An Expressive Hybrid Model for the Composition of Cardinal Directions

An Expressive Hyrid Model for the Composition of Crdinl Directions Ah Lin Kor nd Brndon Bennett School of Computing, University of Leeds, Leeds LS2 9JT, UK e-mil:{lin,brndon}@comp.leeds.c.uk Astrct In our previous pper (Kor nd Bennett, 2003), we hve shown how the nine tiles in the projection-sed model for crdinl directions cn e prtitioned into sets sed on horizontl nd verticl constrints (clled Horizontl nd Verticl Constrints Model). In order to come up with n expressive hyrid model for direction reltions etween two-dimensionl single-piece regions (without holes), we integrte the well-known RCC-8 model with the ove-mentioned model. From this expressive hyrid model, we derive 8 tomic inry reltions nd 13 fesile s well s jointly exhustive reltions for the x nd y directions respectively. Bsed on these tomic inry reltions, we derive two seprte 8x8 composition tles for oth the expressive nd wek direction reltions. We introduce formul tht cn e used for the computtion of the composition of expressive nd wek direction reltions etween whole or prt regions. Lstly, we lso show how the expressive hyrid model cn e used to mke severl existentil inferences tht re not possile for existing models. Introduction Ppdis nd Theorodis (1994) descrie topologicl nd direction reltions etween regions using their minimum ounding rectngles (MBRs). However, the lnguge used is not expressive enough to descrie the direction reltions. Additionlly, the MBR technique yields erroneous outcome when the involving regions re not rectngulr in shpe (Goyl nd Egenhofer, 2000). In order to come up with more expressive lnguge for direction reltions, we shll comine mereologicl, topologicl, nd crdinl directions reltions. Typiclly, composition tles re used to infer sptil reltions etween ojects. However, existing composition tles for crdinl reltions(escrig nd Toledo, 1998; Skidopoulos nd Kourkis, 2000) re wek nd not expressive enough. Consequently, these tles cnnot mke some existentil inferences tht will e shown t the lter prt of this pper. Here, we shll show how we come up with n expressive hyrid model for direction reltions. Bsed on this model, we derive two 8x8 composition tles for expressive s well s wek direction reltions. In this pper, we shll descrie the inry reltions in the expressive hyrid model for direction reltions, nd define whole nd prt reltions. This is followed y introducing formul which could e used to compute oth expressive nd wek direction reltions for whole nd prt regions. Finlly, we shll demonstrte how the model could e used to mke severl types of existentil inferences. Horizontl nd Verticl Constrints Model In the projection-sed model for crdinl directions (Frnk, 1992), the plne of n ritrry single-piece region, is prtitioned into nine tiles, North-West, NW(); North, N(); North-Est, NE(); South-West, SW(); South, S(); South-Est, SE(); West, W(); Neutrl Zone, O(); Est, E(). In our previous pper (Kor nd Bennett, 2003), we hve shown how to prtition the nine tiles into sets sed on horizontl nd verticl constrints clled the Horizontl nd Verticl Constrints Model. However, in this pper, we shll renme the sets for esy comprehension purposes. The following re the definitions of the prtitioned regions: WekNorth() is the region tht covers the tiles NW(), N(), nd NE() Horizontl() is the region tht covers the tiles W(), O(), nd E() WekSouth() is the region tht covers the tiles SW(), S(), nd SE() WekWest() is the region tht covers the tiles SW(), W(), nd NW() Verticl() is the region tht covers the tiles S(), O(), nd N() WekEst() is the region tht covers the tiles SE(), E(), nd NE() The set of oundries of the miniml ounding ox for region is could e represented s {X min (), X mx (), Y min (), Y mx ()}. Figure 1 depicts the horizontl nd verticl sets of tiles for. Horizontl set Verticl set Figure 1: Horizontl nd Verticl Sets of Tiles RCC-8 Model The RCC-8 (Rndell et l, 1992) model consists of 8 tomic inry reltions: PO(,), TPP(,), NTPP(,),

EQ(,), NTPPi(,), TPPi(,), EC(,), nd DC(,) which shll e defined lter in this pper. Expressive Hyrid Model In order to come up with more expressive model for the composition of crdinl directions, we integrte the RCC- 8 nd the Horizontl nd Verticl Constrints Model. We shll consider two-dimensionl single-piece regions without hole. We shll consider the x nd y dimensions seprtely. Definitions If there is referent region, nd nother ritrry region, the possile tomic inry reltions etween them cn e defined s follows: WekNorth(,) - WekNorth() Horizontl(,) - Horizontl() WekSouth(,) - WekSouth() WekEst(,) - WekEst() Verticl(,) - Verticl() WekWest(,) - WekWest() DCy(,) y-dimension of is disconnected from y dimension of EQy(,) y-dimension of is identicl with y dimension of POy(,) y-dimension of prtilly overlps y dimension of ECx(,) y-dimension of is externlly connected to x dimension of TPPy(,) y-dimension of is tngentil proper prt of y dimension of NTPPy(,) y-dimension of is nontngentil proper prt of y dimension of TPPiy(,) y-dimension of is tngentil proper prt of y dimension of NTPPiy(,) y-dimension of is nontngentil proper prt of y dimension of Atomic Binry Reltions of the Hyrid Model In this section, we shll demonstrte how we come up with ll possile inry direction reltions for the hyrid model. All the possile tomic inry reltions for ech horizontl set re shown in Figure 2. The nottions tht will e used in this section re: RELy(,Z) is ny tomic inry reltion etween nd the horizontlly prtitioned region, Z RELx(,Z) is ny tomic inry reltion etween nd the verticlly prtitioned region, Z Bsed on Tle 1, the totl numer of possile inry reltions for the hyrid model in the y-direction is [(2+4+2) + (2x4) + (2x2) + (4x2) + (2x4x2)] which equls 44 cses. However, due to the single-piece condition, the following rules pply: Rule 1: (WekNorth() WekSouth()) Rule 2: Assume U to e {WekNorth(), Horizontl(), WekSouth()}. If NTPPy(,R) where R U then [[NTPPy(,R) RELy(,S)] [NTPPy(,R) RELy(,S ) RELy(,T where S U - R, T U S. Rule 3: Assume U to e {WekNorth(), WekSouth()}. If [TPPy(,Horizontl()) ECy(,R)] where R U then [TPPy(,Horizontl) ECy(,R) RELy(,S)] where S U - R. Bsed on the rules ove, the totl numer of fesile inry reltions for single-piece regions in the y-direction is (44 4 23 4) which equls 13 cses. The thirteen fesile nd jointly exhustive inry reltions for the hyrid model re depicted in Figure 3. This mens tht in the hyrid model, the numer of jointly exhustive inry reltions (in oth the x nd y directions) tht hold etween two single-piece regions will e 13x13. This concurs with the 13x13 tomic reltions in the Rectngle Alger Model (Blini et l. 1998). Comined Mereologicl, Topologicl nd Crdinl Direction Reltions In this section, we shll mke two distinctions: whole nd prt crdinl directions, s well s wek nd expressive reltions. We shll rewrite the nottions used in our previous pper (Kor nd Bennett, 2003). P R (,) mens tht only prt of the destintion extended region,, is in tile R(). The direction reltion A R (,) mens tht whole destintion extended region,, is in the tile R(). As n exmple, when is completely in the South-Est tile of, this direction reltion cn e represented s elow: A SE (,) = P N (,) P NE (,) P NW (,) P S (,) P SE (,) P SW (,) P W (,) P E (,) P O (,) The whole nd wek direction reltions re defined in terms of horizontl nd verticl sets. A N (,) WekNorth(,) Verticl(,) A NE (,) WekNorth(,) WekEst(,) A NW (,) WekNorth(,) WekWest(,) A S (,) WekSouth(,) Verticl(,) A SE (,) WekSouth(,) WekEst(,) A SW (,) WekSouth(,) WekWest(,) A E (,) Horizontl(,) Verticl(,) A W (,) Horizontl(,) WekEst(,) A O (,) Horizontlh(,) WekWest(,) The whole nd expressive direction reltions re defined in terms of expressive horizontl nd verticl sets. A generl form of such direction reltion cn e represented s follows: RELy(,H)[A R (,)] RELy(,H) RELx(,V) (1) RELx(,V) where H nd V re horizontlly nd verticlly prtitioned regions for respectively, nd R() H R() V. Composition Tle The composition tles computed for the horizontl nd verticl sets re shown in Tle 1 nd Tle 2. The composition of tomic inry reltions in the expressive model cn e collpsed into wek reltions s shown in the ove-mentioned tles (with shded oxes).

Region WekNorth() Atomic inry reltions TPPy(, WekNorth()) ECy(,Horizontl()) NTPPy(, WekNorth()) TPPy(, WekNorth()) ECy(,Horizontl()) NTPPy(, WekNorth()) TPPy(, Horizontl()) ECy(,WekNorth()) Horizontl() TPPy(, Horizontl()) ECy(,WekSouth()) NTPPy(, Horizontl()) EQy(, Horizontl()) TPPy(, Horizontl()) ECy(,WekNorth()) TPPy(, Horizontl()) ECy(,WekSouth()) TPPy(, WekSouth()) ECy(,Horizontl()) NTPPy(, WekSouth()) WekSouth() NTPPy(, Horizontl()) EQy(, Horizontl()) POy(, WekNorthl()) POy(, Horizontll()) DCy(,WekSouth()) POy(, WekNorthl()) POy(, Horizontll()) ECy(,WekSouth()) POy(, WekNorthl()) POy(, WekSouthl()) NTPPiy(,Horizontl()) TPPy(, WekSouth()) ECy(,Horizontl()) NTPPy(, WekSouth()) Figure 2: Possile tomic inry reltions for ech horizontlly prtitioned region Composition of Regions with Prts In our previous pper (Kor nd Bennett, 2003), the method for the computtion of prt regions is not roust enough ecuse it does not hold for ll cses. In order to ddress such prolem, we introduce formul (Eqution 2.). The sis of the formul is to consider the direction reltion etween nd ech individul prt of followed y the direction reltion etween ech individul prt of nd c. Assume tht the region covers one or more thn one tile of region while region c encompsses one or more thn one tile of region. The formul for the composition of wek direction reltions cn e written s follows: P R (,) P R (c,) [P R1,) P R2 (,) P Rk ( k,)] [P R (c,)] Figure 3: Thirteen fesile nd jointly exhustive inry reltions in the y-direction for the hyrid model Firstly, estlish the direction reltion etween ech individul prt of nd c. The ove composition cn e rewritten s follows: : POy(, WekSouth()) POy(, Horizontll()) DCy(,WekNorth()) POy(, WekSouthl()) POy(, Horizontll()) ECy(,WekNorth()),)] [P R11 ) P R12 ) P R1m, 1 [[P R2,)] [P R21 ) P R22 ) P R2m, 2 [[P Rk ( k,)][p Rk1, k ) P Rk2, k ) P Rkm, k (2) where 1 k 9, nd 1 m 9 Consider the composition of direction for ech individul prt of nd c. Eqution (1) ecomes:

,) P R11 )] [P R1,) P R12 )] [P R1,) P R1m, 1 [[P R2,) P R21 )] [P R2,) P R22 )] : [P R2,) P R2m, 2 [[P Rk ( k,) P Rk1, k )] [P Rk ( k,) P Rk2, k )] [P Rk ( k,) P Rkm, k (2.) By pplying distriutive lw we hve the following eqution:,) [P R11 ) P R12 ) P R1m, 1 [[P R2,) [P R21 ) P R22 ) P R2m, 2 : [[P Rk ( k,) [P Rk1, k ) P Rk2, k ) P Rkm, k (2.) Firstly, we shll demonstrte how to pply the formul for the composition of wek direction reltions followed y more expressive direction reltions. Composition of Wek Direction Reltions Type 1: A R (,) A R (c,) Find the composition of A O (,) A SW (c,) Use Eqution 2. with k = 1, nd m = 1. P R1,) P R11 ) P O (,) P SW (c,) [Horizontl(,) Verticl(,)] [WekSouth(c,) WekWest(c,)] [Horizontl(,) WekSouth(c,)] [Verticl(,) WekWest(c,)] The outcome of the composition is: [Horizontl(c,) WekSouth(c,)] [Verticl(c,) WekWest(c,)] This mens tht the region c O()W()S()SW(). Type 2: A R (,) P R (c,) Find the composition of A E (,) [P NW (c,) P N (c,)] Use Eqution 2. with k = 1, nd 1 m 2.,) P R11 )] [P R1,) P R12 [[P E (,) P NW (,c 1 )] [P E (,) P N (,c 2 [[Horizontl(,) WekEst(,)] [WekNorth,) WekWest, [[Horizontl(,) WekEst(,)] [WekNorth,) Verticl,)] [[Horizontl(,) WekNorth,)] [WekEst(,) WekWest, [[Horizontl(,) WekNorth,)] [WekEst(,) Verticl, The outcome of the composition is: [[Horizontl,) WekNorth,)] [WekEst,) Verticl,) WekWest, [[Horizontl,) WekNorth,)] [WekEst, Both c 1 c nd c 2 c, so the ove outcome cn e written s: [[Horizontl(c,) WekNorth(c,)] [WekEst(c,) Horizontl(c,) WekWest(c, This mens tht the region c E()O()W() NE()N()NW(). Type 3: P R (,) A R (c,) Find the composition of [P O,) P N,)] A NE (c,) Estlish the reltionship etween c nd ech individul prt of. In this cse, when A NE (c,), P NE (c ) nd P NE (c ) holds (this is not necessrily true for ll cses). Use Eqution 2. with 1 k 2 nd m = 1.,)] [P R11 [[P R2,)] [P R21 [[P O,)] [P NE (c, [[P N,)] [P NE (c, Therefore, the ove composition cn e rewritten s: [[P O,)] [P NE (c [[P N,)] [P NE (c [[Horizontl,) Verticl,)] [WekNorth(c ) WekEst(c [[WekNorth,) Verticl,)] [WekNorth(c ) WekEst(c )] [[Horizontl,) WekNorth(c )] [Verticl,) WekEst(c [[WekNorth,) WekNorth(c )] [Verticl,) WekEst(c )] The outcome of the composition is: [[Horizontl(c,) WekNorth(c,)] [WekEst(c,) Verticl(c, [[NTPPy(c, WekNorth()] [WekEst(c,) Verticl(c, = [[NTPPy(c, WekNorth()] [WekEst(c,) Verticl(c, This mens tht the Y min (c) of the miniml ounding ox for region c is greter thn Y mx () of the miniml ounding ox for region nd c NE()N().

Type 4: P R (,) P R (c,) Find the composition of [P O,) P NE,)] [P O (c,) P W (c,) P SW (c,)] The digrm Figure 4 hs een drwn for this exmple. c Figure 4: An exmple Estlish the direction reltion etween ech individul prt of nd c. Use Eqution 2. with 1 k 2, the vlue of m 1 for 1 is 1 m 1 4. while the vlue m 2 for 2 is 1 m 2 7.,) P R11 )] [P R1,) P R12 )] [P R1,) P R13 )] [P R1,) P R14 (c 4 [[P R2,) P R21 )] [P R2,) P R22 )] [P R2,) P R23, )] [P (,) P (c, )] 2 R2 2 R24 4 2 [P R2,) P R25 (c 5, )] [P (,) P (c, )] 2 R2 2 R26 6 2 [P R2,) P R27 (c 7, 2 [[P O,) P S )] [P O,) P SW )] c [P O,) P W )] [P O,) P O (c 4 [[P NE,) P NE )] [P NE,) P N )] [P NE,) P NW, )] [P (,) P (c, )] 2 NE 2 E 4 2 [P NE,) P O (c 5, )] [P (,) P (c, )] 2 NE 2 W 6 2 [P NE,) P SW (c 7, 2 Apply the distriutive lw s in Eqution 2., nd we will get: [[P NE,) [P S ) P SW ) P W ) P O (c 4 prt(1) [[P O,) [P NE ) P N ) P NW, ) 2 P E (c 4, ) P (c, ) P (c, ) P (c, prt(2) 2 O 5 2 W 6 2 SW 7 2 c In prt(1) of composition, c 1, c 2, c 3, c 4 c. To simplify the composition, we consider the comined horizontl nd verticl sets of ll the prts of c. Thus we hve the following: [WekNorth,) WekEst,)] 2 1 Boundries of miniml ounding ox for region Boundries of miniml ounding ox for region [[Horizontl(c ) WekSouth(c )] [Verticl(c ) WekWest(c [[WekNorth,)][Horizontl(c ) WekSouth(c [[WekEst,)][Verticl(c ) WekWest(c = WekNorth(c, ) Horizontl(c, ) WekSouth(c, )] [WekEst(c,) Verticl(c, ) WekWest(c, )] In prt(2) of composition, c 1, c 2, c 3, c 4, c 5, c 6, c 7 c. The simplified version of the composition is s follows: [Horizontl,) Verticl,)] [[WekNorth(c ) Horizontl(c ) WekSouth(c )] [WekEst(c ) Verticl(c ) WekWest(c [[Horizontl,)][WekNorth(c ) Horizontl(c ) WekSouth(c [[Verticl,)][WekEst(c ) Verticl(c ) WekWest(c = [[WekNorth(c, ) Horizontl(c, ) WekSouth(c, )] [WekEst(c, ) Verticl(c, ) WekWest(c, The finl outcome of the composition is prt(1)prt(2) is equivlent to: [WekNorth(c, ) Horizontl(c, ) WekSouth(c, )] [WekEst(c, ) Verticl(c, ) WekWest(c, )] This mens tht the region c U which is the union of ll the 9 tiles of region. However, sed on Figure 4, region c SW(). Composition of Expressive Direction Reltions We shll use the following nottions to represent the 13 inry y-direction reltions: REL1y(,) NTPPy(,WekNorth()) REL2y(,) TPPy(,WekNorth())ECy(,Horizontl()) REL3y(,)) TPPy(,Horizontl())ECy(,WekNorth()) REL4y(,)) TPPy(,Horizontl())ECy(,WekSouth()) REL5y(,) NTPPy(,Horizontl()) REL6y(,) EQy(,Horizontl()) REL7y(,) NTPPy(,WekSouth()) REL8y(,) TPPy(,WekSouth())ECy(,Horizontl()) REL9y(,) POy(,WekNorth())POy(,Horizontl()) DCy(,WekSouth()) REL10y(,) POy(,WekNorth())POy(,Horizontl()) ECy(,WekSouth()) REL11y(,) POy(,WekNorth())POy(,WekSouth()) NTPPiy(,Horizontl()) REL12y(,) POy(,WekSouth())POy(,Horizontl()) DCy(,WekNorth()) REL13y(,) POy(,WekSouth())POy(,Horizontl()) ECy(,WekNorth()) Similr nottions will e used to represent the 13 inry x-direction reltions (WekNorth is replced y WekEst, Horizontl with Verticl nd WekSouth y WekWest).

Exmple 1: Find the composition of the following: [ REL3y(,) [P O (,)] REL3x(1,) REL2y(,) [PN E (,)] REL2x(1,) ] [ REL3y(c,) [A N (c,)] REL3x(c,) ] Estlish the direction reltion etween c nd ech individul prt of. Use Eqution 2., with 1 k 2 nd 1 m 1 2, nd 1 m 2 2.,)] [P R11 ) P R12 [[P R2,)] [P R21 ) P R22 Use Eqution (1), nd the ove composition cn e rewritten in the following expressive form: [[REL3y,)REL3x,[[REL1y )REL3x )] [REL1y )REL2x [[REL2y,)REL2x,[[REL2y )REL4x )] [REL2y )REL8x [REL3y,)[REL1y )REL1y [REL3x,)[REL3x )REL2x [REL2y,)[REL2y )REL2y [REL2x,)[REL4x )REL8x Use Tles 1 nd 2, nd c 1 c nd c 2 c. Thus, the outcome of the composition cn e written s follows: REL1y(c,)[REL2x(c,)REL3x(c,)] REL1y(c,)[REL2x(c,)REL3x(c,) REL6x(c,)REL13x(c,)] = REL1y(c,)[REL2x(c,)REL3x(c,)] The outcome of the composition is: NTPPy(c,WekNorth()) [TPPx(c,WekEst())ECy(c,Horizontl()) TPPx(c,Verticl())ECy(c, Horizontl())] Exmple 2: This exmple is similr to the fourth exmple in the previous section of this pper. Find the composition of [P O,) P NE,)] [P O (c,) P W (c,) P SW (c,)] Estlish the direction reltion etween c nd ech individul prt of. Use Eqution 2., with 1 k 2 nd 1 m 1 4, nd 1 m 2 7. The composition in expressive form will e s follows: For prt 1 [[REL2y,)REL2x, [[REL4y )REL4x )][REL8y )REL4x )] [REL4y )REL8x )][REL8y(c 4 )REL8x(c 4 The regions c 1,c 2,c 3,c 4 c, the ove composition cn e written s follows: [REL2y,)[REL4y(c )REL8y(c ) REL4y(c )REL8y(c [REL2x,)[REL4x(c )REL4x(c ) REL8x(c )REL8x(c = [REL2y(c,)REL3y(c,)REL6y(c,)REL13y(c,)] [REL2x(c,)REL3x(c,)REL6x(c,)REL13x(c,)] (3) For prt 2 [[REL3y,)REL3x, [[REL8y )REL7x )][REL6y )REL8x )] [REL2y )REL8x )][REL2y(c 4 )REL6x(c 4 )] [REL3y(c 5 )REL6x(c 5 )][REL3y(c 6 )REL2x(c 6 )] [REL2y(c 7 )REL2x(c 7 = [REL2y(c,)REL3y(c,)REL5y(c,)REL12y(c,)] [REL2x(c,)REL3x(c,)REL4x(c,)REL5x(c,) REL7x(c,)REL8x(c,)REL12x(c,)] (3) The finl outcome of the composition is the composition of prt 1 (Eqution 3.) prt 2 (Eqution 3.). Apply Rule 3 from the erlier prt of the pper nd we will get the following: [REL2y(c,)REL3y(c,)REL6y(c,)REL13y(c,)] [REL2y(c,)REL3y(c,) REL12y(c,)] [REL2x(c,)REL3x(c,)REL4x(c,) REL8x(c,)REL12x(c,)] [REL2x(c,)REL3x(c,)REL6x(c,)REL13x(c,)] (4) We collpse some of the disjunction of reltions: REL6y(c,)REL13y(c,) = REL13y(c,) REL4x(c,) REL8x(c,)REL12x(c,) = REL12y(c,) REL6x(c,)REL13x(c,) = REL13x(c,) Eqution 4 ecomes: [REL2y(c,)REL3y(c,) REL13y(c,)] [REL2y(c,)REL3y(c,) REL12y(c,)] [REL2x(c,)REL3x(c,)REL12x(c,)] [REL2x(c,)REL3x(c,) REL13x(c,)] (4.) Region c is single-piece. Therefore, Eqution 4. ecomes: [POy(c,WekNorth())POy(c,WekSouth()) NTPPiy(c,Horizontl())] [POx(c,WekEst())POx(c,WekWest()) NTPPix(c,Verticl())] This mens tht the region c U which is the union of ll the 9 tiles of region. As mentioned erlier, sed on Figure 4, region c SW(). Thus, the outcome of the composition for wek reltions (in the previous section) yields the sme result s this composition. However, the computtion for the ltter is more tedious nd complex when involving regions with mny prts.

Existentil Inference In this pper, we shll demonstrte how the expressive hyrid crdinl direction model could e used to mke severl existentil inferences which re not possile in existing models.. Exmple 1: Find R(,) such tht c WekNorth() nd c WekNorth() Skidopoulos model (2001) which is not expressive enough, cnnot nswer such query ecuse the composition tle computed is not existentil. To nswer this query, we must first specify the expressive reltion etween nd c. There re two possile reltions: TPPy(c,WekNorth()) or WekNorth(c,). If it is the former then composition is is WekNorth(,)WekNorth(c,) which mens R(,) is WekNorth(,). If it is the ltter, there re severl comintions: WekNorth(,)Horizontl(c,) WekNorth(,)WekSouth(c,) Horizontl(,)WekNorth(c,) WekSouth(,)WekNorth(c,) The first reltion in ech composition listed ove is R(,). Exmple 2: Find R(,) nd S(c,) such tht T(,c) is [TPPy(c,Horizontl()) ECy(c,WekSouth())] Bsed on Tle 1, 9 different compositions will yield the following outcome: TPPy(c,Horizontl())ECy(c,WekSouth())] The set of possile compositions, Q, is: {REL1y(,)REL7y(c,), REL2y(,)REL7y(c,), REL3y(,)REL7y(c,), REL3y(,)REL8y(c,), REL5y(,)REL7y(c,), REL5y(,)REL8y(c,), REL6y(,)REL4y(c,), REL7y(,)REL1y(c,), REL8y(,)REL12(c,)} If U equls 8 x 8 tomic inry direction reltions, then the set of ll possile ordered pirs of R nd S which stisfy the ove query will e U Q. composition of wek or expressive reltions etween whole nd prt regions. We hve lso demonstrted how the composition tle with expressive direction reltions could e used to mke severl difficult existentil inferences. Acknowledgement Mny thnks to Syhmnt Hzrik for his vlule comments. References Blini, P., Condott, J., & del Cerro, L. F. (1998). A Model for Resoning out Bidimensionl Temporl Reltions. Proceedings of the Sixth Interntionl Conference on Knowledge Representtion nd Resoning. Escrig, M.T. nd Toledo, F. (1998). Qulittive Sptil Resoning: Theory nd Prctice: Appliction to Root Nvigtion. IOS Press: Amsterdm. Frnk, A. (1992). Qulittive Sptil Resoning with Crdinl Directions. Journl of Visul Lnguges nd Computing, Vol(3), pp.343-371. Goyl, R. nd Egenhofer, M. (2000). Consistent Queries over Crdinl Directions cross Different Levels of Detil. 11 th Interntionl Workshop on Dtse nd Expert Systems Applictions, Greenwich, UK. Kor, A.L. & Bennett, B. (2003). Composition for Crdinl Directions y Decomposing Horizontl nd Verticl Constrints. Proceedings of AAAI 2003 Spring Symposium, Mrch 24-26, Stnford University, Cliforni. Ppdis, D. nd Theodoridis, Y. (1994). Sptil Reltions, Minimum Bounding Rectngles, nd Sptil Dt Structures. University of Cliforni, Technicl Report KDBSLAB-TR-94-04. Rndell, D., Cui, Z. & Cohn, A.G. (1992). A Sptil Logic sed on Regions nd Connection. Proceedings of the Third Interntionl Conference on Knowledge Representtion nd Resoning. Skidopoulos, S. & Kourkis, M. (2001). Composing Crdinl Direction Reltions. Proceedings of SSTD-01, Redondo Bech, CA, USA, July 2001. Exmple 3: Find R(,) nd S(c,) such tht T(,c) is POy(c,WekSouth())POy(c,Horizontl()) ECy(c,WekNorth()) Bsed on Tle 1, we hve 4 pirs of R nd S which stisfy T. They re: REL1y(,)REL7y(c,),REL2y(,) REL8y(c,), REL7y(,)REL1y(c,), REL7y(,)REL2y(c,). Conclusion In this pper, we hve shown how topologicl nd direction reltions cn e integrted to produce more expressive hyrid model for crdinl directions. The composition tle derived from this model could e used to infer oth wek nd expressive direction reltions etween regions. We hve lso introduced nd demonstrted how to use formul to compute the

WekNort(,) Horizontl(,) WekNorth(c,) Horizontl(c,) WekSouth(c,) NTPPy(c,WekNorth()) TPPy(c,WekNorth()) ECy(c,Horizontl()) TPPy(c,Horizontl()) ECy(c,WekNorth()) TPPy(c,Horizontl()) ECy(c,WekSouth()) NTPPy(c,Horizontl()) EQy(c,Horizontl()) NTPPy(c,WekSouth()) TPPy(c,WekSouth()) ECy(c,Horizontl()) NTPPy(,WekNorth()) NTPPy(c,WekNorth()) NTPPy(c,WekNorth()) NTPPy(c,WekNorth()) NTPPy(c,WekNorth()) NTPPy(c,WekNorth()) NTPPy(c,WekNorth()) U 13 reltions NTPPy(c,WekNorth()) TPPy(c,WekNorth()) ECy(c,Horizontl()) POy(c,WekNorth()) POy(,Horizontl()) DCy(c,WekSouth()) POy(c,WekNorth()) POy(,Horizontl()) ECy(c,WekSouth()) POy(c,WekNorth()) POy(,WekSouth()) NTPPiy(c,Horizontl()) TPPy(,WekNorth()) ECy(,Horizontl()) NTPPy(c,WekNorth()) NTPPy(c,WekNorth()) NTPPy(c,WekNorth()) TPPy(c,WekNorth()) ECy(c,Horizontl()) NTPPy(c,WekNorth()) TPPy(c,WekNorth()) ECy(c,Horizontl()) NTPPy(c,Horizontl()) TPPy(c,Horizontl()) ECy(c,WekSouth()) TPPy(c,WekSouth()) ECy(c,Horizontl()) NTPPy(c,WekSouth()) POy(c,WekSouth()) POy(,Horizontl()) DCy(c,WekNorth()) TPPy(c,Horizontl()) ECy(c,WekNorth()) EQy(c,Horizontl()) POy(c,WekSouth()) POy(,Horizontl()) ECy(c,WekNorth()) WekNorth(,) NTPPy(c,WekNorth()) WekNorth(c,) WekNorth(c,), Horizontl(c,) WekSouth(c,), TPPy(,Horizontl()) NTPPy(c,Horizontl()) NTPPy(c,Horizontl()) ECy(,WekNorth()) NTPPy(c,WekNorth()) TPPy(c,WekNorth()) ECy(c,Horizontl()) TPPy(c,Horizontl()) ECy(c,WekNorth()) NTPPy(c,Horizontl()) NTPPy(c,Horizontl()) TPPy(c,Horizontl()) ECy(c,WekNorth()) TPPy(c,Horizontl()) ECy(c,WekSouth()) TPPy(c,WekSouth()) ECy(c,Horizontl()) NTPPy(c,WekSouth()) POy(c,WekSouth()) & POy(,Horizontl()) & DCy(c,WekNorth()) TPPy(c,Horizontl()) ECy(c,WekSouth()) POy(c,WekSouth()) POy(,Horizontl()) DCy(c,WekNorth()) TPPy(,Horizontl()) ECy(,WekSouth()) NTPPy(c,Horizontl()) TPPy(c,Horizontl()) ECy(c,WekNorth()) TPPy(c,WekNorth()) ECy(c,Horizontl()) NTPPy(c,WekNorth()) POy(c,WekNorth()) POy(,Horizontl()) DCy(c,WekSouth()) NTPPy(c,Horizontl()) TPPy(c,Horizontl()) ECy(c,WekNorth()) POy(c,WekNorth()) POy(,Horizontl()) DCy(c,WekSouth()) NTPPy(c,Horizontl()) TPPy(c,Horizontl()) ECy(c,WekNorth()) NTPPy(c,Horizontl()) TPPy(c,Horizontl()) ECy(c,WekNorth()) NTPPy(c,WekSouth()) TPPy(c,WekSouth()) ECy(c,Horizontl())

WekNorth(c,) Horizontl(c,) WekSouth(c,) NTPPy(c,WekNorth()) TPPy(c,WekNorth()) ECy(c,Horizontl()) TPPy(c,Horizontl()) ECy(c,WekNorth()) TPPy(c,Horizontl()) ECy(c,WekSouth()) NTPPy(c,Horizontl()) EQy(c,Horizontl()) NTPPy(c,WekSouth()) TPPy(c,WekSouth()) ECy(c,Horizontl()) Horizontl(,) NTPPy(,Horizontl()) NTPPy(c,Horizontl()) TPPy(c,Horizontl()) ECy(c,WekNorth()) TPPy(c,WekNorth()) ECy(c,Horizontl()) NTPPy(c,WekNorth()) POy(c,WekNorth()) POy(,Horizontl()) DCy(c,WekSouth()) NTPPy(c,Horizontl()) TPPy(c,Horizontl()) ECy(c,WekNorth()) POy(c,WekNorth()) POy(,Horizontl()) DCy(c,WekSouth()) NTPPy(c,Horizontl()) NTPPy(c,Horizontl()) NTPPy(c,Horizontl()) NTPPy(c,Horizontl()) NTPPy(c,Horizontl()) TPPy(c,Horizontl()) ECy(c,WekSouth()) TPPy(c,WekSouth()) ECy(c,Horizontl()) NTPPy(c,WekSouth()) POy(c,WekSouth()) POy(,Horizontl()) DCy(c,WekNorth()) NTPPy(c,Horizontl()) TPPy(c,Horizontl()) ECy(c,WekSouth()) POy(c,WekSouth()) POy(,Horizontl()) DCy(c,WekNorth()) EQy(,Horizontl()) NTPPy(c,WekNorth()) TPPy(c,WekNorth()) ECy(c,Horizontl()) TPPy(c,Horizontl()) ECy(c,WekNorth()) TPPy(c,Horizontl()) ECy(c,WekSouth()) NTPPy(c,Horizontl()) EQy(c,Horizontl()) NTPPy(c,WekSouth()) TPPy(c,WekSouth()) ECy(c,Horizontl()) WekSouth(,) Horizontl(,) WekNorth(c,), Horizontl(c,) Horizontl(c,) Horizontl(c,) WekSouth(c,) NTPPy(,WekSouth()) U 13 reltions NTPPy(c,WekSouth()) NTPPy(c,WekSouth()) NTPPy(c,WekSouth()) NTPPy(c,WekSouth()) NTPPy(c,WekSouth()) NTPPy(c,WekSouth()) NTPPy(c,WekSouth()) TPPy(c,WekSouth()) ECy(c,Horizontl()) POy(c,WekSouth()) POy(,Horizontl()) DCy(c,WekNorth()) POy(c,WekSouth()) POy(,Horizontl()) ECy(c,WekNorth()) POy(c,WekSouth()) POy(,WekNorth()) NTPPiy(c,Horizontl()) TPPy(,WekSouth()) & ECy(,Horizontl()) NTPPy(c,Horizontl()) TPPy(c,Horizontl()) ECy(c,WekNorth()) TPPy(c,WekNorth()) ECy(c,Horizontl()) NTPPy(c,WekNorth()) POy(c,WekNorth()) POy(,Horizontl()) DCy(c,WekSouth()) TPPy(c,Horizontl()) ECy(c,WekSouth()) EQy(c,Horizontl()) POy(c,WekNorth()) POy(,Horizontl()) ECy(c,WekSouth()) TPPy(c,WekSouth()) ECy(c,Horizontl()) NTPPy(c,WekSouth()) NTPPy(c,WekSouth()) TPPy(c,WekSouth()) ECy(c,Horizontl()) NTPPy(c,WekSouth()) NTPPy(c,WekSouth()) WekNorth(c,), Horizontl(c,) WekSouth(c,), WekSouth(c,), NTPPy(c,WekSouth()) Note: The shded oxes re for wek composition of reltions Tle 1: Composition of inry reltions in the y-direction for the hyrid model

WekEst(,) Verticl(,) WekEst(c,) Verticl(c,) WekWest(c,) NTPPx(c,WekEst()) TPPx(c,WekEst()) ECx(c,Verticl()) TPPx(c,Verticl()) ECx(c,WekEst()) TPPx(c,Verticl()) ECx(c,WekWest()) NTPPx(c,Verticl()) EQx(c,Verticl()) NTPPx(c,WekWest()) TPPx(c,WekWest()) ECx(c,Verticl()) NTPPx(,WekEst()) NTPPx(c,WekEst()) NTPPx(c,WekEst()) NTPPx(c,WekEst()) NTPPx(c,WekEst()) NTPPx(c,WekEst()) NTPPx(c,WekEst()) U 13 reltions NTPPx(c,WekEst()) TPPx(c,WekEst()) ECx(c,Verticl()) POx(c,WekEst()) POx(,Verticl()) DCy(c,WekWest()) POx(c,WekEst()) POx(,Verticl()) ECx(c,WekWest()) POx(c,WekEst()) POx(,WekWest()) NTPPix(c,Verticl()) TPPx(,WekEst()) ECx(,Verticl()) NTPPx(c,WekEst()) NTPPx(c,WekEst()) NTPPx(c,WekEst()) TPPx(c,WekEst()) ECx(c,Verticl()) NTPPx(c,WekEst()) TPPx(c,WekEst()) ECx(c,Verticl()) NTPPx(c,Verticl()) TPPx(c,Verticl()) ECx(c,WekWest()) TPPx(c,WekWest()) ECx(c,Verticl()) NTPPx(c,WekWest()) POx(c,WekWest()) POx(,Verticl()) DCy(c,WekEst()) TPPx(c,Verticl()) ECx(c,WekEst()) EQx(c,Verticl()) POx(c,WekWest()) POx(,Verticl()) ECx(c,WekEst()) WekEst(,) NTPPx(c,WekEst()) WekEst(c,) WekEst(c,) Verticl(c,) WekWest(c,) TPPx(,Verticl()) NTPPx(c,Verticl()) NTPPx(c,Verticl()) ECx(,WekEst()) NTPPx(c,WekEst()) TPPx(c,WekEst()) ECx(c,Verticl()) TPPx(c,Verticl()) ECx(c,WekEst()) NTPPx(c,Verticl()) NTPPx(c,Verticl()) TPPx(c,Verticl()) ECx(c,WekEst()) TPPx(c,Verticl()) ECx(c,WekWest()) TPPx(c,WekWest()) ECx(c,Verticl()) NTPPx(c,WekWest()) POx(c,WekWest()) POx(,Verticl()) DCy(c,WekEst()) TPPx(c,Verticl()) ECx(c,WekWest()) POx(c,WekWest()) POx(,Verticl()) DCy(c,WekEst()) TPPx(,Verticl()) ECx(,WekWest()) NTPPx(c,Verticl()) TPPx(c,Verticl()) ECx(c,WekEst()) TPPx(c,WekEst()) ECx(c,Verticl()) NTPPx(c,WekEst()) POx(c,WekEst()) POx(,Verticl()) DCy(c,WekWest()) NTPPx(c,Verticl()) TPPx(c,Verticl()) ECx(c,WekEst()) POx(c,WekEst()) POx(,Verticl()) DCy(c,WekWest()) NTPPx(c,Verticl()) TPPx(c,Verticl()) ECx(c,WekEst()) NTPPx(c,Verticl()) TPPx(c,Verticl()) ECx(c,WekEst()) NTPPx(c,WekWest()) TPPx(c,WekWest()) ECx(c,Verticl())

WekEst(c,) Verticl(c,) WekWest(c,) NTPPx(c,WekEst()) TPPx(c,WekEst()) TPPx(c,Verticl()) TPPx(c,Verticl()) NTPPx(c,Verticl()) EQx(c,Verticl()) NTPPx(c,WekWest()) TPPx(c,WekWest()) ECx(c,Verticl()) ECx(c,WekEst()) ECx(c,WekWest()) ECx(c,Verticl()) Verticl(,) NTPPx(,Verticl()) NTPPx(c,Verticl()) TPPx(c,Verticl()) ECx(c,WekEst()) TPPx(c,WekEst()) ECx(c,Verticl()) NTPPx(c,WekEst()) POx(c,WekEst()) POx(,Verticl()) DCy(c,WekWest()) NTPPx(c,Verticl()) TPPx(c,Verticl()) ECx(c,WekEst()) POx(c,WekEst()) POx(,Verticl()) DCy(c,WekWest()) NTPPx(c,Verticl()) NTPPx(c,Verticl()) NTPPx(c,Verticl()) NTPPx(c,Verticl()) NTPPx(c,Verticl()) TPPx(c,Verticl()) ECx(c,WekWest()) TPPx(c,WekWest()) ECx(c,Verticl()) NTPPx(c,WekWest()) POx(c,WekWest()) POx(,Verticl()) DCy(c,WekEst()) NTPPx(c,Verticl()) TPPx(c,Verticl()) ECx(c,WekWest()) POx(c,WekWest()) POx(,Verticl()) DCy(c,WekEst()) EQx(,Verticl()) NTPPx(c,WekEst()) TPPx(c,WekEst()) ECx(c,Verticl()) TPPx(c,Verticl()) ECx(c,WekEst()) TPPx(c,Verticl()) ECx(c,WekWest()) NTPPx(c,Verticl()) EQx(c,Verticl()) NTPPx(c,WekWest()) TPPx(c,WekWest()) ECx(c,Verticl()) WekWest(,) Verticl(,) WekEst(c,) Verticl(c,) Verticl(c,) Verticl(c,) WekWest(c,) NTPPx(,WekWest()) U 13 reltions NTPPx(c,WekWest()) NTPPx(c,WekWest()) NTPPx(c,WekWest()) NTPPx(c,WekWest()) NTPPx(c,WekWest()) NTPPx(c,WekWest()) NTPPx(c,WekWest()) TPPx(c,WekWest()) ECx(c,Verticl()) POx(c,WekWest()) POx(,Verticl()) DCy(c,WekEst()) POx(c,WekWest()) POx(,Verticl()) ECx(c,WekEst()) POx(c,WekWest()) POx(,WekEst()) NTPPix(c,Verticl()) TPPx(,WekWest()) & ECx(,Verticl()) NTPPx(c,Verticl()) TPPx(c,Verticl()) ECx(c,WekEst()) TPPx(c,WekEst()) ECx(c,Verticl()) NTPPx(c,WekEst()) POx(c,WekEst()) POx(,Verticl()) DCy(c,WekWest()) TPPx(c,Verticl()) ECx(c,WekWest()) EQx(c,Verticl()) POx(c,WekEst()) POx(,Verticl()) ECx(c,WekWest()) TPPx(c,WekWest()) ECx(c,Verticl()) NTPPx(c,WekWest()) NTPPx(c,WekWest()) TPPx(c,WekWest()) ECx(c,Verticl()) NTPPx(c,WekWest()) NTPPx(c,WekWest()) WekWest(,) WekEst(c,) Verticl(c,) WekWest(c,) WekWest(c,) NTPPx(c,WekWest()) Note: The shded oxes re for wek composition of reltions Tle 2: Composition of inry reltions in the x-direction for the hyrid model