A Hybrid Reasoning Model for “Whole and Part” Cardinal Direction Relations

Abstract

We have shown how the nine tiles in the projection-based model for cardinal directions can be partitioned into sets based on horizontal and vertical constraints (called Horizontal and Vertical Constraints Model) in our previous papers (Kor and Bennett, 2003 and 2010). In order to come up with an expressive hybrid model for direction relations between two-dimensional single-piece regions (without holes), we integrate the well-known RCC-8 model with the above-mentioned model. From this expressive hybrid model, we derive 8 basic binary relations and 13 feasible as well as jointly exhaustive relations for the x- and y-directions, respectively. Based on these basic binary relations, we derive two separate 8×8 composition tables for both the expressive and weak direction relations. We introduce a formula that can be used for the computation of the composition of expressive and weak direction relations between “whole or part” regions. Lastly, we also show how the expressive hybrid model can be used to make several existential inferences that are not possible for existing models.

1. Introduction

Relative positions of regions in large-scale spaces, and particularly in the geographic domain, are often described by relations referring to cardinal directions. These relations specify the direction from one region to another in terms of the familiar compass bearings: north, south, east, and west. The intermediate directions northwest, northeast, southwest, and southeast are also often used. Some models for reasoning with cardinal directions are the cone-shaped [1, 2], projection-based models (ibid), and direction matrix [3–5].

Papadias and Theodoridis [6] describe topological and direction relations between regions using their minimum bounding rectangles (MBRs). However, the language used is not expressive enough to describe direction relations. Additionally, the MBR technique yields erroneous outcome when involving regions that are not rectangular in shape [4] Some work has been done on hybrid direction models. Escrig and Toledo [7] and Clementini et al. [8] integrated qualitative orientation and distance to obtain positional information. Isli [9] combined Frank’s [1, 2] cardinal direction relations model and Freksa’s [10] orientation model to facilitate a more expressive reasoning mechanism. Sharma and Flewelling [11] infer spatial relations from integrated topological and cardinal direction relations. Liu and colleagues [12] have developed reasoning algorithms which combine RCC-8 [13] for topological relations (discussed in Section 4) and the cardinal direction calculus (CDC [3–5], discussed in Section 2) for direction relations. Li and colleagues’ work [14, 15] focuses on the development and evaluation of an efficient reasoning mechanism for RCC-8 and RA (Rectangle Algebra, and further explanation can be found in [16, 17]) which is employed to solve the satisfiability problem of these two joint constraint networks.

Typically, composition tables are used to infer spatial relations between objects. They have been employed to make different inferences about cardinal directions relations [3, 19–24]. One of the advantages of composition tables is that they can lead to tractable computation of inferences [25]. In this paper, we have developed an expressive hybrid model for direction relations. We will describe the binary relations in the model and define “whole and part” relations. Based on this model, we derive two 8×8 composition tables for expressive and weak direction relations. This is followed by introducing a formula which could be used to compute both expressive and weak direction relations for “whole and part” regions. Finally, we will demonstrate how the model could be used to make several types of existential inferences.

2. Cardinal Direction Models

Frank [1, 2] defines cardinal directions as cones which are related to the angular direction between an observer’s position (in the form of a point) and a destination point. The cone-shaped cardinal direction model could have 4, 8, or more partitions (look at Figure 1).

Figure 1 

Cone-shaped direction model with 4 or 8 partitions (ibid).

Frank defines the four major cardinal directions (north, south, east, and west) as pair-wise opposites and half planes. When the two sets of half planes are combined, it yields four intermediate cardinal directions (northeast, northwest, southeast, and southwest) which are depicted in Figure 2. Ligozat [21] applies the model to points in a two-dimensional space. Thus, the referent object, Point B, will be given the four major directions. However, the relations between two objects will be denoted by one of the following basic relations: N, S, E, W, NE, NW, SE, SW, or EQ.

Figure 2 

Cardinal directions defined by half-planes.

Frank [1, 2] extends the half-planes to tiles for regions (as shown in Figure 3). In this projection-based model, the plane of an arbitrary single-piece region a is partitioned into nine tiles, North-West, NW(a); North, N(a); North-East, NE(a); South-West, SW(a); South, S(a); South-East, SE(a); West, W(a); Neutral Zone, O(a); East, E(a). According to Frank, the O tile is considered a neutral zone, because in this tile, the relative cardinal direction between two regions cannot be determined due to their proximity.

Figure 3 

Cardinal directions defined by tiles for extended objects [1, 2].

Frank compares and contrasts reasoning with the cone-shaped and the projection-based models for cardinal directions. The reasoning capability for both the systems is limited and weak though they do not differ substantially in their reasoning outcomes. In order to create a more expressive reasoning model, Isli [26] integrates the Frank’s cone-shaped and projection-based models to facilitate reasoning about relative position of points of the 2-dimensional space. This hybrid model is well suited for applications of large-scale high-level vision, such as, for example, satellite-like surveillance of a geographic area.

The cardinal direction calculus (CDC) [3–5] is a very expressive qualitative calculus for directional information of extended objects. A direction relation matrix (DRM) in (1) is used to represent direction relations between connected plane regions. Liu and colleagues [27, 28] have shown that consistency checking of complete networks of basic CDC constraints is tractable, while reasoning with the CDC in general is NP hard. However, if some constraints are unspecified, then consistency checking of incomplete networks of basic CDC constraints is intractable.

The cardinal direction of a target object (region b) to a referent object (region a) as shown in Figure 4 is described by recording those tiles covered by the target object. According to Goyal and Egenhofer [4], a 3×3 matrix is employed to register the intersections between the target object and the tiles of the referent object (see (1)). The elements in the direction-relation matrix correspond to the tiles of the referent object, region a (in Figure 4).

Figure 4 

Nine tiles with regions 𝑎 (as the referent object) and 𝑏 (as the target object) [4, 5].

In (1), the symbol ∅ represents empty tile while ¬∅ represents nonempty tile. These are used to describe cardinal directions at a coarse granularity level. In Figure 4, region 𝑏 occupies the N, NW, and E tiles of region 𝑎. Thus, these three tiles are considered nonempty while the rest are considered empty (as shown in (1)).

Goyal and Egenhofer [5] extend the direction relation matrix, so that it will be more expressive. Instead of using the empty and nonempty notations, it registers how much (in terms of proportion) the target region occupies each tile (see (2)). The expressive direction relation matrix in (2) has 6 elements of zero and three nonzero elements which sum up to 1.0. If the matrix has only one nonzero element then it is known as a single element direction relation matrix while a matrix with more than one nonzero element is called a multielement direction relation matrix (ibid).

Coarse direction relation matrix [4]: dir_𝑅𝑅()=(𝑎)(𝑎)(𝑎)W(𝑎)(𝑎)(𝑎)(𝑎)(𝑎)(𝑎)𝑎,𝑏,NW∩𝑏N∩𝑏NE∩𝑏∩𝑏O∩𝑏E∩𝑏SW∩𝑏S∩𝑏SE∩𝑏dir_𝑅𝑅(𝑎,𝑏)=.∅¬∅¬∅∅∅¬∅∅∅∅(1)

Expressive direction relation matrix [5]: dir_𝑅𝑅()=((𝑎))𝑎,𝑏areaNW∩𝑏(N(𝑎))areaof𝑏area∩𝑏((𝑎))areaof𝑏areaNE∩𝑏(W(𝑎))areaof𝑏area∩𝑏(O(𝑎))areaof𝑏area∩𝑏(E(𝑎))areaof𝑏area∩𝑏((𝑎))areaof𝑏areaSW∩𝑏(S(𝑎))areaof𝑏area∩𝑏((𝑎))areaof𝑏areaSE∩𝑏,areaof𝑏dir_𝑅𝑅(𝑎,𝑏)=.00.050.45000.50000(2)

3. Horizontal and Vertical Constraints Model

Every region has a minimal bounding box with specific minimum and maximum 𝑥 (and 𝑦) values. The boundaries of the minimal bounding box of a region 𝑎 are depicted in Figure 5. The set of boundaries of the minimal bounding box for region 𝑎 could be represented as {𝑋_min(𝑎), 𝑋_max(𝑎), 𝑌_min(𝑎), 𝑌_max(𝑎)}, and these values will be employed to define each tile.

Figure 5 

Horizontal and vertical sets of tiles for a.

The definition of the nine tiles in terms of the boundaries of the minimal bounding box is listed as below. Note, in this paper, all the tiles are regarded as mutually exclusive. Thus neighboring tiles cannot share common boundaries:(i)N(𝑎)≡{⟨𝑥,𝑦⟩∣𝑋_min(𝑎)≤𝑥<𝑋_max(𝑎)∧𝑦≥𝑌_max(𝑎)},(ii)NE(𝑎)≡{⟨𝑥,𝑦⟩∣𝑥≥𝑋_max(𝑎)∧𝑦≥𝑌_max(𝑎)},(iii)NW(𝑎)≡{⟨𝑥,𝑦⟩∣𝑥<𝑋_min(𝑎)∧𝑦≥𝑌_max(𝑎)},(iv)S(𝑎)≡{⟨𝑥,𝑦⟩∣𝑋_min(𝑎)≤𝑥<𝑋_max(𝑎)∧𝑦<𝑌_min(𝑎)},(v)SE(𝑎)≡{⟨𝑥,𝑦⟩∣𝑥≥𝑋_max(𝑎)∧𝑦<𝑌_min(𝑎)},(vi)SW(𝑎)≡{⟨𝑥,𝑦⟩∣𝑥<𝑋_min(𝑎)∧𝑦<𝑌_min(𝑎)},(vii)E(𝑎)≡{⟨𝑥,𝑦⟩∣𝑥≥𝑋_max(𝑎)∧𝑌_min(𝑎)≤𝑦<𝑌_max(𝑎)},(viii)W(𝑎)≡{⟨𝑥,𝑦⟩∣𝑥<𝑋_min(𝑎)∧𝑌_min(𝑎)≤𝑦<𝑌_max(𝑎)},(ix)O(𝑎)≡{⟨𝑥,𝑦⟩∣𝑋_min(𝑎)≤𝑥<𝑋_max(𝑎)∧𝑌_min(𝑎)≤𝑦<𝑌_max(𝑎)}.

In our previous papers [29, 30], we have shown how to partition the nine tiles (in Figure 5) into sets based on horizontal and vertical constraints called the Horizontal and Vertical Constraints Model. However, in this paper, we shall rename the sets for easy comprehension purposes. The following are the definitions of the partitioned regions.(i)WeakNorth(a) is the region that covers the tiles NW(a), N(a), and NE(a). WeakNorth(a) ≡ NW(a) ∪ N(a) ∪ NE(a).(ii)Horizontal(a) is the region that covers the tiles W(a), O(a), and E(a). Horizontal(a) ≡ W(a) ∪ O(a) ∪ E(a).(iii)WeakSouth(a) is the region that covers the tiles SW(a), S(a), and SE(a). WeakSouth(a) ≡ SW(a) ∪ S(a) ∪ SE(a).(iv)WeakWest(a) is the region that covers the tiles SW(a), W(a), and NW(a). WeakWest(a) ≡ SW(a) ∪ W(a) ∪ NW(a).(v)Vertical(a) is the region that covers the tiles S(a), O(a), and N(a). Vertical(a) ≡ S(a) ∪ O(a) ∪ N(a).(vi)WeakEast(a) is the region that covers the tiles SE(a), E(a), and NE(a). WeakEast(a) ≡ SE(a) ∪ E(a) ∪ NE(a).

4. RCC Model

RCC stands for region connection calculus [13, 18, 31]. It is a first-order theory employed for qualitative spatial representation as well as reasoning and is based on Clarke’s logic of connection [32, 33]. The connection predicate, C(𝑎,𝑏), which means “region a is connected with region b”, is the only primitive predicate for RCC. This dyadic relation is both reflexive and symmetric and holds whenever regions 𝑎 and 𝑏 are “connected.” The two main axioms expressing reflexivity and symmetry [18] are as follows: ∀_𝑎[C()∀𝑎,𝑎)](reflexive_𝑎∀_𝑏[C(𝑎,𝑏)⟶C(𝑏,𝑎)]symmetric.(3) Based on this primitive, a basic set of dyadic relations are defined as shown in Table 1.

Table 1 

Spatial relations defined in terms of C(𝑎,𝑏) [18].

The relations P, PP, TPP, and NTPP are nonsymmetrical and will have their respective inverses (Pi, PPi, TPPi, and NTPPi). Of all the listed relations, only 8 relations in the following set {DC, EC, PO, EQ, TPP, NTPP, TPPi, NTPPi} are provably jointly exhaustive and pairwise disjoint (JEPD—which means any two regions are related by exactly one of these eight relations [34, 35]). Randell and colleagues [13] refer this set of relations as RCC-8, and they are depicted in Figure 6.

Figure 6 

8 basic JEPD RCC binary relations [13].

5. Expressive Hybrid Model

In our expressive hybrid model, we have combined our Horizontal and Vertical Constraints Model [29, 30] and RCC-8 [13].

5.1. Definitions

If there is a referent region 𝑎 and another arbitrary region 𝑏, the possible basic binary relations between them can be defined as below.

In terms of weak relations,(i)WeakNorth(b,a): b ⊆ WeakNorth(a),(ii)Horizontal(b,a): b ⊆ Horizontal(a),(iii)WeakSouth(b,a): b ⊆ WeakSouth(a),(iv)WeakEast(b,a): b ⊆ WeakEast(a),(v)Vertical(b,a): b ⊆ Vertical(a),(vi)WeakWest(b,a): b ⊆ WeakWest(a).

In terms of RCC-8 relations,(i)DCy(a,b): y-dimension of 𝑎 is disconnected from y-dimension of b, (ii)EQy(a,b): y-dimension of 𝑎 is identical with y-dimension of b, (iii)POy(a,b): y-dimension of 𝑎 partially overlaps y-dimension of b, (iv)ECy(a,b): y-dimension of 𝑎 is externally connected to y-dimension of b, (v)TPPy(a,b): y-dimension of 𝑎 is a tangential proper part of y-dimension of b, (vi)NTPPy(a,b): y-dimension of 𝑎 is a nontangential proper part of y-dimension of b, (vii)TPPiy(a,b): y-dimension of 𝑏 is a tangential proper part of y-dimension of a, (viii)NTPPiy(a,b): y-dimension of 𝑏 is a non-tangential proper part of y-dimension of a, (ix)DCx(a,b): x-dimension of 𝑎 is disconnected from x-dimension of b, (x)EQx(a,b): x-dimension of 𝑎 is identical with x-dimension of b, (xi)POx(a,b): x-dimension of 𝑎 partially overlaps x-dimension of b, (xii)ECx(a,b): x-dimension of 𝑎 is externally connected to x-dimension of b, (xiii)TPPx(a,b): x-dimension of 𝑎 is a tangential proper part of x-dimension of b, (xiv)NTPPx(a,b): x-dimension of 𝑎 is a non-tangential proper part of x-dimension of b, (xv)TPPix(a,b): x-dimension of 𝑏 is a tangential proper part of x-dimension of a, (xvi)NTPPix(a,b): x-dimension of 𝑏 is a non-tangential proper part of x-dimension of a.

5.2. Basic Binary Relations of the Hybrid Model

In this section, we shall demonstrate how we come up with all possible binary direction relations for the hybrid model. All the possible basic binary relations for each horizontal set are shown in Figure 7. The notations that will be used in this section are as follows.(i)RELy(b,Z) is any basic binary relation between 𝑏 and the horizontally partitioned region, 𝑍. (ii)RELx(b,Z) is any basic binary relation between 𝑏 and the vertically partitioned region, 𝑍.

Figure 7 

Possible basic binary relations for each horizontally partitioned region (note: it will be similar for vertically partitioned region).

Based on Figure 7, the total number of possible binary relations for the hybrid model in the y-direction is [(2+4+2)+(2×4)+(2×2)+(4×2)+(2×4×2)] which equals 44 cases. However, due to the single-piece condition, the following rules apply.

Rule 1. ¬(b ⊆ WeakNorth(a) ∧𝑏⊆ WeakSouth(a)).

Rule 2. Assume 𝑈 to be {WeakNorth(a), Horizontal(a), WeakSouth(a)}.  If NTPP𝑦(𝑏,𝑅) where 𝑅∈𝑈 then ¬(NTPP𝑦(𝑏,𝑅) ∧ REL𝑦(𝑏,𝑆)), where 𝑆∈𝑈−𝑅, or ¬(NTPP𝑦(𝑏,𝑅)∧ REL𝑦(𝑏,𝑆) ∧ REL𝑦(𝑏,𝑇)), where 𝑇∈𝑈−𝑆.

Rule 3. Assume 𝑈 to be {WeakNorth(a), WeakSouth(a)}.
If (TPP𝑦(𝑏,Horizontal(𝑎)) ∧ EC𝑦(𝑏,𝑅)), where 𝑅∈𝑈,
then ¬(TPP𝑦(𝑏,Horizontal) ∧ EC𝑦(𝑏,𝑅)∧ REL𝑦(𝑏,𝑆)),
where 𝑆∈𝑈−𝑅.

Based on the rules above, the total number of feasible binary relations for single-piece regions in the y-direction is (44-4-23-4) which equals 13 cases. The thirteen feasible and jointly exhaustive binary relations for the hybrid model are depicted in Figure 8. This means that, in the hybrid model, the number of jointly exhaustive binary relations (in both the 𝑥- and 𝑦-directions) that hold between two single-piece regions will be 13×13. This concurs with the 13×13 basic relations in the Rectangle Algebra Model [16, 17].

Figure 8 

Thirteen feasible and jointly exhaustive binary relations in the 𝑦-direction for the hybrid model (note: this will be similar for 𝑥-direction for the model).

6. Combined Mereological, Topological, and Cardinal Direction Relations

Mereology (from the Greek μερος, “part”) is the theory of parthood relations: of the relations of part to whole and the relations, of part to part within a whole [36]. In this section, we shall make two distinctions: “whole and part” cardinal directions, as well as “weak and expressive” relations. We shall rewrite the notations used in our previous paper [29]. 𝑃_R(𝑏,𝑎)means that only part of the destination extended region, 𝑏, is in tile R(𝑎). The direction relation 𝐴_R(𝑏,𝑎)means that whole destination extended region, 𝑏, is in the tile R(𝑎). As an example, when 𝑏 is completely in the South-East tile of 𝑎, this direction relation can be represented as shown below: 𝐴_SE(𝑏,𝑎)=¬𝑃_N(𝑏,𝑎)∧¬𝑃_NE(𝑏,𝑎)∧¬𝑃_NW(𝑏,𝑎)∧¬𝑃_S(𝑏,𝑎)∧𝑃_SE(𝑏,𝑎)∧¬𝑃_SW(𝑏,𝑎)∧¬𝑃_W(𝑏,𝑎)∧¬𝑃_E(𝑏,𝑎)∧¬𝑃_O(𝑏,𝑎).(4)

The “whole and weak” direction relations are defined in terms of horizontal and vertical sets:(i)𝐴_N(𝑏,𝑎) ≡ WeakNorth(𝑏,𝑎) ∧ Vertical(𝑏,𝑎),(ii)𝐴_NE(𝑏,𝑎) ≡ WeakNorth(𝑏,𝑎) ∧ WeakEast(𝑏,𝑎),(iii)𝐴_NW(𝑏,𝑎) ≡ WeakNorth(𝑏,𝑎) ∧ WeakWest(𝑏,𝑎),(iv)𝐴_S(𝑏,𝑎) ≡ WeakSouth(𝑏,𝑎) ∧ Vertical(𝑏,𝑎),(v)𝐴_SE(𝑏,𝑎) ≡ WeakSouth(𝑏,𝑎) ∧ WeakEast(𝑏,𝑎),(vi)𝐴_SW(𝑏,𝑎) ≡ WeakSouth(𝑏,𝑎) ∧ WeakWest(𝑏,𝑎),(vii)𝐴_E(𝑏,𝑎)≡ Horizontal(𝑏,𝑎) ∧ WeakEast(𝑏,𝑎),(viii)𝐴_W(𝑏,𝑎)≡ Horizontal(𝑏,𝑎) ∧ WeakWest(𝑏,𝑎),(ix)𝐴_O(𝑏,𝑎)≡ Horizontal(𝑏,𝑎) ∧ Vertical(𝑏,𝑎).

The “whole and expressive” direction relations are defined in terms of expressive horizontal and vertical sets. A general form of such direction relation can be represented as follows: ^{REL𝑦(𝑏,H(𝑎))}𝐴_𝑅(𝑏,𝑎)_{REL𝑥(𝑏,V(𝑎))}≡REL𝑦(𝑏,H(𝑎))∧REL𝑥(𝑏,V(𝑎)),(5) where H(a) and V(a) are horizontally and vertically partitioned regions for 𝑎, respectively, where 𝑏⊆R(𝑎) and R(a) ⊆ (H(a) ∩ V(a)).

7. Composition Table

Composition is a common inference mechanism for a wide range of relations and has been exploited for automated reasoning. It has been employed for reasoning about temporal descriptions of events based on intervals [37], topological relations [5, 38–42], direction relations [1, 24, 29, 30], and combined topological relations with cardinal direction relations [19]. To reiterate, one of the main advantages of using composition tables is that they can lead to tractable computation of significant classes of inference [25].

Given the relation between 𝑎 and 𝑏 and the relation between 𝑏 and 𝑐, a composition table allows for concluding the relation between 𝑎 and 𝑐. Bennett [41] defines the concept of the composition of two binary relations as follows.   Given a theory Θ which is used to define a set β of mutually exhaustive and pairwise disjoint dyadic relations (i.e., a basis set), the composition, Comp(𝑅₁, 𝑅₂), of two relations 𝑅₁ and 𝑅₂ which are taken from ß is defined to be the disjunction of all relations 𝑅₃ in ß, such that, for arbitrary constants 𝑎, 𝑏, and 𝑐, the formula 𝑅₁(𝑎,𝑏)∧𝑅₂(𝑏,𝑐)∧𝑅₃(𝑎,𝑐) is consistent with Θ.

7.1. Composition of Regions with Parts

In our previous paper [29], the method for computing the composition of cardinal direction relations for part regions is not robust enough, because it does not hold for all cases. In order to address this problem, we introduce a formula (obtained through case analyses) for computing the composition of cardinal direction relations. The basis of the formula is to consider the direction relation between 𝑎 and each individual part of 𝑏 followed by the direction relation between each individual part of 𝑏 and 𝑐.

Assume that the region covers one or more tiles of region 𝑎 while region 𝑐 covers one or more tiles of 𝑏.The direction relation between 𝑎 and 𝑏 is 𝑅(𝑏,𝑎) while the direction relation between 𝑏 and c is 𝑆(𝑐,𝑏). The composition of direction relations could be written as follows: 𝑅(𝑏,𝑎)∧𝑆(𝑐,𝑏).(6) Firstly, establish the direction relation between 𝑎 and each individual part of b: 𝑅()()≡𝑅𝑏,𝑎∧𝑆𝑐,𝑏₁𝑏₁,𝑎∧𝑅₂𝑏₂,𝑎⋯∧𝑅_𝑘𝑏_𝑘∧[𝑆(,,𝑎𝑐,𝑏)](7) where 1≤𝑘≤9.

Consider the direction relation of each individual part of 𝑏 and 𝑐. Equation (7) becomes 𝑅₁𝑏₁,𝑎∧𝑆₁₁𝑐₁,𝑏₁∨𝑅₁𝑏₁,𝑎∧𝑆₁₂𝑐₂,𝑏₁⋯∨𝑅₁𝑏₁,𝑎∧𝑆_1𝑚𝑐_1𝑚,𝑏₁∧𝑅₂𝑏₂,𝑎∧𝑆₂₁𝑐₁,𝑏₂∨𝑅₂𝑏₂,𝑎∧𝑆₂₂𝑐₂,𝑏₂⋯∨𝑅₂𝑏₂,𝑎∧𝑆_2𝑚𝑐_2𝑚,𝑏₂∧⋯𝑅_𝑘𝑏_𝑘,𝑎∧𝑆_𝑘1𝑐₁,𝑏_𝑘∨𝑅_𝑘𝑏_𝑘,𝑎∧𝑆_𝑘2𝑐₂,𝑏_𝑘⋯∨𝑅_𝑘𝑏_𝑘,𝑎∧𝑆_𝑘𝑚𝑐_𝑘𝑚,𝑏_𝑘,(8) where 1≤𝑘,𝑚≤9.

7.2. Composition of Weak Direction Relations

Firstly, we shall demonstrate how to apply the formula for the composition of weak direction relations followed by more expressive direction relations.

Type 1. 𝐴_𝑅(𝑏,𝑎)∧𝐴_𝑅(𝑐,𝑏).
Find the composition of 𝐴_O(𝑏,𝑎)∧𝐴_SW(𝑐,𝑏).
Use (8) with 𝑘=1 and 𝑚=1: 𝑅₁𝑏₁,𝑎∧𝑆₁₁𝑐₁,𝑏₁≡𝐴_O(𝑏,𝑎)∧𝐴_SW()≡[()(∧[()(≡[()(∧[()(.𝑐,𝑏Horizontal𝑏,𝑎∧Vertical𝑏,𝑎)]WeakSouth𝑐,𝑏∧WeakWest𝑐,𝑏)]Horizontal𝑏,𝑎∧WeakSouth𝑐,𝑏)]Vertical𝑏,𝑎∧WeakWest𝑐,𝑏)](9)
The outcome of the composition is [()(∧[()(.Horizontal𝑐,𝑎∨WeakSouth𝑐,𝑎)]Vertical𝑐,𝑎∨WeakWest𝑐,𝑎)](10)
This means that the region c ⊆ O(a) ∨ W(a) ∨ S(a) ∨ SW(a).

Type 2. 𝐴_𝑅(𝑏,𝑎)∧𝑃_𝑅(𝑐,𝑏).
Find the composition of 𝐴_E(𝑏,𝑎)∧[𝑃_NW(𝑐,𝑏)∧𝑃_N(𝑐,𝑏)].
Use (8) with 𝑘=1, and 1≤𝑚≤2: 𝑅₁𝑏₁,𝑎∧𝑆₁₁𝑐₁,𝑏₁∨𝑅₁𝑏₁,𝑎∧𝑆₁₂𝑐₂,𝑏₁≡𝐴_𝐸(𝑏,𝑎)∧𝐴_𝑁𝑊𝑐₁,𝑏∨𝐴_𝐸(𝑏,𝑎)∧𝐴_𝑁𝑐₂≡()(∧𝑐,𝑏[[Horizontal𝑏,𝑎∧WeakEast𝑏,𝑎)]WeakNorth₁,𝑏∧WeakWest𝑐₁∨()(∧𝑐,𝑏[[Horizontal𝑏,𝑎∧WeakEast𝑏,𝑎)]WeakNorth₂,𝑏∧Vertical𝑐₂≡()𝑐,𝑏Horizontal𝑏,𝑎∧WeakNorth₁∧()𝑐,𝑏WeakEast𝑏,𝑎∧WeakWest₁∨()𝑐,𝑏Horizontal𝑏,𝑎∧WeakNorth₂∧()𝑐,𝑏WeakEast𝑏,𝑎∧Vertical₂,𝑏.(11)
The outcome of the composition is Horizontal𝑐₁,𝑎∨WeakNorth𝑐₁∧𝑐,𝑎WeakEast₁,𝑎∨Vertical𝑐₁,𝑎∨WeakWest𝑐₁∨𝑐,𝑎Horizontal₂,𝑎∨WeakNorth𝑐₂∧𝑐,𝑎WeakEast₂,𝑎.(12)
Viewing the fact that 𝑐₁⊂𝑐 and 𝑐₂⊂𝑐, the above outcome can be written as ()(∧[()()(.[[Horizontal𝑐,𝑎∨WeakNorth𝑐,𝑎)]WeakEast𝑐,𝑎∨Vertical𝑐,𝑎∨WeakWest𝑐,𝑎)]](13)
This means that the region 𝑐 ⊆ E(a) ∨ O(a) ∨ W(a) ∨ NE(a) ∨ N(a) ∨ NW(a).

Type 3. 𝑃_𝑅(𝑏,𝑎) ∧𝐴_𝑅(𝑐,𝑏).
Find the composition of [𝑃_O(𝑏₁,𝑎)∧𝑃_N(𝑏₂,𝑎)]∧𝐴_NE(𝑐,𝑏). Establish the relationship between 𝑐 and each individual part of 𝑏. In this case, 𝐴_NE(𝑐,𝑏), 𝑃_NE(𝑐,𝑏₁) and 𝑃_NE(𝑐,𝑏₂) holds (this is not necessarily true for all cases).
Use (8) with 1≤𝑘≤2 and 𝑚=1. 𝑃_𝑅1𝑏₁,𝑎∧𝑃_𝑅11𝑐₁,𝑏₁∧𝑃_𝑅2𝑏₂,𝑎∧𝑃_𝑅21𝑐₁,𝑏₂≡𝑃_O𝑏₁,𝑎∧𝑃_NE(∧𝑃𝑐,𝑏)_N𝑏₂,𝑎∧𝑃_NE(𝑐,𝑏).(14)
Therefore, the above composition can be rewritten as 𝑃_O𝑏₁,𝑎∧𝑃_NE𝑐,𝑏₁∧𝑃_N𝑏₂,𝑎∧𝑃_NE𝑐,𝑏₂≡Horizontal𝑏₁,𝑎∧Vertical𝑏₁∧,𝑎WeakNorth𝑐,𝑏₁∧WeakEast𝑐,𝑏₁∧WeakNorth𝑏₂,𝑎∧Vertical𝑏₂∧,𝑎WeakNorth𝑐,𝑏₂∧WeakEast𝑐,𝑏₂≡Horizontal𝑏₁,𝑎∧WeakNorth𝑐,𝑏₁∧Vertical𝑏₁,𝑎∧WeakEast𝑐,𝑏₁∧WeakNorth𝑏₂,𝑎∧WeakNorth𝑐,𝑏₂∧Vertical𝑏₂,𝑎∧WeakEast𝑐,𝑏₂.(15)
The outcome of the composition is ()(∧[()(∧[[Horizontal𝑐,𝑎∨WeakNorth𝑐,𝑎)]WeakEast𝑐,𝑎∨Vertical𝑐,𝑎)]]NTPP_𝑦((𝑎∧[()(=𝑐,WeakNorth))WeakEast𝑐,𝑎∨Vertical𝑐,𝑎)]NTPP_𝑦((𝑎∧[()(.𝑐,WeakNorth))WeakEast𝑐,𝑎∨Vertical𝑐,𝑎)](16)
This means that the 𝑌_min(𝑐) of the minimal bounding box for region 𝑐 is greater than 𝑌_max(𝑎) of the minimal bounding box for region 𝑎 and 𝑐⊆NE(𝑎)∨N(𝑎).

Type 4. 𝑃_𝑅(𝑏,𝑎)∧𝑃_𝑅(𝑐,𝑏).
Find the composition of 𝑃_O𝑏₁,𝑎∧𝑃_NE𝑏₂,𝑎∧𝑃_O(𝑐,𝑏)∧𝑃_W(𝑐,𝑏)∧𝑃_SW(𝑐,𝑏).(17)
Figure 9 has been drawn for this example. Establish the direction relation between each individual part of 𝑏 and 𝑐.
Use (8) with 1≤𝑘≤2; the value of 𝑚₁ for 𝑏₁ is 1≤𝑚₁≤4, while the value 𝑚₂ for 𝑏₂ is 1≤𝑚₂≤7: 𝑃_𝑅1𝑏₁,𝑎∧𝑃_𝑅11𝑐₁,𝑏₁∨𝑃_𝑅1𝑏₁,𝑎∧𝑃_𝑅12𝑐₂,𝑏₁∨𝑃_𝑅1𝑏₁,𝑎∧𝑃_𝑅13𝑐₃,𝑏₁∨𝑃_𝑅1𝑏₁,𝑎∧𝑃_𝑅14𝑐₄,𝑏₁∧𝑃_𝑅2𝑏₂,𝑎∧𝑃_𝑅21𝑐₁,𝑏₂∨𝑃_𝑅2𝑏₂,𝑎∧𝑃_𝑅22𝑐₂,𝑏₂∨𝑃_𝑅2𝑏₂,𝑎∧𝑃_𝑅25𝑐₅,𝑏₂∨𝑃_𝑅2𝑏₂,𝑎∧𝑃_𝑅26𝑐₆,𝑏₂∨𝑃_𝑅2𝑏₂,𝑎∧𝑃_𝑅27𝑐₇,𝑏₂≡𝑃_O𝑏₁,𝑎∧𝑃_S𝑐₁,𝑏₁∨𝑃_O𝑏₁,𝑎∧𝑃_SW𝑐₂,𝑏₁∨𝑃_O𝑏₁,𝑎∧𝑃_W𝑐₃,𝑏₁∨𝑃_O𝑏₁,𝑎∧𝑃_O𝑐₄,𝑏₁∧𝑃_NE𝑏₂,𝑎∧𝑃_NE𝑐₁,𝑏₂∨𝑃_NE𝑏₂,𝑎∧𝑃_N𝑐₂,𝑏₂∨𝑃_NE𝑏₂,𝑎∧𝑃_NW𝑐₃,𝑏₂∨𝑃_NE𝑏₂,𝑎∧𝑃_E𝑐₄,𝑏₂∨𝑃_NE𝑏₂,𝑎∧𝑃_O𝑐₅,𝑏₂∨𝑃_NE𝑏₂,𝑎∧𝑃_W𝑐₆,𝑏₂∨𝑃_NE𝑏₂,𝑎∧𝑃_SW𝑐₇,𝑏₂.(18)

Figure 9 

An example.

In part (1) of the above composition, 𝑐₁,𝑐₂,𝑐₃,𝑐₄⊂𝑐. To simplify the composition, we consider the combined horizontal and vertical sets of all the parts of 𝑐.Thus, we have the following: WeakNorth𝑏₁,𝑎∧WeakEast𝑏₁∧,𝑎Horizontal𝑐,𝑏₁∨WeakSouth𝑐,𝑏₁∧Vertical𝑐,𝑏₁∨WeakWest𝑐,𝑏₁≡WeakNorth𝑏₁∧,𝑎Horizontal𝑐,𝑏₁∨WeakSouth𝑐,𝑏₁∧WeakEast𝑏₁∧,𝑎Vertical𝑐,𝑏₁∨WeakWest𝑐,𝑏₁=[()()(∧[()()(.WeakNorth𝑐,𝑎∨Horizontal𝑐,𝑎∨WeakSouth𝑐,𝑎)]WeakEast𝑐,𝑎∨Vertical𝑐,𝑎∨WeakWest𝑐,𝑎)](19)

In part (2) of the above composition, 𝑐₁, 𝑐₂, 𝑐₃, 𝑐₄, 𝑐₅, 𝑐₆, 𝑐₇⊂𝑐. The simplified version of the composition is as follows: Horizontal𝑏₂,𝑎∧Vertical𝑏₂∧,𝑎WeakNorth𝑐,𝑏₂∨Horizontal𝑐,𝑏₂∨WeakSouth𝑐,𝑏₂∧WeakEast𝑐,𝑏₂∨Vertical𝑐,𝑏₂∨WeakWest𝑐,𝑏₂≡Horizontal𝑏₂∧,𝑎WeakNorth𝑐,𝑏₂∨Horizontal𝑐,𝑏₂∨WeakSouth𝑐,𝑏₂∧Vertical𝑏₂∧,𝑎WeakEast𝑐,𝑏₂∨Vertical𝑐,𝑏₂∨WeakWest𝑐,𝑏₂=()()(∧[()()(.[[WeakNorth𝑐,𝑎∨Horizontal𝑐,𝑎∨WeakSouth𝑐,𝑎)]]WeakEast𝑐,𝑎∨Vertical𝑐,𝑎∨WeakWest𝑐,𝑎)]](20)

The final outcome of the composition is part (1)∧ part (2) is equivalent to [()()(∧[()()(.WeakNorth𝑐,𝑎∨Horizontal𝑐,𝑎∨WeakSouth𝑐,𝑎)]WeakEast𝑐,𝑎∨Vertical𝑐,𝑎∨WeakWest𝑐,𝑎)](21)

This means that the region 𝑐⊆𝑈which is the union of all the 9 tiles of region 𝑎. However, based on Figure 9, region 𝑐⊄SW(𝑎).

7.3. Composition of Expressive Direction Relations

We shall use the following notations to represent the 13 binary y-direction relations:(i)REL1𝑦(𝑏,𝑎)-NTPP𝑦(𝑏,WeakNorth(𝑎)), (ii)REL2𝑦(𝑏,𝑎)-TPP𝑦(𝑏,WeakNorth(𝑎)) ∧ EC𝑦(𝑏,Horizontal(𝑎)),(iii)REL3𝑦(𝑏,𝑎)-TPP𝑦(𝑏,Horizontal(𝑎)) ∧ EC𝑦(𝑏,WeakNorth(𝑎)),(iv)REL4𝑦(𝑏,𝑎)-TPP𝑦(𝑏,Horizontal(𝑎)) ∧ EC𝑦(𝑏,WeakSouth(𝑎)),(v)REL5𝑦(𝑏,𝑎)-NTPP𝑦(𝑏,Horizontal(𝑎)),(vi)REL6𝑦(𝑏,𝑎)-EQy(𝑏,Horizontal(𝑎)),(vii)REL7𝑦(𝑏,𝑎)-NTPP𝑦(𝑏,WeakSouth(𝑎)),(viii)REL8𝑦(𝑏,𝑎)-TPP𝑦(𝑏,WeakSouth(𝑎)) ∧ EC𝑦(𝑏,Horizontal(𝑎)),(ix)REL9𝑦(𝑏,𝑎)-PO𝑦(𝑏,WeakNorth(𝑎)) ∧ PO𝑦(𝑏,Horizontal(𝑎)) ∧  DC𝑦(𝑏,WeakSouth(𝑎)),(x)REL10𝑦(𝑏,𝑎)-PO𝑦(𝑏,WeakNorth(𝑎)) ∧ PO𝑦(𝑏,Horizontal(𝑎)) ∧  EC𝑦(𝑏,WeakSouth(𝑎)),(xi)REL11𝑦(𝑏,𝑎)-PO𝑦(𝑏,WeakNorth(𝑎)) ∧ PO𝑦(𝑏,WeakSouth(𝑎)) ∧  NTPPi𝑦(𝑏,Horizontal(𝑎)),(xii)REL12𝑦(𝑏,𝑎)-PO𝑦(𝑏,WeakSouth(𝑎)) ∧ PO𝑦(𝑏,Horizontal(𝑎)) ∧  DC𝑦(𝑏,WeakNorth(𝑎)),(xiii)REL13𝑦(𝑏,𝑎)-PO𝑦(𝑏,WeakSouth(𝑎)) ∧ PO𝑦(𝑏,Horizontal(𝑎)) ∧  EC𝑦(𝑏,WeakNorth(𝑎)).

Similar notations will be used to represent the 13 binary 𝑥-direction relations (WeakNorth is replaced by WeakEast, Horizontal with Vertical, and WeakSouth by WeakWest).

Example 1. Find the composition of the following: ^{REL3𝑦(𝑏1,𝑎)}𝑃_O𝑏₁,𝑎_{REL3𝑥(𝑏1,𝑎)}∧^{REL2𝑦(𝑏2,𝑎)}𝑃_NE𝑏₂,𝑎_{REL2𝑥(𝑏2,𝑎)}∧^{REL1𝑦(𝑐,𝑏)}𝐴_N(𝑐,𝑏)_{REL5𝑥(𝑐,𝑏)}.(22)
Establish the direction relation between 𝑐 and each individual part of 𝑏. Use (8), with 1≤𝑘≤2 and 1≤𝑚₁≤2, and 1≤𝑚₂≤2: 𝑃_𝑅1𝑏₁,𝑎∧𝑃_𝑅11𝑐₁,𝑏₁∨𝑃_𝑅12𝑐₂,𝑏₁∧𝑃_𝑅2𝑏₂,𝑎∧𝑃_𝑅21𝑐₁,𝑏₂∨𝑃_𝑅22𝑐₂,𝑏₂.(23)
Use (5), and the above composition can be rewritten in the following expressive form: REL3𝑦𝑏₁,𝑎∧REL3𝑥𝑏₁∧𝑐,𝑎REL1𝑦₁,𝑏₁∧REL3𝑥𝑐₁,𝑏₁∨REL1𝑦𝑐₂,𝑏₁∧REL2𝑥𝑐₂,𝑏₁∧REL2𝑦𝑏₂,𝑎∧REL2𝑥𝑏₂∧𝑐,𝑎REL1𝑦₁,𝑏₂∧REL8𝑥𝑐₁,𝑏₂∨REL1𝑦𝑐₂,𝑏₂∧REL4𝑥𝑐₂,𝑏₂≡REL3𝑦𝑏₁,𝑎∧REL1𝑦𝑐₁,𝑏₁∨REL1𝑦𝑐₂,𝑏₁∧REL3𝑥𝑏₁,𝑎∧REL3𝑥𝑐₁,𝑏₁∨REL2𝑥𝑐₂,𝑏₁∧REL2𝑦𝑏₂,𝑎∧REL1𝑦𝑐₁,𝑏₂∨REL1𝑦𝑐₂,𝑏₂∧REL2𝑥𝑏₂,𝑎∧REL8𝑥𝑐₁,𝑏₂∨REL4𝑥𝑐₂,𝑏₂.(24)
Use Tables 2 and 3, and 𝑐₁⊂𝑐 and 𝑐₂⊂𝑐. Thus, the outcome of the composition can be written as follows: ()∧[()(()∧[()()()(()∧[()(.REL1𝑦𝑐,𝑎REL2𝑥𝑐,𝑎∨REL3𝑥𝑐,𝑎)]∧REL1𝑦𝑐,𝑎REL2𝑥𝑐,𝑎∨REL3𝑥𝑐,𝑎∨REL6𝑥𝑐,𝑎∨REL13𝑥𝑐,𝑎)]=REL1𝑦𝑐,𝑎REL2𝑥𝑐,𝑎∨REL3𝑥𝑐,𝑎)](25)
The outcome of the composition is: ((𝑎∧((𝑎((𝑎((𝑎((𝑎.NTPP𝑦𝑐,WeakNorth))TPP𝑥𝑐,WeakEast))∧EC𝑦𝑐,Horizontal))∨TPP𝑥𝑐,Vertical))∧EC𝑦𝑐,Horizontal))(26)

Table 2 

(a) Composition of binary relations in the 𝑦-direction for the hybrid model (composed relations of {WN} and {WN,H}). (b) Composition of binary relations in the 𝑦-direction for the hybrid model (composed relations of {H} and {WN,H}). (c) Composition of binary relations in the 𝑦-direction for the hybrid model (composed relations of {S} and {WN,H}). (d) Composition of binary relations in the 𝑦-direction for the hybrid model (composed relations of {WN,H,S} and {WS}).

Table 3 

(a) Composition of binary relations in the 𝑥-direction for the hybrid model (composed relations of {WE} and {WE,V,WW}). (b) Composition of binary relations in the 𝑥-direction for the hybrid model (part 1: composed relations of {V} and {WE,V,WW}). (c) Composition of binary relations in the 𝑥-direction for the hybrid model (part 2: composed relations of {V} and {WE,V,WW}). (d) Composition of binary relations in the 𝑥-direction for the hybrid model (part 2: composed relations of {WW} and {WE,V,WW}).

Example 2. This example is similar to the fourth example in the previous section of this paper.
Find the composition of 𝑃_O𝑏₁,𝑎∧𝑃_NE𝑏₂∧𝑃,𝑎_O(𝑐,𝑏)∧𝑃_W(𝑐,𝑏)∧𝑃_SW(𝑐,𝑏).(27)
Establish the direction relation between 𝑐 and each individual part of 𝑏.Use (8), with 1≤𝑘≤2 and 1≤𝑚₁≤4, and 1≤𝑚₂≤7.
The composition in expressive form will be as follows.

For part 𝑏₁, REL2𝑦𝑏₁,𝑎∧REL2𝑥𝑏₁∧𝑐,𝑎REL4𝑦₁,𝑏₁∧REL4𝑥𝑐₁,𝑏₁∨REL8𝑦𝑐₂,𝑏₁∧REL4𝑥𝑐₂,𝑏₁∨REL4𝑦𝑐₃,𝑏₁∧REL8𝑥𝑐₃,𝑏₁∨REL8𝑦𝑐₄,𝑏₁∧REL8𝑥𝑐₄,𝑏₁.(28)

The regions 𝑐₁, 𝑐₂, 𝑐₃, 𝑐₄⊂𝑐; the above composition can be written as follows: REL2𝑦𝑏₁∧,𝑎REL4𝑦𝑐,𝑏₁∨REL8𝑦𝑐,𝑏₁∨REL4𝑦𝑐,𝑏₁∨REL8𝑦𝑐,𝑏₁∧REL2𝑥𝑏₁∧,𝑎REL4𝑥𝑐,𝑏₁∨REL4𝑥𝑐,𝑏₁∨REL8𝑥𝑐,𝑏₁∨REL8𝑥𝑐,𝑏₁=()()()(∧[()()()(.REL2𝑦𝑐,𝑎∨REL3𝑦𝑐,𝑎∨REL6𝑦𝑐,𝑎∨REL13𝑦𝑐,𝑎)REL2𝑥𝑐,𝑎∨REL3𝑥𝑐,𝑎∨REL6𝑥𝑐,𝑎∨REL13𝑥𝑐,𝑎)](29a)

For part 𝑏₂, REL3𝑦𝑏₂,𝑎∧REL3𝑥𝑏₂∧𝑐,𝑎REL8𝑦₁,𝑏₂∧REL7𝑥𝑐₁,𝑏₂∨REL6𝑦𝑐₂,𝑏₂∧REL8𝑥𝑐₂,𝑏₂∨REL2𝑦𝑐₃,𝑏₂∧REL8𝑥𝑐₃,𝑏₂∨REL2𝑦𝑐₄,𝑏₂∧REL6𝑥𝑐₄,𝑏₂∨REL3𝑦𝑐₅,𝑏₂∧REL6𝑥𝑐₅,𝑏₂∨REL3𝑦𝑐₆,𝑏₂∧REL2𝑥𝑐₆,𝑏₂∨REL2𝑦𝑐₇,𝑏₂∧REL2𝑥𝑐₇,𝑏₂=()()()(∧[()()()()()()(.REL2𝑦𝑐,𝑎∨REL3𝑦𝑐,𝑎∨REL5𝑦𝑐,𝑎∨REL12𝑦𝑐,𝑎)REL2𝑥𝑐,𝑎∨REL3𝑥𝑐,𝑎∨REL4𝑥𝑐,𝑎∨REL5𝑥𝑐,𝑎∨REL7𝑥𝑐,𝑎∨REL8𝑥𝑐,𝑎∨REL12𝑥𝑐,𝑎)]](29b) The final outcome of the composition is the composition of part 𝑏₁ (29a) ∧ part 𝑏₂ (29b).

Apply Rule 3 from the earlier part of the paper, and we will get the following: ()()()(∧()()(∧[()()()()(∧[()()()(.REL2𝑦𝑐,𝑎∨REL3𝑦𝑐,𝑎∨REL6𝑦𝑐,𝑎∨REL13𝑦𝑐,𝑎)REL2𝑦𝑐,𝑎∨REL3𝑦𝑐,𝑎∨REL12𝑦𝑐,𝑎)REL2𝑥𝑐,𝑎∨REL3𝑥𝑐,𝑎∨REL4𝑥𝑐,𝑎∨REL8𝑥𝑐,𝑎∨REL12𝑥𝑐,𝑎)]REL2𝑥𝑐,𝑎∨REL3𝑥𝑐,𝑎∨REL6𝑥𝑐,𝑎∨REL13𝑥𝑐,𝑎)](30) We collapse some of the disjunction of relations: ()()()()()()()()()().REL6𝑦𝑐,𝑎∨REL13𝑦𝑐,𝑎=REL13𝑦𝑐,𝑎REL4𝑥𝑐,𝑎∨REL8𝑥𝑐,𝑎∨REL12𝑥𝑐,𝑎=REL12𝑦𝑐,𝑎REL6𝑥𝑐,𝑎∨REL13𝑥𝑐,𝑎=REL13𝑥𝑐,𝑎(31) Equation (30) becomes ()()(∧()()(∧[()()(∧[()()(.REL2𝑦𝑐,𝑎∨REL3𝑦𝑐,𝑎∨REL13𝑦𝑐,𝑎)REL2𝑦𝑐,𝑎∨REL3𝑦𝑐,𝑎∨REL12𝑦𝑐,𝑎)REL2𝑥𝑐,𝑎∨REL3𝑥𝑐,𝑎∨REL12𝑥𝑐,𝑎)]REL2𝑥𝑐,𝑎∨REL3𝑥𝑐,𝑎∨REL13𝑥𝑐,𝑎)](32) Region 𝑐 is single piece. Therefore, (32) becomes ((𝑎((𝑎((𝑎∧[((𝑎((𝑎((𝑎.PO𝑦𝑐,WeakNorth))∧PO𝑦𝑐,WeakSouth))∧NTPP𝑖𝑦𝑐,Horizontal))(33)PO𝑥𝑐,WeakEast))∧PO𝑥𝑐,WeakWest))∧NTPP𝑖𝑥𝑐,Vertical))](34) This means that the region 𝑐⊆𝑈which is the union of all the 9 tiles of region 𝑎. As mentioned earlier, based on Figure 9, region 𝑐⊄SW(𝑎). Thus, the outcome of the composition for weak relations (in the previous section) yields the same result as this composition. However, the computation for the latter is more tedious and complex when involving regions with many parts.

8. Existential Inference

The composition table in Table 2 is the result of transitive inferences made about regions 𝑎 and 𝑐, given the hybrid cardinal direction relations for regions 𝑎 and 𝑏 as well as regions 𝑏 and 𝑐. In the context of this paper, an existential inference is the inference made about the spatial relation between 𝑎 and 𝑏, given the relations between 𝑐 and 𝑎 or/and the given relations between 𝑐 and 𝑏. We shall demonstrate how our expressive hybrid cardinal direction model could be used to make several existential inferences which are not possible in existing models.

Example 3 (Find 𝑅(𝑏,𝑎) such that 𝑐⊂ WeakNorth(𝑏) and 𝑐⊂ WeakNorth(𝑎)). To answer this query, we must first specify the expressive relation between 𝑎 and 𝑐.
There are two possible relations: TPP𝑦(𝑐,WeakNorth(𝑎)) or WeakNorth(𝑐,𝑎). If it is the former then composition is WeakNorth(𝑏,𝑎) ∧ WeakNorth(𝑐,𝑏) which means R(𝑏,𝑎) is WeakNorth(𝑏,𝑎). If it is the latter, there are several combinations: (i) WeakNorth(𝑏,𝑎) ∧ Horizontal(𝑐,𝑏)(ii) WeakNorth(𝑏,𝑎) ∧ WeakSouth(𝑐,𝑏)(iii) Horizontal(𝑏,𝑎) ∧ WeakNorth(𝑐,𝑏)(iv) WeakSouth(𝑏,𝑎) ∧ WeakNorth(𝑐,𝑏).
This means 𝑅(𝑏,𝑎) are Horizontal(𝑏,𝑎) or WeakSouth(𝑏,𝑎) when 𝑐⊂ WeakNorth(𝑏) and 𝑐⊂ WeakNorth(𝑎).

Example 4 (Find 𝑅(𝑏,𝑎) and 𝑆(𝑐,𝑏) such that 𝑇(𝑎,𝑐) is ¬[TPP𝑦(c,Horizontal(𝑎)) ∧ EC𝑦(𝑐,WeakSouth(𝑎))]). Based on Table 2, 9 different compositions will yield the following outcome:
TPP𝑦(𝑐,Horizontal(𝑎)) ∧ EC𝑦(𝑐,WeakSouth(𝑎))
The set of possible compositions, 𝑄, is:
{REL1𝑦(𝑏,𝑎) ∧ REL7𝑦(𝑐,𝑏), REL2𝑦(𝑏,𝑎) ∧ REL7𝑦(𝑐,𝑏),
REL3𝑦(𝑏,𝑎) ∧ REL7𝑦(𝑐,𝑏), REL3𝑦(𝑏,𝑎) ∧ REL8𝑦(𝑐,𝑏),
REL5𝑦(𝑏,𝑎) ∧ REL7𝑦(𝑐,𝑏), REL5𝑦(𝑏,𝑎) ∧ REL8𝑦(𝑐,𝑏),
REL6𝑦(𝑏,𝑎) ∧ REL4𝑦(𝑐,𝑏), REL7𝑦(𝑏,𝑎) ∧ REL1𝑦(𝑐,𝑏),
REL8𝑦(𝑏,𝑎) ∧ REL12(𝑐,𝑏)}.
If 𝑈 equals 8 × 8 basic binary direction relations, then the set of all possible ordered pairs of 𝑅 and 𝑆 which satisfy the above query will be 𝑈−𝑄.

Example 5 (Find R(b,a) and S(c,b) such that T(a,c) is PO𝑦(c,WeakSouth(a)) ∧ PO𝑦(c,Horizontal(a)) ∧ EC𝑦(c,WeakNorth(a))). Based on Table 2, we have 4 pairs of 𝑅 and 𝑆 which satisfy 𝑇. They are: REL1y(b,a) ∧REL7y(c,b), REL2y(b,a) ∧ REL8y(c,b), REL7y(b,a) ∧ REL1y(c,b), REL7y(𝑏,𝑎) ∧ REL2y(𝑐,𝑏).

9. Conclusion

In this paper, we have shown how topological and direction relations can be integrated to produce a more expressive hybrid model for cardinal directions. The composition table derived from this model could be used to infer both weak and expressive direction relations between regions. We have also introduced and demonstrated how to use a formula to compute the composition of weak or expressive relations between “whole and part” regions. We have also demonstrated how the composition table with expressive direction relations could be used to make several difficult existential inferences.