From the Egg-Yolk to the Scrambled-Egg Theory

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1 From: FLIRS-02 Proceedings. Copyright 2002, I ( ll rights reserved. From the Egg-olk to the Scrambled-Egg Theory Hans W. Guesgen Computer Science Department, University of uckland Private Bag 92019, uckland, New Zealand hanscs.auckland.ac.nz bstract The way we deal with space in many everyday situations is on a qualitative basis, allowing for imprecision in spatial descriptions when we interact with each other. This is often achieved by specifying the spatial relations between the objects or regions that we talk about. In artificial intelligence, a variety of formalisms have been developed that deal with space on the basis of relations between objects or regions that objects might occupy. One of these formalisms is the RCC theory, which is based on a primitive relation, called connectedness, and uses a set of topological relations, defined on the basis of connectedness, to provide a framework to reason about regions. This paper discusses an extension of the RCC theory, which deals with vagueness in spatial representations. The extension is based on the egg-yolk theory, but unlike the egg-yolk theory uses fuzzy logic to express vagueness. Motivation There are many ways of dealing with spatial information, but researchers have been arguing successfully that human beings often do so in qualitative way. Therefore, computer systems should support such a form of reasoning (see, e.g., (Hernández 1991)). In (Guesgen & Hertzberg 1993), for example, we introduce a form of spatial reasoning that extends llen s temporal logic (llen 1983) to the three dimensions of space by applying very simple methods for constructing higher-dimensional models and for reasoning about them, namely combination (i.e., building tuples of one-dimensional relations) and projection (i.e., extracting one-dimensional aspects from the tuples). There are other approaches that proceed in more or less the same way(freksa 1990; Hernández 1991; Mukerjee & Joe 1990). However, it seems that more recently the RCC theory (Randell, Cui, & Cohn 1992) has gained a particular interest in the research community. This first-order theory is based on aprimitive relation, called connectedness, and uses eight topological relations, defined on the basis of connectedness, to provide a framework to reason about regions. The original RCC theory has been designed to deal with precisely defined regions, but later on has been extended to cope with vagueness in spatial representations, in particularvague or indeterminate boundaries (Cohn & Gotts 1996). Copyright cfl 2002, merican ssociation for rtificial Intelligence ( ll rights reserved. Point in the region Vague point Point outside the region Figure 1: vague region represented by two crisp regions (the egg and the yolk). The extension, called the egg-yolk theory, uses two crisp regions, the egg and the yolk, to characterize a vague region. ll points within the yolk are considered to be in the region, whereas all points outside the egg are outside the region. The white characterizes the pointsthatmayor may not belong to the region (see Figure 1). In this paper, we introduce an extension of the egg-yolk theory, called the scrambled-egg theory. The idea is to utilize the neighborhood structure that is inherent in the RCC theory and to define fuzzy sets for the relations between regions based on this neighborhood structure. Rather than distinguishing between points that are definitely in the region (the yolk) and those that are possibly in the region (the white), we mix the two sets of points into one region (the scrambledegg). Our approach is to some degree related to the one described in (Egenhofer & l-taha 1992), which uses the concept of 9-intersection instead of the RCC theory as basis. Egenhofer and l-tahu do not use fuzzy logic, but they utilize astructure similar to the RCC neighborhood structure, called the closest topological relationship graph, to deal with indeterminate boundaries. The paper is organized as follows. We start with a brief review of the RCC theory and its extension to regions with indeterminate boundaries. We then show a way of associating fuzzy sets with the relations in the RCC theory, which can be viewed as an alternative for representing relations between regions with indeterminate boundaries. Finally, we will sketch an algorithm to reason about these fuzzy sets. The RCC Theory Revisited The idea of using relations to reason about spatio-temporal information dates back at leasttothebeginning of the eight- 476 FLIRS 2002

2 y d y y 6 Φ Φ Φ Φ x d x - x Figure 2: The relations between two rectangles with respect to the x-axis and y-axis, where d denotes the llen relation during. ies, when llen (1983) introduced an interval logic for reasoning about relations between time intervals. lthough llen s logic can be used to reason about three-dimensional space (Guesgen 1989), it often leads to counterintuitive results, inparticular if rectangular objects are not aligned to the chosen axes (see Figure 2). The RCC theory (Randell, Cui, & Cohn 1992) avoids this problem by defining the relation between two regions based on their topological properties and therefore independently of any coordinate system. The basis of the RCC theory is the connection relation, which is a reflexive and symmetric relation, satisfying the following axioms: 1. For each region : C(; ) 2. For each pair of regions, : C(; )! C(; ) From this relation, additional relations can be derived, which include the eight jointly exhaustive and pairwise disjoint RCC8 relations: 1 RCC8 = fdc; EC; PO; EQ; TPP; TPPi; NTPP; NTPPig Reasoning about space is achieved in the RCC theory by applying a composition table to pairs of relations, similar to the composition table in llen s logic. Given the relation R 1 between the regions and,andtherelation R 2 between the regions and Z, the composition table determines the relation R 3 between the regions and Z,i.e., R 3 = R 1 ffir 2. In the case of a set of regions with more than three regions, the composition table can be applied repeatedly to three-element subsets of until no more relations can be updated, resulting in a set of relations that is locally consistent. This is basically the idea behind llen s algorithm. Imprecision in Spatial Relations Reasoning about space often has to deal with some form of imprecision. For example, when we talk about a region like 1 See Figure 3 for an interpretation and a graphical illustration of the RCC8 relations. Relation/Interpretation Illustration DC(,) disconnected from EC(,) externally connected to PO(,) partially overlaps EQ(,) identical with χfi TPP(,) tangential proper part of χfi TPPi(,) tangential proper part of χfi NTPP(,) nontangential proper part of χfi NTPP(,) nontangential proper part of Figure 3: n illustration of the RCC8 relations. the city of uckland, we usually do not know exactly where the boundaries are for that regions. Nevertheless, we are perfectly capable to reason about such a region. Or if we hear on the radio that a cold front is moving in from ntarctica, we can estimate when this front connects to New Zealand, although we might not be able to decide with certainty whether the cold front is still disconnected from (DC), externally connected to (EC), or already partially overlapping (PO)NewZealand. Lehmann and Cohn (1994) have introduced an extension to the RCC theory, called the egg-yolk theory, which deals with imprecision in spatial representations by using two crisp regions to characterize an imprecise region. One of these regions is called the yolk, the other one the egg. ll points within the yolk are considered to be in the region, whereas all points outside the egg are outside the region. FLIRS

3 The white (i.e., the egg without the yolk) characterizes the points that may or may not belong to the region. The egg-yolk theory uses a set of five base relations, called RCC5, instead of the eight base relations in RCC8: RCC5 = fdr; PO; EQ; PP; PPig Given two imprecise regions e and e,thercc5 relations are used to describe the relationship between (1) the egg of e and the egg of e,(2)theyolk of e and the yolk of e,(3) the egg of e and the yolk of e,and(4)the yolk of e and the egg of e,resulting in 46 possible relationships between e and e. s Lehmann and Cohn point out, it is possible to use more than two regions to describe an imprecise region. We follow this idea here and combine it with an approach that we used before to introduce imprecise reasoning into llen s logic. The approach is based on the concept of conceptual neighborhoods, which was first introduced by Freksa (1992) for llen relations and later applied to the RCC theory (Cohn et al. 1997; Cohn & Gotts 1996). Conceptual Neighborhoods and Fuzzy Sets Two relations on regions and are conceptual neighbors if the shape of or can be continuously deformed such that one relation is transformed into the other relation without passing through a third relation. Figure 4 shows the conceptual neighbors for the RCC8 relations. The notion of conceptual neighbors can be used to introduce imprecision into reasoning about spatial relations (Guesgen & Hertzberg 1996). For that purpose, we first represent each RCC8 relation by a characteristic function as follows: μ R : RCC8! f0; 1g The function yields a value of 1 if and only if the argument is equal to the RCC8 relation denoted by the characteristic function: ρ 1; if μ R (R 0 R 0 = R )= The next step towards the introduction of imprecision is to transform the RCC8 relations into fuzzy sets. fuzzy set R ~ of a domain D is a set of ordered pairs, (d; μ ~R (d)), where d is an element of the underlying domain D and μ ~R : D! [0; 1] is the membership function of R.Inother ~ words, instead of specifying whether an element d belongs to a subset R of D or not, we assign a grade of membership to d. The membership function replaces the characteristic function of a classical subset of D. In the context of the RCC8 relations, this means that each RCC8 relation is represented as a set of pairs, each pair consisting of an element of RCC8 (which is the underlying domain) and the value of the characteristic function of the relation applied to that element. For example, if two regions and are externally connected (i.e., EC(; )), we use the characteristic function of the relation EC to convert this statement into the following: f(r;μ EC (R)) j R 2 RCC8g(; ) = f(ec; 1); (DC; 0); (PO; 0);:::g(; ) Instead of having two classes, one with the accepted relations where μ EC results in 1 and another with the discarded relations where μ EC results in 0, we now assign acceptance grades (or membership grades, to use the term from fuzzy set theory) with the relations. If the relation is EC, weas- sign the membership grade 1; if the relation is a neighbor of EC, wechoose a membership grade ff 1 with 1 ff 1 0; if therelation is a neighbor of a neighbor of EC, weassign a grade ff 2 with ff 1 ff 2 0; andso on. Since there is no general formula for determining the membership grades ff 1 ;ff 2 ;:::;,choosing the right grade for each degree of neighborhood can be a problem. On the other hand, there are experiments showing that fuzzy membership grades are quite robust, which means that it is not necessary to have precise estimations of these grades (Bloch 2000). The explanation given for this observation is twofold: first, fuzzy membership grades are used to describe imprecise information and therefore do not have to be precise, and second, each individual fuzzy membership grade plays only aminor role in the whole reasoning process, as it is usually combined with several other membership grades. If the membership grades are combined using the min/max combination scheme, as it is the caseintherest of this paper, we do not even need numeric values for the alphas. In this case, reasoning can be performed on symbolic values, provided that there is a total order on the alphas. Non-atomic RCC8 relations (i.e., disjunctions of RCC8 relations) can be transformed into fuzzy RCC8 relations by using the same technique. non-atomic RCC8 relation is givenbya set ofatomic RCC8 relations, whichis interpreted in a disjunctive way. We therefore transform each atomic relation in the set intoafuzzy RCC8 relation and compute the fuzzy union of the resulting sets. There are different ways of computing the union of fuzzy sets. Here, we choose the one introduced in (Zadeh 1965), which associates with each element in the resulting fuzzy set the maximum of the membership grades that the element has in the original fuzzy sets. Formally, a fuzzy RCC8 relation R e can be defined by using a function that denotes the conceptional distance between the relation R and a relation R 0,i.e., results in 1 if R is aneighbor of R 0,in2ifRis aneighbor of a neighbor of R 0,andso on: :RCC8 RCC8! f0; 1; 2;:::g can be defined recursively as follows: 1. If R = R 0,then (R; R 0 )=0 2. Otherwise, (R; R 0 )=minf (R; R 00 )+1j R 00 neighbor of R 0 g Given a sequence of membership grades, 1=ff 0 ff 1 ff 2 0,thefunction can be used to associate RCC8 relations with membership grades, depending on some given RCC8 relation R (see Figure 5 for an example). In particular, we can define a membership function μ as follows: er μ er : RCC8! [0; 1] μ er (R0 )=ff (R;R 0 ) 478 FLIRS 2002

4 DC(; ) EC(; ) PO(; ) TPP(; χfi ) NTPP(; χfi ) χfi EQ(; ) χfi TPPi(; ) NTPPi(; ) Figure 4: The RCC8 relations arranged in a graphs showing the conceptual neighbors. ff χfi 2 ff χfi 3 ff 2 ff 1 1 ff 1 χfi χfi ff 2 ff 3 Figure 5: The assignment of membershipgrades to the RCC8 relations with EC(; ) as referencerelation. With this definition, the fuzzy RCC8 relation e R of a relation R 2 RCC8 is given by the following: er = f(r 0 ;μ er (R0 )) j R 0 2 RCC8g We now extend the formulation of RCC8 relations as characteristic functions to the composition of RCC8 relations, starting again with crisp relations and continuing with fuzzy relations. In the crisp case, the composition table can be represented as a set of characteristic functions of the following form: μ R1ffiR2 : RCC8! f0; 1g The function yields a value of 1 for arguments that are elements of the corresponding entry in the composition table; otherwise, a value of 0: μ R1ffiR2 (R) =ρ 1; if R R 1 ffi R 2 For example, if R 1 = EC and R 2 = TPPi, thenthe characteristic function of the relation R 1 ffi R 2 = EC ffi TPPi = fec; DCg is defined as follows: ρ 1; if R 2fEC; DCg μ ECffiTPPi (R) = dopting the min/max combination scheme from fuzzy set theory, we can now define the fuzzy composition e R1 ffi e R2 of two fuzzy RCC8 relations e R1 and e R2 as the following fuzzy RCC8 relation: f(r;μ (r)) j R 2 RCC8g er1ffier2 where μ is given by the following: er1ffier2 μ (r) = max fminfμ er1ffier2 er1 (R0 1);μ er2 (R0 2)gg R 0 1;R 0 22RCC8 μ R 0 1 ffir0 2 (r)=1 The fuzzy composition of relations plays a central role in a number of algorithms for reasoning about fuzzy RCC8 FLIRS

5 relations. One of these algorithms is an llen-type algorithm for computing local consistency in networks of fuzzy RCC8 relations. Input to this algorithm is a set of regions and a set of (not necessarily atomic) fuzzy RCC8 relations. The aim of the algorithm is to transform the given relations into a set of relations that are consistent with each other. This is achieved through an iterative process that repeatedly looks at three regions,,andz,andtheirfuzzy relations er 1 (; ), e R2 (; Z),ande R3 (; Z),computes the composition of two of the relations, and compares the result with the third relation: er 3 (; Z) ψ e R3 (; Z) [e R1 (; ) ffi e R2 (; Z)] Unlike llen s original algorithm, the fuzzy version of the algorithm does not make a yes/no decision about whether a relation is admissible or not, but computes a new membership grade for that relation. The new membership grade is compared with the initial membership grade of the relation. If the new grade is smaller than the initial grade, the membership grade of the relation is updated with the new grade. Summary This paper introduces an extension of the RCC theory, which is based on work done by Bennett, Cohn, Gooday, Gotts, and others at the University of Leeds. The main idea is to associate fuzzy sets with the relations in the RCC theory, utilizing the notion of conceptual neighborhoods. Unlike the egg-yolk theory, our approach encodes vagueness in the relations between regions, rather than the regions themselves. In other words, we do not distinguish between egg white and yolk, but view each region as a scrambled region that has a set of relations, each to a certain degree, with another region. Freksa, C Temporal reasoning based on semiintervals. rtificial Intelligence 54: Guesgen, H., and Hertzberg, J constraint-based approach to spatiotemporal reasoning. pplied Intelligence (Special Issue on pplications of Temporal Models) 3: Guesgen, H., and Hertzberg, J Spatial persistence. pplied Intelligence (Special Issue on Spatial and Temporal Reasoning) 6: Guesgen, H Spatial reasoning based on llen s temporal logic. Technical Report TR , ICSI, Berkeley, California. Hernández, D Relative representation of spatial knowledge: The 2-D case. In Mark, D., and Frank,., eds., Cognitive and Linguistic spects of Geographic Space.Dordrecht, The Netherlands: Kluwer Lehmann, F., and Cohn, The EGG/OLK reliability hierarchy: Semantic data integration using sorts with prototypes. In Proc. 3rd International Conference on Information and Knowledge Management (CIKM-94), Mukerjee,., and Joe, G qualitative model for space. In Proc. I-90, Randell, D.; Cui, Z.; and Cohn, spatial logic based on regions and connection. In Proc. KR-92, Zadeh, L Fuzzy sets. Information and Control 8: References llen, J Maintaining knowledge about temporal intervals. Communications of the CM 26: Bloch, I Spatial representation of spatial relationship knowledge. In Proc. KR-00, Cohn,., and Gotts, N The Egg-olk representation ofregions with indeterminate boundaries. In Burrough, P., and Frank,., eds., Geographical Objects with Undetermined Boundaries, GISDTSeries No. 2.London, England: Taylor and Francis Cohn,.; Bennett, B.; Gooday, J.; and Gotts, N Representing and reasoning with qualitative spatial relations about regions. In Stock, O., ed., Spatial and Temporal Reasoning.Dordrecht, The Netherlands: Kluwer Egenhofer, M., and l-taha, K Reasoning about gradual changes of topological relationships. In Frank,.; Campari, I.; and Formentini, U., eds., Theories and Methods of Spatio-Temporal Reasoning in Geographic Space, Lecture Notes in Computer Science 639. Berlin, Germany: Springer Freksa, C Qualitative spatial reasoning. In Proc. Workshop RUM, FLIRS 2002