Isli, A., Museros Cabedo, L., Barkowsky, T., & Moratz, R. (2000). A topological calculus for cartographic entities. In C. Freksa, W. Brauer, C. Habel, & K. F. Wender (Eds.), Spatial Cognition II - Integrating abstract theories, empirical studies, formal models, and practical applications (pp. 225-238). Berlin: Springer.

A Topological Calculus for Cartographic Entities

1

Amar Isli , Lledó Museros Cabedo†, Thomas Barkowsky , and Reinhard Moratz *

*

*

*

University of Hamburg, Department for Informatics, Vogt-Kölln-Str. 30, 22527 Hamburg, Germany

{isli, barkowsky, moratz}@informatik.uni-hamburg.de †

Universitat Jaume I, Departamento de Informática, Campus del Riu SEc, TI-1126-DD, Castellon, Spain [email protected]

Abstract. Qualitative spatial reasoning (QSR) has many and varied applications among which reasoning about cartographic entities. We focus on reasoning about topological relations for which two approaches can be found in the literature: region-based approaches, for which the basic spatial entity is the spatial region; and point-set approaches, for which spatial regions are viewed as sets of points. We will follow the latter approach and provide a calculus for reasoning about point-like, linear and areal entities in geographic maps. The calculus consists of a constraint-based approach to the calculus-based method (CBM) in (Clementini et al., 1993). It is presented as an algebra alike to Allen's (1983) temporal interval algebra. One advantage of presenting the CBM calculus in this way is that Allen’s incremental constraint propagation algorithm can then be used to reason about knowledge expressed in the calculus. The algorithm is guided by composition tables and a converse table provided in this contribution.

1

Introduction

Geographic maps are generally represented as a quantitative database consisting of facts describing the different regions of the map. For many applications, however, the use of such maps is restricted to the topological relations between the different regions; in other words, such applications abstract from the metric considerations of the map. Topological relations are related to the connection between spatial objects. Their important characteristic is that they remain invariant under topological transformations, such as rotation, translation, and scaling. A variety of topological approaches to spatial reasoning can be found in the literature, among which the following two trends: 1. Approaches for which the basic spatial entity is the spatial region. These approaches consider spatial regions as non-empty, regular regions; therefore, points, lines, and boundaries cannot be considered as spatial regions (Randell et al., 1992; 1

This work is supported by the Deutsche Forschungsgemeinschaft (DFG) in the framework of the Spatial Cognition Priority Program (grants Fr 806-7 and Fr 806-8).

Gotts, 1996; Bennett, 1994; Renz & Nebel, 1998). The basic relation for these is C(x,y), i.e. x connects with y. C(x,y) holds when the topological closures of x and y share a point. Using the C relation, the approaches define a set of atomic topological relations, which are generally mutually exclusive and complete: for any two regions, one and only one of the atomic topological relations holds between them. One important theory in this group is the region connection calculus (RCC) developed in (Randell et al., 1992). 2. Approaches which consider a region as a set of points. The basic entities for these approaches consist of points, lines and areas. The topological relations between regions are defined in terms of the intersections of the interiors and boundaries of the corresponding sets of points (Egenhofer & Franzosa, 1991; Pullar & Egenhofer, 1988; Egenhofer, 1991; Clementini & Di Felice, 1995). Each approach has its own set of atomic topological relations. The aim of this work is to develop a calculus suitable for reasoning about topological relations between point-like, linear, and areal cartographic entities. We cannot follow the first trend since it does not consider point-like and linear entities as regions. We will indeed follow the second trend. An important work in the literature on reasoning about topological relations of cartographic entities is the CBM calculus (calculus-based method) described in (Clementini & Di Felice, 1995), originating from Egenhofer’s work on intersections of the interiors and boundaries of sets of points (Egenhofer & Franzosa, 1991; Pullar & Egenhofer, 1988; Egenhofer, 1991). The objects (features) manipulated by the calculus consist of points, lines and areas commonly used in geographic information systems (GIS); that means that all kinds of features consist of closed (contain all their accumulation points) and connected (do not consist of the union of two or more separated features) sets. The knowledge about such entities is represented in the calculus as facts describing topological relations on pairs of the entities. The relation of two entities could be disjoint, touch, overlap, cross, or in. In geographic maps, it is often the case that one has to distinguish between the situation that a region is completely inside another region, the situation in which it touches it from inside, and the situation in which the two regions are equal. For instance, a GIS user might want to know whether Hamburg is strictly inside Germany or at (i.e., touches) the boundary of the country with the rest of the earth. The CBM calculus makes use of a boundary operator which, when combined with the in relation, allows for the distinction to be made. A close look at the calculus shows that, indeed, it is suited for conjunctive-fact queries of geographic databases (i.e. queries consisting of facts describing either of the five atomic relations or a boundary operator on a pair of features). In this contribution, we provide an Allen style approach (Allen, 1983) to the calculus. Specifically, we present an algebra which will have nine atomic relations resulting from the combination of the five atomic relations and the boundary operator of the CBM calculus. Our main motivation is that we can then benefit from Allen style reasoning: 1. We can make use of Allen’s constraint propagation algorithm to reason about knowledge expressed in the algebra. This means that composition tables recording the composition of every pair of the atomic relations have to be provided for the algebra, as well as a converse table.

2. The algebra will benefit from the incrementality of the propagation algorithm: knowledge may be added without having to revise all the processing steps achieved so far. 3. Disjunctive knowledge will be expressed in an elegant way, using subsets of the set of all nine atomic relations: such a disjunctive relation hold on a pair of features if and only if either of the atomic relations holds on the features. This is particularly important for expressing uncertain knowledge, which is closely related to the notion of conceptual neighborhoods (Freksa, 1992).

2

The Calculus

As mentioned in the introduction, our aim is to propose a constraint-based approach to the CBM calculus developed by Clementini, Di Felice, and Oosterom (Clementini & Di Felice, 1995; Clementini et al., 1993). For that purpose, we will develop an algebra as the one Allen (1983) presented for temporal intervals, of which the atomic relations will be the three relations resulting from the refinement of the in relation, together with the other four atomic relations of the CBM calculus. We will provide the result of applying the converse and the composition operations to the atomic relations: this will be given as a converse table and composition tables. These tables in turn will play the central role in propagating knowledge expressed in the algebra using Allen’s constraint propagation algorithm (Allen, 1983). We will use the topological concepts of boundary, interior, and dimension of a (point-like, linear or areal) feature. We thus provide some background on these concepts, taken from (Clementini et al., 1993). The boundary of a feature h is denoted by δh; it is defined for each of the feature types as follows: 1. δP: we consider the boundary of a point-like feature to be always empty. 2. δL: the boundary of a linear feature is the empty set in the case of a circular line, the two distinct endpoints otherwise. 3. δA: the boundary of an area is the circular line consisting of all the accumulation points of the area. The interior of a feature h is denoted by h°. It is defined as h° = h - δh. Note that the interior of a point-like entity is equal to the feature itself. The function dim, which returns the dimension of a feature of either of the types we consider, or of the intersection of two or more such features, is defined as follows (the symbol ∅ represents the empty set): If S•∅ then dim(S) =

0 if S contains at least a point and no lines and no areas 1 if S contains at least a line and no areas 2 if S contains at least an area

else dim(S) is undefined.

2.1

The Relations

We can now define the topological relations of our algebra. We use the original relations touch, overlap, cross and disjoint of the CBM calculus. As we cannot use the boundary operator which allows the CBM calculus to distinguish between the three subrelations equal, completely-inside, and touching-from-inside of the in relation (which are not explicitly present in the calculus) the three subrelations will replace the superrelation in the list of atomic relations. In addition, the new relations completelyinside and touching-from-inside being asymmetric, we need two other atomic relations corresponding to their respective converses, namely completely-insidei and touching-from-insidei. The definitions of the relations are given below. The topological relation r between two features h1 and h2, denoted by (h1, r, h2), is defined on the right hand side of the equivalence sign in the form of a point-set expression. Definition 1. The touch relation: (h1, touch, h2) ↔ h°1 ∩ h°2 = ∅ ∧ h1 ∩ h2 • ∅ Definition 2. The cross relation: (h1, cross, h2) ↔ dim(h°1 ∩ h°2) = max(dim(h°1), dim(h°2)) - 1 ∧ h1 ∩ h2 • h1 ∧ h1 ∩ h2 • h2 Definition 3. The overlap relation: (h1, overlap, h2) ↔ dim(h°1) = dim(h°2) = dim(h°1 ∩ h°2) ∧ h1 ∩ h2 • h1 ∧ h1 ∩ h2 • h2 Definition 4. The disjoint relation: (h1, disjoint, h2) ↔ h1 ∩ h2 = ∅ Definition 5. We define the equal, completely-inside, and touching-from-inside relations using the formal definition of the in relation: (h1, in, h2) ↔ h1 ∩ h2 = h1 ∧ h°1 ∩ h°2 • ∅ Given that (h1, in, h2) holds, the following algorithm distinguishes between the completely-inside, the touching-from-inside, and the equal relations: if (h2, in, h1) then (h1, equal, h2) else if h1 ∩ δh2 • ∅ then (h1, touching-from-inside, h2) else (h1, completely-inside, h2) Definition 6. The completely-insidei relation: (h1, completely-insidei, h2) ↔ (h2, completely-inside, h1) Definition 7. The touching-from-insidei relation: (h1, touching-from-insidei, h2) ↔ (h2, touching-from-inside, h1) Figure 1 shows examples of the completely-inside and touching-from-inside situations. At this point we have defined the atomic relations of the new calculus which are touch, cross, overlap, disjoint, equal, completely-inside, touching-from-inside, com-

pletely-insidei, and touching-from-insidei. Now, we will prove that these relations are mutually exclusive, that is, it cannot be the case that two different relations hold between two features. Furthermore, we will prove that they form a full covering of all possible topological situations, that is, given two features, the relation between them must be one of the nine defined here. To prove these two characteristics we construct the topological relation decision tree depicted in Fig. 2. a)

b)

c)

h2

h2

h1

e)

d) h2

h2

h1

h1

h2 h1

h1

Fig. 1. Examples of the completely-inside and touching-from-inside situations: examples with two areal entities, a) representing (h1, touching-from-inside, h2) and b) (h1, completely-inside, h2); examples of situations with a linear entity and an areal entity, c) representing (h1, completely-inside, h2), and d) and e) representing (h1, touching-from-inside, h2) situations h°1 ∩ h°2 = φ T

F

h 1 ∩ h2 ? φ T touch

h1 ∩ h2 = h1 F

disjoint

T

F

h1 ∩ h2 = h2 T equal

h1 ∩ h2 = h2

F

T

T completelyinside

dim(h°1 ∩ h°2) = max(dim(h°1), dim(h°2)) - 1

δh1 ∩ h2 = φ

h1 ∩ δh2 = φ F touchingfrom-inside

T completelyinsidei

F

F touchingfrom-insidei

T cross

F overlap

Fig. 2. Topological relation decision tree

Proof. Every internal node in this topological relation decision tree represents a Boolean predicate of a certain topological situation. If the predicate evaluates to true then the left branch is followed, otherwise the right branch is followed. This process is repeated until a leaf node is reached which will indicate which of the atomic topological relations this situation corresponds to. Two different relations cannot hold between two given features, because there is only one path to be taken in the topological relation decision tree to reach a particular topological relation. And there can be no cases outside the new calculus, because every internal node has two branches, so for every Boolean value of the predicate there is an appropriate path and every leaf node has a label that correspond to one of the atomic topological relations. ‘ Definition 8. A general relation of the calculus is any subset of the set of all atomic relations. Such a relation, say R, is defined as follows:

(∀h1,h2) ((h1, R, h2) ⇔

∨ (h , r, h ))

r∈R

1

2

The next step is to define the operations that we can apply to these relations, namely converse and composition. 2.2

The Operations

Definition 9. The converse of a general relation R is denoted as R∪. It is defined as:

(∀h1, h2 ) ((h1, R, h2) ⇔ (h2, R∪, h1))

Definition 10. The composition R1 ⊗ R2 of two general relations R1 and R2 is the most specific relation R such that: (∀h1, h2, h3) ((h1, R1, h2) ∧ (h2, R2, h3) ⇒ (h1, R, h3)) Table 1 and Tables 5 - 22 provide the converse and the composition for the atomic relations of the algebra. For general relations R1 and R2 we have: R1∪

=

∪ r∪

r∈R1

R1 ⊗ R2 =

∪r

r1∈R1 r ∈R2

1

⊗ r2

Table 1. The converse table

r Overlap Touch Cross Disjoint Completely-inside Touching-from-inside Completely-insidei Touching-from-insidei Equal

r∪ Overlap Touch Cross Disjoint Completely-insidei Touching-from-insidei Completely-inside Touching-from-inside Equal

We denote by XY-U, with X and Y belonging to {P, L, A}, the universal relation, i.e. the set of all possible atomic relations, between a feature h1 of type X and a feature h2 of type Y (we use P for a point-like feature, L for a linear feature, and A for an areal feature). For instance, PP-U is the set of all possible topological relations between two point-like entities. These universal relations are as follows: PP-U = equal, disjoint PL-U = touch, disjoint, completely-inside PA-U = touch, disjoint, completely-inside

LP-U = touch, disjoint, completely-insidei LL-U = touch, disjoint, overlap, cross, equal, touching-from-inside, completely-inside, touching-from-insidei, completely-insidei LA-U = touch, cross, disjoint, touching-from-inside, completely-inside AP-U = touch, disjoint, completely-insidei AL-U = touch, cross, disjoint, touching-from-insidei, completely-insidei AA-U = touch, overlap, disjoint, equal, touching-from-inside, completely- inside, touching-from-insidei, completely-insidei Note that the sets LP-U, AP-U, and AL-U are the converse sets of PL-U, PA-U, and LA-U, respectively. Given any three features h1, h2, and h3 such that (h1, r1, h2) and (h2, r2, h3), the composition tables should be able to provide the most specific implied relation R between the extreme features, i.e. between h1 and h3. If we consider all possibilities with h1, h2, and h3 being a point-like feature, a linear feature, or an areal feature, we would need 3 27 (3 ) tables. However, we construct only 18 of these tables from which the other 9 can be obtained. The 18 tables to be constructed split into 6 for h2=point-like feature, 6 for h2=linear feature and 6 for h2=areal feature: when feature h1 is of type X, feature h2 of type Y, and feature h3 of type Z, with X, Y, and Z belonging to {P, L, A}, the corresponding composition table will be referred to as the XYZ table. In tables 2, 3, and 4 we show the tables constructed and their numbers of entries. Table 2. Number of entries of the constructed tables for h2=point-like entity

TABLE PPP table PPL table PPA table LPL table LPA table APA table TOTAL NUMBER OF ENTRIES:

NUMBER OF ENTRIES |PP-U| x |PP-U|= 4 |PP-U| x |PL-U|= 6 |PP-U| x |PA-U|= 6 |LP-U| x |PL-U|= 9 |LP-U| x |PA-U|=9 |AP-U| x |PA-U|=9 43

Table 3. Number of entries of the constructed tables for h2=linear entity

TABLES PLP table PLL table PLA table LLL table LLA table ALA table TOTAL NUMBER OF ENTRIES:

NUMBER OF ENTRIES |PL-U| x |LP-U|= 9 |PL-U| x |LL-U|= 27 |PL-U| x |LA-U|= 15 |LL-U| x |LL-U|= 81 |LL-U| x |LA-U|= 45 |AL-U| x |LA-U|= 25 202

Table 4. Number of entries of the constructed tables for h2=areal entity

TABLES PAP table PAL table PAA table LAL table LAA table AAA table TOTAL NUMBER OF ENTRIES:

NUMBER OF ENTRIES |PA-U| x |AP-U|= 9 |PA-U| x |AL-U|= 15 |PA-U| x |AA-U|= 24 |LA-U| x |AL-U|= 25 |LA-U| x |AA-U|= 40 |AA-U| x |AA-U|= 64 177

Let us consider the case h2=linear entity. The six tables to be constructed for this case are the PLP, PLL, PLA, LLL, LLA, and ALA tables. From these six tables, we can get the other three, namely the LLP, ALP, and ALL tables. We illustrate this by showing how to get the r1⊗r2 entry of the LLP table from the PLL table. This means that we have to find the most specific relation R such that for any two linear features L1 and L2, and any point-like feature P, if (L1, r1, L2) and (L2, r2, P) then (L1, R, P). We can represent this as: L1

r1

L2 R

r2

P From the converse table we can get the converses r1∪ and r2∪ of r1 and r2, respectively. The converse R∪ of R is clearly the composition r2∪⊗r1∪ of r2∪ and r1∪, which can be obtained from the PLL table: r1∪ L1

L2 r2∪

R∪ P

Now R is the converse of R∪: R = (R∪)∪. Below we present the composition tables, in which the relation touch is denoted by T, cross by C, overlap by O, disjoint by D, completely-inside by CI, touching-frominside by TFI, equal by E, completely-insidei by CIi, and touching-from-insidei by TFIi.

Table 5. The PPP composition table r2 r1

E D

E

D

E D

D E, D

Table 6. The PPL composition table r2 r1

E D

T

D

CI

T PL-U

D PL-U

CI PL-U

T

D

CI

T PA-U

D PA-U

CI PA-U

T

D

CI

T, C, TFI LA-U T, C

T, C, D LA-U T, C, D

C, TFI, CI LA-U C, CI, TFI

Table 7. The PPA composition table r2 r1

E D

Table 8. The LPA composition table r2 r1

T D CIi

Table 9. The LPL composition table r2

T

D

CI

T, O, C, E, TFI, TFIi T, D, O, C, CI, TFI T, O, C, TFIi, CIi 

T, D, O, C , TFIi, CIi LL- U T, D, O, C, TFIi, CIi

T, O, C, TFI, CI T, D, O, C, TFI, CI O, C, E, TFI, CI, TFIi, CIi

r1

T D CIi

Table 10. The APA composition table r2

T

D

CI

T, O, E, TFI, TFIi T, O, D, TFI, CI O, TFIi, CIi

T, O, D, CIi, TFIi AA-U T, O, D, CIi, TFIi

O, TFI, CI T, O, D, CI, TFI O, E, TFI, CI, TFIi, CIi

r1

T D CIi

Table 11. The PLP composition table r2 r1

T D CI

T

D

CIi

PP-U D D

D PP-U D

D D PP-U

Table 12. The PLA composition table r2 r1

T D CI

T

C

D

CI

TFI

T, D PA-U T, D

PA-U PA-U PA-U

D PA-U D

CI PA-U CI

T, CI PA-U T, CI

Table 13. The PLL composition table r2 r1

T D CI

T

C

O

D

E

CI

TFI

CIi

TFIi

PL-U PL-U T, D

PL-U PL-U PL-U

PL-U PL-U PL-U

D PL-U D

T D CI

CI PL-U CI

T, D PL-U CI

D D PL-U

T, D D PL-U

Table 14. The LLA composition table r2 r1

T D O C E TFI CI TFIi CIi

T

C

D

TFI

CI

T, C, D, TFI LA-U T, C, D T, C, D T T, D T, D T, C T, C

LA-U LA-U LA-U LA-U C LA-U LA-U C C

T, C, D LA-U T, C, D T, C, D D D D T, C, D T, C, D

T, C, TFI, CI LA-U C, TFI, CI C, TFI, CI TFI TFI, CI TFI, CI TFI, CI TFI, CI

C, TFI, CI LA-U C, TFI, CI C, TFI, CI CI CI CI C, TFI, CI C, TFI, CI

Table 15. The ALA composition table r2 r1

T C D CIi TFIi

T

C

D

CI

TFI

AA-U\CI, CIi AA-U\E, CI, CII AA-U\E, CIi, TFIi AA-U\T, D, E, CI, TFI AA-U\D, E, CI, TFI

AA-U\E, CI, CIi AA-U

AA-U\E, CI, TFI AA-U\E, CI, TFI AA-U

AA-U\T, D, E, CIi, TFIi O, TFI, CI

AA-U\D, E, CIi, TFIi O, TFI, CI

AA-U\E, CIi, TFIi AA-U\T, D O, TFI, CI

AA-U\E, CIi, TFIi O,TFIi, CIi

AA-U\E, CIi, TFIi O, TFIi, CIi O, TFIi, CIi

AA-U\E, CI, TFI AA-U\E, CI, TFI

O, E, TFI, TFIi

Table 16. The LLL composition table r2

T

C

O

D

E

CI

TFI

CIi

TFIi

T

LL-U\ CIi

T, C, O, D, TFIi, CIi LL-U\ E, CI, TFI LL-U\ E, CI, TFI LL-U

T, C, O, TFI, CI C, O, TFI, CI O, TFI, CI

T, C, O, TFI, CI C, O, TFI, CI O, TFI, CI

D

T, D

T, C, O, D, TFIi, CIi T, C, O, D, TFIi, CIi T, C, O, D, TFI, CI T D

T, C, O, D, TFI, CI LL-U\ E, CIi, TFIi LL-U

T

C

T, C, O, D, TFI, CI LL-U

T, C, D

T, C, D

D

E CI

LL-U \E, CIi, TFIi TFI CI

T, O, D, TFIi, CIi D

D D

LL-U \E, CIi, TFIi CI CI

T, O, D, TFIi, CIi D

CIi LL-U \C

D

TFI

LL-U \C, E

TFI, CI

LL-U\ E, CI, TFI LL-U\ E, CI, TFI

CIi

LLU\ T, C, D O, TFI, CI

O, TFII, CIi

TFIi T, O, D, TFI, CI LL-U \C, CI, CIi CIi

r1

O

LL-U\ E, CI, TFI LL-U\ E, CIi, TFIi C T, C, D

TFI

T, D

T, C, D

CIi

T, C, O, TFIi, CIi T, C, O, TFIi, CIi

C, O, TFII, CIi

LL-U\ E, CIi, TFIi O T, O, D, TFI, CI T, O, D, TFI, CI O, TFIi, CIi

C, O, TFIi, CIi

O, TFIi, CIi

D

E CI

TFIi

C

O

TFIi

O, E, TFI, TFIi

T, O, D, TFIi, CIi CIi

CIi

Table 17. The PAP composition table r2 r1

T D CI

T

D

CIi

PP-U D D

D PP-U D

D D PP-U

CI, CIi

Table 18. The PAL composition table r2

T

C

D

CIi

TFIi

PL-U PL-U D

PL-U PL-U PL-U

D PL-U D

D D PL-U

T, D D PL-U

r1

T D CI

Table 19. The PAA composition table r2 r1

T D CI

T

O

D

E

CI

TFI

CIi

TFIi

T, D PA-U D

PA-U PA-U PA-U

D PA-U D

T D CI

CI PA-U CI

T, CI PA-U CI

D D PA-U

T, D D PA-U

Table 20. The LAL composition table r2

T

C

D

CIi

TFIi

T

LL-U

T, D, O

T, D, C, O, TFIi, CIi D

T, D, C, O, TFIi, CIi D

CI

T, D, C, O, TFIi, CII T, D, C, O, TFI, CI D

T, D, C, O, TFIi, CIi T, D, C, O, TFIi, CIi LL-U

D

C

T, D, C, O, TFI, CI LL-U

D

LL-U

TFI

T, D, O

D

T, D, C, O, TFIi, CIi 

T, D, C, O, TFI, CI T, D, C, O, E, TFI, TFIi

r1

D

T, D, C, O, TFI, CI T, D, C, O, TFI, CI T, D, C, O, TFI, CI

Table 21. The LAA composition table r2

T

O

D

E

CI

TFI

CIi

TFIi

T, D, C T, D, C LA-U

T, C, CI, TFI C, CI, TFI LA-U

D

T, D

D

C, CI, TFI C, CI, TFI LA-U

T, C, D D

T, C, D D

CI

D

D

CI

CI

CI

LA-U

LA-U

TFI

T, D

LAU LAU LAU LAU LAU

T

D

T, D, C, TFI T, D, C LA-U

D

TFI

CI

CI, TFI

T, C, D

C, T, D, TFI

r1

T C

C

Table 22. The AAA composition table r2 r1

T

O

D

E CI

3

T

O

D

E

CI

TFI

CIi

TFIi

T, D, O, E, TFI, TFIi T, D, O, TFIi, CIi T, D, O, TFI, CI T D

T, D, O, TFI, CI AA-U

T, D, O, TFIi, CIi T, D, O, TFIi, CIi AA-U

T

O, CI, TFI

O, T, CI, TFI

D

D, T

O

O, CI, TFI

O, TFI, CI

D

T, D, O, TFI, CI

T, D, O, TFIi, CIi D

D D

E CI

CI CI

T, D, O, TFI, CI TFI CI

T, D, O, TFIi, CIi D

CIi AA-U

TFIi T, D, O, CI, TFI

D

TFI

CI

CI, TFI

E, O, CI, TFI, CIi, TFIi O, CI, TFI

O, TFIi, CIi

T, D, O, TFIi, CII CIi

T, D, O, E, TFIi, TFI CIi

CIi

TFIi, CIi

T, D, O, TFI, CI O T, D, O, TFI, CI T, D, O, CI, TFI

TFI

T, D

CII

O, CIi, TFIi

O, CIi, TFIi

T, D, O, CIi, TFIi

CIi

TFIi

T, O, CIi, TFIi

O, CIi, TFII

T, D, O, CIi, TFIi

TFIi

O, E, TFI, TFIi

Conclusions

We have proposed a constraint-based approach to the CBM calculus in (Clementini et al., 1993). What we have obtained is a calculus consisting of atomic relations and of the algebraic operations of converse and composition. As such, the calculus is an algebra in the same style as the one provided by Allen (1983) for temporal intervals. The objects manipulated by the calculus are point-like, linear and areal features, contrary to most constraint-based frameworks in the qualitative spatial and temporal reasoning literature, which deal with only one type of feature (for instance, intervals in (Allen, 1983)). One problem raised by this was that the calculus had 27 composition tables, fortunately of moderate sizes. We have shown in this work that the use of 18 of these tables is sufficient, as from these 18 we can derive the other nine.

Reasoning about knowledge expressed in the presented calculus can be done using a constraint propagation algorithm alike to the one in (Allen, 1983), guided by the 18 composition tables and the converse table. Such an algorithm has the advantage of being incremental: knowledge may be added without having to revise the processing steps achieved so far.

Acknowledgments We would like to thank Christian Freksa, Teresa Escrig, and Julio Pacheco for helpful discussions and suggestions during the preparation of this work. We thank the anonymous reviewer for the critical comments. This work has been partially supported by CICYT under grant number TAP99-0590-C02-02.

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