DIMENSIONS OF QUALITATIVE SPATIAL REASONING

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1 In Qualitative Reasoning in Decision Technologies, Proc. QUARDET 93, N. Piera Carreté & M.G. Singh, eds., CIMNE Barcelona 1993, pp DIMENSIONS OF QUALITATIVE SPATIAL REASONING Christian Freksa and Ralf Röhrig Fachbereich Informatik Universität Hamburg Bodenstedtstraße Hamburg 50, Germany {freksa, Abstract Qualitative knowledge can be viewed as that aspect of knowledge which critically influences decisions. Reasoning with qualitative knowledge has been studied extensively for the temporal domain. It turns out that the spatial domain is considerably richer than the temporal domain. The richness of the spatial domain is illustrated by showing the design choices for qualitative reasoning systems. After a general discussion of these dimensions of spatial knowledge nine research projects are presented for a comparison of their adaptation of qualitative methods to the domain of space. A classification with respect to the dimensions of spatial knowledge is presented and discussed. 1. QUALITATIVE REASONING IN ARTIFICIAL INTELLIGENCE Qualitative reasoning is becoming increasingly popular in Artificial Intelligence (AI) and its areas of application. Various factors contribute to this development, including the following insights: 1) High-precision quantitative measurements are not as universally useful for the analysis of complex systems as was believed in the beginning of the computer age, when great advances in the natural sciences and in engineering were made as a result of the availability of high-precision measurement and computation equipment; 2) Much experience and confidence have been gained in AI regarding the representation and processing of non-numerical knowledge; 3) Although storage capacity is much less an issue today than it used to be, it has been recognized that decision making on the basis of large amounts of quantitative knowledge can be computationally expensive (see, for example, connectionist approaches); as a consequence, a reduction of data without loss of information remains an important goal; 4) It has become apparent that higher cognitive mechanisms employ qualitative rather than quantitative mechanisms even if the knowledge originally is available in quantitative form through perception. Qualitative knowledge can be viewed as that aspect of knowledge which critically influences decisions. Which is the critical aspect of the available knowledge may depend much on the particular situation, of course. Thus, it is not surprising that different aspects are considered in different knowledge and/or application areas. On the other hand, there may be universal principles structuring a domain and its features. It is the purpose of this paper to look at dimensions and features which are universal to the domain of physical space.

2 Qualitative reasoning has been studied extensively for the temporal domain. The representational primitives used are (time) points [McDermott 1982] or temporal intervals [Allen 1983]. Although Allen's approach has various deficiencies [Freksa 1992], it has become a paradigm for qualitative reasoning; several researchers have adapted this paradigm to the domain of spatial knowledge. It turns out that the spatial domain is considerably richer than the temporal domain; for this reason there is a variety of ways in which the adaptation from the temporal domain to the spatial domain can be done. In the present paper we analyze a selection of the resulting approaches and compare them with one another. We illustrate the richness of the spatial domain by showing the design choices for qualitative reasoning systems. 2. DIMENSIONS OF SPATIAL KNOWLEDGE We have selected nine research projects on qualitative spatial reasoning for this comparative study. Each of them approaches spatial reasoning from at least one of four areas of concern: the approaches of Cohn, Güsgen, and Schlieder 1 are based upon formal concepts (logic-based, geometrical, topological, resp.); the approaches of Freksa, Hernández, and Mukerjee & Joe are largely motivated by cognitive considerations; the work of Egenhofer and Frank is driven by issues in geographical reasoning; and the work of Jungert has roots in data-base oriented research concerned with encoding pictorial information. Despite these differences of concern, the approaches converge remarkably with respect to some basic issues. Qualitative spatial representation systems can be compared with regards to the reference system used (Cartesian, polar), the scope of their reference system (local, global), the alignment of the reference system (internal, external), their representational primitives (point objects, spatially extended objects), the represented aspects (topology, arrangement, orientation, distance), the aspects represented qualitatively, the dimensionality of the represented space, their granularity (fine, coarse, flexible), the modularity of the aspects represented, their ability to represent incomplete and imprecise knowledge, the flexibility of integrating various kinds of spatial knowledge, their ability of incremental knowledge enrichment, among others. We will briefly introduce some of these notions in a general manner before we discuss how they are reflected in the nine approaches analyzed. In quantitative representations of space, the reference system is specified by 'rulers' which measure spatial objects and their locations. In a three-dimensional Cartesian reference system, for example, three rulers are arranged orthogonally; objects and locations are specified by a measurement on each of these rulers. Similarly, in a planar polar coordinate system, objects and locations are specified by a distance and angle measurement. In qualitative representations, we do not require scales on the rulers since they do not employ metrics; however, we still need a specification of the dimensions. Which scope does the reference system have? Is there one global reference system to which all descriptions refer or are there many local reference systems that are used in smaller vicinities? In quantitative representations, both approaches are used, dependent on possible advantages of having reference to local entities: comparisons across measurements may be carried out more easily in a global reference system, but the measurements themselves may be easier in local reference systems. Similarly, in qualitative representations, comparisons are carried out among entities; this can be done more easily if they are related to the same reference system; on the other hand, the 1 Selected references to publications on the projects under study are given at the end of the article.

3 entities themselves are described in terms of comparisons; these comparisons can be carried out directly, if the reference system is linked to the entities to be compared. What determines the alignment of the reference system? In quantitative representations, the scales are defined outside the problem space; the reference system containing the scales can be external to the problem space or it can be aligned to internal objects or structures of the problem space. In qualitative representations, features are typically compared within the problem space; but the feature dimensions may be aligned either to internal (e.g. egocentric or allocentric) or to external (e.g. geocentric) entities. Should we use points or spatially extended objects as representational primitives? Point locations are as non-spatial as time points are non-temporal since spatiality and temporality are due to spatial and temporal extension, respectively. This suggests that only extended entities should be used in these domains. On the other hand, abstraction from these fundamental aspects has proven useful for describing space and time in physics and geometry. The issue is related to the question how much abstraction is useful and how much abstraction misses the important properties of the respective domains. Practically speaking, the use of spatially extended objects limits the generality of spatial inferences, since there is a multitude of qualitatively different shapes; these result in a multitude of different spatial situations. The use of point objects, on the other hand, may miss crucial aspects of spatiality (for example: shape). Which are the important aspects of space which we have to consider? Do the interactions of spatial objects (as describable through topological relations) suffice or do we need more specific knowledge about relative arrangement, orientations, or distances? Topological relations completely abstract from aspects of shape and orientation; arrangement information adds some rudimentary aspects of orientation; full orientation knowledge constrains topology rather severely; and complete distance information hardly abstracts at all and leaves little or no freedom for a physical instantiation. These considerations raise a question - which is strongly linked to the external internal issue - should all aspects be represented qualitatively or can we profit from interactions with a quantitative approach? An issue which so far has been given little treatment in the AI literature is the question how the choice of approaches is affected by the dimensionalities of the problem domain and of its surrounding space. The more abstract approaches to representation appear to be less affected by the dimensionality than the more concrete approaches. The approaches considered in this paper are restricted to two or three spatial dimensions, respectively. Representational modularity concerns the question whether the treatment of different aspects of space is integrated within one approach or whether they are treated highly independently of one another. Granularity refers to the resolution of spatial features or decision criteria and to the transition between different levels of resolution: what can we infer from coarse spatial knowledge under consideration on a fine level of resolution and vice versa? Incomplete and imprecise knowledge has to be dealt with in all real-world situations. The notion 'incomplete' may refer to two distinguishable situations: it may mean that there is a 'hole' in the acquired knowledge (missing facts) or it may mean that the available knowledge does not have sufficient precision, i.e. each question about a situation can be answered but not necessarily with sufficient detail.

4 Flexibility addresses the question how easily a given representation scheme can deal with spatial knowledge of different granularity, completeness, precision, etc. The ability of incremental knowledge enrichment, finally, refers to the ability of a representation and inference system to integrate new facts and procedures incrementally without requiring recomputation of already derived knowledge. In the following section we will look at specific approaches to qualitative spatial knowledge representation. 3. QUALITATIVE REASONING IN THE DOMAIN OF SPACE Qualitative reasoning is based on comparative knowledge rather than on metric information. The simplest comparison that can be performed in the spatial domain is the comparison of points in space with respect to their equality. Simple comparison of points in space can be extended in several ways: first, adopting the concept of orientation, space obtains a structure which can be used to further classify unequal locations of points; second, instead of points in space one might be interested in spatially extended objects. Many relations are conceivable for qualitative comparison of spatially extended objects. The following analysis of approaches to qualitative spatial representation will reflect the distinction between the comparison of spatially extended objects and the exploration of orientation. When examining systems for the representation of orientation, a major question is how the spatial axes described by the formalism are aligned to the domain. Reasons for a specific placement of the axes may be independent of the properties of the objects under consideration. We will call these kinds of decision factors external. On the other hand, internal properties of the domain may guide the alignment of the reference system. comparison of points spatially extendedobjects orientation spatially extendedobjects & orientation integrated external internal external internal Egenhofer Cohn, Cui, Randell Hernández (T) Frank Hernández (O) Schlieder Freksa Güsgen Jungert Mukerjee, Joe Figure 1: Classification of approaches to qualitative spatial reasoning. We classify the nine research projects under consideration into three coarse and five finer categories, depending on their use of spatially extended objects and/or

5 orientation information and on the reference to external or internal entities, respectively. This classification is depicted in Figure l. Some of the approaches selected for this analysis of spatial reasoning cover both concepts, that of spatially extended objects and that of orientation. The approaches falling into the same class with regard to the distinctions described above will be further analyzed and differentiated with respect to other, more specific, features. 3.1 From points to spatially extended objects For spatially extended objects in the domain of space we can qualitatively distinguish the interior, the boundary, and the exterior of the object, without bothering about the concrete shape or size of the object. Comparison of extended objects can then be reduced to the comparison of the parts of the objects. A set theoretical analysis of the possible relations between objects based on the above partition is provided by the work of Egenhofer and Franzosa. For the comparison of objects they examine the intersection of their parts. Most fundamental in this setting is the distinction of the values empty and non-empty for the intersection. Since there are three parts (including its exterior) for each object that are compared with three parts of another object, there are 2 9 = 512 set theoretically distinguishable relations. Egenhofer showed that only eight of them have a meaningful interpretation in physical space. These eight relations are labeled disjoint, meet, equal, inside, covered by, contains, covers, and overlap. Some variants of this theory were developed by Cohn and his coworkers. Cohn abandons the distinction of the interior and the boundary of an object. He defines the same eight topological relations that where provided by Egenhofer from the single binary relation 'is connected to'. Furthermore Cohn extends the theory to handle concave objects by distinguishing the regions inside and outside of the convex hull of objects. In this way many more relations can be defined, for example, an object may be inside, partially inside, or outside another object's convex hull. An application of the topological relations to two-dimensional layout plans for offices was done by Hemàndez. Describing qualitative positions of object in 2D, he distinguishes topological relations and orientation relations (cf. Figure 1, Hernández (T) and Hernández (O)). Since he is dealing with projections of objects, he can use the eight topological relations applicable to convex objects. He shows some important properties of these relations, for example, that there is a neighborhood structure among them, and how this structure influences the reasoning process. Another observation is that there are pairs of converse relations (covers / covered by and inside / contains), whereas the other relations are symmetric. 3.2 From points to orientations An orientation can be viewed as a labeled axis, an orientation system as a system of labeled axes. Besides the questions, how many axes are to be used, whether the Cartesian or a polar system is preferred, or whether to define the axes globally or locally, it has to be stated how the axes are aligned to the domain, i.e., where the axes are to be located with respect to the objects under consideration. Externally aligned orientations There are different reasons for aligning an orientation system, which are independent of the properties of the objects in the domain, i.e. are external. For example, in large scale navigation often the system of cardinal points is used to describe orientation. This system is neither aligned with a moving object nor with a landmark in the scene. The north-south direction is always aligned with the North and South Pole

6 and the east-west direction is always orthogonal to the north-south axis. Two examples of systems of cardinal directions are given by Frank. Instead of axes, he uses either cones or half-planes to define cardinal directions. Regardless of the shape of the regions and the number of directions used, either the four directions north, east, west, south, or at a finer level of granularity north, north-east, east, south-east, south, south-west, west, and north-west, the center of the northern region will always point to the North Pole and is not committed to any object of the domain. Another possible external reference for aligning an orientation system is the surrounding object or space. This kind of referencing is often used in small scale space or in in-door space descriptions. Specific properties of the room containing the scene are used to align a system of orientations. These properties can, for example, be the door through which a room was entered, or a black board at one side of the room. In his application of theoretical results in the domain of lay-out plans for offices Hernández uses the door of the office as a reference for his back orientation. Having fixed one orientation, he can define the other orientations with respect to the first one. For example, the orientation front is always opposite to the orientation back. Depending on the level of granularity, other orientations are defined at certain angles to the former ones. Since Hernández offers a representation that allows to explicitly state a reference frame for any orientation relation between two objects, he also provides a method for transforming the orientation information from one reference frame to another. Reasoning processes are always based on one global reference frame, the one aligned with the surrounding room. Internally aligned orientations Alignment of the orientation system to the domain with regard to properties of the object in the domain, i.e. with regard to internal reasons, can be based on mainly two different considerations: first, there might be a deictic view, i.e., the relative orientation of two objects is given with respect to the position of another object; second, some nonspatial properties of the objects may be used to define a reference orientation for the alignment of the orientation system. Since this section is dedicated to the comparison of points, and points do not have non-spatial properties that could be used to define a reference orientation, the discussion of this kind of internal alignment is given in the next section, where the concept of orientation is analyzed in the light of spatially extended objects. To apply the deictic method to alignment of the orientation system means to take a third object or location into account. In natural language processing, this object is called a second reference object. For example, in the context of route descriptions, the actual or the imagined location of the speaker is taken as the second reference object to interpretate phrases like 'there is a church on the left'. In general, this way of aligning the orientation system results in orientation relations over three objects, the reference object of the relation, the entity to be related to the reference object, an the second reference object. The simplest way to qualitatively describe the arrangement of a set of points in two-dimensional space is to use the orientation of triangles, clockwise or counterclockwise. The orientation of triangles can be viewed as a relation over the three vertices of the triangle. Schlieder called the orientation of triangles global knowledge in his approach. Local knowledge is collected through perception processes and is therefore bound to a single point, in which perception takes place. Schlieder showed, that simply looking around and determining the order of visible landmarks in every point of the scene is not sufficient to reconstruct the global arrangement information. His idea was to augment the perceived landmarks by recording the opposite of each landmark as well. The resulting structure is called a panorama graph and consists of a cyclic list of the landmarks as well as their opposites, appearing in the order they occur when looking around in one point of the scene. Schlieder showed, that arrangement knowledge may be reconstructed from these panorama graphs. Determining the order of

7 landmarks in a point of perception can again be viewed as a set of orientation relations over three points, each relating two neighboring points with the point of perception. An approach that explicitly works with relations over three points is provided by Freksa. Relative orientation is described by relations between a point and a vector. Qualitatively different relations are left, right and on the line. These relations correspond to the orientation of triangles, clockwise and counter-clockwise, in Schlieder's approach. For a further differentiation of the left and right side of the vector, Freksa uses a biologically motivated concept of perception. People are not only able to distinguish their left hand side from their right hand side, but are definitely able to distinguish a front region from what is behind them. Freksa represents this front/back dichotomy by a bound orthogonal to the bound between the left and the right region. Placing this bound in each of the two end points of the vector being the reference for orientation relations, one obtains a structure (double cross) that separates the two dimensional plane into 15 disjoint regions. As shown by Latecki and Röhrig [1993], using the concept of oriented triangles for the description of arrangement of points in a two dimensional plane is not sufficient to get all interesting inferences, when only partial knowledge is available. Knowing that three points A, B, and C form a clockwise oriented triangle, and that points B, C, and a fourth point D form a clockwise oriented triangle, too, it is not possible to infer anything about the orientation of the points A, B, and D. But, augmenting the knowledge with the concept of qualitative angles, i.e., allowing a distinction of acute and obtuse angles, it is possible to define a composition of any two orientation relations that results in a new orientation relation. Latecki and Röhrig mention that the concept of qualitative angles is very similar to the front/back dichotomy introduced by Freksa. This concept leads to a finer subdivision of the plane again and has some nice algebraic properties that can improve the reasoning process. 3.3 Combining extended objects with orientation As mentioned above, Allen's approach to qualitative temporal reasoning has become a paradigm for several methods to qualitatively represent and reason about space. The primitives used by Allen are time intervals, i.e., temporally extended objects. Since intervals are convex by definition, the same eight topological relations are applicable to distinguish qualitatively different situations, as used by Cohn, Egenhofer and Hernández. Due to the inherent orientation of time five of the topological relations fall into two further distinguishable relations: disjoint becomes before and after; meets gets extended by met by; started by is separated into started by and finishes; and overlap is extended by overlapped by. Thus, the combination of (temporally) extended objects with the orientation of time results in 13 disjoint relations, which exhaustively describe the qualitatively distinguishable connections between two temporal intervals. There are two major problems in adapting this paradigm to the domain of spatial knowledge. First, space is of higher dimensionally than time. Therefore, all approaches that transfer the temporal structure to space use one dimensional projections of spatially extended, but connected objects. Second, space lacks an inherent orientation. This problem is the general problem of using orientation systems: the problem of alignment of a system of axes to the domain. Externally aligned orientations A straightforward application of this qualitative reasoning paradigm to twodimensional space was described by Güsgen. He uses a Cartesian system of three X-, Y- and Z- axes. Objects are projected to theses axes, and compared by means of eight qualitative relations (distinct from Cohn / Hernández / Egenhofer's) along each of the

8 axes separately. So, relative position of two objects is given by a tuple of qualitative relations. The same method of transferring temporal relations to the two dimensional spatial domain was used by Jungert. He uses the geo system with the cardinal directions north/south and east/west to align the orientation system in which the objects are to be compared. While in Güsgen's approach only such rectangular objects are compared, that are aligned to the system of X-, Y, and Z-axes, in the formalism of Jungert it is possible to split objects of arbitrary shape and to represent the relative position of the parts to other objects or object parts. The parts of a single object have a defined neighborhood relationship which is maintained by different methods than the 13 qualitative relations per axis. Internally aligned orientation When looking at internal reasons for alignment of an orientation system to the domain, beside the deictic method described in the section above, it is possible to make use of non-spatial properties of the objects. Several properties induce a front side of an object. For example, for moving objects it is easy to determine the front. The same holds for objects that have a visual apparatus, like a camera or an animal. Other objects get their fronts from functional considerations, like desk or a wardrobe. In the approach of Mukerjee and Joe, these non-spatial properties are used to predefine a special direction, called the front. This direction is used as a reference for the definition of the orthogonal directions left, back, and right. Relative orientation is given by the comparison of the reference frames of two objects. The front direction is also used to define a line of travel. This line serves as the axis to which all objects are projected. These projections can then be compared with the object at hand by means of the 13 qualitative relations provided by Allen. In order to describe relative position between two objects, the comparison of projections is done on the line of travel for each object separately. A spatial relation is then constructed from a relative orientation and a relative position. 4. CONCLUSIONS We have studied nine research projects on qualitative spatial reasoning which originated from rather different areas of concern. Nevertheless, these approaches converge remarkably with respect to some basic issues. This suggests that qualitative spatial reasoning has matured to a stage where we may develop a general theory of representing spatial knowledge which integrates a wide range of views. The development of such a theory first of all requires a clarification and unification of the applicable terminology. This includes a clarification of the question which kind of domains and dimensions are essential for describing spatial knowledge. Furthermore, a connection to the linguistic literature on spatial concepts should be drawn. For example, in the field of linguistic approaches to automated text understanding, there is a distinction of external and internal reference systems for the interpretation of spatial prepositions and within the internal ones there is a further distinction of deictic and intrinsic reference frames [Retz-Schmidt 88].

9 ACKNOWLEDGMENTS AND DISCLAIMER We thank the participants of the space inference seminar at the University of Hamburg for discussions and scrutinization of the approaches reflected in this report; we also acknowledge numerous discussions with the authors of these approaches; the authors of the present paper, however, are solely responsible for possible misrepresentations of the approaches analyzed. SELECTED REFERENCES ON THE RESEARCH PROJECTS STUDIED Cui, Z., Cohn, A.G., Randell, D.A.: Qualitative simulation based on a locical formalism of space and time. Proc.of AAAI92, AAAI Press, 1992 Cui, Z., Cohn, A.G., Randell, D.A.: Qualitative simulation based on a locic of space and time. Proc.AISB Workshop on Qualitative and Causal Reasoning, University of Birmingham, March, Cui, Z., Cohn, A.G., Randell, D.A.: A taxonomy of logical defined qualitative spatial relations. In: Guarino, N., Poli, R. (eds), Interna. Workshop on Formal Ontology, Padova, Italy, March, Egenhofer, M.: A formal definition of binary topological relationships. In W. Litwin and H.-J. Schek, editors, Third Intern.Conference on Foundations of Data Organization and Algorithms vol. 367 of Lecture Notes in Computer Science, Springer, Berlin Egenhofer, M.: Reasoning about binary topological relations. Günter, O., Schek, H.-J. (eds): Advances in Spatial Databases - Second Symposium, SSD '91, Lecture Notes in Computer Science, Vol. 525, pp , Springer, New York, Egenhofer, M., Al-Taha, K.: Reasoning about gradual changes of topological relationship. In: A. U. Frank, I. Campari, U. Formentini (eds.) Theories and Methods of Spatio-Temporal Reasoning in geographic space, Springer, Berlin, Frank, A.U.: Qualitative spatial reasoning with cardinal directions. Proc. Seventh Austrian Conf. on Artificial lntelligence, Wien, , Springer, Berlin Freksa, C.: Temporal reasoning based on semi-intervals. Artificial Intelligence 54 (1992) Freksa, C.: Using orientation information for qualitative spatial reasoning. In: A. U. Frank, I. Campari, U. Formentini (eds.) Theories and Methods of Spatio- Temporal Reasoning in geographic space, Springer, Berlin, Freksa, C., Zimmermann, K.: On the utilization of spatial structures for cognitively plausible and efficient reasoning, Proc. IEEE Intern. Conf. on Systems, Man, and Cybernetics. Chicago, October, Güsgen, H.W.: Spatial reasoning based on Allen's temporal logic. TR , International Computer Science Institute, Berkeley Hernández, D.: Relative representation of spatial knowledge: The 2-D case. Mark, D.M., Frank, A. (eds), Cognitive and Linguistic Aspects of Geograpgic Space, NATO Advanced Studies Institute. Kluwer, Dordrecht, pp ,1991.

10 Hernández, D.: Qualitative representation of spatial knowledge, Doctoral Dissertation, Technische Universität München, Jungert, E.: Extended symbolic projections as a knowledge structure for spatial reasoning. Pattern Recognition, Kittler, J. (ed), Cambridge, U.K., March Chang, S.-K., Jungert, E.: Pictorial data management based upon the theory of symbolic projections, J. of Visual Languages and Computing (1991) 2, pp , Jungert, E.: The observers point of view: An extension of symbolic projection. In: A. U. Frank, I. Campari, U. Formentini (eds.) Theories and Methods of Spatio- Temporal Reasoning in geographic space, Springer, Berlin, Mukerjee, A., Joe, G.: A qualitative model for space, Proc. AAAI-90, , Schlieder, C.: Anordnung. Eine Fallstudie zur Semantik bildhafter Repräsentation. In: C. Freksa & C. Habel (eds.) Repräsentation und Verarbeitung räumlichen Wissens, , Springer, Berlin Schlieder, C.: Anordnng und Sichtbarkeit. Eine Charakterisierung unvollständigen räumlichen Wissens. Doctoral Dissertation, Universität Hamburg, GENERAL REFERENCES Allen, J.F., Maintaining knowledge about temporal intervals, CACM 26 (11) (1983) Latecki, L., Röhrig, R.: Arrangment and qualitative angles for spatial reasoning, IJCAI '93, to appear. McDermott, D.: A temporal logic for reasoning about processes and plans. Cognitive Science 6, (1982), Retz-Schmidt, G.: Various views on spatial prepositions. AI Magazine, Vol.9 #2, pp , 1988.