On the Utilization of Spatial Structures for Cognitively Plausible and Efficient Reasoning

Transcription

1 To pper in: Proeeings of the IEEE Interntionl Conferene on Systems, Mn, n Cyernetis. Chigo, Otoer On the Utiliztion of Sptil Strutures for Cognitively Plusile n Effiient Resoning Christin Freks n Ki Zimmermnn Fhereih Informtik, Universität Hmurg Boenstetstr. 16, 2000 Hmurg 50, Germny [email protected] [email protected] Astrt We present n pproh to qulittive sptil resoning whih is se on iretionl orienttion informtion s ville through pereptul proesses. Qulittive orienttions in 2-imensionl spe re given y the reltion etween point n vetor. The pper presents our si ioni nottion for sptil orienttion reltions whih exploits the sptil struture of the omin n explores vriety of wys in whih these reltions n e mnipulte n omine for sptil resoning. I. INTRODUCTION Our knowlege out physil spe iffers from ll other knowlege in very signifint wy: we n pereive spe iretly through vrious hnnels onveying istint molities. Unlike in the se of other pereivle omins, sptil knowlege otine through one hnnel n e verifie or refute through the other hnnels. As onsequene, we re isproportionlly onfient out wht we know out spe: we tke for grnte tht it is s we pereive it. Our reserh on sptil representtions n resoning is motivte y the intuition tht eling with spe shoul e viewe s ognitively more funmentl thn strt resoning. Afterll, one of the very first tsks we lern to omplish is to orient ourselves in the environment. The use of sptil metphors in lnguge n prolem solving tsks lso inites tht there might e speilise, mye less expressive, ut optimize, sptil inferene mehnism. Why else woul we trnslte prolem into the speilise omin of spe if it oul e hnle y generl inferene mehnism? As onsequene, we wnt to unerstn eite sptil resoning efore we onstrut generl strt resoning engines. The gol of this reserh is the oneption of sptil inferene engine whih els with Mnusript reeive July 24, sptil knowlege in wy more similr to iologil systems thn systems se on strt logi lnguges. Sptil orienttion informtion, or more speifilly, iretionl informtion out the environment, is iretly ville to nimls n humn eings through pereption, n is ruil for estlishing sptil lotion n for pth fining. Suh informtion typilly is impreise, prtil, n sujetive. In orer to el with this kin of sptil informtion we nee methos for equtely representing n proessing the knowlege involve. In this pper we present n pproh to representing n proessing qulittive orienttion informtion whih is motivte y ognitive onsiertions out the knowlege quisition proess. Consier simple loliztion tsk: you wlk stright long ro, turn to the right, wlk stright, turn left, n wlk stright gin. Now you woul like to know where you re lote with respet to the first ro you wlke on. Tsks like this re very funmentl for lmost ll nimls n humn eings. We mostly rry them out suonsiously exept when we fer getting lost in unergroun wlkwys. In the following we esrie how we represent this knowlege for moelling sptil resoning. II. THE REPRESENTATION STRUCTURE We introue n orienttion gri for representing qulittive orienttion informtion. The gri is ligne to the orienttion etermine y two points in spe, the strt point n the en point of movement, e.g. the points n. First we n istinguish three ifferent possile positions of n itionl point with respet to (w.r.t.) the imgine line through n. Point might e left of the line, on the line or right of the line. We ll these reltions left, stright, n Pge 1 of 6

2 Christin Freks, Ki Zimmermnn: On the Utiliztion of Sptil Strutures for Cognitively Plusile n Effiient Resoning right. But sine we onsier n to e the strt point n the en point of movement, respetively, we n not only refer to the line through n, iviing the whole plne, ut to the vetor enoting the oriente pth from to. With this in min we n impose new qulittive istintions in the reltive position of point w.r.t. the vetor, lle front, neutrl n k, eh relte to one of the lines through the en points n orthogonl to, where front mens the iretion to whih the vetor points n k its inverse. Fig. 1 shows the lines tht form the orienttion gri n Fig. 1 n 1 show the ioni representtion we use to enote the 15 possile positions n orienttions of tht n e istinguishe y mens of the gri. Although people n most nimls o not hve pereption system for expliit front/k or forwr/kwr isrimintion s they o for left/right isrimintion, the segmenttion of the plne into front n k semi-plne lso is meningful from ognitive point of view: we oneptulize people, nimls, roots, houses, et. s hving n intrinsi front sie (ompre Prienow [1990], Mukerjee & Joe [1990]); this results in n impliit ihotomy etween front region n k region n forwr n kwr orienttion. ) ) ) Fig.1: ) The si position n orienttion gri; ) the ioni representtion of point w.r.t. vetor superimpose on the gri; ) reltion of point w.r.t. vetor without the si gri. Comprison to Existing Approhes A vriety of pprohes to qulittive sptil resoning hs een propose. Güsgen [1989] pte Allen s [1983] qulittive temporl resoning pproh to the sptil omin y ggregting multiple imensions into Crtesin frmework. Güsgen s pproh is strightforwr ut it fils to equtely pture the sptil interreltionships etween the iniviul oorintes. The pproh hs severe limittion: only retngulr ojets ligne with their Crtesin referene frme n e represente in this sheme. Sine we only represent the reltive position n orienttion informtion of points s strtions of rel ojet positions we o not mke ny ssumptions on the shpe of the ojets n we re not restrite to one speifi retngulr oorinte system tht hs to e pplie to ll ojets. Rnell [1991] ttks the prolem of representing qulittive reltionships of onve ojets. He introues ling film funtion for generting onvex hulls of onve ojets; he then lists ll qulittively ifferent reltions etween n ojet ontining t most one onvity n onvex ojet. Egenhofer & Frnzos [1991] evelop forml pproh to esriing sptil reltions etween point sets in terms of the intersetions of their ounries n interiors. They o not use orienttion informtion. Hernánez [1992] onsiers 2-imensionl projetions of 3-imensionl sptil senes. He ttempts to overome some efiienies of Güsgen s pproh y introuing projetion n orienttion reltions. For the imension of projetion he opts n extens the ies of Egenhofer [1989], i.e. the inry topologil reltionship etween two res in the plne. But he omines the topologil informtion with reltive orienttion informtion tht n e efine on multiple levels of grnulrity. Nevertheless, he is still esriing senes within stti referene system. Freks [1991] suggests pereption-se pproh to qulittive sptil resoning; mjor gol of this pproh is to fin nturl n effiient wy for eling with inomplete n fuzzy knowlege. Shlieer [1990] evelops n pproh whih is not se on the reltion etween extene ojets or onnete point sets. Shlieer investigtes the properties of projetions from 2-D to 1-D n speifies the requirements for qulittively reonstruting the 2-imensionl sene from set of projetions yieling prtil rrngement informtion. Frnk [1991] isusses the use of orienttion gris ( rinl iretions ) for sptil resoning. The investigte pprohes yiel pproximte results, ut the egree of preision is not esily ontrolle. Mukerjee & Joe [1990] present truly qulittive pproh to higher-imensionl sptil resoning out oriente ojets. Orienttion n retngulr extension of the ojets re use to efine their referene frmes. Our representtion lso llows us to esrie orienttion n position qulittively, ut it oes not el with the shpes of ojets n is not restrite to ojets of ertin shpes. Furthermore the opertions on our representtion o not yiel pproximte vlues ut orret rnges of vlues. One of the mjor ifferenes to previous pprohes is tht the reltive Pge 2 of 6

3 Christin Freks, Ki Zimmermnn: On the Utiliztion of Sptil Strutures for Cognitively Plusile n Effiient Resoning positions of other ojets re not esrie w.r.t. to one position ut w.r.t. to vetor tht esries the movement etween two positions. Our pproh is motivte y ognitive onsiertions out the vilility of sptil informtion through pereption proesses [Freks 1991]. A mjor gol of this pproh is to fin nturl n effiient wy for eling with inomplete n fuzzy knowlege. The opertions pplile on this kin of representtion re esrie elow. tsk t its mximum preision. Fig. 2 shows the resulting reltions of the inversion opertion. III. THE OPERATIONS So fr we hve only introue the new representtion sheme for position n orienttion informtion. Let us now onsier the opertions tht we n pply to sptil informtion. First we will isuss the opertions working on single reltion, i.e. the reltive position of one point w.r.t. to one vetor, n then the opertion of omining two or more vetors: omposition. Inversion The first opertion we will onsier is the inversion (INV ) of the vetor. This orrespons to the tsk of turning k n sking the question "Wht woul the sptil orienttion of e if I were to wlk k from to?" Thus from the knowlege out the reltion of w.r.t. the vetor, i.e. :, we eue the reltion of w.r.t. the inverte vetor, i.e. :. The inversion opertion yiels preise result. All of the following opertions o not; rther they sometimes yiel isjuntion of possile reltions. The preision of the solution is result of the ft tht when pplying the inversion opertion left simply eomes right, front eomes k, n vie vers. Unlike in the se of liner imensions, inrementing quntittive orienttion les k to previous orienttions, i.e. INV (I NV (z:xy))=z:xy. In this sense, orienttion is irulr imension. Although the ove sttement seems well known n trivil, existing pprohes o not el with perioiity of orienttion expliitly. Perioiity is either eliminte y not mitting ertin orienttions, s in [Shätz 1990] or it is ignore y treting ifferent orienttions s inepenent entities, s in Frnk [1991]. In the exmple in setion IV we will show tht the opertion of inversion is essentil in euing the resulting isjuntion of reltions in n inferene Fig. 2: The inversion opertion n the resulting reltions in their orresponing positions, e.g. Inv( )=. Homing Another wy to eue new reltions etween point n vetor is homing (HM). In this se we re intereste in nswering the question: "Where is the strt point if I ontinue my wy y wlking from to?" Formlly this eution step n e expresse s fining the reltion of w.r.t., i.e. :, from the given reltion :. Fig. 3 shows this grphilly. Fig. 3: Orienting the pst point to when ontinuing the pth from to. Unlike the inversion opertion the opertion of homing oes not yiel preise results in the sense tht only single possile reltion lwys results from single se reltion. Inste for three input reltions the pplition of HM results in isjuntion of possile reltions. Fig. 4 shows the results. Pge 3 of 6

4 Christin Freks, Ki Zimmermnn: On the Utiliztion of Sptil Strutures for Cognitively Plusile n Effiient Resoning is neessry, euse ll the reltions foun in the HM tle re foun in the SC tle, only in ifferent orer. Fig. 6 shows the orresponene etween the two opertions. Fig. 4: The left tle shows the resulting reltions from the HM opertion. The tle on the right epits its inverse, HMI, i.e. INV (H M(z:xy)) Notie the retngle, olumn two, row two. HM( ) =, i.e., the result is solutely unetermine; eh sptil reltion is vli nite for solution. This is euse if you re wlking from to n then k to, mye with some other steps in etween, the referene vetor for point eomes, i.e. the null vetor. But on the other hn, the unetermine reltion estlishes ontinuity of the opertor w.r.t. the resulting topologil reltions. As we n esily see if we move from one reltion to its iret neighor, only neighoring reltions re e or remove from the resulting set. This ehvior of oneptully neighoring reltions hs een stuie y Freks [1992] for the omin of time n for the omin of spe, using the ove introue representtion [Freks 1992]. In Hernánez & Zimmermnn [1992] it is shown how these neighoring strutures n e pplie to n omine with efult resoning methos. Shortut The thir possiility for reomining the points within one given referene frme is lle shortut (SC). This is the eution of the reltion of w.r.t., from the given reltion :, see Fig. 5. This llows us to solve two very ommon prolems in sptil resoning. First we my eue the reltions etween ojets if we re relly intereste in fining shortut. On the other hn, shortut llows us to speify the reltive position of ojets tht re not on our pth ut to the sie of it. Thus, we my relte oserve ojets on the sie of our pth, e.g., to the iret pth from to n use this kin of informtion in the lter resoning proess, using only one omintion opertor. One n see the strong inner resemlne etween the homing n the shortut opertions, for whih only one tle Fig. 5: The shortut opertion n the resulting sets of possile reltions for SC, left tle, n its inverse SCI (z:xy) = INV (SC( z:xy)), right tle. HM SC Fig. 6: Corresponene etween the opertions HM n SC. Algeri Comintion of Opertions Fig. 7 shows how the opertions n e omine lgerilly. This kin of omintion is not ommuttive, ut it is ssoitive. The ssoitivity llows us, for exmple, to pply generl n possily prllel onstrint propgtion lgorithm in whih the temporl orer of omintion oes not mtter. If the omintion were not ssoitive, we woul e restrite to n orere omputtion, e.g., kwr hining. o ID INV SC SCI HM HMI ID ID INV SC SCI HM HMI INV INV ID HM HMI SC SCI SC SC SCI ID INV HMI HM SCI SCI SC HMI HM ID INV HM HM HMI INV ID SCI SC HMI HMI HM SCI SC INV ID Fig. 7: The lgeri omintion of opertions. The result of HM(SC(z:xy)) HMI (z:xy) n e foun in the fourth row, olumn six. Pge 4 of 6

5 Christin Freks, Ki Zimmermnn: On the Utiliztion of Sptil Strutures for Cognitively Plusile n Effiient Resoning If we re given two segments of pth, we n omine them. If, for exmple, we hve : n :, we n eue :, i.e., we n try to relte lter sttions of pth to erlier segments. This opertion oes not, in generl, proue unique results, e.g., omining ' is in the right front of ' when ' is in the right front of ', results in isjuntion initing tht n e nywhere to the right of, see Fig. 8. uiling, tht we try to lote the lnmrk gin. Most of the time we hve very ler intuition out the generl iretion in whih we hve to serh for it. We will now show tht it is not neessry to efine new omposition tles for suh tsks. The solution to this kin of question n e eue lgerilly in gol-oriente kwr hining system or through simple onstrint propgtion mehnism whih woul orrespon to n unoriente forwr hining tsk. Consier the following route esription, epite in Fig. 9: Wlk own the ro (). You will see hurh () in front of you on the left. Before you reh the hurh turn own the pth tht les forwr to the right (). The question one might sk on his wy own the pth is Where is the hurh w.r.t. me?. More formlly we n stte this s: Given : n :, eue :. Fig. 8: Composing n les to the result. The tle showing the results of omining the 15x15=225 possile pirs n e foun in Freks [1992] ompnie y n extensive isussion of the resulting neighorhoo struture of the hosen representtion. This isussion will not e repete here. There exists proof tht with the three si opertions INV, HM, n SC, plus omposition, we n uil n eue reltion for every possile omintion of points w.r.t. to ny vetor. The proof is not presente here, euse we re urrently investigting whether the result hieve is t its mximum shrpness, i.e., whether the pproh is omplete in the mthemtil sense. This hs yet to e prove. IV. EXAMPLE Up to now only the representtion n the four funmentl opertions tht my e performe on it hve een isusse. We will now show tht these four opertions n e omine suessfully to nswer questions out the orienttion of ritrry points w.r.t. ny line in 2D spe. Sine it is known tht lnmrks ply very importnt role in humn orienttion n pth fining [Lynh 1960], we onsier s ognitively funmentl the tsk of orienting the new position not only w.r.t. ples rehe y the pth, ut lso with respet to lnmrks off the route. It is very ommon phenomenon tht if lnmrk ws visile from one point in the pth n then eomes osure, e.g., y ^ # L Fig. 9: A hurh left to the pth, hien y some trees. In Fig. 10 we see the result otine y iretly euing : through omposition. As n esily e seen from the figure, this is not the result we expete; it is ovious tht the result shoul e, The hurh is somewhere to my left. This more pproprite result my e otine if we o not sk for the iret omposition tht les to :, ut if we sk for the omposition tht gives us the inverte reltion : inste (see Fig. 11). Thus, in n implementtion we hve to ensure tht eh possile wy to eue smller isjuntion of reltions is explore to proue result t its optimum. SCI HM Fig. 10: Asking for the omposition of : iretly only yiels: The hurh is somewhere to my left or ehin me. Pge 5 of 6

6 Christin Freks, Ki Zimmermnn: On the Utiliztion of Sptil Strutures for Cognitively Plusile n Effiient Resoning HMI INV Fig. 11: Asking for the omposition of the inverse of : inste les to the expete improve result: The hurh is somewhere to my left. V. OPEN QUESTIONS Is this inferene system omplete? Up to now we hve only proven tht we n eue reltion etween n ritrry point w.r.t. to ny line. We re urrently trying to prove tht these eue reltions re t their mximum shrpness. Wht is the overll omplexity of the omplete inferene mehnism? We n prove tht the numer of possile reltions of points n lines inreses monotonilly with O(k 3 ), where k is the numer of points. In generl we woul expet the runtime of onstrint propgtion system tht mkes expliit ll the impliit reltions to e of exponentil orer. But for the omin of time intervl logi there exists proof, see Vilin, Kutz & vn Beek [1989], tht if only onvex sets of reltions re fe into the system, the lgorithm is of polynomil orer. Informlly speking this mens only oneptully neighoring reltions without holes shoul e speifie for the orienttion. Does there exist preferre wy of rrying out the omputtion? We hve seen tht the result of eution epens on the wy the result is erive n whih reltions re use. We re urrently investigting whether there exists preferre wy, whih woul spee up the inferene system, euse we woul hve to tke into ount only the preferre suset of possile omintions. ACKNOWLEDGMENT Most of the presente ies hve een isusse in the ontext of the Spe Inferene Engine projet t the University of Hmurg with Mro Homnn, Crsten Wiegn, Antje Wulf, n Rolf Sner. Their vlule ritiism n omments helpe improve the pproh onsierly. We thnk Clyton MMilln for proofreing the mnusript. INV REFERENCES Allen, J.F., Mintining knowlege out temporl intervls, CACM 26 (11) (1983) Chng, S.K., Jungert, E.: A sptil knowlege struture for imge informtion systems using symoli projetions. Pro. of the Ntionl Computer Conferene, Dlls, Texs, Novemer 2-6, 1986, Egenhofer, M.: A forml efinition of inry topologil reltionships. In W. Litwin n H.-J. Shek, eitors, Thir Interntionl Conferene on Fountions of Dt Orgniztion n Algorithms, vol. 367 of Leture Notes in Computer Siene, Springer, Berlin Egenhofer, M., Frnzos, R.: Point-set topologil sptil reltions. Intern.. Journl of Geogrphil Informtion Systems, vol.5, n.2, , Frnk, A.U.: Qulittive sptil resoning with rinl iretions. Pro. Seventh Austrin Conferene on Artifiil Intelligene, Wien, , Springer, Berlin Freks, C.: Qulittive sptil resoning. In: D.M. Mrk & A.U. Frnk (es.) Cognitive n Linguisti Aspets of Geogrphi Spe, Kluwer, Dorreht Freks, C.: Temporl resoning se on semi-intervls. Artifiil Intelligene 54 (1992) Freks, C.: Using orienttion informtion for qulittive sptil resoning. Pro. of the Interntionl Conferene GIS - From Spe to Territory. Theories n Methos of Sptio-Temporl Resoning. Pis (Itly), Septemer Springer, New York, In press. Furh, U., Dirlih, G., Freks, C.: Towrs theory of knowlege representtion systems. In: W. Biel & B. Petkoff (es.) Artifiil Intelligene Methoology, Systems, Applitions (pp ). Amsterm: North-Holln Güsgen, H.W.: Sptil resoning se on Allen s temporl logi. TR , Interntionl Computer Siene Institute, Berkeley Hernánez, D.: Digrmmtil spets of qulittive representtions of spe. Proeeings of the AAAI Spring Symposium on Resoning with Digrmmti Representtions Stnfor Hernánez, D., K. Zimmermnn: Defult resoning n the qulittive representtion of knowlege. Report, Tehnishe Universität Münhen. Lee, S.-Y., Hsu, F.-J., Piture lger for sptil resoning of ioni imges represente in 2D C-string. Pttern Reognition Letters 12 (1991) Lynh, K.: The Imge of the ity. The MIT Press & Hrvr University Press, Cmrige/Mss Mukerjee, A., Joe, G.: A qulittive moel for spe, Pro. AAAI-90, Prienow, S.: Interktion von propositionlen un ilhften Repräsenttionen. In: C. Freks & C. Hel (es.) Repräsenttion un Verreitung räumlihen Wissens, , Springer, Berlin Rnell, D.: Anlysing the fmilir: logil representtion of spe n time. 3 r Interntionl Workshop Semntis of Time, Spe, n Movement, Toulouse Shätz, B.: Ein konnektionistisher Anstz zur verteilten Drstellung von Ojektpositionen. In: W. Hoeppner (e.) Workshop 'Räumlihe Alltgsumgeungen es Menshen', , Kolenz Shlieer, C.: Anornung. Eine Fllstuie zur Semntik ilhfter Repräsenttion. In: C. Freks & C. Hel (es.) Repräsenttion un Verreitung räumlihen Wissens, , Springer, Berlin Vilin, M., H. Kutz, P.G. vn Beek: Constrint propgtion lgorithms for temporl resoning: revise report. In D.S. Wel n J. e Kleer, es. Reings in qulittive resoning out physil systems, , Morgn Kufmnn, Sn Mteo, CA, Zimmermnn, K.: A proposl for representing ojet sizes. Pro. of the workshop 'Sptil onepts: Conneting ognitive theories with forml representtions' t the ECAI 1992, Wien. Pge 6 of 6