Syllogistic Reasoning for Cardinal Direction Relations

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1 Syllogistic Reasoning for Cardinal Direction Relations Ah Lian Kor 1,and Brandon Bennett 2 1 School of Computing, Creative Technologies, and Engineering, Faculty of Arts, Environment and Technology, Leeds Metropolitan University, Headingley Campus, LS6 3QS, Leeds, UK. 2 School of Computing, University of Leeds, Woodhouse Lane, LS2 9JT, Leeds, UK. A.Kor@leedsmet.ac.uk, Brandon@comp.leeds.ac.uk Abstract - Syllogistics is a type of logical reasoning which involves quantifiers such as All, and Some. Here, we explore the use of syllogisms to formalize quantified direction relations by incorporated them into the classical Projection-based Model for cardinal directions [Frank,1992], a two-dimensional Euclidean space relative to an arbitrary single-piece region, a, is partitioned into the following nine tiles: North-West, nw(a); North, n(a); North-East, ne(a); West, w(a); Neutral Zone, o(a); East, e(a); South-West, sw(a); South, s(a); and South-East, se(a). Typically, only these tiles are employed for reasoning about cardinal direction relations [Ligozat,1988; Goyal and Egenhofer, 2000;.Skiadopoulos and Koubarakis, 2001,2004]. Chen et. al (2007) have integrated cardinal direction relations and RCC-5 and RCC-8 to represent directional and topological information. They have investigated the mutual dependencies them, and discussed the composition of the hybrid relations. However, in this paper, we have developed formalisms to facilitate an expressive reasoning mechanism for spatial databases. These formalisms is the outcome of the integration of RCC-5 [Cohn, et.al,1997] for topological relations, the Projectbased Model for cardinal direction relations and syllogisms. In the latter part of the paper, we demonstrate how to compute the composition of such hybridized cardinal direction relations. Keywords: Syllogistic reasoning, cardinal direction, composition table, formal reasoning, quantifier, knowledge representation 1.Introduction According to Egenhofer and Mark (1995), Naïve Geography encompasses formal but yet commonsense models which form the basis for the design of Geographic Information Systems. They view qualitative spatial and temporal reasoning as the crux of Naïve Geography because it deals with partial information and helps simplify spatial queries in GIS [ibid; Frank, 1992] that is prevalent in spatial applications. Little work has been done on the formalization and efficient processing of qualitative reasoning for cardinal directions [Theorodis, et. al. 1998] which describe relative positions of regions in large-scale spaces. 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 north-west, north-east, etc. are also often used. The models for reasoning with cardinal directions are the cone-shaped, projection-based models [Frank, 1992], and relative orientation model [Freksa, 1992]. Isli [2004] has developed an expressive model by combining these two models. Papadias and Theorodis [1997] 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 [Goyal and Egenhofer, 2000]. Some work has been done on hybrid direction models. Qualitative orientation has been integrated with distance to obtain positional information [Escrig et.al, 1998; Clementini et.al,1997; Moratz et. al, 2012]. Gerevini and Renz [2002] combined topological and metric size information for spatial reasoning. Sharma and Flewelling [1995] infer spatial relations from integrated topological and cardinal direction relations and Sharma et. al [1994] has developed a prototype known as Qualitative Spatial Reasoner which integrates the two and also approximate distances. Kor and Bennett [2003, 2020, in press] have developed expressive models cardinal direction relations. Liu et. al [2009] analyzed the interaction between topological and directional constraints for extended spatial regions. They have shown that the JSP over basic RCC-8 and basic Rectangle Algebra (RA and note that it is a two-dimensional counterpart of Interval Algebra) networks can be solved in polynomial time while the JSP over basic RCC-8 and basic Cardinal Direction Calculus (CDC) networks is NP-Complete. On the other hand, Li [2006, 2007] combined RCC-5 and RCC-8 with DIR9 (employed to express directional information and is a subclass of RA) respectively. He discussed the Constraints Satisfaction Problem (CSP) and also the reasoning complexity. Chen, et. al [2007] also investigated the mutual dependencies between basic relations of RCC-5 and cardinal direction relations, enhanced the constraint propagation algorithm to enforce path consistency and examining the consistency of the hybrid constraints. Quantifiers have been applied to temporal reasoning [Pratt and Francez, 1998] but not cardinal directions. Thus, our novel contribution is the integration of topological binary relations in RCC-5, direction relations and syllogisms. Syllogism is a form of deductive reasoning (Bucciarelli & Johnson-Laird, 1999) which comprises a small set of inferences that are based on premises which conatin quantifiers (All, Some, and No). We have employed the composition technique making the inferences and also used visualization to confirm the validity of the syllogistic reasoning for cardinal directions. 2.Cardinal Direction Relations Model In the projection-based model for cardinal directions, a two-dimensional Euclidean space of an arbitrary singlepiece region, a, is partitioned into nine tiles as shown in Figure 1. They are North-West, nw(a); North, n(a);

2 North-East, ne(a); West, w(a); Neutral Zone, o(a); East, e(a); South-West, sw(a); South, s(a); and South-East, se(a). 5. Syllogistic Reasoning Model for Direction Relations In this paper, we shall demonstrate how the premises shown in Table 1 could be integrated with RCC-5 relations in Table 2 to facilitate reasoning about direction relations. 1. All b is in n(a) is represented by: [in(b, n(a))] pp(b,n(a)) eq(b,n(a)) Several possible graphical representations are in Figure 2. Figure 1. A Projection-based Cardinal Direction Model 3.Syllogistic Reasoning Syllogistic reasoning syllogistic reasoning is viewed as a form of deductive reasoning which involves reasoning with a pair of premises containing quantifiers All, and Some (Bucciarelli and Johnson-Laird, 1999; Johnson- Laird, 1983). Syllogisms were first analysed by Aristotle and the four types of premises are: All x is y (a universal and affirmative assertion); Some x is y (a particular and affirmative assertion); No x is y (a universal and negative assertion); and Some x is not y (a particular and negative assertion). We shall employ the quantifiers All, and Some to extend existing spatial language for expressing direction relations. Johnson-Laird [1983] used Euler Circles to represent the 4 types of assertions diagrammatically (Table 1). Table 1. Diagrammatic Representation for Assertions with the Quantifiers All, and Some 4.Formal Representation for topological relations We shall first begin with the parthood relation, p(x,y) which means x is a part of y. It is the primitive for 'x overlaps y' and most of the RCC-5 [Cohn et.al, 1997] JEPD binary topological relations for regions. A diagrammatic representation of the RCC-5 topological relations is shown in Table 2. D1.ovr(x,y) def z [ p(z,x) p(z,y) ] D2.eq(x, y) def [ x=y ] D3.pp(x, y) def [ p(x,y) eq(x,y) ] D4.ppi(x, y) def [ p(y,x) eq(x,y) ] D5.po(x, y) def [ οvr(x,y) p(x,y) p(y,x) ] D6.dr(x,y) def [ ovr(x,y) ] Figure 2. Several Graphical Representations for 'All b is in n(a)' Assume that a region is represented by x and any tile of the reference region is y. All x is in y is represented by: [in(x,y)] pp(x,y) eq(x,y) 2. Some b is in n(a) is represented by: [in(b, n(a))] eq(b,n(a)) pp(b,n(a)) po(b,n(a)) ppi(b,n(a)) The following graphical representation is not exhaustive. Figure 3. Graphical Representation for 'Some b is in n(a)' Assume that a region is represented by x and any tile of the reference region is y. Some x is in y is represented by: [in(x,y)] pp(x,y) eq(x,y) po(x,y) ppi(x,y) 3. No b is in n(a) can be represented as: [ in(b, n(a))] dr(b,n(a)) or [in(b, n(a))] dr(b,n(a)) The non-exhaustive graphical representation is as below: Figure 4. Graphical Representation for 'No b is in n(a)' No x is in y is represented by: [ in(x,y)] dr(x,y) or [in(x,y)] dr(x,y) 4. Some n(a) is not in b can be defined as: [ in(b, n(a))] dr(b,n(a)) po(b,n(a)) ppi(b,n(a)) The non-exhaustive graphical representation is as follows: Table 2. Diagrammatic Representation for RCC-5 relations Figure 5. Graphical Representation for 'Some n(a) is not in b ' Some x is not in y is represented by: [ in(x,y)] dr(x,y) po(x,y) ppi(x,y) Assume R = {n(a), ne(a), nw(a), w(a), o(a), e(a), s(a), se(a), sw(a)}. The general definitions of the above mentioned assertions are as follows:

3 D7.All b is in a tile t where t R (a tile of a region a) is represented as: [in(b, t)] pp(b,t) eq(b,t) D8.Some b is in a tile t where t R (a tile of a region a) is represented as: [in(b, t)] pp(b,t) eq(b,t) po(b,t) ppi(b,t) D9.No b is in a tile t where t R (a tile of a region a) is represented as: [ in(b, t)] dr(b,t) or [in(b, t)] dr(b,t) D10. Some b is not in a tile t where t R (a tile of a region a) is represented as: [ in(b, t)] dr(b,t) po(b,t) ppi(b,t) Assume U is a universal set of the RCC-5 relations between a region b and a tile t of region a. Thus, U={dr(b,t),pp(b,t) eq(b,t) po(b,t) ppi(b,t)}. Based on definitions D7 to D10 above, i. Some b or No b is in a tile t where t R (a tile of a region a) is represented as: [in(b,t)] [in(b,t)] pp(b,t) eq(b,t) po(b,t) ppi(b,t) dr(b,t) U ii. Some b is not or All b is in a tile t where t R (a tile of a region a) is represented as: [ in(b, t)] [in(b, t)] pp(b,t) eq(b,t) po(b,t) ppi(b,t) dr(b,t) U Thus, the quantifiers Some b is a complement of No b and vice versa. On the other hand, Some b is not is a complement of All b and vice versa. 6.Composition Direction Relations Based on the Syllogistic Reasoning Model Skiadopoulos and Koubarakis [2004] have provided a formal presentation of the composition of cardinal direction relations based on Goyal and Egenhofer s direction model [2001]. In our previous paper [Kor and Bennett, 2010], we have developed a formula to compute the composition of weak direction relations between whole and part extended regions. Chen et. al [2007] have discussed the composition of RCC-5 topological relations and cardinal direction relations. However, in this paper, we have integrated RCC-5, cardinal direction relations and syllogism. We shall first start by establishing the possible topological relation/s between each individual tile of b and that particular tile of a when given a topological relation between region b and a tile of region a. Such relations have been tabulated in Table 3 (note the table is in three parts).

4

5 Bucciarelli et.al [1999] has composed quantified relations (known as syllogistic reasoning in Cognitive Science) between two general entities. However, the procedure for composing two quantified direction relations is quite different. We use the generated topological relations between the tiles of regions a and b in Table 3 and the composition table (Table 4) for the RCC-5 relations [Bennett, 1994], to compute the composition of quantified cardinal direction relations. We shall verify the outcome of the composition diagrammatically. Table 4. Composition Table for RCC-5 Relations [Bennett, 1994] 6.1 Examples of Syllogistics Reasoning about Hybrid Cardinal Direction Relations Example 1. All b are in ne(a) and All c are in ne(b) Use D7, and the boolean algebraic expression for the above composition is as follows: [in(b, ne(a))] [in(c, ne(b))] [pp(b, ne(a)) eq(b, ne(a))] [pp(c, ne(b)) eq(c, ne(b)) (1) Given the relations pp(b, ne(a)) and eq(b, ne(a)), use Table 3 to establish the relationship between ne(b) and ne(a) in order to make the appropriate transitive inferences. Based on the information in Table 3, substitute the following into Equation (1): pp(b, ne(a)) with pp(ne(b),ne(a)); and eq(b,ne(a)) with pp(ne(b),ne(a)). After substitution, Equation (1) will be as follows: [pp(ne(b),ne(a)) pp(ne(b),ne(a))] [pp(c,ne(b)) eq(c,ne(b))] (1.1) Use Idempotent Law for union, and Equation (1.1) [pp(ne(b),ne(a))] [pp(c, ne(b)) eq(c, ne(b))] (1.2) Use the Distributive Law and Equation (1.2) is expanded as follows: [pp(ne(b),ne(a))] [pp(c,ne(b)) eq(c,ne(b))] [pp(ne(b),ne(a)) pp(c,ne(b))] [pp(ne(b),ne(a)) eq(c,ne(b))] (1.3) Use the composition table in Table 4 and Equation (1.3) [pp(c, ne(b))] [pp(c, ne(b))] (1.4) [pp(c, ne(b))] (1.5) {note: use Idempotent Law for union again} Figure 6: All b is in ne(a) and All c is in ne(b) Compare the relation in Equation (1.5) with the aforementioned assertion D7 and we obtain this quantified relation: All c is in the ne(a) tile. When we compare the relation with assertion D8, then we have the following conclusion: Some c is in the ne(a) tile. This is confirmed by the diagrammatic representation in Figure 6.

6 Example 2. All b are in se(a) and All c are in n(b) Use D7, and the boolean algebraic expression for the above composition is as follows: [in(b, se(a))] [in(c, n(b))] [pp(b, se(a)) eq(b, se(a))] [pp(c, n(b)) eq(c,n(b))] (2) Given the relations pp(b, se(a)) and eq(b, se(a)), use Table 3 to establish the relationship between n(b) and se(a) in order to make the appropriate transitive inferences. Based on the information in Table 3, substitute the following into Equation (2): pp(b, se(a)) with {po dr}(n(b),se(a)); and eq(b, se(a)) with {po dr}(n(b),se(a)). After substitution, Equation (2) will be as follows: [{po dr}(n(b),se(a)) {po dr}(n(b),se(a))] [pp(c,n(b)) eq(c,n(b))] (2.1) Use Idempotent Law for union, and Equation (2.1) [{po dr}(n(b),se(a))] [pp(c, n(b)) eq(c, n(b))] (2.2) [po(n(b),se(a)) dr(n(b),se(a))] [pp(c, n(b)) eq(c, n(b))] Use the Distributive Law and Equation (2.2) is expanded as follows: {[po(n(b),se(a))] [pp(c, n(b)) eq(c, n(b))]} {[dr(n(b),se(a))] [pp(c, n(b)) eq(c, n(b))]} {[po(n(b),se(a)) pp(c,n(b))] [po(n(b),se(a)) eq(c, n(b))]} {[dr(n(b),se(a)) pp(c,n(b))] [dr(n(b),se(a)) eq(c, n(b))]} (2.3) Use the composition table in Table 4 and Equation (2.3) {po pp}(c,se(a)) po(c,se(a)) {dr po pp}(c,se(a)) dr(c,se(a)) (2.4) {dr po pp}(c,se(a)) (2.5) {note: use Idempotent Law for union again} The outcome of the composition is {(dr po pp)}(c,se(a)). Compare the relation with the aforementioned assertions (D7 to D10) and the following conclusions hold: All c is in se(a); Some c is in se(a); No c is in se(a); Some c is in not in se(a);. This is confirmed by the diagrammatic representation in Figure 7 (note: the representation for each category is not exhaustive). Figure 7. All b is in se(a) and All c is in n(b) Example 3. No b are in w(a) and All c are in n(b) Use D7 and D9, and the boolean algebraic expression for the above composition is as follows: [ in(b, w(a))] [in(c, n(b))] [dr(c, w(b))] [pp(c, n(b)) eq(c, n(b))] (3) Given the relation dr(b, w(a)), use Table 3 to establish the relationship between n(b) and w(a) in order to make the appropriate transitive inferences. Based on the information in Table 3, substitute the following into Equation (3): dr(b, w(a)) with {po dr ppi}(n(b),w(a)). After substitution, Equation (3) will be as follows: [{po dr ppi}(n(b),w(a))] [pp(c, n(b)) eq(c, n(b))] (3.1) [po(n(b),w(a)) dr(n(b),w(a)) ppi(n(b),w(a))] [pp(c, n(b)) eq(c, n(b))] (3.2) Use the Distributive Law and Equation (3.2) is expanded as follows: [po(n(b),w(a))] [pp(c, n(b)) eq(c, n(b))] [dr(n(b),w(a))] [pp(c, n(b)) eq(c, n(b))] [ppi(n(b),w(a))] [pp(c, n(b)) eq(c, n(b))] (3.3) {[[po(n(b),w(a))] [pp(c,n(b))]] [[po(n(b),w(a))] [eq(c,n(b))]] {[[dr(n(b),w(a))] [pp(c,n(b))]] [[dr(n(b),w(a))] [eq(c,n(b))]]} {[[ppi(n(b),w(a))] [pp(c,n(b))]] [[ppi(n(b),w(a))] [eq(c,n(b))]]} (3.4) Use the composition table in Table 3 and Equation (3.4) {po pp}(c, w(a)) po(c, w(a)) {dr po pp}(c, w(a)) dr(c, w(a)) ovr(c, w(a)) ppi(c, w(a)) (3.5) Use Idempotent Law for union, and Equation (3.5) {po pp dr ppi}(c, w(a)) ovr(c, w(a)) (3.6) Based on D.1, the topological relationship ovr(c, w(a)) {po pp eq ppi}(c, w(a)). Substitute this into Equation (3.6), use Idempotent Law for union again, and Equation (3.6) {po pp eq dr ppi}(c, w(a)) {note this is the universal set of relations for RCC-5} Compare the relation with D7 to D10 and we obtain the following quantified relations: union of D7 and D10 Some c is not or All c is in w(a) tile; union of D8 and D9 Some c is or No c is in w(a) tile. This concurs with the diagrammatic representation in Figure 8 (note: the representation for each category is not exhaustive). Figure 8. No b is in w(a) and All c is in n(b) 7.Conclusions In this paper, we have employed RCC-5 and quantifiers (All and Some) to formalize quantified direction relations. We have developed a procedure for computing the composition of such quantified direction relations. The correctness of the outcome of the composition is further verified diagrammatically. However, based on the worked out examples, the conclusions derived from the inferences made are rather weak. 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