 2.a. [ x ] = [ x Σ x] = [ x] [ Σ x].b. [ Σ x] = [ x ] [ x] f [ x] [0]

4 .c. [ x] = [ x ] [ Σ x] f [ Σx] [0].a. [ Σ x] = Σ [ x] f Σ[ x] [?].b. [ x] = [ x] f [ x] [?].a [ x ] = [ x x ] = [ x x ] = [ x x ] = [( x2 x1 ( x2 + x1] = [ x Σ x] = [ x] [ Σ x] (Prop. 3..b. usng.a and Prop. 1. snce[ x] [?] ( x s real.c. smlar o.b..a. [ Σ x] = [ x2 + x1] [ x2] + [ x1] = Σ [ x] (Prop. 3. Implyng [ Σ x] = Σ [ x] snce Σ[ x] [?] (Prop. 1..b. smlar o.a The consrans n Thm. 3, are useful o compare real valued varables such as me duraons, funcons a a pon, or change of a funcons n an nerval. Nex, we descrbe how we can compare funcons hroughou nervals. Defnon 5. Comparson Funcons on he Same Doman For wo funcons f 1 and f 2, we consruc he comparson funcons f, Σ f, and f on a common nerval I as follows : f f + f ( I. Σ ( = def 2 ( 1 (. f ( = def f2 ( f1 (. f ( = def f2 ( f1 ( ( I ( I Defnon 6. Same Doman Ponwse Comparson Sgns Gven wo funcons f1 and f 2 and her comparson funcons Σ f, f, and f on a common nerval I, f we le F 1, F 2, Σ F, F, and F be her mages on I, he well-defned sgn expressons [ F], [ Σ F], and [ F ] are called he ponwse comparson sgns. Specfcally, [ F ] s called he ponwse magnude comparson sgn. Moreover, we call [ F 1] and [ F 2 ] he smple sgn consans and usng hem we defne he compound sgn consans as: Σ [ F] = def [ F2 ] + [ F1 ] and [ F ] = def [ F2 ] [ F1 ]. Smlarly, for wo real numbers x 1 and x 2, we wll call he well-defned expressons [ x], [ Σ x], and [ x ] he real comparson sgns and [ x 1], [ x 2], [ x], and Σ [ x], he real sgn consans. Proposon 5. Trval Ponwse Comparson Consrans Gven wo funcons f 1, f 2 compared on a common nerval I, he followng consrans hold:. (a [ F1 ] = [ F2 ] = [0] (b [ F ] = [0] (c Σ [ F] = [0] (d [ F ] = [ F ] = [ Σ F ] = [0]. [ ΣF ] Σ [ F ]. [ F] [ F] v. (a [ F1 ] [?] [ F2 ] [?] (b [ F] [?] Σ[ F] [?] (c [ F] [?] [ ΣF ] [?]. Σ f ( = f2 ( + f1 ( [ ΣF] [ F1 ] + [ F2 ] (Thm. 2. smlar o The nex heorem esablshes he relaon beween he comparson sgns [ F], [ Σ F], and [ F ], gven he sgn consans. Theorem 4. Ponwse Comparson Consrans Gven wo funcons f 1, f 2 compared on a common nerval I,. [ F ] = [ F] [ Σ F] f [ F] [?] Σ[ F] [?]. [ Σ F ] = Σ [ F ] f Σ[ F] [?]. [ F] = [ F ] f [ F] [?] v. [ Σ F ] = [ F ] [ F] f [ F] [?] v. [ F] = [ F ] Σ [ F] f Σ[ F] [?]. For wo funcons f 1 and f 2 we ge ( I : ( ( ( [ f ] = [ f ] [ Σ f ] (Thm.3..a, Def. 5 Proved by Prop. 4 usng (b(c n Prop.5.v,.by Proposon 5, usng Proposon 1. v. [ F] = [ F ] [?] usng [ F] = [ F] = [0] s rval by Prop.5.. oherwse: Snce [ F] [0],[?] [ f ( ] [0] I For f 1 and f 2 we ge ( I : ( ( ( [ Σ f ] = [ f ] [ f ] and, (Theorem 3..b proved by Prop. 4 snce [ F] = [ F ] [?] In applcaon of Comparson Calculus we wll descrbe n hs paper, when we compare wo funcons f 1 and f 2 on I, he sgn consans [ F 1], [ F 2 ], and herefore [ F ], Σ [ F] wll be rvally exracable from a QSIM behavor. Moreover, he comparson consrans are usually useful when compared funcons f 1 and f 2 don change sgns on I ([ F1 ],[ F2 ] [?]. If ha s he case, eher Σ[ F] [?] or [ F] [?] wll be rue (Prop 5.v. (a(b ensurng ha some of he consrans n Theorem 4 wll always be applcable. Closer nspecon of hese consrans reveals ha, gven he sgn consans, he wo comparson sgns [ F] and [ Σ F] can be calculaed from [ F ]. Therefore, obanng [ F ] wll be more mporan hen obanng [ F] and [ Σ F]. Theorem 5.Qualave Fundamenal Theorem Of Calculus For a funcon h wh a connuous dervave h, f H and H are her mages on he nerval I = (,, we b e

5 ge: [ H ] [ h ] [ H ] b + ( hb h ( b = The nex heorem esablshes he lnk beween he comparson sgns of funcons wh he comparson sgns of her dervave. Theorem 6. Comparson Propagaon Over Dervave For wo funcons f 1 and f 2 wh connuous dervaves f 1 and f 2, we ge:. [ F] [ f ] [ F ] b + ( fb f ( b2 f ( b1 Σ Σ b + Σ ( fb f ( b2 f ( b1. [ F ] [ f ] [ F ] = Σ = + Proved by replacng h n Thm. 5 wh f and Σ f So far, we have assumed ha he funcons we compare are defned on he same nerval. Our sraegy for comparng wo funcons on dfferen nervals wll be o consruc a sngle sandard comparson nerval, and o ransform he compared funcons so ha hey are defned on. Defnon 7. Mnmum Inerval Gven wo nervals I 1 = ( b1, e1 and I 2 = ( b2, e2 we defne he mnmum nerval of I 1 and I 2 as I = (0,, where mn ( 1, 2 wo nervals. f e1 f b1 f e2 f b2 = and 1, 2 are he lenghs he f1 ( f1 ( f b1 f2 ( f2 ( f b2 f b2 f b1 f2 f 1 ( b1 e1 0 0 ( b2 e2 0 0 Fgure 3. Consrucng Dreconal Funcons Defnon 8. Dreconal Funcons Gven a par of funcons f 1, f 2 on he domans I 1 = ( b1, e1, I 2 = ( b2, e2, for each f we defne wo funcons, he forward dreconal funcon f, and he backward dreconal funcon f on he mnmum nerval I of I 1 and I2 as follows:. f ( = f ( b +. f ( f ( e = For a funcon f, he funcon f sars a he nal pon of f and races s values n he forward drecon for he duraon of he mnmum nerval. Smlarly, f sars a he end pon of f and races backwards. Fgure 3 shows wo funcons f1 and f 2 wh her correspondng dreconal funcons. By comparng each dreconal funcon of f 1 wh each dreconal funcon of f 2, we ge four dfferen ses of comparson funcons. To dsngush hese, we add he drecons of he dreconal funcons as subscrps o he comparson funcons. Defnon 9. Comparson Funcons on Dfferen Domans Gven a par of funcons f 1, and f 2 on I 1, and I 2, we defne he dreconal comparson funcons f (, Σ f ( and f ( on he mnmum nerval I o be he comparson funcons of he dreconal funcons f1 α and f2 β on I where α and β are wo drecons from he se {, } such ha we ge: f = f f. ( 2 ( 1 ( β α. Σ f ( = f2 ( + f1 ( β α. f ( f2 ( f1 ( = β α Table 1. Dreconal Comparson Funcons and Sgns f 1, f 2 drecon comparson sgns, forward [ F ] [ Σ F ] [ F ], backward [ F ] [ Σ F ] [ F ], ouward [ F ] [ Σ F ] [ F ], nward [ F ] [ Σ F ] [ F ] For example, f we compare f1 funcon of 1 and f2, he backward f, we wll f, wh he forward funcon of 2 ge he ouward comparson funcons and sgns, whch have he subscrp, a smplfcaon of,. The names and smplfed subscrps of all comparson sgns are gven n Table 1. In Fgure 4, we compare he dreconal funcons from Fgure 3 n he forward and nward drecons. For (a, we ge: [ F ] =[ ], [ F ] =[ ], [ Σ F ] =[+] and for (b we ge: [ F ] =[?], [ F ] =[?], [ Σ F ] =[+]. Nex, we wll presen he man ponwse comparson

6 consran ha s key o he knd of reasonng descrbed n second secon. We frs explan he knd of reasonng we wan o oban, on a smple example. f1 ( f ( f1( 2 I 0 f 2 ( I 0 ( a ( b Fgure 4. Dreconal Ponwse Comparson Example Le us assume we observe wo cars wh posons x 1, x 2 and veloces v 1, v 2 on he nervals I 1, I 2. Assume ha we know: [ ] =[+]: he speed of he second car s ponwse greaer n he mnmum nerval [ ] [?], [0] : The cars don sop or change drecon [ x ] = [ ] : The frs car has raveled a longer dsance. The second car should ravel more dsance durng he mnmum nerval I n he frs par of he comparson (snce [ ] =[+]. On he oher hand, he frs car should ravel more n oal durng I 1 ([ x ] = [ ]. Snce he cars canno sop or change drecon ( [ ] [?], [0], afer he end of I, he frs car should connue o ravel for overcomng he dsance raveled by he second car. As a resul, we can conclude ha he frs car has raveled longer, ha s: [ ] = [ ]. Theorem 7. Man Ponwse Comparson Consran Le x 1, x 2 be wo funcons and v 1, v 2 her connuous dervaves on I 1, I 2. For wo drecons α and β, we have: [ x ] [ ] + [ ] (f [ ] [?],[0] =1, 2 We can ge he same resul ha we obaned nuvely above, usng our new consran. Snce we have [ ] =[+], [ ] = [ ], and snce he precondon of x he consran s sasfed, we ge [ ] [ + ] + [ ] whch mples [ ] = [ ]. Alhough he man ponwse comparson consran looks somewha smlar o he one ha we have presened n (Könk and Say 1998, ha consran canno be appled n hs suaon. I requres nformaon abou he average velocy comparson [ v ] and, we show n (Könk 2000 ha n hs suaon more han one possble value for [ v ] s conssen wh he avalable nformaon. Smlarly, f we knew [ ] =[+] n he accelerang elevaor problem, snce we could deec [ x ] = [0] by observng he QSIM behavor, we could conclude [ ] = [ ]. [ A ] =[+] s easy o deec because acceleraon a any pon n he second nerval s greaer han he acceleraon a any pon n he frs nerval, and hs fac can be exraced from a QSIM race by ordnal comparson of he end pons of a. Theorem 6 s he lnk beween he comparson sgn of a funcon and s dervave, and Theorem 4 s useful for conversons beween he hree comparson sgn ypes. Usng hem, we wll show how one can oban he value of [ ] from he value of [ A ] =[+]. The only remanng problem s ha hese wo heorems are no saed n erms of dreconal ponwse comparson sgns n Table 1. The nex heorem flls n hs gap. ( = (, ( ( 1 1 α α 2 Theorem 8. Mappng Beween Dreconal Funcons and Dervave Rules If we le f 1 and f 2 be he dreconal funcons of v 1 and v 2 on I n he drecons α 1 and α 2 such ha f v f = v and a 1, a 2 are he dervaves of v 1 and v 2, we wll ge:. f ( ( ( a f α = ' ' = a f α = ' ' Inpus:. f b f ' ' b = v α = { v f α = ' '. e [ F ] = [ A ] α1 α2 v. [ F ] = [ ] f [ ] [?] v. [ F ] [ ] f ' ' and [ ] [?] = A α = A { [ A ] f α = ' ' and [ A ] [?] Oupu: v. [ ] = [ F ] α1 α2 v ( b + = a ( b + = a ( f α = ' '. f ( = v ( e = a ( e = a ( f α = ' '. f b = ( ( ( ( v 0 = v b = vb f α = ' ' v 0 = v e = ve f α = ' ' = usng herefore we ge. We have f ( a ( α he resul usng: f ( f2 ( f1 ( a2 ( a1 ( = a ( α2 α1 α1 α2 = = v. Usng Prop. 1. wh [ F ] [ ] (snce F = α

7 v. Usng and [ A ] [?] we ge: [ A ] = [ A ] f α = ' ' [ F ] = [ A ] = [ A ] f α = ' ' We are fnally ready o apply our echnques o exrac he desred duraon facs from he accelerang elevaor problem (Fgure 1. We le: I 1 = ( 0, 1 and I 2 = ( 1, 2 and make an ouward comparson. The followng quanes can be exraced from he behavor n a sraghforward way (Fgure 2. [ ve1] = [ vb2] = [ v 1 ] = [0], [ 1] = [ + ], [ 2 ] = [ ], ( [ A 1] = [ ], [ A 2 ] = [ ], [ A ] = [ + ] To apply he mappng n Thm. 8, we le ( α1, α 2 = (, : f1 ( = v1 ( and f2 ( = v2 ( and we ge: [ fb1] = [ ve1] = [0], [ fb2] = [ vb2 ] = [0] ( [ F ] = [ ] = [ + ] ( A [ F1 ] = [ 1 ] = [ + ], [ F2 ] = [ 2 ] = [ ] (v [ F1 ] = [ A1 ] = [ + ], [ F2 ] = [ A2 ] = [ ] (v Usng hese values we can apply he rules derved earler and hey fre as depced n Table 2. Table 2. Dervave Ponwse Comparson Rules Frng Example Fred Rule Reason Resul Def. 6 [ f 1] = [ f 2] = [0] Σ [ ] = [0] b b Thm.3..a Σ [ f b ] = [0] [ Σ f b ] = [0] Def. 6 [ F 1 ] = [ + ] and [ F 2 ] = [ ] [ F ] = [ ] Thm.4.v [ F ] = [ + ], [ F ] = [ ] [ Σ F ] = [ ] Thm.6. [ Σ f b ] = [0], [ Σ F ] = [ ] [ Σ F] = [ ] Def. 6 [ F 1] = [ + ], [ F 2] = [ ] [ F ] = [ ] Thm.4. [ F ] = [ ] [ F ] = [ ] Thm.4. [ Σ F] = [ ], [ F ] = [ ] [ F ] = [ + ] The oupu obaned s [ ] = [ F ] = [ + ] (Theorem 8.v. Snce we also know ha he dsances raveled n he wo nervals are he same ([ x ] =[0] and snce he oher requremens of Theorem 7 are sasfed, we can smplfy he consran [ x ] [ ] + [ ] o [0] [ + ] + [ ], whch mples [ ] =[ ], leadng o he exracon of he duraon fac 2 1 < 1 0. To calculae he comparson sgn [ ], we have used he comparson sgn [ A ] and he sgn consans [ v 1], f b e [ v b2 ], [ 1], [ 2], [ A 1], and [ A 2 ]. The sgn consans can be easly exraced from he behavor. A ponwse comparson sgn s eher calculaed from a oal comparson as n he case of [ A ], or, as n he case of [ ], s value can propagae over a dervave consran by applyng he comparson consrans (Theorem 3, Theorem 4 and Theorem 6 ogeher wh he mappng n Theorem 8. I s also possble for a ponwse comparson sgn o be compued from s hgher order dervaves usng he above scheme n a recursve way. Moreover, ponwse comparson can propagae over oher consrans such as addon, mulplcaon, M + and M - relaons. (Könk 2000 Concluson We have presened a mehod for exracng relave duraon facs from pure qualave smulaon oupus whch, when combned wh oher duraon exracon mehods, may lead o spurous behavor elmnaon. As a useful sde effec, we have developed a new sgn algebra and a comparson formalsm, whch can be appled o oher qualave reasonng problems such as comparave analyss as well. In hs paper, we have focused on he formal descrpon and conssency proofs of he duraon fac exracon sysem. Expermens o es he effecveness of hs sysem for elmnang spurous behavors wll be performed n furher sages of our research. In (Könk 2000 we sared o formalze a more general mahemacal framework, comparson flers. I combnes he ponwse comparson echnques descrbed here and he symmery exracon mehods descrbed n (Könk and Say 1998 n a unfed framework based on Comparson Calculus and SR1* Algebra. In (Könk 2000, we have proven conssency of addonal consrans and obaned some heorecal resuls abou her usefulness. Relave duraon fac exracon was frs mplemened by Çv (1992, who presens a posprocessor whch annoaes QSIM oupus wh deduced emporal nerval comparsons for some fxed models. Çv s work does no deal wh spurous behavors. Weld s(1988 dfferenal qualave (DQ analyss echnque nvolves concepually comparng wo behavors of he same varable for purposes of perurbaon analyss. When comparng mulple raversals of he same nerval, we use smlar mahemacal foundaons, albe for a dfferen purpose. Some heorems and conceps of DQ analyss are specal nsances of Comparson Calculus, and some are complemenary o he heorems ha we have proved here. Specfcally, DQ analyss does no deal wh comparson n dfferen drecons, whle Comparson Calculus does no deal wh comparson wh respec o a varable oher han me. In (Könk 2000, we showed how Comparson Calculus can solve a perurbaon analyss problem ha DQ canno solve. Some of he smulaons mproved by he duraon conssency fler nvolve occurrence branchng, n whch

8 mulple branches are added o he behavor ree o represen dfferen possble me-orderngs of wo unrelaed varables reachng her respecve landmarks. Hsory - based reasoners lke Wllams TCP (1986 were desgned wh he purpose of elmnang hs phenomenon. There has been some work (Clancy and Kupers 1993 (Tokuda 1996 o modfy he QSIM framework n hs drecon. Our approach would be useful n cases where he dsncons creaed by he global sae -based branchng mechansms are relevan from he user s pon of vew, and ncorrec predcons n hs forma need o be mnmzed. Hybrd qualave-quanave reasoners (Berlean and Kupers 1997 (Kay 1996 enable he assocaon of numercal values wh he me-pons n he qualave smulaon oupu, renderng he comparson of nerval lenghs rval. Our work shows ha such comparsons are possble and useful n pure qualave smulaon as well. Snce he nroducon of pure QSIM n (Kupers 1986, several oher global flers (Kupers 1994 (Say and Kuru 1993 (Say 1998 (Say 2001 have been added o he reperory, dealng wh dfferen classes of spurous predcons. The duraon conssency fler elmnaes he se of predcons from whch nconssen conclusons abou he relave lenghs of he me nervals can be drawn, mprovng he predcve performance of he overall smulaor, and ncreasng he level of complexy of he sysems ha can be reasoned abou. The work repored here furher mproves he power of duraon conssency fler. Avalably of he program The PROLOG source code of our mplemenaon of QSIM wh he duraon conssency fler s avalable o neresed researchers. Conac: konk@umch.edu Acknowledgemens We hank he anonymous revewers for her helpful commens. Ths work was parally suppored by he Boğazç Unversy Research Fund. (Gran no: 97HA101 References Berlean, D., and Kupers, B Qualave and Quanave Smulaon: Brdgng he Gap. Arfcal Inellgence 95: Clancy, D. J., and Kupers, B. J Behavoral Absracon for Tracable Smulaon. In Proc. 7 h In. Workshop on Qualave Reasonng Abou Physcal Sysems, Clarke, E., and Zhao, X Analyca: A Theorem Prover for Mahemaca, Techncal Repor, CMU-CS , School of Compuer Scence, Carnege Mellon Unv. Çv, H Duraon Analyss n QSIM and Exenson of QSIM o Dscree Tme Sysems. M.S. hess, Compuer Scence, Boğazç Unv. Kay, H Refnng Imprecse Models and Ther Behavors. Ph.D. dss., Dep. of Compuer. Scence, Unv. of Texas. Könk, T Explong Relave Duraons n Qualave Smulaon. M.S. hess, Sysems and Conrol Eng., Boğazç Unv. (hp:// Könk, T., and Say, A. C. C Exracng and usng relave duraon nformaon n pure qualave smulaon. In Qualave Reasonng: The Twelfh Inernaonal Workshop, , Techncal Repor WS-98-01, Menlo Park, Calf.: AAAI Press. Kupers, B. J Qualave Smulaon. Arfcal Inellgence 29: Kupers, B. J Qualave Reasonng: Modelng and Smulaon wh Incomplee Knowledge, Mass.: MIT Press. Say, A. C. C L Hôpal s Fler for QSIM. IEEE Transacons on Paern Analyss and Machne Inellgence 20:1-8. Say, A. C. C Improved Reasonng Abou Infny Usng Qualave Smulaon. Compung and Informacs, 20: Say, A. C. C., and Kuru, S Improved flerng for he QSIM algorhm. IEEE Trans. on Paern Analyss and Machne Inellgence 15: Tokuda, L Managng Occurrence Branchng n Qualave Smulaon. In Proc. 13 h Naonal Conference on Arfcal Inellgence, Menlo Park, Calf.: AAAI Press. Weld, D. S Comparave Analyss. Arfcal Inellgence Journal 36: Weld, D. S., and Kleer, J. D. eds Readngs n Qualave Reasonng Abou Physcal Sysems. Calf.: Morgan Kaufmann. Wllams, B. C Dong Tme: Pung Qualave Reasonng on Frmer Ground. In Proceedngs of he Ffh Naonal Conference on Arfcal Inellgence, San Maeo, Calf: Morgan Kaufmann. Wllams, B. C A Theory of Ineracons: Unfyng Qualave and Quanave Algebrac Reasonng. Arfcal Inellgence 51(1-3:39-94.