When ancient Egyptians said that the width of the door was five Cubits, they meant that five connected objects (each has the size of âRoyal Egyptian. Cubitâ) can ...
Journal of Philosophical Logic (2007) DOI: 10.1007/s10992-007-9074-y
#
Springer 2007
TIANSI DONG
A COMMENT ON RCC: FROM RCC TO RCC++ Received 25 January 2006 ABSTRACT. The Region Connection Calculus (RCC theory) is a well-known
spatial representation of topological relations between regions. It claims that the connection relation is primitive in the spatial domain. We argue that the connection relation is indeed primitive to the spatial relations, although in RCC theory there is no room for distance relations. We first analyze some aspects of the RCC theory, e.g. the two axioms in the RCC theory are not strong enough to govern the connection relation, regions in the RCC theory cannot be points, the uniqueness of the operation in the theory is not guaranteed, etc. To solve
some of the problems, we propose an extension to the RCC theory by introducing the notion of region category and adding a new axiom which governs the characteristic property of the connection relation. The extended theory is named as RCCþþ . We support the claim that the connection relation is primitive to spatial domain by showing how distance relations, size relations are developed in RCCþþ . At last we revisit a sub-family of un-intended models in RCC theory, argue that RCCþþ is more suitable than RCC with regards to its original intended model, and discuss the representation limitation of the RCC, as well as RCCþþ . 1. INTRODUCTION The Region Connection Calculus (RCC) is first proposed in [22] and much progress has been made since then, [1, 2, 5, 6, 16, 17], etc. It is now accepted as a fundamental qualitative spatial representation in the community. In this section we review the basic theory of RCC. 1.1. The Formalism of the RCC Theory In RCC theory regions rather than points are taken as primitive units. It is assumed that regions are more natural than points to represent indefiniteness that is germane to qualitative representation and that spaces occupied by any real physical objects are always regions, [7]. The RCC theory has two axioms for the connection relation ‘C’ between regions: (1) for any region x, x connects with itself; (2) for any region x
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and y, if x connects with y, then y also connects with x. The two axioms are formalized as follows. 8x½Cðx; xÞ 8xy½Cðx; yÞ ! Cðy; xÞ Based on the two axioms, other spatial relations can be defined. For example, the relation that region x disconnects from region y, DCðx; yÞ, is defined as that x does not connect with y, i.e., :Cðx; yÞ. The relation that x is a part of y; Pðx; yÞ, is defined as that for any region z, if z connects with x, then z connects with y, i.e., 8z½Cðz; xÞ ! Cðz; yÞ. Other spatial relations between regions, see [22], are listed as follows and illustrated in Figure 1. That region x is a proper part of region y, PPðx; yÞ, is defined as x is a part of y and y is not a part of x, i.e., PPðx; yÞdef ¼ Pðx; yÞ ^ :Pðy; xÞ
That region x is identical to region y, EQðx; yÞ (or x ¼ y), is defined as x is a part of y and y is a part of x, i.e., EQðx; yÞdef ¼ Pðx; yÞ ^ Pðy; xÞ
That region x overlaps region y, Oðx; yÞ, is defined as there is region z such that z is a part of x and y, i.e., Oðx; yÞdef ¼ 9z ½Pðz; xÞ ^ Pðz; yÞ
That region x partially overlaps region y, POðx; yÞ, is defined as x overlaps and x is not a part of y, and y is not a part of x, i.e.,
y,
POðx; yÞdef ¼ ½Oðx; yÞ ^ :Pðx; yÞ ^ :Pðy; xÞ
x
y
x
DC (x, y) y
x
y
x
EC (x, y) x
y
x
y
y
PO (x, y) y
x
TPP -1(x, y) TPP (x, y) NTPP -1(x, y) NTPP (x, y)
x y EQ (x, y)
Figure 1. The eight relations between regions in RCC theory
A COMMENT ON RCC: FROM RCC TO RCC++
That region x externally connects with region y, ECðx; yÞ, is defined as x connects with y and x does not overlap y, i.e., ECðx; yÞdef ¼ ½Cðx; yÞ ^ :Oðx; yÞ
The relation that region x is a tangential proper part of region y, is defined with the following two conditions: (1) x is a proper (2) there is region z such that z externally connects with x and y, i.e.,
TPPðx; yÞ, part of y;
TPPðx; yÞdef ¼ ½PPðx; yÞ ^ 9z½ECðz; xÞ ^ ECðz; yÞ
That region x is a non-tangential proper part of region y, NTPPðx; yÞ, is defined with the following two conditions: (1) x is a proper part of y; (2) there is no region z such that z externally connects with x and y, i.e., NTPPðx; yÞdef ¼ ½PPðx; yÞ ^ :9z½ECðz; xÞ ^ ECðz; yÞ
The inverse of TPPðx; yÞ is written as TPP1 ðx; yÞ, i.e., TPP1 ðx; yÞdef ¼ TPPðy; xÞ
The inverse of NTPPðx; yÞ is written as NTPP1 ðx; yÞ, i.e., NTPP1 ðx; yÞdef ¼ NTPPðy; xÞ
1.2. The Intended Models of RCC Theory The RCC theory is based on Clarke’s theory, [3, 4]. Its novelty is the interpretation of the connection relation. In Clarke’s theory that two regions x and y are connected means that x and y share a common point. This interpretation leads to the fact that region x does not connect with the complement of itself. In RCC theory this is taken as a violation of the commonsense understanding of space. Accordingly, RCC theory changes the interpretation of the connection relation as follows: the relation that two regions x and y are connected means that topological closures of x and y share a common point. Therefore, in RCC theory region x connects with the complement of itself. This connection relation is weaker than that of Clarke’s theory. In RCC theory when two regions connect, either they share a common point, or their closures share a common point, or the distance between them is zero, [8]. As a fundamental theory of spatial relations for extended objects, a question is raised: Is there room for other spatial relations in the RCC theory? For example, can we use the connection relation in the RCC theory to define
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distance relations, if it is indeed the primitive relation in the spatial domain? The rest part is structured as follows: Section 2 comments on the RCC theory: Section 2.1 argues that regions cannot be points; Section 2.2 shows region equivalent classes which is hidden in the theory; Section 2.3 presents a mystery of the operator used in the theory; Section 2.4 shows an unintended region of the theory; Section 2.5 argues that there is no room for distance relations; Section 2.5.1 presents an un-intended model of RCC; Section 2.5.2 shows a family of models of RCC. Section 3 proposes a possible extension to RCC: RCCþþ : Section 3.1 presents an understanding of distance relations; Section 3.2 introduces the first extension to RCC: the notion of region category; Section 3.3 presents an understanding of the connection relation; Section 3.4 introduces an axiom which characterizes the connection relation; Section 3.5 introduces the descriptor which fits for the region reference under the interpretation of the connection relation; Section 3.6 describes the intended region of the theory; Section 3.7 introduces the notion of near extension; Section 3.8 develops size relations between regions; Section 3.9 develops the distance relation between regions; Section 3.10 argues that orientation relations can be understood as distance comparison relations. Section 4 visits models of RCCþþ : Section 4.1 revisits a family of un-intended models of RCC discussed in Section 2.5.1; Section 4.2 argues RCCþþ is more suitable than RCC with regards to the intended model; Section 4.3 shows the representation limitations the RCC theory.
2. SOME COMMENTS
ON THE
RCC THEORY
2.1. Points are not Regions Although the interpretation of the connection relation requires the notion of point, a region in RCC theory cannot be a point. We examine definitions of O and EC to show this. That two regions overlap (O) is defined as that there exists a region which is part of both of the two regions (shared by the two regions). That two regions are externally connected (EC) is defined as that they are connected and do not overlap. That is, their closures share a point and there is no region which is shared by both of the regions. This follows that for every region it is not shared by them. If RCC theory allows a region to be a point, this will lead to the fact that for every point it is not shared by the two regions. This is equivalent to that the two regions are disconnected. That is, if a region can be a point in RCC theory, the definition of O will make the definition of EC self-contradictory. Therefore, a region in the RCC theory cannot be a point. And further more, the concept of a point, if exists in the RCC theory, shall not be a region.
A COMMENT ON RCC: FROM RCC TO RCC++
2.2. Region Equivalent Classes It is easy to prove that two regions x and y are identical, EQðx; yÞ or x ¼ y, in the RCC theory, if and only if the following formula holds 8z½Cðz; xÞ $ Cðz; yÞ
As Cðx; yÞ is interpreted as closures of x and y share a common point, two regions are identical if and only if their closures are coincided. For example, in 2-D Euclidean space, region x2 þ y2 4 and region x2 þ y2
