Representation and Reasoning with Uncertain Temporal Relations
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1 Rprsntaton and Rasonng wth Uncrtan Tmporal Rlatons Vladmr Ryaov (*) Sppo Puuronn (*) Vagan Trzyan (**) (*) Dpartmnt of Computr Scnc and Informaton Systms Unvrsty of Jyvaskyla P.O.Box 5 SF-4051 Jyvaskyla FINLAND -mal: vlad@jytko.jyu.f sp@jytko.jyu.f (**) Dpartmnt of Artfcal Intllgnc and Informaton Systms Kharkov Stat Tchncal Unvrsty of Radolctroncs Lnna Av. 14 UA-1076 Kharkov UKRAINE -mal: vagan@mla.kharkov.ua Astract Rprsntaton and rasonng wth uncrtan rlatons twn tmporal ponts s th man goal of ths papr. Oftn humans hav to dal wth uncrtan knowldg. Bascally uncrtanty ncluds two man aspcts: nxactnss (proalstc aspct) and nconsstncy. Tmporal ara s not an xcpton. In ths papr w suggst on way to rprsnt uncrtan rlatons twn tmporal ponts. Ths rprsntaton allows to stmat th dgr of xactnss n tmporal rlatons y provdng proalts for possl tmporal rlatons and also to drv th structur of possl nconsstnt rlaton. W consdr nconsstnt rlaton as conflctng knowldg whn dscrng th sam rlaton. Th asc vctor wth svn paramtrs that rprsnts a rlaton twn two tmporal ponts conssts of two parts: nxact forth and nconsstnt trad. Th frst part dstruts proalts among asc rlatons "" "" "" and proalty of nconsstnt rlaton. Th scond part rprsnts th composton for possl nconsstnt rlaton: prcntag of "" of "" and of "" wthn th nconsstnt rlaton. Th rasonng mchansm proposd n ths papr allows to compos nvrs and add such tmporal rlatons y rcalculatng th valus of th rsultng vctor. Such rprsntaton maks t possl to valuat rasonng rsult gvng xact masurs for nxact and nconsstnt parts of a rsultng rlaton. 1. Introducton Th prolm of rprsntaton and rasonng wth tmporal knowldg arss n a wd rang of dscplns ncludng computr scnc phlosophy psychology and lngustcs. In computr scnc t s a cor prolm of nformaton systms program vrfcaton artfcal ntllgnc and othr aras nvolvng modlng procss. Durng th arly 80s som gnral pcs of work amd at provdng gnral thors of tm and acton appard such as McDrmott s tmporal logc [] Alln s thory of acton and tm [1] Vlan s thory of tm [5]. Ths proposals wr good to stalsh th two man contndrs as tmporal ontologcal prmtvs (pont and ntrval) to mak ntal proposals on rprsntatonal ssus and rasonng algorthms to pont out th gnral prolms (th rasonng y dfault th ntracton of actons th us of a tmporal rasonr n applcaton) and for showng that a mor asc machnry has to uld for dfnng a gnral thory of tm [6]. In ths papr w consdr rlatons twn tmporal ponts and w tak tmporal ponts as ontologcal prmtvs. Oftn humans hav to dal wth uncrtanty whch ncluds two man aspcts: nxactnss (proalstc) and nconsstncy. Tmporal ara s not an xcpton. In ths papr w suggst on way to rprsnt uncrtan tmporal rlatons twn ponts. Ths rprsntaton allows to stmat th dgr of xactnss n tmporal rlatons and also to drv th structur of possl nconsstnt rlaton. Th proposd approach can consdrd consstng of two parts: frst on dfns th rprsntaton modl and th scond part dals wth rasonng mchansm whch uss th proposd rprsntaton modl. Approachs to tmporal rasonng dal wth nxact tmporal knowldg n th followng way. It s supposd that nxact tmporal rlaton s a dsjuncton of two or mor xact rlatons. If tmporal nformaton s nxact n som applcatons ths approachs procss t wthout consdraton of th proalts of ach of asc rlatons n nxact on. Ths sms to a waknss f w ar spakng aout
2 dcson support systms or systms whr th proalty of ach altrnatv plays an mportant rol. Th rally mportant task that arss n such systms s not only to prdct th rsult rlaton ut also to provd proalts for ach of altrnatvs. Hnc on of th goals w hav statd to achvd n proposd rprsntaton modl s to nclud th alty to provd proalts for possl tmporal rlatons. In many stuatons thr s a nd to rason wth nconsstnt knowldg []. Ths nconsstncs may occur for xampl du to sourcs of nformaton that ar not fully rlal and thus nformaton contans contradcton. Multpl xprts opnons s such stuaton whn nconsstncy may occur. W consdr nconsstnt rlaton as conflctng knowldg whn dscrng th sam rlaton. It mght so that such knowldg was otand from svral knowldg sourcs. Th wdly usd dfnton of consstncy s th followng: th consstnt knowldg supposs asnc of contradcton and nconsstnt knowldg contans contradcton []. On can ask why t s ncssary to formalz nconsstncy t smply dtrmns som knd of rror and w should thnk how to avod t ut not to dfn. Inconsstncy surrounds us vrywhr. Inconsstncy n nformaton s th norm and w should fl happy to al to formalz t [7]. In all aras of human havor on hav to rsolv nconsstncs that occur vry oftn. But popl usually don t vn notc that th nformaton thy hav s nconsstnt. Thy just us t and apply human rasonng mchansm for makng dcson. Th dffrnc twn artfcal and ral (human) ntllgnc havor whn an nconsstncy occurs concrns ntrprtaton of t. To a human rsolvng nconsstncs s not ncssarly don y rstorng consstncy ut y supplyng ruls tllng on how to act whn th nconsstncy arss [7]. For artfcal ntllgnc thr s an urgnt nd to rvs th vw that nconsstncy s a ad thng and nstad vw t as mostly a good thng. W oftn hav to dal wth nconsstnt tmporal nformaton whn w mak dcson aout tmporal rlatons whn w dal wth amalgamaton of tmporal dataass plannng undr uncrtanty ntrprtaton of natural languag and so on. But th logc for dalng wth occurrd nconsstncy almost always us classcal logc approach whch s amd drctly for rstorng consstncy. Rstorng consstncy oftn mans lmnaton of a sourc of nconsstncy on gloal or local lvl. Th pulshd approachs ar amd to fnd consstncy consdr only consstnt part of knowldg and ths mans loss of nformaton from nconsstnt parts. W argu that n many applcaton aras for xampl dcson makng systms t s ssntal to hav complt nformaton aout rlaton vn f t contans contradcton. Any loss of nformaton may caus drvng ncorrct knowldg and hnc to wrong dcson. Ths pont of vw s cntral n our consdraton. In ths papr w propos mchansm for rprsntng uncrtan tmporal rlatons whch ncluds alty to rprsnt oth nxact and nconsstnt rlatons and rasonng wth thm.. Rprsntaton of Uncrtan Rlatons Ths scton dals wth rprsntaton modl for uncrtan rlatons twn tmporal ponts. Frst w wll gv som dfntons that srv as a ackground for asc concpts usd throughout th papr. And w start wth th dfnton of tmporal rlatons w ar dalng wth. Dfnton.1. Basc rlatons that can hold twn tmporal ponts ar and. W wll call thm xact tmporal rlatons twn ponts. Possl dsjuncton of ths rlatons namly ( or ) ( or ) ( or ) and? ( or or ) w wll call nxact tmporal rlatons twn ponts. So thr ar xact rlatons and 4 nxact rlatons. That was usual consdraton n th pulshd ltratur aout tmporal rlatons. But w should valuat ths dfnton from th prspctv of uncrtanty rprsntaton. What knd of uncrtan rlatons can drvd at all and dos ths dfnton s al to formalz thm. Lt us rmnd that w consdr uncrtanty can otand y two ways: nxactnss n dfnng nformaton and nconsstncy. Th Dfnton.1 s al to spcfy all th nxact rlatons twn two tmporal ponts ut s dos not hav any da how to formalz nconsstncy. Morovr t was not supposd at all to do ths n pulshd approachs n ths ara. In ths papr w ntrprt an nconsstnt tmporal rlaton n th followng way.
3 Dfnton.. Inconsstnt tmporal rlaton s conjuncton of two or mor asc tmporal rlatons and t nhrts all th tmporal manngs of th asc tmporal rlatons ncludd. Inconsstnt rlaton ncluds conflctng manngs of nformaton whn dscrng th sam rlaton. For xampl f th on xprt says: Ths rlaton s and th anothr on says: Ths rlaton s. Th common opnon s th rlaton and and t s dnotd as nconsstnt rlaton and. In othr words w assum that f w ar gvn contradct nformaton aout th sam rlaton w wll dfn ths rlaton usng all th gvn nformaton. It s supposd to stor th nconsstnt knowldg ut not to try to rstor consstncy. By ths w ar gong to dstngush twn nconsstnt rlatons takng nto account how thy wr otand. Ths can achvd y supposng that ach nconsstnt rlaton has t s own structur. Ths structur s dfnd consstng of asc rlatons that hav composd th nconsstnt rlaton as t s shown y th followng dfnton. Dfnton.. An nconsstnt rlaton s composd of asc rlatons.g.. W dfn th composton of nconsstnt rlaton as followng trad: [d d d ] whr d d d dnot th prcntag contnt for ach of asc rlatons wthn th rlaton twn two tmporal ponts and d d d 1. Dfnton.4. Valu of xactnss of any of asc rlatons twn tmporal ponts s th proalty that xactly ths rlaton holds twn th gvn two tmporal ponts. Snc w hav thr asc rlatons ( ) plus possl nconsstnt rlaton twn tmporal ponts w hav th followng xactnss varals: valus of xactnss of rlatons and nconsstnt rlaton rspctvly. Th sum of ths varals s qual to 1 snc thy nclud proalts of all possl rlatons that can hold twn th two tmporal ponts 1. Dfnton.5. Rprsntaton of any rlaton twn two tmporal ponts a and taks nto account th approprat valus of xactnss and th composton of nconsstncy and t s dfnd y th followng vctor: ( [ d d d ]) a whr I 1 and d d d 1 and valus d d d ar dfnd only n th cas whn 0. Dfnton.6. Th ntal xactnss valus n th cas of? rlaton twn two tmporal ponts ar qual rspctvly to and Lt us consdr thr xampls that llustrat th usalty of proposd rprsntaton modl. Exampl 1. W consdr a rlaton twn two tmporal ponts a and (Fg.1). Sourc a l l 1 1 L a 0 0 Fgur 1. Rprsntaton of nxact rlaton
4 In ths xampl w hav only on sourc of nformaton and th rlatons that s provdd s nxact rlaton and hnc should dfnd through valus of xactnss of rprsntaton vctor. Th valus of nconsstnt group ar not dfnd wth an accordanc to th Dfnton.5 snc proalty of nconsstnt rlaton s qual to 0. Ths s xampl of tmporal rlaton wthout any nconsstncy. Exampl. Th Fgur shows th rprsntaton of nconsstnt tmporal rlaton. Not that th caus of nconsstncy s svral sourcs of nformaton that gv us contradct nformaton. Sourc 1 a l Sourc Sourc l L a Fgur. Rprsntaton of nconsstnt rlaton Hr w assum that th ntal proalts ar qual to ach othr and hnc thy qual to. Ths mpacts th dstruton nsd nconsstnt group. By othr valus of th valus of varals nsd nconsstnt group wll anothr. Exampl. Now lt us consdr uncrtan rlaton. In Fgur two sourcs of nformaton provd us nxact knowldg. Sourc 1 a l Sourc l Fgur. Rprsntaton of uncrtan rlaton Frst w spcfy rprsntaton vctors for nformaton from Sourc 1 and Sourc. Th vctors look 1 1 lk: for Sourc 1-00 for Sourc Thn w hav som-how to comn ths vctors to drv th common knowldg. To do ths w nd to hav a rasonng mchansm that would us th rprsntaton vctors and conssts of thr opratons: nvrs composton and addton that ar dfnd undr uncrtan rlatons twn tmporal ponts. Th nxt scton wll prsnt such mchansm and th Exampl wll contnud n th nd of Scton.. Rasonng wth Uncrtan Rlatons Th rasonng mchansm ncluds thr opratons: nvrs composton and addton. Frst two opratons ar classcal for all mchansms that ar ntndd for prformng rasonng undr tmporal rlatons. Th thrd usual opraton that s dscrd n pulshd approachs s ntrscton. But n th proposd n ths papr mchansm w hav rplacd t wth addton opraton. Hr w hav to dstngush twn ths opratons. Th dffrnc twn thm sms to lk th dffrnc twn ways to
5 handl nconsstncy whn t has occurrd. Th ntrscton opraton s amd to fnd out th common part n rlatons to ntrsctd. Ths lfts no chancs for occurrng nconsstncy ut dscnd potntal trouls that wr mntond n Scton. Th addton opraton s ntndd for summarzng all th nformaton provdd n rlatons undr opraton. Whn contradcton s drvd th nconsstnt part of rprsntaton vctor s changd. Th proposd n prvous scton rprsntaton modl s usd n all ths opratons. Bcaus of complcatd proof w would not provd xhaustv formalsms ut only th ncsstat for undrstandng. Th nxt dfnton gvs us th noton of rlaton twn any tmporal ponts. W wll nd t furthr whl dfnng opratons for rasonng. Dfnton.1. Lt us suppos that w ar gvn two tmporal ponts a and. and w hav a rlaton L that holds twn ths ponts. Thn th prdcat of truth as follows: tru f rlaton L holds twn tmporal ponts a and ; Pa ( L ) fals othrws Dfnton.. Th nvrs opraton (Fg.4) s dnotd ~ and dfnd y th followng quaton: P(a L ) P( L ~ a a a) whr a ar tmporal ponts [ d d d ] s orgnal rlaton twn ponts a and L ~ a s th rsult of nvrson L a ( ) rprsntd y th rlaton L a ( r r r r dr dr dr ) [ ]. Thn w suggst th followng formulas to rcalculat th varals n rprsntaton vctor: d d 1 d d 1 d d 1. r r r r r r L ~ L a a l L a L c a l L a l a l l c L L L ac a c Fgur 4. Invrs opraton Fgur 5. Composton opraton Dfnton.. Th composton opraton (Fg.5) s dnotd * and dfnd y th followng quaton: P( a La ) P( L c c) P( a La * L c c) whr a and c ar tmporal ponts L a ( ) ponts a and L c ( [ d d d ]) L a L c [ ] s th frst orgnal rlaton twn s th scond orgnal rlaton twn ponts and c * s th rsult of composton rprsntd y th rlaton L ac ( r r r r [ dr dr dr ]) twn ponts a and c. Thn w suggst th followng formulas to rcalculat th varals n rprsntaton vctor: r r r r
6 d r d r d r whr 1 d 1 1 d 1 d1 1 1d 0 1d d 1d d 1 1 d 0 d 1 1 d d 1 1 d 0. 1 d 1 1d 01d d0 d 1 1 d 0. 1 d 1 1d 01 d1d 01 d d d 1 1 d d 1 1 d 0 d 1 1 d d 1 1 d 0. Ths formulas wr otand y fndng all possl compostons twn all lmnts takn from oth vctors. W us th composton tal for tmporal ponts that had n proposd y Vlan and Kautz n [4] (Fg. 6). *????????????????? Fgur 6. Composton tal Fgur 7. Intrscton tal Usng ths tal w s th followng cass support ths proalty: (1) * ; () * ; () * and also (4) * and (5) *. Proalts of th frst thr cass fully support th r. Th valu of 0 dfns th parts of support for cass (4) and (5) whch ar long to th support of r. Thn t follows: r Smlarly: cas 1 cas cas r cas 4 cas r Accordng to th Dfnton. d r s th prcntag valu of th rlaton wthn th nconsstnt on. Basd on composton tal t follows that th followng cass support ths prcntag valu: (1) (n. group 1)* (nc. group ); () (n. group 1)* (nc. group ); () (n. group 1)* (nc. group ); (4) (n. group 1)* (nc. group ); (5) (n. group 1)* (nc. group ); (6) (nc. group 1)* (n. group ); (7) (nc. group 1)* (n. group ); (8) (nc. group 1)* (n. group ); (9) (nc. group 1) * (n. group ); (10) (nc. group 1)* (n. group ); (11) (nc. group 1)* (nc. group ); (1) (nc. group 1)* (nc. group ); (1) (nc. group 1)* (nc. group ); (14) (nc. group 1)* (nc. group ); (15) (nconsstnt group 1)* (nconsstnt group ) whr nc. group 1 s [ ] n. group 1 s ( ) nc. group s [ d d d ] and n.. In cass w us partal support dfnd y th proalty 0. group s ( ) Thus th valu of support for th rsultng valu d r s calculatd as th sum of all cass. Smlarly w otan xprssons for and. Th total support of nconsstnt rlaton r s qual to caus t asd on proalts for all cass of nconsstncy.
7 Fnal valus for d d d ar calculatd usng th aov support valus towards satsfyng th rqurmnt d d d 1 as follows: d r r r r r r r d r d r. Dfnton.4. Th addton opraton (Fg.8) s dnotd and dfnd y th followng quaton: P a L ) P( a L ) P( a L L ) ( 1 1 whr a ar tmporal ponts L 1 ( ) a and L ( d d d ) s th rsult of addton rprsntd y th rlaton L a ( r r r r dr dr dr ) [ ] s th frst orgnal rlaton twn ponts [ ] s th scond orgnal rlaton twn ponts a and L L La L L 1 [ ]. 1 a l L 1 L l Fgur 8. Addton opraton for rlatons twn tmporal ponts Thn w suggst th followng formulas to rcalculat th varals n rprsntaton vctor: r 1 r 1 r 1 r d r d r d r whr d d d d d d d d d d d d 1 d 1 d d d d d d d 1 1d 1 d 1 d 1 d 1 d 1 d
8 d d d 1 1 d d d d d d d d d d d d d d d d d 1 1d 1 d 1 d 1 d d d 1d d d d d d d 1 d d d d d d d d d. 1 d d d d d 1 1 d d 1 1 To otan addton rsult w nd to consdr all possl comnatons twn all lmnts takn from oth vctors and dcd whch rsultng valu should supportd y th proalty of vry comnaton. Accordng to th Dfnton.4 th valu r s th proalty that xactly th rlaton holds twn tmporal ponts a and. Accordng to ntrscton tal (Fg.7) th only cas supportng ths proalty s:. Thus r s calculatd as follows: r 1. Smlarly r 1 r 1. On can s that th only cass supportng nxact group rqur th sam oprands takn from oth orgnal vctors. Accordng to th Dfnton. of nconsstnt rlaton all othr cass gv support to nconsstnt group of rsultng rlaton. Accordng to th Dfnton. d r s th prcntag valu of th rlaton wthn th nconsstnt group of th rlaton L r twn tmporal ponts a and. Basd on ntrscton tal t follows that th followng cass support ths prcntag valu: (1) (nxact group 1) (nxact group ); () (nxact group 1) (nxact group ); () (nxact group 1) (nxact group ); (4) (nxact group 1) (nxact group ); (5) (nxact group 1) (nconsstnt group ); (6) (nxact group 1) (nconsstnt group ); (7) (nxact group 1) (nconsstnt group ); (8) (nxact group 1) (nconsstnt group ); (9) (nxact group 1) (nconsstnt group ); (10) (nconsstnt group 1) (nxact group ); (11) (nconsstnt group 1) (nxact group ); (1) (nconsstnt group 1) (nxact group ); (1) (nconsstnt group 1) (nxact group ); (14) (nconsstnt group 1) (nxact group );
9 (15) (nconsstnt group 1) (nconsstnt group ); (16) (nconsstnt group 1) (nconsstnt group ); (17) (nconsstnt group 1) (nconsstnt group ); (18) (nconsstnt group 1) (nconsstnt group ); (19) (nconsstnt group 1) (nconsstnt group ) whr nconsstnt group 1 s [ d d d ] nconsstnt group s nxact group 1 s ( ) and nxact group s ( ) [ d d d ]. In cass w us full support of th cas to th rsultng valu. In othr cass th dvd th support n proporton twn th proalts of approprat valus. Thus th valu of support for th rsultng valu d r s calculatd as follows: d cas 5 cas 1 cas 1 1 cas 1 1 cas d d 1 d d d d d 1 d d 1 1 d cas 6 cas 7 cas 8 1 d d d d d d d cas 11 cas 1 cas 1 cas 9 d d 1 1 d d d d d d d d d d cas cas 15 cas 16 d d d 1 1 d d d d d d cas 18 cas 19 cas 17 Smlarly w otan xprssons for and. Th total support of nconsstnt rlaton r cas 10 s qual to caus t asd on proalts for all cass of nconsstncy. Fnal valus for r r r d d d ar calculatd usng th sam formulas as for composton opraton. Exampl (contnud). Now w can fnsh th xampl from Scton. Lt us rmnd that w hav two 1 1 vctors: 00 and that rprsnt th sam rlaton twn two tmporal ponts. W us th addton opraton to comn thm nto on and wth an accordanc to th formulas from Dfnton.4 w hav: It mans that: 1) wth th proalty 1 4 th rlaton twn a and s ; wth th proalty 4 of th asc rlatons nsd nconsstnt on s qual to t s nconsstnt rlaton and th prcntag of ach 1.
10 Concluson Rprsntaton and rasonng wth uncrtan tmporal rlatons ar th man goals of ths papr. W try to show on way to tak nto account valus of all possl altrnatvs wthn on tmporal rlaton as proalts for asc rlatons. Also w consdr th structur of possl nconsstncy n tmporal rlaton. Th asc vctor wth svn paramtrs that rprsnts a rlaton twn two tmporal ponts conssts of two parts: nxact forth and nconsstnt trad. Th frst part dstruts proalts among asc rlatons "" "" "" and proalty of nconsstncy. Th scond part rprsnts th composton for possl nconsstnt rlaton: prcntag of "" of "" and of "" wthn th nconsstnt rlaton. Th rasonng mchansm allows to compos and fnd out addton of such tmporal rlatons y rcalculatng valus of vctors. Such rprsntaton maks t possl to valuat fnal rlaton y provdng xact masurs for nxact and nconsstnt parts. Rfrncs [1] Alln J Towards a gnral thory of acton and tm. Artfcal Intllgnc [] McDrmott D A tmporal logc for rasonng aout procsss and plans. Cogntv Scnc [] Roos N A Logc for Rasonng wth Inconsstnt Knowldg. Artfcal Intllgnc [4] Vlan N. Kautz H Constrant propagaton algorthms for tmporal rasonng. In Procdngs of Ffth Natonal Confrnc of th Amrcan Assocaton for Artfcal Intllgnc Phladlpha USA [5] Vlan M A systm for rasonng aout tm. In Procdngs of AAAI [6] Vla L A Survy on Tmporal Rasonng n Artfcal Intllgnc AI Communcatons [7] Gaay D. Huntr A Makng nconsstncy rspctal 1: A logcal framwork for nconsstncy n rasonng n Foundatons of Artfcal Intllgnc Rsarch dtd y Ph. Jorrand and J. Klmn Lctur Nots n Computr Scnc volum
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