A family of directional relation models for extended objects

Transcription

1 1 A fmily of diretionl reltion models for extended ojets Spiros Skidopoulos, Nikos Srks, Timos Sellis, Mnolis Kourkis Astrt In this pper, we introdue fmily of expressive models for qulittive sptil resoning with diretions. The proposed fmily is sed on the ognitive plusile one-sed model. We formlly define the diretionl reltions tht n e expressed in eh model of the fmily. Then, we use our forml frmework to study two interesting prolems: omputing the inverse of diretionl reltion nd omposing two diretionl reltions. For the omposition opertor, in prtiulr, we onentrte on two ommonly used definitions, nmely onsistenysed nd existentil omposition. Our forml frmework llows us to prove tht our solutions re orret. The presented solutions re hndled in uniform mnner nd pply to ll the models of the fmily. Index Terms Sptil dtses nd GIS, one-sed diretionl reltions, inverse nd omposition opertors. I. INTRODUCTION The sujet of this pper elongs to the roder reserh re of Qulittive Sptil Resoning (QSR). The gol of QSR is to pproh ommonsense knowledge nd resoning out spe using symoli nd not numeril methods. It is no surprise tht QSR hs found pplitions in mny diverse sientifi res tht onentrte on uilding suessful intelligent systems: Geogrphi Informtion Systems [1], [2], Artifiil Intelligene [3], [4], Dtses [5], [6] nd Multimedi [7] just to nme few. Most reserhers hve onentrted on the three min spets of spe, nmely topology [8], [3], [1], [4], distne [2], [9] nd orienttion [10], [11], [12], [13], [14], [15], [16], [17]. The uttermost im in these lines of reserh is to define new tegories of sptil opertors, s well s to uild effiient lgorithms for the utomti proessing of queries using suh opertors. In this pper, we onsider extended ojets nd onentrte on inry diretionl reltions. Suh reltions desrie how primry ojet is pled reltive to referene ojet utilizing o-ordinte system (e.g., ojet is north of ojet ). Erly qulittive models for diretionl reltions This work hs een prtilly supported y PENED 03, projet o-funded y the Europen Soil Fund (75%) nd the Generl Seretrit of Reserh nd Tehnology (25%). S. Skidopoulos is with the Deprtment of Computer Siene nd Tehnology, University of Peloponnese, Kriskki Str., 22100, Tripoli, Hells (Greee). E-mil:spiros@uop.gr N. Srks is with the Deprtment of Computer Siene University of Toronto 10 King s College Rod M5S 3G4, Toronto, ON, Cnd. E-mil: nsrks@s.toronto.edu T. Sellis is with the Shool of Eletril nd Computer Engineering Ntionl Tehnil University of Athens 9, Iroon Polytehneiou Str., Zogrphou 15773, Athens, Hells (Greee). E-mil: timos@dl.ee.ntu.gr. M. Kourkis is with the Deprtment of Informtis nd Teleommunitions, Ntionl nd Kpodistrin University of Athens, Pnepistimiopolis, Ilisi, Athens 15784, Hells (Greee). E-mil: kourk@di.uo.gr pproximte n extended sptil ojet y representtive point (most ommonly the entroid) [10], [2], [12], [16]. Typilly, suh models prtition the spe round the referene ojet into numer of mutully exlusive res. For instne, the projetion-sed model prtitions the spe using lines prllel to the xes (Fig. 1) while the one-sed model prtitions the spe using lines with n origin ngle φ (Fig. 1). Depending on the dopted model the reltion etween two ojets my hnge. For instne, onsider Fig. 1: ording to the projetion-sed model is northest of (Fig. 1) while ording to the one-sed model is north of (Fig. 1). Lter models pproximte n ojet y representtive re (most ommonly the minimum ounding ox) nd express diretions on these pproximtions [14], [15]. Unfortuntely, models tht pproximte oth the primry nd the referene ojet my give misleding diretionl reltions when ojets re overlpping, intertwined, or horseshoe-shped (for n extended disussion see [18]). NW W SW Fig. 1. N S () NE E SE W N () Models of diretionl reltions S More reently, Goyl [18] nd Skidopoulos nd Kourkis [17], [19] studied model tht expresses the diretionl reltion y only pproximting the referene ojet (using its minimum ounding ox) while using the ext shpe of the primry ojet. Intuitively, this model () prtitions the plne round the referene ojet into 9 res similrly to the projetion-sed model (these res orrespond to diretionl reltions suh s north, northest et.) nd () reords the res oupied y the primry ojet. These res provide the diretionl reltion etween the primry nd the referene ojet. For instne, in Fig. 1, ojet is prtly NE nd prtly E of ojet. We denote this model y PDR (Projetionsed Diretionl Reltions). Clerly, the PDR model offers more preise nd expressive model thn previous pprohes tht pproximte ojets using points or oxes [18]. However, the PDR model is not without weknesses. The numer of reltions tht n e expressed in the model is very lrge (511 reltions). Furthermore, the PDR model prtitions the referene spe similrly to the projetion-sed model E NW W SW N S () NE E SE

2 2 using lines prllel to the xes (Fig. 1). Most people do not find this prtition nturl. Typilly, people tend to orgnize surrounding spe using lines with n origin ngle similrly to the one-sed model (Fig. 1). Hene, most people find the one-sed prtition more intuitive nd desriptive. The ognitive plusiility of the one-sed model hs een verified y studies in the field of Cognitive Sienes (see for instne [20], [21]). Moreover, the one-sed prtition is typil pproximtion for the field of view of the humn eye nd mer lenses [22], [23]. For the ove resons, onesed models hve een used in Computer Vision [22], [24], Root Nvigtion [25] nd Geogrphi Informtion Systems [11], [26]. In this pper, we propose CDR (Cone-sed Diretionl Reltions), n lterntive fmily of diretionl reltion models tht is sed on the ognitive plusile one-sed model. In the CDR fmily of models, only the referene ojet is pproximted y its minimum ounding ox (s in PDR), ut the spe round the referene ojet is prtitioned into 5 res using the one-sed model. The fmily ontins n infinite numer of models. Eh model in the CDR fmily is identified y unique vlue for φ (0 <φ<90 ) tht defines the origin ngle of the spe prtitioning lines (see lso Fig. 2). In other words, for eh prtiulr pplition, y hoosing suitle vlue for φ, we n find n pproprite model in the CDR fmily. Moreover, CDR models result in set of 31 reltions, signifintly smller set ompred to PDR whih hs 511 reltions. We formlly define the reltions tht n e expressed in the CDR fmily nd fous on two interesting prolems: omputing the inverse of diretionl reltion nd omposing two diretionl reltions. The inverse nd omposition opertions, for vrious kinds of sptil reltions, hve reeived onsiderle ttention in the literture [27], [1], [2], [15], [4], [12]. For the omposition opertor, in prtiulr, reserh hs minly onentrted on two definitions, nmely existentil nd onsisteny-sed omposition [8], [28]. Existentil omposition is the stndrd definition of omposition from set theory [8], [28], [15], [4]. Consisteny-sed omposition is weker interprettion useful in severl domins [29], [30]. The inverse nd omposition opertions re used s mehnisms for inferring new sptil reltions from existing ones. Suh mehnisms re importnt s they re in the hert of ny system tht retrieves olletions of ojets similrly relted to eh other using sptil reltions. For instne, these inferene mehnisms re very helpful when we need to detet inonsistenies in given set of sptil reltions [16], [4] or preproess sptil queries nd prune the serh spe [31]. Inverse nd omposition re lso n essentil prt of Reltion Algers [32], [33], [34] so their forml study is prerequisite to ny lgeri pproh to sptil resoning. Moreover, omposition is often used to identify lsses of reltions tht hve trtle onsisteny prolem [27], [4], [35]. The tehnil ontriutions of this pper n e summrized s follows: We propose the CDR fmily of diretionl reltion models. The reltions tht n e expressed in eh model of the fmily re formlly defined. CDR models re sed on the ognitive plusile one-sed model nd n e ustomized to serve wide vriety of pplitions. Finlly, CDR offers smll nd esy to mnge set of reltions. We onsider the inverse opertion for diretionl reltions in the CDR fmily. We present method to ompute the inverse of reltion nd formlly prove its orretness. We study the prolem of omposing two diretionl reltions of the CDR fmily. We first present method for onsisteny-sed omposition. To this end, we onsider progressively more expressive lsses of diretionl reltions nd present onsisteny-sed omposition lgorithms for these lsses. Our theoretil frmeworks llows us to prove formlly tht our lgorithms re orret. Finlly, we onsider the existentil definition of omposition. Contrry to onsisteny-sed omposition, we show tht the inry reltion resulting from the existentil omposition of two diretionl reltions nnot lwys e expressed using the reltions of the CDR fmily. The rest of the pper is orgnized s follows. Setion II defines the CDR fmily of diretionl reltion models. In Setion III, we study the inverse nd omposition prolem for the diretionl reltions in the CDR fmily. Finlly, Setion IV offers onlusions nd disusses future reserh diretions. II. A FAMILY OF DIRECTIONAL RELATION MODELS In this setion, we present the CDR fmily of diretionl reltion models. We onsider the Euliden spe R 2. Ojets re defined s non-empty nd ounded sets of points in R 2. Let e n ojet. The minimum ounding ox of ojet, denoted y m(), is the smllest retngle, ligned with the xes, tht enloses the ojet (Fig. 2). Throughout this pper, we will onsider ojets tht re formed y finite unions of ojets tht re homeomorphi to the losed unit disk [17]. This set of ojets is denoted y REG. Ojets in REG n e disonneted nd hve holes. However, lss REG exludes points, lines nd ojets with emnting lines. A thorough disussion out REG nd the wy ojets re modeled in REG ppers in [36], [19]. To define reltion in CDR etween primry ojet nd referene ojet, we onsider the minimum ounding ox of ojet nd four rys originting from its four verties. We denote y r 1 () (respetively r 2 (), r 3 (), r 4 ()) the ry tht origintes from the upper right (respetively the upper left, lower left, lower right) vertex of m() (see lso Fig. 2). The origin ngle of eh ry is presented in Fig. 2. Note tht the origin ngle hs the sme vlue for ll rys, suh tht the plne is prtitioned symmetrilly. This ngle, denoted y φ, is lled the hrteristi ngle of the model nd n hve vlues in the intervl (0, 90 ) 1. Every possile vlue of φ speifies new model in the CDR fmily. Suh model will e denoted y CDR(φ). For instne, CDR(30 ) denotes the model of CDR where 1 For vlues φ =0 nd φ =90 the model degenertes to suse of PDR.

3 3 Fig. 2. r 2 () φ N() r 1 () φ W() B() E() 2 φ φ m() r S() 3 () r 4 () () () () Referene tiles nd reltions φ =30. Notie tht the vlue of φ is fixed within ertin model CDR(φ) nd is not llowed to vry for different ojets. The nlysis tht we present in this pper is vlid for ny model CDR(φ) in the CDR fmily (0 <φ<90 ). Wherever neessry in the mteril tht follows, the vlue of φ ppers s prmeter. The minimum ounding ox of the referene ojet, long with the four rys divide the plne into 5 res whih we ll tiles (Fig. 2). The peripherl tiles orrespond to the four diretionl reltions north, west, south nd est. These tiles will e denoted y N(), W (), S() nd E() respetively. The entrl re orresponds to the ojet s minimum ounding ox nd is denoted y B(). Notie tht (i) ll tiles re losed, (ii) ll tiles ut B() re unounded, (iii) the union of ll 5 tiles is R 2 nd (iv) two distint tiles hve disjoint interiors ut my shre points in their oundries, for instne, W () nd B() shre the left-side of the minimum ounding ox of. Even though tiles shre some points long their orders, there is no miguity in defining reltions in the CDR fmily euse lss REG does not ontin ojets tht ould lie entirely on the orderline (like lines, points nd ojets with emnting lines). Informlly, if primry ojet is inluded (in the settheoreti sense) in tile S() of some referene ojet (Fig. 2), then we sy tht is south of nd we write S. Similrly, we n define north (N), west (W ), est (E) nd ounding ox (B) reltions. If primry ojet lies prtly in tile N() nd prtly in tile W () of some referene ojet (Fig. 2) then we sy tht is prtly north nd prtly west of nd we write N:W. The generl definition of si diretionl reltion in our frmework is s follows. Definition 1: A si diretionl reltion is n expression R 1 : :R k where (i) 1 k 5, (ii) R 1,...,R k {N, W, S, E, B} nd (iii) R i R j for every i, j suh tht 1 i, j k nd i j. A si diretionl reltion R 1 : :R k is lled single-tile if k =1; otherwise it is lled multi-tile. Exmple 1: Expressions S nd N:W re si diretionl reltions. The first is single-tile reltion, while the seond is multi-tile. Ojets involved in these reltions re shown in Fig. 2 nd Fig. 2 respetively. In order to void onfusion, we will write the singletile elements of si diretionl reltion ording to the following order: N, W, S, E nd B. Thus, we lwys write N:W :B insted of W :B:N or N:B:W. Moreover, for reltion suh s N:W :B we will often refer to N, W nd B s its tiles. 1 The set of si diretionl reltions (single or multi-tile) in every CDR model ontins 5 ( 5 i=1 i) =31elements. We will use B to denote this set. Reltions in B re jointly exhustive nd pirwise disjoint, nd n e used to represent definite informtion out diretions. Thus, reltions in B express preise knowledge like ojet is north of, denoted y N. Using the reltions of B s our sis, we n define the powerset 2 B of B whih ontins 2 31 reltions. Elements of 2 B re lled diretionl reltions nd n e used to represent not only definite ut lso indefinite informtion out diretions. Thus, reltions in 2 B lso express impreise knowledge like ojet is either prtly north nd prtly west or entirely west of ojet, denoted y {N:W, W }. In generl, expression n i=1 R i denotes tht ojet is relted to with some reltion mong R 1,...,R n. We will use Q, Q 1, Q 2,... to denote diretionl reltions nd R, R 1, R 2,... to denote si diretionl reltions, either single-tile or multi-tile. Let us now highlight the dvntges of the proposed model. Cognitive plusiility. CDR models re sed on the onesed prtition of spe tht is lose to the humn pereption of diretion s shown y Cognitive Siene studies [20], [21]. Informlly, the one-se prtition is typil pproximtion for the field of view of mer lenses nd the humn eye [22], [23]. Cone-sed models hve een used in Computer Vision [22], [24], Root Nvigtion [25] nd Geogrphi Informtion Systems [11], [26]. Appliility. The CDR models n e used in wide set of pplitions tht use diretions like Geogrphi Informtion Systems nd Root Nvigtion. In this pper, we hve foused on the Geogrphi Informtion Systems prdigm nd use rdinl diretion reltions (like West). The model n lso e used in other pplitions y simply renming the pproprite reltions (for instne using Left insted of West). For ompleteness, let us lso give root nvigtion exmple [25]. Consider set of roots, equipped with pereptul pilities, exploring n unknown re. This proess n e optimized if roots move towrds unexplored res. Thus, every root should know the position nd the explored re of every other root. To this end, the roots ould ompose nd rodst omplete metri mp of their viinity. This solution is ostly minly euse metri mps re hrd to ompute nd rodst. Alterntively, every root ould () lote the lndmrks nd the roots in its viinity, () identify their reltions (using the CDR model) nd () rodst this informtion to the other roots. This qulittive solution does not require full metri mpping pilities, requires signifintly smller ndwidth nd typilly is suffiient for the exploring tsk. Similrly, roots n use CDR reltions to retin their formtion. Customiztion. The CDR models re distinguished ording to the prmeter φ (0 <φ<90 ) tht defines the origin ngle of the spe seprting lines (Fig. 2). For eh prtiulr pplition, y hoosing suitle φ, we n find n pproprite model in the CDR fmily. For instne, in Root Nvigtion pplition where the field of view ngle of the lenses used in root s vision system is 30, we my hoose to use the CDR(30 ) model. Smll set of reltions. The CDR models n express smll

4 4 Fig. 3. r 2 () W() r 3 () B() S() () N() r 1 () E() r 4 () The minimum ounding otgon A'() ε 5 B() ε 1 ε 2 A() B'() ε 6 () D'() ε 7 D() ε 8 C'() C() nd esy to use set of 31 jointly exhustive nd pirwise disjoint reltions. This numer is signifintly smller thn the respetive set of PDR reltions tht ontins 511 reltions. The next setion defines formlly the reltions tht n e expressed in the CDR fmily. A. Defining diretionl reltions formlly Intuitively, in order to derive the si diretionl reltion etween primry ojet nd referene ojet, one needs to identify the tiles of the plne indued y where ojet lies. However, suh n intuitive (ut informl) definition is generlly indequte for the study of sptil model. For the PDR model, two ojets re relted through single-tile diretionl reltion, iff their minimum ounding oxes re relted with the sme reltion. This oservtion llows the definition of single-tile reltions using sets of onditions involving the vertex oordintes of the ojets minimum ounding oxes [17]. This more elorte definition ws susequently used to study the inverse, omposition nd onsisteny heking prolems for tht prtiulr model. We will ttempt to derive suh definition for the proposed fmily of models. We n esily demonstrte tht the ove oservtion does not hold for the proposed CDR fmily. For instne, in Fig. 3 notie tht while N,wehve m() N:W. Thus, the minimum ounding ox provides rude pproximtion of n ojet in the CDR fmily. However, more preise pproximtion exists nd n e onstruted s follows. Let us onsider n ritrry model CDR(φ). We further refine the minimum ounding ox of n ojet y using lines prllel to rys r 1,...,r 4. After we form the minimum ounding ox round the ojet, we lso form four lines, tngent to the ojet nd prllel to the rys. We use those lines to lip the orners of the minimum ounding ox. By doing so, we ome up with new pproximtion whih we ll minimum ounding otgon (Fig. 3). The minimum ounding otgon is formlly defined in Definition 2 nd we will lter see tht is pproprite for defining reltions in CDR(φ). Notie tht the minimum ounding otgon is speil eight-orner pproximtion elonging to the generl lss of minimum ounding n-orner pproximtions, for n =8[37]. Nottion 1: We denote y: O x (respetively O y ) the x (respetively y) oordinte of point O. ε 1 ε 2 the intersetion point of lines ε 1 nd ε 2. ε 4 ε 3 Fig. 4. () ε A(s)=A'(s) B(s)=B'(s) s () D(s)=D'(s) C(s)=C'(s) Tngeny nd degenerted minimum ounding otgon (ε) x (respetively (ε) y ) the x oordinte (respetively y oordinte) of the intersetion point of line ε with the x- xis (respetively y-xis), i.e., (ε) x =(ε x) x nd (ε) y = (ε y) y. Sine set REG inludes disonneted ojets, we lso need n pproprite definition of tngeny. So, line is tngent to n ojet in REG if it is tngent to one of its omponents nd the whole ojet lies on single side of the line. For exmple, in Fig. 4 line ε is tngent to the omposite ojet. Definition 2: Let CDR(φ) e n ritrry model in the CDR fmily (where φ is the hrteristi ngle of the model). The minimum ounding otgon of n ojet REG, denoted y mo φ (), is the polygon reted y lines ε 1,...,ε 8, where: (i) ε 1,...,ε 8 re tngentil to. (ii) ε 1, ε 2 re prllel to the y xis nd (ε 1 ) x < (ε 2 ) x. (iii) ε 3, ε 4 re prllel to the x xis nd (ε 3 ) y < (ε 4 ) y. (iv) ε 5, ε 6 form ngle φ with the x xis nd (ε 5 ) x < (ε 6 ) x. (v) ε 7, ε 8 form ngle 180 φ with the x xis nd (ε 7 ) x < (ε 8 ) x. The points forming the polygon presented in ounterlokwise order re A() =ε 4 ε 5, A () =ε 1 ε 5, B() =ε 1 ε 7, B () =ε 3 ε 7, C() =ε 3 ε 6, C () =ε 2 ε 6, D() =ε 2 ε 8 nd D () =ε 4 ε 8 (Fig. 3). Exmple 2: In ertin ses, depending on the shpe of ojet, mo φ () n degenerte to polygon hving 3, 4, 5, 6 or 7 verties. For instne, in Fig. 4 the minimum ounding otgon of ojet s hs only 4 verties. This ft does not ffet our nlysis. To define the minimum ounding otgon, we only need to speify 4 points (insted of 8) 2. More speifilly, verties A, B, C nd D n e omputed using verties A, B, C nd D s follows: A x = B x A y = A y tn φ(a x B x ) B x = B x + 1 tn φ (B y C y ) B y = C y C x = D x C y = C y +tnφ(d x C x ) D x = D x 1 tn φ (A y D y ) D y = A y Note lso tht from the minimum ounding otgon we n esily ompute the minimum ounding ox. For instne, in Fig. 3, the m() is the ox (with its sides ligned with the xes) speified y points (B x (),C y ()) nd (D x (),A y ()). Using the minimum ounding otgon, we n formlly define single-tile reltions. Definition 3: Let (i) CDR(φ) e n ritrry model in the CDR fmily (0 <φ<90 ), (ii) nd e two ojets in 2 Similrly to the minimum ounding ox, where we need 2 points (insted of 4).

5 5 REG nd (iii) A(), B(), C(), D() nd A(), B(), C(), D() the verties of mo φ () nd mo φ () respetively. Reltions N, W, S, E nd B re defined s follows. N iff tn(φ)b x () +B y () tn(φ)b x () + A y (), tn(φ)c x () C y () tn(φ)d x () A y () nd C y () A y () W iff tn(φ)d x () +D y () tn(φ)b x () + A y (), tn(φ)c x () C y () tn(φ)b x () C y () nd D x () B x () S iff tn(φ)a x () A y () tn(φ)b x () C y (), tn(φ)d x () + D y () tn(φ)d x ()+C y () nd A y () C y () E iff tn(φ)a x () A y () tn(φ)d x () A y (), tn(φ)b x () + B y () tn(φ)d x ()+C y () nd B x () D x () B iff C y () C y (), A y () A y (), B x () B x () nd D x () D x () A single-tile reltion etween primry ojet nd referene ojet n lso e defined using the minimum ounding otgon of (mo φ ()) nd the minimum ounding ox of (m()). Notie tht the ove definition is essentilly equivlent to Definition 3 sine, s we hve previously seen, m() n e esily omputed from mo φ (). We hve expressed Definition 3 using only minimum ounding otgons for two resons. First, using the sme type of pproximtion for oth ojets nd results in more simple nd uniform definition. More importntly, the minimum ounding otgon of the referene ojet (mo φ ()) will e more useful in susequent omputtions. For instne, it is esy to verify tht m() n only e used to ompute reltions where ts s referene ojet while mo φ () n e used to ompute ny reltion involving (regrdless if ts s primry or referene ojet). Multi-tile diretionl reltions re defined s follows: Definition 4: Let nd e two ojets in REG nd R = R 1 : :R k multi-tile diretionl reltion. Then, R 1 : :R k holds iff there exist ojets 1,..., k REG suh tht 1 R 1,..., k R k nd = 1 k. Exmple 3: In Fig. 2, we hve N:W sine there exist ojets 1 nd 2 in REG suh tht 1 N, 2 W nd = 1 2. To void overloding Fig. 2, we hve not illustrted mo φ (), mo φ ( 1 ) nd mo φ ( 2 ). In the rest of the pper, we will lso omit the minimum ounding otgon of the primry ojet whenever the reltion n e esily seen. Finlly, in Definition 4, notie tht for every i, j suh tht 1 i, j k nd i j, i nd j hve disjoint interiors ut my shre points in their oundries. In the following setion, we will study the prolems of omputing the inverse of diretionl reltion nd the omposition of two diretionl reltions. Our results re vlid for every model CDR(φ) in the CDR fmily. III. INVERSE AND COMPOSITION In this setion, we will study the prolem of omputing the inverse nd the omposition of diretionl reltions in the CDR fmily. We first present method for omputing the inverse of CDR reltion nd then method for omposing two CDR reltions. The presented solutions re hndled in uniform mnner nd pply to ll the CDR(φ) models of the CDR fmily. Let us first define the inverse of reltion. Definition 5: Let Q e diretionl reltion in 2 B. The inverse of reltion Q, denoted y inv(q), is nother diretionl reltion whih stisfies the following. For ritrry ojets, REG, inv(q) holds, iff Qholds. Two definitions of omposition pper in the literture. The first one is the stndrd existentil definition from set theory [8], [4]. Definition 6: Let Q 1 nd Q 2 e diretionl reltions in 2 B. The existentil omposition of reltions Q 1 nd Q 2, denoted y Q 1 ; Q 2, is nother diretionl reltion from 2 B whih stisfies the following. For ritrry ojets nd, Q 1 ; Q 2 holds if nd only if there exists n ojet suh tht Q 1 nd Q 2 hold. The seond definition is s follows [8], [28]. Definition 7: Let Q 1 nd Q 2 e diretionl reltions in 2 B. The onsisteny-sed omposition of reltions Q 1 nd Q 2, denoted y Q 1 Q 2, is nother diretionl reltion from 2 B whih stisfies the following. Q 1 Q 2 ontins ll reltions R B suh tht there exist ojets,, REG suh tht Q 1, Q 2 nd Rhold. The onsisteny-sed definition of omposition is weker tht the existentil definition. Oserve tht R 1 ; R 2 R 1 R 2 holds. The ove definitions re importnt nd hve ttrted the interest of mny reserhers sine they n e used s mehnism for inferring new informtion from existing one [8], [28], [4]. In this setion, we first present method to ompute the inverse of CDR reltion (Lemmt 1, 2 nd Theorem 1). Then, we study onsisteny-sed omposition. We onsider progressively more expressive lsses of diretionl reltions nd give onsisteny-sed omposition lgorithms for these lsses (Lemmt 3, 4, 5 nd Theorem 2). Finlly, we onsider the existentil definition of omposition nd show tht the inry reltion resulting from the existentil omposition of some diretionl reltions nnot e expressed using the CDR reltions. Our theoretil frmework llows us to prove formlly tht our solutions re orret. As we disussed in Setion II-A, reltions in the CDR fmily re defined using the minimum ounding otgon while PDR reltions re defined using the minimum ounding ox. When hndling the inverse nd omposition prolems, this differene is ruil nd renders unpplile the m-sed tehnique developed in [17] for the PDR model. This led us to develop new mo-sed strtegy for hndling the inverse nd omposition prolems for the CDR fmily. During the study of the inverse nd omposition prolems, we will use the informl, inlusion-sed definition of si diretionl reltions. The forml definition involving the minimum ounding otgons of ojets is used impliitly. We note tht the mo-sed definition is not without use, s () it is impliitly used nd () it is n integrl prt of the frmework tht will e required for the further study of the proposed models (e.g., for the study of the onsisteny heking nd vrile elimintion prolems).

6 6 Before we present our results, we introdue the neessry nottion. Nottion 2: Let R 1,...,R k e single-tile diretionl reltions. We denote y δ(r 1,...,R k ) the disjuntion of ll si diretionl reltions tht n e onstruted y omining the single tile reltions R 1,...,R k. For instne, δ(n,w,b) stnds for the following diretionl reltion: {N, W, B, N:W, N :B, W:B, N:W :B} Moreover, we define: δ(r 1 : :R k )=δ(r 1,...,R k ) nd δ(δ(r 11,...,R 1k1 ),...,δ(r m1,...,r mkm )) = δ(r 11,...,R 1k1,...,R m1,...,r mkm ). We denote y U dir the universl diretionl reltion, i.e., U dir = δ(n,w,s,e,b). Nottion 3: Let R {N,W,S,E} e single tile. We denote y R (respetively R, R ) the tile tht we meet y moving ounter-lokwise (respetively lokwise, dimetrilly) from tile R. Given reltion R expressions R, R nd R re defined in the following tle. R N W S E R W S E N R E N W S R S E N W Nottion 4: Let R 1,...,R k e si diretionl reltions. The tile-union of R 1,...,R k, denoted y tile-union(r 1,...,R k ), is the si diretionl reltion tht onsists of ll the tiles in reltions R 1,...,R k. Furthermore, we denote y Comine(Q 1,...,Q k ) (where Q 1,...,Q k 2 B ) the diretionl reltion {R B : R = tile-union(s 1,...,s k ) s 1 Q 1 s k Q k }. Exmple 4: Consider two si diretionl reltions, N:W nd N:E:B. Then, tile-union(n:w, N :E:B) =N:W :E:B. Furthermore, onsider two diretionl reltions, {N,N:W} nd {S, S:E}. Then, Comine({N, N:W }, {S, S:E}) = tile-union(n,s), N:S, tile-union(n,s:e), N:S:E, =. tile-union(n:w, S), N:W :S, tile-union(n:w, S:E) N:W :S:E A. Computing the inverse of diretionl reltion Before we proeed, we present useful proposition. Proposition 1 revels the inherent symmetry in the CDR fmily nd simplifies the proofs of Lemmt 1 nd 2 tht follow. Proposition 1: Consider si diretionl reltion R 1 : :R k. Let us ssume tht its inverse is diretionl reltion tht n e represented s funtion of the five tiles N,W,S,E,B, i.e., inv(r 1 : :R k ) = f(n,w,s,e,b). Then: (i) inv(r1 : :Rk ) = f(n,w,s,e,b) = f(w, S, E, N, B) (ii) inv(r1 : :Rk ) = f(n,w,s,e,b) = f(e,n,w,s,b) 1 () Fig. 5. Proving Proposition 1 1 (iii) inv(r 1 : :R k ) = f(n,w,s,e,b) = f(s, E, N, W, B) Proof: Cse (i). This is due to the symmetry of the diretionl reltions of CDR. To give n exmple, let us oserve Fig. 5. For this sptil onfigurtion, we hve tht 1 N:W 1, 1 S:E 1 nd, thus S:E inv(n:w ). Now, onsider rotting the onfigurtion of Fig. 5 y 90 ounterlokwise (Fig. 5). The effet of this rottion is tht the tiles in our reltions hve lso een rotted. Speifilly, N eme W (N ), W eme S (W ), S eme E (S ) nd E eme N (E ). Notie tht, in Fig. 5, we hve tht 2 W :S 2, 2 N:E 2 nd, thus, N:E inv(w :S). Note tht these expressions n e derived diretly from the expressions 1 N:W 1, 1 S:E 1 nd S:E inv(n:w ) onerning Fig. 5 y diretly pplying the forementioned sustitutions. Cses (ii) nd (iii). These ses lso hold due to the symmetry of diretionl reltions. To verify this, we hve to rotte the plne lokwise y 90 nd 180 respetively. We now present nd formlly prove Lemm 1, for omputing the inverse of single-tile reltions. Lemm 1: Let R {N,W,S,E} e single-tile diretionl reltion. Then: (i) inv(r) =δ(r,r,r ) {R,R } (ii) inv(b) =U dir {N,W,S,E} Proof: Cse (i). We will first prove tht the expression of Lemm 1(i) holds for R = N, i.e., inv(n) =δ(w, S, E) {W, E}. Let, e two ojets in REG suh tht N. Sine is north of, we n intuitively understnd nd esily verify tht no prt of n in turn lie north or inside the minimum ounding ox of ojet. Therefore, no prt of n lie inside tiles N() nd B() nd onsequently, tiles N nd B nnot pper in diretionl reltion inv(n), whih implies tht inv(n) δ(w, S, E) ={W, S, E, W :S, W :E,S:E,W:S:E} Let us now onsider every si reltion in δ(w, S, E) nd hek whether it elongs to inv(n) or not. 1) Reltion S: Fig. 6 demonstrtes tht Sis possile, thus, S inv(n). 2) Reltion W :S: Fig. 6 shows tht W:S is possile, thus, W :S inv(n). 3) Reltion S:E: Similrly with reltion W :S, we n show tht S:E inv(n). 4) Reltion W :E: Fig. 6 depits the not so ovious possiility tht W:E, thus, W :E inv(n). 5) Reltion W :S:E: Fig. 6d shows tht W:S:E is possile, thus, W :S:E inv(n). 2 () 2

7 7 S W:S W:E W:S:E E E:B S:E S:E:B () () () (d) () () () (d) Fig. 6. Proving Lemm 1(i) for R = N S S:B B 6) Reltion W : It is not possile to rete sptil onfigurtion suh tht W, thus, W inv(n). 7) Reltion E: Similrly with reltion W, it is not possile to rete sptil onfigurtion suh tht End, thus, E inv(n). Therefore, we hve tht inv(n) =δ(w, S, E) {W, E} (M). By pplying Proposition 1 to Expression (M), we lso hve: inv(w )=δ(s, N, E) {S, N}, inv(s) =δ(e,w,n) {E,W} nd inv(e) =δ(n,s,w) {N,S}. The ove expressions nd Expression (M) re ptured y Lemm 1(i). Cse (ii). In order to ompute inv(b) we pply the sme proedure s with Cse (i). Let, e two ojets in REG suh tht B. In this se, we nnot eliminte ny tiles from inv(b), so s strting point we will onsider tht inv(b) U dir By exmining every si reltion in U dir, in the sme wy we did while proving Cse (i), we onlude tht inv(b) = U dir {N,W,S,E}. The following Lemm n e used to ompute the inverse of multi-tile reltions. Lemm 2: Let R = R 1 : :R k (2 k 5) e multitile diretionl reltion. Let lso R = {N,W,S,E,B} {R 1,...,R k }. Then: (i) inv(r) =δ(r), if B {R 1,...,R k } (ii) inv(r) =δ(r, B) R, if B {R 1,...,R k } Proof: Cse (i). We will first prove tht the expression of Lemm 2(i) holds for R = N:W, i.e., inv(n:w ) = δ(s, E, B). Let, e two ojets in REG suh tht N:W. Sine is north nd west of, we n intuitively understnd nd esily verify tht no prt of n in turn lie north or west of ojet. Therefore, no prt of n lie within tiles N() nd W (). Consequently, we n exlude tiles N nd W from diretionl reltion inv(n:w ). Thus, s strting point we n use the following expression: inv(n:w ) δ(s, E, B) = {S, E, B, S:E,S:B,E:B,S:E:B}. Let us now onsider every si reltion in δ(s, E, B) nd hek whether it elongs to inv(n:w ) or not. 1) Reltion E: Fig. 7 shows tht Eis possile, thus, E inv(n:w ). 2) Reltion E:B: Fig. 7 illustrtes tht E:B is possile, thus, E:B inv(n:w ). Fig. 7. (e) (f) (e) Proving Lemm 2(i) for R = N:W (g) (e) 3) Reltion S:E: Fig. 7 shows tht S:E is lso fesile, thus, S:E inv(n:w ). 4) Reltion S:E:B: Fig. 7d shows tht S:E:B is fesile, thus, S:E:B inv(n:w ). 5) Reltion S: Fig. 7e illustrtes tht Sis possile, thus, S inv(n:w ). 6) Reltion S:B: Fig. 7f shows tht S:B is fesile, thus, S:B inv(n:w ). 7) Reltion B: Fig. 7g illustrtes tht Bis possile, thus, B inv(n:w ). Therefore, inv(n:w ) = δ(s, E, B) (T1). By pplying Proposition 1 to Expression (T1), we hve: inv(w :S) =δ(n, E, B), inv(n:e) =δ(w, S, B) nd inv(s:e) =δ(n,w,b). The ove expressions verify tht Lemm 2(i) lso holds for reltions W :S, N:E nd S:E. We will now prove tht the expression of Lemm 2(i) holds for R = N:S, i.e., inv(n:s) =δ(w, E, B). To this end, we will follow the sme proedure s with reltion N:W. Let nd e two ojets in REG suh tht N:S. Sine no prt of ojet n lie inside tiles N() nd S(),we n exlude tiles N nd S from diretionl reltion inv(n:s). Therefore, s strting point we n use expression: inv(n:s) δ(w, E, B) = {W, E, B, W :E,W:B,E:B,W:E:B}. By exmining every reltion in δ(w, E, B), we onlude tht inv(n:s) =δ(w, E, B) (T2). Furthermore, y pplying Proposition 1 to Expression (T2), we n prove tht Lemm 2(i) lso holds for reltion W :E. Let us now prove tht the expression of Lemm 2(i) holds for R = N:W :E, i.e., inv(n:w :E) = δ(s, B). We will follow the sme proedure s with reltion N:W. Let nd e two ojets in REG suh tht N:W :E. Sine no prt of ojet n lie inside tiles N(), W () nd E(), we n exlude tiles N, W nd E from diretionl reltion inv(n:w :E). Therefore, s our strting point we n use expression: inv(n:w :E) δ(s, B) ={S, B, S:B}.

8 8 By exmining every reltion in δ(s, B), we onlude tht inv(n:w :E) = δ(s, B) (T3). Furthermore, y pplying Proposition 1 to Expression (T3), we n prove tht Lemm 2(i) lso holds for reltions N:W :S, N:S:E nd W :S:E. Lstly, it is esy to verify tht the expression of Lemm 2(i) lso holds for R = N:W :S:E, i.e., inv(n:w :S:E) = δ(b) =B. Summrizing, we hve proven tht Lemm 2(i) holds for ll multi-tile reltions tht do not ontin tile B. Cse (ii). This se n e proven y pplying the sme proedure s with Cse (i). We strt y verifying tht the expression of Lemm 2(ii) holds for reltions N:B, N:W :B, N:S:B, N:W :E:B nd N:W :S:E:B. Then, y pplying Proposition 1, we n verify tht Lemm 2(ii) holds for ll multi-tile reltions tht inlude tile B. To ompute the inverse of n ritrry diretionl reltion we n use the following theorem in omintion with Lemmt 1 nd 2. Theorem 1: Let Q = k i=1 R i e diretionl reltion in 2 B, where R i re si diretionl reltions. Then, inv(q) = k i=1 (inv(r i)). Note tht inv(r i ) n e omputed using Lemmt 1 nd 2. Proof: From the definition of inverse (Definition 5), we hve: inv(q) ={R B :(, REG )( Q R)}. Sine Q = k i=1 R i,wehveq= R 1 R k. Therefore, inv(q) ={R B :(, REG ) ( R 1 R k ) R}. By distriuting over, wehve: inv(q) ={R B :(, REG ) ( R 1 R) ( R k R)}. Thus, inv(q) = k i=1 (inv(r i)) holds. B. Computing the omposition of two diretionl reltions Before we ddress the omposition prolem, we present Proposition 2, whih serves the sme purpose s Proposition 1 did while studying the inverse prolem, i.e., it revels the inherent symmetry in the CDR fmily nd simplifies the proofs of the relevnt Lemmt. Proposition 2: Consider two si diretionl reltions R 1 = R 11 : :R 1k nd R 2 = R 21 : :R 2m. Let us ssume tht their omposition is diretionl reltion tht n e represented s funtion of the five tiles N,W,S,E,B, i.e., R 1 R 2 = f(n,w,s,e,b). Then: (i) R11: :R1k R 21: :R2m = f(n,w,s,e,b)=f(w, S, E, N, B) (ii) R11 : :R 1k R 21 : :R 2m = f(n,w,s,e,b)=f(e,n,w,s,b) (iii) R 11 : :R 1k R 21 : :R 2m = f(n,w,s,e,b)=f(s, E, N, W, B) Proof: This is due to the symmetry of the diretionl reltions of CDR. We will ddress the omposition prolem one step t time. First, we onsider the se of omposing two singletile reltions. Lemm 3: Let R {N,W,S,E} e single-tile diretionl reltion. Then: (i) R R = R (ii) R R = U dir (iii) R R = R R = R B = δ(r, R,R,B) (iv) B R = δ(r, R,R ) (v) B B = B Proof: Cse (i). We will first prove tht the expression of Lemm 3(i) holds for R = N, i.e., N N = N. Let, nd e three ojets in REG suh tht Nnd N holds. Fig. 8 presents ojets nd suh tht N. Sine N, ojet lies inside tile N() (the dotted re of Fig. 8). Notie tht tile N() n only lie inside tile N() nd, onsequently, ojet n only lie inside tile N(). More formlly, we hve tht N() N(). In other words, if Nnd Nthen N, thus, N N = N (S1) holds. By pplying Proposition 2 to Expression (S1), we lso hve: W W = W, E E = E nd S S = S. All the ove expressions nd Expression (S1) re ptured y Lemm 3(i). Cse (ii). We will first prove tht the expression of Lemm 3(ii) holds for R = N, i.e., N S = U dir. Let, nd e three ojets in REG suh tht Nnd Sholds. Fig. 8 presents ojets nd suh tht S. Ojet lies inside tile N() (the dotted re of Fig. 8). Notie tht re N() n interset with ll five tiles of ojet, nmely N(), W (), S(), E() nd B(). Consequently, ojet n lie within ny of these five tiles or ny omintion of them. In other words, if Nnd Sthen δ(n,w,s,e,b), or U dir nd, thus, N S = U dir (S2) holds. By pplying Proposition 2 to Expression (S2), we n esily verify tht Lemm 3(ii) holds. Cse (iii). We will first prove tht expression R R = δ(r, R,R,B) holds for R {N,W,S,E}. Fig. 8 helps us verify tht the expression holds for R = N, i.e., N W = δ(n,w,e,b) (S3). Then, y pplying Proposition 2 to Expression (S3), we n prove tht the expression holds for every R {N,W,S,E}. In similr mnner, we n lso prove tht R R = δ(r, R,R,B) holds for R {N,W,S,E}. We will now prove tht expression R B = δ(r, R,R,B) holds for R {N,W,S,E}. Fig. 8d helps us verify tht the expression holds for R = N, i.e., N B = δ(n,w,e,b) (S4). Then, y pplying Proposition 2 to Expression (S4), we n prove tht the expression holds for every R {N,W,S,E}. Therefore, we hve proven tht Lemm 3(iii) holds. Cse (iv). Fig. 8e helps us verify tht expression B R = δ(r, R,R ) holds for R = N, i.e., B N = δ(n,w,e) (S5). By pplying Proposition 2 to Expression (S5), we n verify tht Lemm 3(iv) holds. Cse (v). This se is trivil nd Fig. 8f helps us verify tht B B = B. We will now turn our ttention to the omposition of single-tile with multi-tile diretionl reltion. To this end, we use Algorithm COMPOSE SM (Fig. 9). The lgorithm

9 9 if N nd N then N () if N nd B then δ(n,w,e,b) (d) Fig. 8. Proving Lemm 3 if N nd S then U dir () if B nd N then δ(n,w,e) (e) if N nd W then δ(n,w,e,b) () (f) if B nd B then B tkes s inputs single-tile diretionl reltion R 1 nd multi-tile reltion R 2 = R 21 : :R 2k (k 2) nd returns the omposition R 1 R 2. The following is n exmple of Algorithm COMPOSE SM in opertion. Exmple 5: Let R 1 = N nd R 2 = N:B = R 1 :B. Using Algorithm COMPOSE SM (Line 3), we hve N N:B = N. This n e verified using Fig. 10. Algorithm COMPOSE SM Input: A single-tile reltion R 1 nd multi-tile reltion R 2 = R 21: :R 2k, 2 k. Output: The omposition R 1 R 2. Method: 1. If (R 1 = B nd R 2 = R 21:B) Return δ(r 21,R 21,R 21,B) 2. If (R 1 = B) Return U dir 3. If (R 2 = R 1:B) Return R 1 4. If (R 2 = R 1:R 1 or R2 = R1:R 1 :B) Return δ(r1,r 1 ) 5. If (R 2 = R 1:R 1 or R2 = R1:R 1 :B) Return δ(r1,r 1 ) 6. If ({R 1,R 1 } R2 or {R1,R 1,R 1 } R2) Return δ(r1,r 1,R 1 ) 7. If (B R 2) Return δ(r 1,R 1,R 1,B) 8. Return U dir Fig. 9. Algorithm COMPOSE SM The following lemm estlishes the orretness Algorithm COMPOSE SM. Lemm 4: Let R 1 e single-tile nd R 2 = R 21 : :R 2k e multi-tile diretionl reltion. Then, R 1 R 2 n e omputed y Algorithm COMPOSE SM. Proof: Every line of the Algorithm omputes the omposition of set of pirs of si diretionl reltions. Prtiulrly, Lines 1 nd 2 ompute the omposition for R 1 = B. The rest of the Algorithm omputes the omposition for R 1 {N,W,S,E}. Therefore, we will exmine eh line of the lgorithm individully nd verify tht it orretly omputes the relevnt omposition. Line 1. This If sttement sttes tht B R 21 :B = δ(r 21,R21,R 21,B) where R 21 {N,W,S,E}. We will first prove tht this expression holds for R 21 = N, i.e., B N:B = δ(n,w,e,b). Let, nd e three ojets in REG suh tht Bnd N:B holds. Fig. 10 presents ojets nd suh tht N:B. Sine B, ojet lies inside tile B() (the dotted re of Fig. 10). Notie tht re N() n interset with tiles N(), W (), E() nd B(). Consequently, ojet n lie within ny of these four tiles or ny omintion of them. In other words, if Bnd N:B then δ(n,w,e,b). Thus, N N:B = δ(n,w,e,b) (E1) holds. By pplying Proposition 2 to Expression (E1), we lso hve: B W :B = δ(n,w,s,b), B E:B = δ(n,s,e,b) nd B S:B = δ(w, S, E, B) All the ove expressions nd Expression (E1) re ptured y Line 1 of Algorithm COMPOSE SM. Line 2. The ondition of this If sttement is stisfied when R 1 = B nd R 2 {N:B,W:B, S:B, E:B}, otherwise the ondition of Line 1 would hve een stisfied. Line 2 holds for R 2 = N:W, i.e., B N:W = U dir. This n e verified using Fig. 10. By lso using Fig. 10, we n verify tht the omposition of B with ny reltion tht is mde up of t lest two djent tiles (i.e., R 2 {N:B,W:B, S:B, E:B}) is equl to U dir Line 3. This If sttement sttes tht R 1 R 1 :B = R 1, R 1 {N,W,S,E}. To prove tht this expression holds for R 1 = N, i.e., N N:B = N (E2), we use Fig. 10. Then, y pplying Proposition 2 to Expression (E2), we n verify tht Line 3 is orret for ll R 1 {N,W,S,E}. Line 4. The ondition of this If sttement is stisfied when R 2 {R 1 :R1,R 1:R1 :B}. We will only prove tht Line 4 is orret for R 2 = R 1 :R1, i.e., R 1 R 1 :R1 = δ(r 1 :R1 ). The proof for R 2 = R 1 :R1 :B is similr. Line 4 holds for R 1 = N, i.e., N N:W = δ(n,w) (E3). This n e verified using Fig. 10d. By pplying Proposition 2 to Expression (E3), we n lso verify tht Line 4 is orret for ll R 1 {N,W,S,E}. Line 5. The proof is similr to the proof of Line 4. Line 6. The ondition of this If sttement is stisfied when reltion R 2 ontins tiles {R 1,R 1 } or {R 1,R1,R 1 }.We will first prove tht Line 6 holds for R 1 = N. Then, R 2 ontins tiles {N,S} or {N,W,E}. We will onentrte on the two most representtive reltions of this group, nmely R 2 = N:S nd R 2 = N:W :E, sine the proofs for the other reltions of the group re lmost identil to one of these two. Fig. 10e helps us verify tht N N:W :E = δ(n,w,e) (E4) nd Fig. 10f tht N N:S = δ(n,w,e) (E5). Then, y pplying Proposition 2 to Expressions (E4) nd (E5), we n verify tht Line 6 holds for ll R 1 {N,W,S,E}. Line 7. The If ondition of this sttement requires tht B {R 21,...,R 2k }. However, for the Algorithm to reh Line 7, the onditions of the If sttements in Lines 3-6 must hve not een stisfied. These four sttements provide the omposition for ll reltions R 2 tht ontin tile R 1. Therefore, reltion R 2 inludes tile B ut not tile R 1. In other words, R 2 {R1 :B, R 1 :B, R 1 :B, R 1 :R 1 :B, R 1 :R 1 :B, R1 :R 1 :B, R 1 :R1 :R 1 :B}. We will onentrte reltions of the form R 2 = R 1 :B nd prove tht R 1 R :B = δ(r 1,R1,R1,B) (E6) holds (the proofs for the rest of the reltions re similr). Expression (E6) holds for R 1 = N, i.e., N S:B = δ(n,w,e,b). This n e verified using Fig. 10g. Then, y pplying Proposition 2 we n prove tht Expression (E6) holds for every R 1 {N,W,S,E}. Line 8. If the exeution of the Algorithm rehes Line 8, then reltion R 2 nnot ontin tiles R 1 nd B, i.e.,

10 10 if B nd N:B then δ(n,w,e,b) () if B nd N:W then U dir () if N nd N:B then N () if N:W nd N:B then Comine(N, δ(n,w,s,b)) δ(n,w,s,b) N 1 δ(n,w,s,b) δ(n,w) δ(n,e) if W:B nd N then Comine(δ(N,W,S,B), δ(n,e)) or Comine(δ(N,W), δ(n,w,e)) 1 δ(n,w,e) if N nd N:W then δ(n,w) if N nd N:W:E then δ(n,w,e) if N nd N:S then δ(n,w,e) () () () (d) if N nd S:B then δ(n,w,e,b) (g) Fig. 10. Proving Lemm 4 (e) (h) (f) if N nd W:E then U dir R 2 {R1 :R 1,R 1 :R 1,R 1 :R 1,R 1 :R 1 :R 1 }. When tile R 1 is present in reltion R 2, the proof is similr to the proof of Lemm 3(ii), whih sttes tht R 1 R 1 = U dir, so we will not onsider these reltions. Insted, we will prove expression R 1 R1 :R 1 = U dir, whih is not pprent. Fig. 10h shows tht the expression holds for R 1 = N, i.e., N W :E = U dir. Then, y pplying Proposition 2, we n prove tht Line 8 is orret for ll R 1 {N,W,S,E}. Summrizing our progress so fr, we hve presented Lemm 3 tht n e used to ompute the omposition of two singletile reltions, nd then Algorithm COMPOSE SM tht provides the omposition of single-tile nd multi-tile reltion. In other words, we re le to ompute the omposition of single-tile nd si (single-tile or multi-tile) diretionl reltion. The next logil step is to ddress the prolem of omposing multi-tile nd si reltion. Let us study two speifi exmples tht will help us understnd the method used to ompute the omposition of suh reltions. Exmple 6: Let us ompute N:W N:B. Let, nd e three ojets in REG suh tht N:W nd N:B.To ompute N:W N:B, we hve to find ll possile reltions etween nd. Aording to Definition 4, N:W implies tht there exist ojets 1 nd 2, suh tht 1 N, 2 W nd = 1 2. We n hndle the omposition prolem for eh of the two omponents of ojet seprtely nd then use the orresponding results to rete the diretionl reltion N:W N:B. Fig. 11 shows two ojets nd suh tht N:B. The hevily dotted re orresponds to tile N(), where ojet 1 lies ( 1 N ) nd the lightly dotted re orresponds to tile W (), where ojet 2 lies ( 2 W ). Sine 1 N Fig. 11. Illustrtions of Exmples 6 nd 7 nd N:B, we hve 1 N N:B nd using Algorithm COMPOSE SM, we n ompute tht 1 N (see lso Fig. 11). Similrly, sine 2 W nd N:B,wehve 2 δ(n,w,s,b). Let us see how we n use these results to lulte the possile reltions etween nd. Sine = 1 2, then if for exmple 1 N nd 2 W :S, we hve tht tile-union(n,w:s) or N:W :S. Likewise, if 1 N nd 2 N:W :B, then tile-union(n,n:w :B) or N:W :B. Therefore, diretionl reltion N:W N:B, inludes ll si reltions reted y tking the tile-union of eh reltion in {N} with every reltion in δ(n,w,s,b). In other words, N:W N:B = Comine(N,δ(N,W,S,B)). The result, s well s the proedure we hve used in order to ompute N:W N:B n e ptured y expression N:W N:B = Comine(N N:B,W N:B). One ould e tempted to generlize this expression nd use it to ompute the omposition of ny two reltions: R 11 : :R 1k R 2 = Comine(R 11 R 2,...,R 1k R 2 ), (C) ut unfortuntely it does not lwys produe the orret result. Let us see nother exmple tht will help lrify why the forementioned expression fils. Exmple 7: Let us ompute W :B N. If we use Expression (C) we hve W :B N = Comine(W N,B N). Sine W :S Comine(W B,W N), it follows from the ove eqution tht there exists sptil onfigurtion suh tht W:B, Nnd W:S. It is esy to verify tht suh onfigurtion does not exist, thus, Expression (C) nnot e pplied to W :B N. To ompute the orret omposition, let, nd e three ojets in REG suh tht W:B nd N. Aording to Definition 4, W:B implies tht there exist ojets 1 nd 2, suh tht 1 W, 2 B nd = 1 2. Fig. 11 nd Fig. 11 depit two sptil onfigurtions involving ojets nd, suh tht N. In oth ses, the lightly dotted re to tile W () (i.e., the re where ojet 1 lies), while the hevily dotted re orresponds to tile B() (i.e., the re where ojet 2 lies). For the onfigurtion of Fig. 11, we hve tht 1 δ(n,w,s,b) nd 2 δ(n,e). Thus, we hve Comine(δ(N,W,S,B),δ(N,E)) nd Comine(δ(N,W,S,B),δ(N,E)) W :B N. For the onfigurtion of Fig. 11, we hve tht 1 δ(n,w) nd 2 δ(n,w,e). Thus, we

11 11 Algorithm COMPOSE M Input: A multi-tile reltion R 1 = R 11: :R 1k (2 k) nd si reltion R 2 Output: The omposition R 1 R 2. Method: R {N,W, S,E} C = Comine(R 11 R 2,...,R 1k R 2) If (R 2 = R) If (R 1 = R :B) Return C {R :R,R :B, R :R :B} If (R 1 = R :B) Return C {R :R,R :B, R :R :B} If (R 2 = R:B) If (R 1 = R :B) Return C {R :R } If (R 1 = R :B) Return C {R :R } If (R 2 {R:R,R:R :B}) If (R 1 = R:R ) Return C {R :B, R :R :B} If (R 1 = R:R ) Return C {R :B, R :R :B} If (R 1 = R :R ) Return C {R :B, R:R :B} If (R 1 = R :R ) Return C {R :B, R:R :B} If (R 1 = R:R ) Return C δ(r,r ) If (R 1 = R:R :R ) Return C δ(r,r,b) {R:R :B, R :R :B} If (R 1 = R:R :R ) Return C δ(r,r,b) {R:R :B, R :R :B} If (R 1 {R:R :B, R:R :R :B, R:R :R :B}) Return C δ(r,r,b) If (R 1 {N:W :S:E, N:W :S:E:B}) Return C δ(r,r,b) If (R 2 {R:R :R,R:R :R :B}) If (R 1 = R :R ) Return C {R } If (R 1 {R :R,R:R :R }) Return C {R :B, R :R :B} If (R 1 {R :R,R:R :R }) Return C {R :B, R :R :B} If (R 1 = R :R :R ) Return C {R,R :B, R :R :B, R :R :B} If (R 1 {R :R :B, R :R :R :B}) Return C {R,R :B} If (R 1 = N:W :S:E) Return C {R :R :B, R :R :B} Return C Fig. 12. Algorithm COMPOSE M hve Comine(δ(N,W),δ(N,W,E)) nd Comine(δ(N,W),δ(N,W,E)) W :B N. Summrizing, we hve Comine(δ(N,W,S,B),δ(N,E)) Comine(δ(N,W),δ(N,W,E)) W :B N. It is not hrd to verify, tht ny other sptil onfigurtion suh tht W:B nd Nwould produe omposition results tht re suset of those produed y the onfigurtions of Fig. 11 nd Fig. 11. Thus, W :B N = Comine(δ(N,W,S,B),δ(N,E)) Comine(δ(N,W),δ(N,W,E)) or equivlently W :B N = Comine(W N,B N) {W :S, W :B,W:S:B}. Summrizing Exmples 6 nd 7, we n distinguish two ses. For some pirs of reltions R 1 = R 11 : :R 1k nd R 2, like N:W nd N:B of Exmple 6, Expression (C) n e pplied diretly. For the other ses, there re pirs, like W :B nd N of Exmple 7, tht Expression (C) nnot e pplied diretly. Fortuntely, s we will see lter, we lwys hve: R 1 R 2 Comine(R 11 R 2,...,R 1k R 2 ) Bsed on this oservtion, we present Algorithm COMPOSE M (Fig. 12), tht n e used to ompute the omposition of multi-tile nd si diretionl reltion. Algorithm COMPOSE M tkes s inputs multi-tile diretionl reltion R 1 = R 11 : :R 1k (k 2) nd si reltion R 2. Initilly, the lgorithm omputes set C = Comine(R 11 R 2,...,R 1k R 2 ). Then, it removes from set C ll reltions tht nnot elong to the omposition R 1 R 2. The following lemm demonstrtes the orretness of Algorithm COMPOSE M. Lemm 5: Let R 1 nd R 2 e two si diretionl reltions. Then R 1 R 2 n e omputed y Algorithm COMPOSE M. Proof: To demonstrte the orretness of Algorithm COMPOSE M, we will present the steps we followed in order to rete it. As we disussed erlier, for some reltion pirs R 1 = R 11 : :R 1k nd R 2 we n diretly ompute their omposition using Expression (C) (Exmple 6), while for other pirs we must ompute their omposition from first prinipls (like in Exmple 7). For these pirs, the omposition is equl to suset of Comine(R 11 R 2,...,R 1k R 2 ) nd therefore n e desried using n expression of the form R 1 R 2 = Comine(R 11 R 2,...,R 1k R 2 ) S, where S is set of si diretionl reltions. Bsed on this oservtion, we present Tle I. This tle presents the omposition of ll 26 multi-tile reltions nd reltions B, N, N:B, N:W, N:W :B, N:W :E, N:W :E:B, N:W :S:E nd N:W :S:E:B. In Tle I, we use str () to denote tht the omposition n e omputed using expression R 1 R 2 = Comine(R 11 R 2,...,R 1k R 2 ).In se where the omposition is omputed using expression R 1 R 2 = Comine(R 11 R 2,...,R 1k ) S, we simply write the set S. The omplete trnsitivity tle n e derived from Tle I using Proposition 2. The struture of Algorithm COMPOSE M reflets the results of Tle I. The omposition of most reltion pirs is equl to C, while hndful of pirs produe results equl to C S. So, the Algorithm is minly list of rules desriing these exeptions. Let us now see how these rules n e derived from Tle I. Consider for instne the omposition of reltions W :B nd N. Aording to Tle I, we hve tht W :B N = Comine(W N,B N) {W :S, W :B,W:S:B}. By pplying Proposition 2 to this expression, we onlude tht: S:B W = Comine(S W, B W ) {S:E,S:B,S:E:B} N:B E = Comine(N E,B E) {N:W, N :B,N:W:B} E:B S = Comine(E S, B S) {N:E,E:B,N:E:B} We n esily verify tht the ove expressions re equivlent to this single expression: R :B R = Comine(R R, B R) {R :R,R :B,R :R :B}, R {N,W,S,E}. The finl version of the Algorithm, s presented in Fig. 12, ontins ll the rules tht n e derived from Tle I s If sttements. The following exmple demonstrtes how Algorithm COMPOSE M is used. Exmple 8: The four outer If sttements of Algorithm COMPOSE M regrd the pttern of reltion R 2. R 2 = R implies tht R 2 onsists of single peripherl tile. R 2 =R:B implies tht R 2 onsists of single peripherl tile nd tile B. Similrly, R 2 {R:R,R:R :B} implies tht R 2 onsists of two non-djent tiles nd possily tile B, while R 2 {R:R :R,R:R :R :B} implies tht R 2 onsists of three djent peripherl tiles nd possily tile B. For instne, the pttern of reltion R 2 = N:S is R:R, where R = N. Hving determined the pttern of reltion R 2 nd ssigned vlue to R, we proeed, if neessry, to the inner If sttements nd sustitute the vlue of R tht we determined. For exmple, let us onsider the omposition N:W N:S. As we mentioned erlier, the pttern of reltion R 2 is R:R, where R = N. As onsequene, the ondition of the third outer If sttement is

12 12 R 1 /R 2 B N N :B N :W, N :W :B N :S N :S:B N :W :E, N :W :E:B N :W :S:E, N :W :S:E:B N :W {W :B, W :S:B} {W :B, W:S:B} N :S {W, E, W :E} {W, E, W :E} N :E {E:B, S:E:B} {E:B, S:E:B} N :B W :S {W :B, N:W :B} {W :B, N:W :B} {S:B, S:E:B} W :E {S} W :B {W :S, W :B W :S:B} {W :S} S:E {E:B, N:E:B} {E:B, N:E:B} {S:B, W :S:B} S:B E:B {S:E, E:B, S:E:B} {S:E} N :W :S {W, E, W :E, W :B, E:B, {W, W :E, W :B, E:B, N :W :B, W:S:B, W:E:B} N :W :B, W:S:B, W :E:B} {S:E:B} N :W :E N :W :B N :S:E {W, E, W :E, W :B, E:B, {E, W :E, W :B, E:B, N :E:B, S:E:B, W :E:B} N :E:B, S:E:B, W:E:B} {W :S:B} N :S:B {W, E, W :E, W:B, {W, E, W :E, W :B, E:B, W:E:B} E:B, W :E:B} N :E:B W :S:E {S, S:B, W :S:B, S:E:B} W :S:B W :E:B {S, S:B} S:E:B N :W :S:E {W, E, W :E, W:B, {W :E, W :B, E:B, W:E:B} E:B, W :E:B} {W :S:B, S:E:B} N :W :S:B {W, E, W :E, W:B, {W, W :E, W:B, E:B, W:E:B} E:B, W :E:B} N :W :E:B N :S:E:B {W, E, W :E, W:B, {E, W:E, W:B, E:B, W:E:B} E:B, W :E:B} W :S:E:B {S, S:B} N :W :S:E:B {W, E, W :E, W:B, E:B, W:E:B} TABLE I PROVING LEMMA 5 {W :E, W:B, E:B, W :E:B} stisfied, so we proeed to the relevnt inner If sttements. By sustituting R N, we notie tht the first inner If sttement is stisfied, sine R 1 = R:R = N:W. Therefore, we hve tht N:W N:S = Comine(N N:S, W N:S) {W :B,W:S:B}. To ompute the omposition of two ritrry diretionl reltions we use the following theorem. Theorem 2: Let Q 1 = k i=1 R 1i nd Q 2 = m j=1 R 2j e two diretionl reltions in 2 B where ll R 1i,R 2j re si diretionl reltions. Then, R 1 R 2 = {R B : R R 1i R 2j }. Note tht R 1i R 2j n e omputed using Lemmt 3, 4 nd 5. Proof: Bsed on Definition 7, we hve tht: Q 1 Q 2 = {R B :(,, REG ) ( Q 1 Q 2 R)} Sine Q 1 = k i=1 R 1i nd Q 2 = m i=1 R 2j, we hve Q 1 = R 11 R 1k nd Q 2 = R 21 R 2m. Therefore, Q 1 Q 2 = {R B :(,, REG )( R 11 R 1k ) ( R 21 R 2m ) R}. Finlly, y distriuting nd, we hve tht: Q 1 Q 2 = {R B :(,, REG ) ( i,j (R 1i R 2j R)} = {R B : R R 1i R 2j } Let us now leve the onsisteny sed definition of omposition nd onsider the stndrd notion of existentil omposition from set theory (Definition 6). Similrly to mny models of sptil reltions [8], [33], [17], the lnguge of CDR is not expressive enough to pture the inry reltion whih is () Fig. 13. Illustrtions of Exmple 9 the result of the existentil omposition of diretionl reltions. This is illustrted y the following exmple. Exmple 9: Consider ojet vriles,, nd diretionl reltions N nd N. The only diretionl reltion implied y these two onstrints is N(see Fig. 13). This is ptured y the ft tht N N = N (see Lemm 3). Let us now ssume tht (N; N) =N lso holds. Then, for eh pir of ojets 0 nd 0 suh tht 0 N 0, there exists n ojet 0 REG suh tht 0 N 0 nd 0 N 0. However, Fig. 13 shows two suh ojets 0 nd 0 suh tht 0 N 0 nd it is impossile to find n ojet 0 REG suh tht 0 N 0 nd 0 N 0. If we onsider Fig. 13, we will notie tht the semntis of existentil omposition imply tht ojet lies ompletely on the north tile of (i.e., Nholds), nd the minimum ounding oxes of ojets nd do not touh. Intuitively, the seond onstrint is not expressile in the lnguge of diretionl reltions presented in Setion II. It is n open question to define n pproprite set of reltions tht ould e used to ugment the lnguge of CDR suh tht the onstrints needed to define the result of existentil omposition re expressile. 0 0 ()

13 13 IV. CONCLUSIONS In this pper, we hve introdued fmily of diretionl reltion models. We hve formlly defined the reltions tht n e expressed in the fmily nd studied the inverse nd the omposition (onsisteny-sed nd existentil) of diretionl reltions. We hve presented methods to ompute the inverse nd onsisteny-sed omposition while we hve demonstrted tht the result of existentil omposition nnot e expressed. The forementioned methods pply to ll the models of the fmily. Further reserh ould onentrte on the extension of CDR lnguge so tht existentil omposition is definle nd the study of lgorithms for (i) omputing the miniml network of set of diretionl onstrints, (ii) for enforing onsisteny nd (iii) performing vrile elimintion ( tsk whih reltes to existentil omposition). REFERENCES [1] M. Egenhofer, Resoning out Binry Topologil Reltionships, in Proeedings of SSD 91, ser. LNCS, vol. 525, 1991, pp [2] A. Frnk, Qulittive Sptil Resoning out Distnes nd Diretions in Geogrphi Spe, Journl of Visul Lnguges nd Computing, vol. 3, pp , [3] Z. Cui, A. Cohn, nd D. Rndell, Qulittive nd Topologil Reltionships in Sptil Dtses, in Proeedings of SSD 93, ser. LNCS, vol. 695, [4] J. Renz nd B. Neel, On the Complexity of Qulittive Sptil Resoning: A Mximl Trtle Frgment of the Region Connetion Clulus, Artifiil Intelligene, vol. 1-2, pp , [5] P. Rigux, M. Sholl, nd A. Voisrd, Sptil Dt Bses. Morgn Kufmn, [6] D. Ppdis, Y. Theodoridis, T. Sellis, nd M. Egenhofer, Topologil Reltions in the World of Minimum Bounding Retngles: A Study with R-trees, in Proeedings of SIGMOD 95, 1995, pp [7] A. Sistl, C. Yu, nd R. Hddd, Resoning Aout Sptil Reltionships in Piture Retrievl Systems, in Proeedings of VLDB 94, 1994, pp [8] B. Bennett, A. Isli, nd A. Cohn, When does Composition Tle Provide Complete nd Trtle Proof Proedure for Reltionl Constrint Lnguge? in Proeedings of the IJCAI 97 workshop on Sptil nd Temporl Resoning, [9] K. Zimmermnn, Enhning Qulittive Sptil Resoning - Comining Orienttion nd Distne, in Proeedings of COSIT 93, ser. LNCS, vol. 716, 1993, pp [10] A. Adelmoty nd H. Willims, Approhes to the Representtion of Qulittive Sptil Reltionships for Geogrphi Dtses, in Proeedings of the Advned Geogrphi Dt Modeling, [11] R. Billen nd E. Clementini, A model for ternry projetive reltions etween regions, in Proeedings of EDBT 04, ser. LNCS, vol. 2992, 2004, pp [12] C. Freks, Using Orienttion Informtion for Qulittive Sptil Resoning, in Proeedings of COSIT 92, ser. LNCS, vol. 639, 1992, pp [13] B. Fltings, Qulittive Sptil Resoning Using Algeri Topology, in Proeedings of COSIT 95, ser. LNCS, vol. 988, [14] A. Mukerjee nd G. Joe, A Qulittive Model for Spe, in Proeedings of AAAI 90, 1990, pp [15] D. Ppdis, Reltion-Bsed Representtion of Sptil Knowledge, Ph.D. disserttion, Dept. of Eletril nd Computer Engineering, Ntionl Tehnil University of Athens, [16] G. Ligozt, Resoning out Crdinl Diretions, Journl of Visul Lnguges nd Computing, vol. 9, pp , [17] S. Skidopoulos nd M. Kourkis, Composing Crdinl Diretion Reltions, Artifiil Intelligene, vol. 152, no. 2, pp , [18] R. Goyl, Similrity Assessment for Crdinl Diretions etween Extended Sptil Ojets, Ph.D. disserttion, Deprtment of Sptil Informtion Siene nd Engineering, University of Mine, April [19] S. Skidopoulos nd M. Kourkis, On the Consisteny of Crdinl Diretions Constrints, Artifiil Intelligene, vol. 163, no. 1, pp , [20] J. Huttenloher, L. Hedges, nd S. Dunn, Ctegories nd prtiulrs: prototype effets in estimting sptil lotion. Psyhologil Review, vol. 98, no. 3, pp , [21] N. Frnklin, L. Henkel, nd T. Zngs, Prsing surrounding spe into regions, Memory nd Cognition, vol. 23, pp , [22] R. Hrtley nd A. Zissermn, Multiple View Geometry in Computer Vision, 2nd ed. Cmridge University Press, [23] Wikipedi, Field of view, 2006, vile t [24] D. Forsyth nd J. Pone, Computer Vision: Modern Approh, 1st ed. Prentie-Hll, [25] J. Kuffner nd J. Ltome, Memory, nd lerning for virtul humns, in Proeedings of Computer Animtion, 1999, pp [26] D. Peuquet nd Z. Ci-Xing, An Algorithm to Determine the Diretionl Reltionship Between Aritrrily-Shped Polygons in the Plne, Pttern Reognition, vol. 20, no. 1, pp , [27] M. Grigni, D. Ppdis, nd C. Ppdimitriou, Topologil Inferene, in Proeedings of IJCAI 95, 1995, pp [28] G. Ligozt, When tles tell it ll: Qulittive sptil nd temporl resoning sed on liner ordering, in Proeedings of COSIT-01, ser. LNCS, vol. 2205, 2001, pp [29] D. Rndell, A. Cohn, nd Z. Cui, Computing Trnsitivity Tles: A Chllenge for Automted Theorem Provers, in Proeedings of CADE 92, ser. LNCS, vol. 607, June 1992, pp [30] M. Egenhofer nd R. Frnzos, Point Set Topologil Reltions, Interntionl Journl of Geogrphil Informtion Systems, vol. 5, pp , [31] D. Ppdis, N. Arkoumnis, nd N. Krpilidis, On The Retrievl of Similr Configurtions, in Proeedings of SDH 98, 1998, pp [32] A. Trski, On the lulus of reltions, Journl of Symoli Logi, vol. 6, pp , [33] I. Düntsh, H. Wng, nd S. MCloskey, A reltion-lgeri pproh to the Region Connetion Clulus, Theoretil Computer Siene, vol. 255, pp , [34] S. Li nd M. Ying, Region Connetion Clulus: Its models nd omposition tle, Artifiil Intelligene, vol. 145, pp , [35] B. Neel nd H.-J. Bürkert, Resoning out Temporl Reltions: A Mximl Trtle Sulss of Allen s Intervl Alger, Journl of the ACM, vol. 42, no. 1, pp , [36] S. Skidopoulos, C. Ginnoukos, N. Srks, P. Vssilidis, T. Sellis, nd M. Kourkis, Computing nd Mnging Crdinl Diretion Reltions, Trnstion on Knowledge nd Dte Engineering, vol. 17, no. 12, pp , [37] T. Brinkhoff, H.-P. Kriegel, nd R. Shneider, Comprison of pproximtions of omplex ojets used for pproximtion-sed query proessing in sptil dtse systems, in Proeedings of ICDE 93, 1993, pp Spiros Skidopoulos reeived his diplom nd Ph.D. from the Ntionl Tehnil University of Athens nd his M.Phil. from UMIST. He is urrently n Assistnt Professor t the Deprtment of Computer Siene nd Tehnology, University of Peloponnese. He hs pulished more thn 25 ppers in the res of sptil nd temporl dtses, onstrint dtses nd resoning, query evlution nd optimiztion, nd dt wrehouses. More informtion is ville t spiros.

14 14 Nikos Srks ompleted his undergrdute studies t the Ntionl Tehnil University of Athens in 2004 nd is urrently working towrds PhD degree t the University of Toronto. His reserh interests inlude sptil resoning, top-k query evlution nd we serh. Timos Sellis reeived his diplom degree in Eletril Engineering in 1982 from the Ntionl Tehnil University of Athens (NTUA), Greee. In 1983 he reeived the M.S. degree from Hrvrd University nd in 1986 the Ph.D. degree from the University of Cliforni t Berkeley, where he ws memer of the INGRES group, oth in Computer Siene. In 1986, he joined the Deprtment of Computer Siene of the University of Mrylnd, College Prk s n Assistnt Professor, nd eme n Assoite Professor in Between 1992 nd 1996 he ws n Assoite Professor t the Computer Siene Division of NTUA, where he is urrently Full Professor. Prof. Sellis is lso the hed of the Knowledge nd Dtse Systems Lortory t NTUA. His reserh interests inlude peer-to-peer dtse systems, dt wrehouses, the integrtion of We nd dtses, nd sptil dtse systems. He hs pulished over 120 rtiles in refereed journls nd interntionl onferenes in the ove res nd hs een invited speker in mjor interntionl events. He ws orgniztion hir for VLDB 97 nd PC Chir for ACM SIGMOD Prof. Sellis is reipient of the prestigious Presidentil Young Investigtor (PYI) wrd given y the President of USA to the most tlented new reserhers (1990), nd of the VLDB Yer Pper Awrd for his work on sptil dtses. He ws the president of the Ntionl Counil for Reserh nd Tehnology of Greee ( ) nd memer of the VLDB Endowment ( ). Mnolis Kourkis is n Assoite Professor in the Dept. of Informtis nd Teleommunitions, Ntionl nd Kpodistrin University of Athens. Prof. Kourkis hs pulished more thn 85 ppers in the res of dtse nd knowledge-se systems, temporl nd sptil resoning, onstrint progrmming, intelligent gents, semnti we, peerto-peer nd grid omputing. His reserh hs een funded y the Europen Commission, the Greek Generl Seretrit for Reserh nd Tehnology nd industril soures. More informtion is ville t kourk.