Y Y Y Y EC(; Y ) T(; Y ) T?1 (; Y ) Y Y Y Y DC(; Y ) PO(; Y ) EQ(; Y ) NT(; Y ) NT?1 (; Y ) Figure 1: Two-dimensiona exampes for the eight base reatio

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1 To appear in Proc. 15th Internationa Joint Conference on Articia Inteigence (IJCAI'97), Nagoya, Japan, August 1997 On the Compexity of Quaitative Spatia Reasoning: A Maxima Tractabe Fragment of the Region Connection Cacuus Jochen Renz Bernhard Nebe Institut fur Informatik, Abert-Ludwigs-Universitat Freiburg Am Fughafen 17, D Freiburg, Germany Abstract The computationa properties of quaitative spatia reasoning have been investigated to some degree. However, the question for the boundary between poynomia and NP-hard reasoning probems has not been addressed yet. In this paper we expore this boundary in the \Region Connection Cacuus" RCC-8. We extend Bennett's encoding of RCC-8 in moda ogic. Based on this encoding, we prove that reasoning is NP-compete in genera and identify a maxima tractabe subset of the reations in RCC-8 that contains a base reations. Further, we show that for this subset path-consistency is sucient for deciding consistency. 1 Introduction When describing a spatia conguration or when reasoning about such a conguration, often it is not possibe or desirabe to obtain precise, quantitative data. In these cases, quaitative reasoning about spatia congurations may be used. One particuar approach in this context has been deveoped by Rande, Cui, and Cohn [1992], the so-caed Region Connection Cacuus (RCC), which is based on binary topoogica reations. One variant of this cacuus, RCC-8, uses eight mutuay exhaustive and pairwise disjoint reations, caed base reations, to describe the topoogica reationship between two regions (see aso Egenhofer [1991]). Some of the computationa properties of this cacuus have been anayzed by Grigni et a. [1995] and Nebe [1995]. However, no attempt has yet been made to determine the boundary between poynomia and NP-hard fragments of RCC-8, as it has been done for Aen's [1983] This research was partiay supported by DFG as part of the project fast-qua-space, which is part of the DFG specia research eort on \Spatia Cognition". interva cacuus [Nebe and Burckert, 1995]. We address this probem and identify a maxima fragment of RCC-8 that is sti tractabe and contains a base reations. As in the case of quaitative tempora reasoning, this proof reies on a computer generated case-anaysis that cannot be reproduced in a research paper. 1 Further, we show that for this fragment path-consistency is sucient for deciding consistency. 2 2 Quaitative Spatia Reasoning with RCC RCC is a topoogica approach to quaitative spatia representation and reasoning where spatia regions are subsets of topoogica space [Rande et a., 1992]. Reationships between spatia regions are dened in terms of the reation C(a; b) which is true i the cosure of region a is connected to the cosure of region b, i.e. if they share a common point. Regions themseves do not have to be internay connected, i.e. a region may consist of dierent disconnected parts. The domain of spatia variabes (denoted as ; Y ; Z) is the whoe topoogica space. In this work we wi focus on RCC-8, but most of our resuts can easiy be appied to RCC-5, a subset of RCC-8 [Bennett, 1994]. RCC-8 uses a set of eight pairwise disjoint and mutuay exhaustive reations, caed base reations, denoted as DC, EC, PO, EQ, T, NT, T?1, and NT?1, with the meaning of DisConnected, Externay Connected, Partia Overap, EQua, Tangentia Proper Part, Non-Tangentia Proper Part, and their converses. Exampes for these reations are shown in Figure 1. In RCC-5 the boundary of a region is not taken into account, i.e. one does not distinguish between DC and EC and between T and NT. These reations are combined to the RCC-5 base reations DR for DiscRete and for Proper Part, respectivey. Sometimes it is not known which of the eight base reations hods between two regions, but it is possibe 1 The programs can be obtained from the authors. 2 Fu proofs can be found in our technica report [Renz and Nebe, 1997].

2 Y Y Y Y EC(; Y ) T(; Y ) T?1 (; Y ) Y Y Y Y DC(; Y ) PO(; Y ) EQ(; Y ) NT(; Y ) NT?1 (; Y ) Figure 1: Two-dimensiona exampes for the eight base reations of RCC-8 to restrict to some of them. In order to represent this, unions of base reations can be used. Since base reations are pairwise disjoint, this resuts in 2 8 dierent reations, incuding the union of a base reations, which is caed universa reation. In the foowing we wi write sets of base reations to denote these unions. Using this notation, DR, e.g., is identica to fdc; ECg. Spatia formuas are written as RY, where R is a spatia reation. Apart from union ([), other operations are dened, namey, converse (^), intersection (\), and composition () of reations. The forma denitions of these operations are: 8; Y : 8; Y : (R [ S)Y (R \ S)Y $ $ RY _ SY, RY ^ SY, 8; Y : R^Y $ Y R, 8; Y : (R S)Y $ 9Z : (RZ ^ ZSY ): The compositions of the eight base reations are shown in Tabe 1. Every entry in the composition tabe species the reation obtained by composing the base reation of the corresponding row with the base reation of the corresponding coumn. Composition of two arbitrary RCC-8 reations can be obtained by computing the union of the composition of the base reations. A spatia conguration can be described by a set of spatia formuas. One important computationa probem is deciding consistency of, i.e. deciding whether it is possibe to assign regions to the spatia variabes in a way that a reations hod. We ca this probem RSAT. When ony reations of a specic set S are used in, the corresponding reasoning probem is denoted RSAT(S). In the foowing bs denotes the cosure of S under composition, intersection, and converse. 3 Encoding of RCC-8 in Moda Logic In this work we use Bennett's [1995] encoding of RCC-8 in propositiona moda ogic. 3 Bennett obtained this encoding by anayzing the reationship of regions to the universe U. He restricted his anaysis to cosed regions 3 We assume in the remainder that the reader is famiiar with moda ogic as presented, e.g., by Fitting [1993]. DC EC PO T NT T -1 NT -1 EQ DR DR DR DR DC * PO PO PO PO DC DC DC DR DR,EQ DR EC PO EC PO PO,T PO PO DR DC EC -1 T -1 DR DR PO PO DR DR PO PO PO * PO PO PO DR DR,EQ DR T DC DR PO NT PO,T PO T T -1-1 DR DR NT DC DC PO NT NT PO * NT DR EC PO PO,EQ PO T -1 PO PO -1 T -1 NT -1 T T -1 DR PO PO PO PO,EQ NT -1 PO NT -1 NT -1 NT EQ DC EC PO T NT T -1 NT -1 EQ Tabe 1: Composition tabe for the eight base reations of RCC-8, where species the universa reation Reation Mode Constraints Entaiment Constraints DC EC :( ^ Y ) :(I ^ IY ) :; :Y :( ^ Y ); :; :Y PO :(I ^ IY );! Y ; Y! ; :; :Y T! Y! IY ; Y! ; :; :Y T?1 NT Y!! IY Y! I;! Y ; :; :Y Y! ; :; :Y NT?1 EQ Y! I! Y ; Y!! Y ; :; :Y :; :Y Tabe 2: Encoding of the base reations in moda ogic that are connected if they share a point and overap if they share an interior point. If, e.g, and Y are disconnected, the compement of the intersection of and Y is equa to the universe. Further, both regions must not be empty, i.e. the compements of both and Y are not equa to the universe. In this way the eight base reations can be represented by constraints of the form (m = U), caed mode constraints, and (m 6= U), caed entaiment constraints, where m is a set-theoretic expression containing perhaps the topoogica interior operator i. Any mode constraint must hod, whereas no entaiment constraint must hod [Bennett, 1994]. The mode and entaiment constraints can be encoded in moda ogic, where spatia variabes correspond to propositiona atoms and the interior operator i to a moda operator I (see Tabe 2). The axioms for i must aso hod for the moda operator I, which resuts in the foowing axioms [Bennett, 1995]: 1: I!, 3: I> $ > (for any tautoogy >), 2: II $ I, 4: I( ^ Y ) $ I ^ IY:

3 Axioms 1 and 2 correspond to the moda ogics T and 4, axioms 3 and 4 aready hod for any moda ogic K, so I is a moda S4-operator. The four axioms specied by Bennett are not sucient to excude non-cosed regions. In order to account for that, we add two formuas for each atom, which correspond to topoogica properties of cosed regions. A cosed region is the cosure of an open region and the compement of a cosed region is an open region: $ :I:I; : $ I:: In order to combine the dierent mode and entaiment constraints, Bennett [1995] uses another moda operator 2. 2m is interpreted as m = U and :2m as m 6= U. Any mode constraint m can be written as 2m and any entaiment constraint as :2m. If 2 is true in a word w of a mode M, written as (M; w j` 2), then must be true in any word of M. So 2 is an S5-operator with the constraint that a words are mutuay accessibe. Therefore Bennett [1995] cas it a strong S5-operator. So the encoding of RCC-8 is done in muti-moda ogic with an S4-operator and a strong S5-operator. Let be a set of RCC-8 formuas and Reg() be the set of spatia variabes used in, then m() species the moda encoding of, where! 0 ^ m() = m 1 (RY ) ^ 1 m 2 () A : RY 2 2Reg() m 1 (RY ) is a disjunction of the conjunctivey connected mode and entaiment constraints for the base reations in R. m 2 resuts from the axioms of the I- operator and the additiona properties of cosed regions: m 2 () = 2(I! ) ^ 2(I! II) ^2(:! I:) ^ 2(! :I:I): 2(II! I), 2(I:! :) and 2(:I:I! ) are entaied by the other formuas and can be ignored. As foows from the work by Bennett [1995], is consistent i m() is satisabe. In order to refer to the singe mode and entaiment constraints, we wi introduce some abbreviations. Denition 3.1 Abbreviations for the mode constraints: xy 2(:( ^ Y )) A xy 2(:(I ^ IY )) xy 2(! Y ) B xy 2(! IY ) xy 2(Y! ) C xy 2(Y! I): As the entaiment constraints are negations of the mode constraints, they wi be abbreviated as negations of the above abbreviations. When it is obvious which atoms are used, the abbreviations wi be written without indices. The abbreviations can be regarded as \propositiona atoms". Then it is possibe to write the moda encoding m 1 (RY ) of every reation R of RCC-8 as a \propositiona formua" of abbreviations. We wi ca this formua the abbreviated form of R. In the remainder we wi use the encoding of m 1 (RY ) such that the abbreviated form is in conjunctive norma form (CNF). 4 Computationa Properties of RCC-8 In this section we prove that reasoning with RCC-8 as we as RCC-5 is NP-hard. A simiar but weaker resut has been proven by Grigni et a. [1995] (see Section 8). In this paper NP-hardness proofs for dierent sets S of RCC-8 reations wi be carried out. A of them use a reduction of a propositiona satisabiity probem to RSAT(S) by constructing a set of spatia formuas for every instance I of the propositiona probem, such that is consistent i I is a positive instance. These satisabiity probems incude 3SAT, NOT-ALL-EQUAL-3SAT where every cause has at east one true and one fase itera, and ONE-IN-THREE-3SAT where exacty one itera in every cause must be true [Garey and Johnson, 1979]. The reductions have in common that every itera as we as every itera occurrence L is reduced to two spatia variabes L and Y L and a reation R = R t [ R f, where R t \ R f = ; and L RY L. L is true i L R t Y L hods and fase i L R f Y L hods. Additiona \poarity" constraints have to be introduced to assure that for the spatia variabes :L and Y :L, corresponding to the negation of L, :L R t Y :L hods i L R f Y L hods, and vice versa. Using these poarity constraints, spatia variabes of negative itera occurrences are connected to the spatia variabes of the corresponding positive itera, and ikewise for positive itera occurrences and negative iteras. Further, \cause" constraints have to be added to assure that the cause requirements of the specic propositiona probem are satised in the reduction. Theorem 4.1 RSAT(RCC-5) is NP-hard. Proof Sketch. Transformation of NOT-ALL-EQUAL- 3SAT to RSAT(RCC-5) (see aso Grigni et a. [1995]). R t = fg and R f = f?1 g. Poarity constraints: L f;?1 g :L ; Y L f;?1 gy :L, L fpogy :L ; Y L fpog :L. Cause constraints for every cause c = fi; j; kg: i f;?1 g j ; j f;?1 g k ; k f;?1 g i, i fpogy k ; j fpogy i ; k fpogy j. Since RCC-5 is a subset of RCC-8, this resut can be easiy appied to RCC-8. Coroary 4.2 RSAT(RCC-8) is NP-hard. In order to identify the borderine between tractabiity and intractabiity, one has to examine a subsets of RCC-8. We imit ourseves to subsets containing a base

4 reations, because these subsets sti aow to express denite knowedge, if it is avaiabe. Additionay, we require the universa reation to be in the subset, so that it is possibe to express compete ignorance. This reduces the search space from subsets to subsets. We proved a property that has ikewise been used in identifying the maxima tractabe subset of Aen's cacuus [Nebe and Burckert, 1995] that can be used to further reduce the search space. Theorem 4.3 RSAT( bs) can be poynomiay reduced to RSAT(S) Coroary 4.4 Let S be a subset of RCC RSAT( bs) 2 P i RSAT(S) 2 P. 2. RSAT(S) is NP-hard i RSAT( bs) is NP-hard. The rst statement of Coroary 4.4 can be used to increase the number of eements of tractabe subsets of RCC-8 consideraby. With the second statement of Coroary 4.4, NP-hardness proofs of RSAT can be used to excude certain reations from being in any tractabe subset of RCC-8. The NP-hardness proof of Theorem 4.1, e.g., ony uses the reations fpog and f;?1 g. So for any subset S with the two reations contained in bs, RSAT(S) is NP-hard. The foowing NP-hardness resuts can be used to excude more reations. Lemma 4.5 Let S be a subset of RCC-8 containing a base reations. If any of the reations ft; NT; T?1 ; NT?1 g, ft; T?1 g, fnt; NT?1 g, fnt; T?1 g or ft; NT?1 g is contained in bs, then RSAT(S) is NP-hard. Proof Sketch. When R f [ R t is repaced by ft; NT; T?1 ; NT?1 g, ft; T?1 g or fnt; NT?1 g, the transformation of Theorem 4.1 can be appied. For fnt; T?1 g and ft; NT?1 g ONE-IN-THREE-3SAT has to be used. By computing the cosure of a sets containing the eight base reations together with one additiona reation, the foowing emma can be obtained. Lemma 4.6 RSAT(S) is NP-hard for any subset S of RCC-8 containing a base reations together with one of the 72 reations of the foowing sets: N 1 = fr j fpog 6 R and (ft; T?1 g R or fnt; NT?1 g R)g; N 2 = fr j fpog 6 R and (ft; NT?1 g R or ft?1 ; NTg R)g: 5 Transformation of RSAT to SAT For transforming RSAT to propositiona satisabiity (SAT) we wi transform every instance of RSAT to a propositiona formua in CNF that is satisabe i is consistent. We wi start from m(), the moda encoding of, and show that whenever m() is satisabe it has a Kripke mode of a specic type. This mode wi then be used to transform m() to a propositiona formua. m() is satisabe if it is true in a word w of a Kripke mode M = hw; fr 1 = W W; R 2 W Wg; i, where W is a set of words, R 1 the accessibiity reation of the 2-operator, R 2 the accessibiity reation of the I-operator, and a truth function that assigns a truth vaue to every atom in every word. The truth conditions for M; w j` m() can be specied as a combination of truth conditions of the singe atoms according to the form of m(). In this way M; w j` I', e.g., can be written as (8u : wr 2 u:m; u j` ') and M; w j` :I' as (9u : wr 2 u:m; u j` :'). We wi ca this form of writing M j` m() the expicit form of m(). Before transforming m() to a propositiona formua, we have to show that there is a Kripke mode of m() that is poynomia in the number of spatia variabes n. Denition 5.1 Let u 2 W be a word of the mode M. u is a word of eve 0 if vr 2 u ony hods for v = u. u is a word of eve + 1 if vr 2 u hods for a word v of eve and there is no word v 6= u of eve >. We assume that every occurrence of a sub-formua of m() of the form :2', where ' contains no 2 operators, introduces a new word of eve 0. As these subformuas correspond to entaiment constraints, the number of words of eve 0 is poynomia in n. For every spatia variabe and every word w there might be sub-formuas that force the existence of a word u with wr 2 u where is true or where : is true. Because there are n dierent spatia variabes, 2n dierent words u with wr 2 u are sucient for each word w. Denition 5.2 An RCC-8-frame F = hw; fr 1 ; R 2 gi has the foowing properties: 1. W contains ony words of eve 0; 1 and For every word w of eve k (k = 0; 1) there are exacty 2n words u of eve k + 1 with wr 2 u. 3. For every word w of eve k there is exacty one word u for every eve 0 k with ur 2 w. An RCC-8-mode is based on an RCC-8-frame. Lemma 5.3 m() is satisabe i M; w j` m() for an RCC-8-mode M with poynomiay many words. Now it is possibe to transform the expicit form of m() to a propositiona formua p(m()) in CNF such that p(m()) is satisabe i m() is satisabe in a poynomia RCC-8-mode M. For this purpose, propositiona atoms w are introduced which stand for the truth of atom in word w of the RCC-8-mode M. Further, universay quantied truth conditions are transformed into conjunctions and existentiay quantied

5 truth conditions are transformed into truth conditions on particuar words, which can be determined using the structure of the RCC-8-frame and the moda formua. Theorem 5.4 RSAT(RCC-8) can be poynomiay reduced to SAT. With Coroary 4.2 this eads to the foowing theorem. Theorem 5.5 RSAT(RCC-8) is NP-compete. 6 Tractabe Subsets of RCC-8 In order to identify a tractabe subset of RCC-8, we anayze which reations can be expressed as propositiona Horn formuas, as satisabiity of Horn formuas (HORN- SAT) is tractabe. Proposition 6.1 Appying the transformation p to the mode and entaiment constraints, to the axioms for I, and to the properties of cosed regions eads to Horn formuas. Since the mode constraints and A are transformed to indenite Horn formuas, the transformation of any disjunction of these constraints with any other constraint is aso Horn. A reations with an abbreviated form using ony abbreviations or disjunctions of abbreviations transformabe to Horn formuas can be transformed to Horn formuas. In this way 64 dierent reations can be transformed to Horn formuas. We ca the subset of RCC-8 containing these reations H 8. Theorem 6.2 RSAT(H 8 ) can be poynomiay reduced to HORNSAT and therefore RSAT( bh 8 ) 2 P. Theorem 6.3 bh 8 contains the foowing 148 reations: bh 8 = RCC-8 n (N 1 [ N 2 [ N 3 ) with N 1 and N 2 as dened in Lemma 4.6 and N 3 = frjfeqg R and ((fntg R; ftg 6 R) or (fnt?1 g R; ft?1 g 6 R))g: For proving that bh 8 is a maxima tractabe subset of RCC-8, we have to show that no reation of N 3 can be added to bh 8 without making RSAT intractabe. Lemma 6.4 The cosure of every set containing bh 8 and one reation of N 3 contains the reation feq; NTg. Therefore it is sucient to prove NP-hardness of RSAT( bh 8 [ feq; NTg) for showing that bh 8 is a maxima tractabe subset of RCC-8. Lemma 6.5 RSAT( bh 8 [ feq; NTg) is NP-hard. Proof Sketch. Transformation of 3SAT to RSAT( bh 8 [ feq; NTg). R t = fntg and R f = feqg. Poarity constraints: L fec; NTg :L ; Y L ftgy :L ; L ft; NTgY :L ; Y L fec; Tg :L ; Cause constraints for each cause c = fi; j; kg: Y i fnt?1 g j ; Y j fnt?1 g k ; Y k fnt?1 g i : Theorem 6.6 bh 8 is a maxima tractabe subset of RCC-8. It has to be noted that there might be other maxima tractabe subsets of RCC-8 that contain a base reations. As bh 8 is tractabe, the intersection of RCC-5 and bh 8 is aso tractabe. We wi ca this subset bh 5. Theorem 6.7 bh 5 is the ony maxima tractabe subset of RCC-5 containing a base reations. 7 Appicabiity of Path-Consistency As shown in the previous section, RSAT( bh 8 ) can be soved in poynomia time by rst transforming a set of bh 8 formuas to a propositiona Horn formua and then deciding it in time inear in the number of iteras. This way of soving RSAT does not appear to be very ecient. As RSAT is a Constraint Satisfaction Probem (CSP) [Mackworth, 1987], where variabes are nodes and reations are arcs of the constraint graph, agorithms for deciding consistency of a CSP can aso be used. A correct but in genera not compete O(n 3 ) agorithm for deciding inconsistency of a CSP is the path-consistency method [Mackworth, 1977] that makes a CSP path-consistent by successivey removing reations from a edges using 8k : R ij R ij is the reation between i and j. If the empty reation occurs whie performing this operation, the CSP is not path-consistent, otherwise it is. In this section we wi prove that path-consistency decides RSAT( bh 8 ). This is done by showing that the pathconsistency method nds an inconsistency whenever positive unit resoution (PUR) resoves the empty cause from the corresponding propositiona formua. As PUR is refutation-compete for Horn formuas, it foows that the path-consistency method decides RSAT(H 8 ). The ony way to get the empty cause is resoving a positive and a negative unit cause of the same variabe. Since the Horn formuas that are used contain ony a few types of dierent causes, there are ony a few ways to resove unit causes using PUR. R ij \ (R ik R kj ), where i; j; k are nodes and Denition 7.1 R K denotes the set of reations of H 8 with the conjunct K appearing in their abbreviated form. R K1;K 2 denotes R K1 [ R K2. R? denotes R [R _ [R A_ [R C [R _C [R A_C. An R K -chain RK (; Y ) is a path from region to region Y, where a reations between successive regions are from R K. Lemma 7.2 Let be a set of H 8 -formuas. A positive unit cause f w g can ony be resoved from fy w g and a cause resuting from R? Y 2. When such a resoution is possibe, R ;A Y cannot hod, so R ;C Y must hod.

6 A negative unit cause f: w g can ony be resoved from fy w g and a cause resuting from R ;A Y 2. Lemma 7.3 If the positive unit cause f w g can be resoved with PUR using an R? -chain from to Y, the path-consistency method resuts in R ;C Y. Using Lemma 7.3, it can be proven that the pathconsistency method decides RSAT(H 8 ). Using the proof of Theorem 4.3, it is possibe to express every reation of bh 8 as a Horn formua. Then the foowing theorem can be proven. Theorem 7.4 The path-consistency method decides RSAT( bh 8 ). Another interesting question is whether the pathconsistency method computes minima possibe reations on bh 8. As the foowing exampe demonstrates, this is not the case even for the set bh 5. AfgD is impossibe athough the constraint graph is path-consistent: 8 Reated Work PO - A B H HH PO _ DR _ PO H DR _ HH? PO _? HHj - C D DR _?1 Nebe [1995] showed that RSAT( bb) can be decided in poynomia time, where B is the set of the RCC-8 base reations. Since B H 8, our resut is more genera. Further, bb contains ony 38 reations, whereas bh 8 contains 148 reations, i.e. about 58% of RCC-8. Grigni et a. [1995] proved NP-hardness of probems simiar to RSAT. For instance, they considered the probem of reationa consistency, which means that there exists a path-consistent renement of a reations to base reations, and showed that this probem is NPhard. Whie our NP-hardness resut on RSAT impies their resut, the converse impication foows ony using the above cited resut by Nebe [1995]. In addition to this syntactic notion of consistency, Grigni et a. [1995] considered a semantic notion of consistency, namey, the reaizabiity of spatia variabes as internay connected panar regions. This notion is much more constraining than our notion of consistency. It is aso computationay much harder. 9 Summary We anayzed the computationa properties of the quaitative spatia cacuus RCC-8 and identied the boundary between poynomia and NP-hard fragments. Using a modication of Bennett's encoding of RCC-8 in a mutimoda propositiona ogic, we transformed the RCC-8 consistency probem to a probem in propositiona ogic and isoated the reations that are representabe as Horn causes. As it turns out, the fragment identied in this way is aso a maxima fragment that contains a base reations and is sti computationay tractabe. Further, we showed that for this fragment path-consistency is suf- cient for deciding consistency. As in the case of quaitative tempora reasoning, our resut aows to check whether the reations that are used in an appication aow for a poynomia reasoning agorithm. Further, if the appication requires an expressive power beyond the poynomia fragment, it can be used to speed up backtracking agorithms. Assuming that the reations are uniformy distributed, the average branching factor is reduced from 4:0 to 1:4375 using bh 8 instead of B to spit the reations (see aso [Nebe, 1997]). References [Aen, 1983] J.F. Aen. Maintaining knowedge about tempora intervas. Comm. of the ACM, 26(11), [Bennett, 1994] B. Bennett. Spatia reasoning with propositiona ogic. In Proc. KR'94, Morgan Kaufmann, [Bennett, 1995] B. Bennett. Moda ogics for quaitative spatia reasoning. Bu. IGPL, 4(1), [Egenhofer, 1991] M.J. Egenhofer. Reasoning about binary topoogica reations. In Proc. SSD'91, LNCS 525, pp.143{ 160. Springer, [Fitting, 1993] M.C. Fitting. Basic moda ogic. In Handbook of Logic in Articia Inteigence and Logic Programming { Vo. 1, pp.365{448, Oxford, [Garey and Johnson, 1979] M.R. Garey and D.S. Johnson. Computers and Intractabiity A Guide to the Theory of NP-Competeness. Freeman, [Grigni et a., 1995] M. Grigni, D. Papadias, and C. Papadimitriou. Topoogica inference. In Proc. IJCAI'95, pp.901{906, [Mackworth, 1977] A.K. Mackworth. Consistency in networks of reations. Articia Inteigence 8, [Mackworth, 1987] A.K. Mackworth. Constraint satisfaction. In S. C. Shapiro (ed), Encycopedia of Articia Inteigence, pp.205{211. Wiey, [Nebe and Burckert, 1995] B. Nebe and H.-J. Burckert. Reasoning about tempora reations: A maxima tractabe subcass of Aen's interva agebra. Journa ACM, 42(1), pp.43{66, [Nebe, 1995] B. Nebe. Computationa properties of quaitative spatia reasoning: First resuts. In KI'95, LNCS 981, pp.233{244, Springer, [Nebe, 1997] B. Nebe. Soving hard quaitative tempora reasoning probems: Evauating the eciency of using the ORD-Horn cass. Constraints, 3(1), pp.175{190, [Rande et a., 1992] D.A. Rande, Z. Cui, and A.G. Cohn. A spatia ogic based on regions and connection. In Proc. KR'92, pp.165{176, [Renz and Nebe, 1997] J. Renz and B. Nebe. On the compexity of quaitative spatia reasoning: A maxima tractabe fragment of the Region Connection Cacuus. Tech. report 87, Inst. f. Informatik, Univ. Freiburg, 1997.