Image-Schemata-Based Spatial Inferences: The Container-Surface Algebra *

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1 Image-Schemata-Based Spatial Inferences: The Container-Surface Algebra * M. Andrea Rodríguez and Max J. Egenhofer National Center for Geographic Information and Analysis and Department of Spatial Information Science and Engineering University of Maine Orono, ME {andrea, max}@spatial.maine.edu Abstract Algebras over spatial relations have become an important aspect of spatial reasoning for retrieving and handling large and complex spatial data sets. While such spatial relations play a fundamental role in specifying constraints in a spatial query language, there has been little concern as to whether existing spatial-relation algebras are cognitively plausible. In order to construct more intuitive and easier-to-use spatial query languages, this work pursues an alternative approach to spatial reasoning based on a small set of operators that derive cognitively plausible spatial relations. The work focuses on a set of operators that are associated with the behavior of image schemata recurrent patterns that people learn through bodily experiences and use to assign meaning to objects and situations. A study of a small-scale space, involving the surface and container schemata, describes objects in this space and the possible spatial relations among them. The informal description is then translated into a formal algebraic specification and generalized for spatial relations in a scene that involves surface and container schemata. This formalization axiomatizes spatial inferences that are then applied and compared to a larger geographic space. As a result, a small set of spatial operators were defined for small- and large-scale spaces; however, these operators show discrepancies when applied to a combination of objects belonging to different scales. It is expected that this study can be useful in exploring new theories for the design of query languages of future geographic information systems. 1. Introduction Research in spatial reasoning has been recently concerned with the construction of algebras to make inferences about spatial relations. Examples are compositions of topological relations (Egenhofer 1994), cardinal directions (Frank 1991), approximate distances (Hong et al. 1995), and their combinations (Sharma and Flewelling 1995). These studies follow the approach given for inferences about interval relations, i.e., they embed spatial relations within a space in which decisions about compositions can be derived. The result is a set of symbols (i.e., spatial relations) and a set of logical rules about the combinations of these symbols (i.e., compositions). While these systems are mathematically sound, it remains an open question whether they describe cognitively plausible settings (Hernández 1994, Hirtle 1991). * Max Egenhofer s work is partially supported by the National Science Foundation through NSF grants IRI and IRI , by the National Center for Geographic Information and Analysis through NSF grant SBR , by Rome Laboratory under grant number F , and by a Massive Digital Data Systems contract sponsored by the Advanced Research and Development Committee of the Community Management Staff and administered by the Office of Research and Development. Andrea Rodríguez is partially supported by a Fulbright Fellowship.

2 Naive Geography (Egenhofer and Mark 1995) promotes an alternative path with the investigation of concepts that are grounded in human experiences and, therefore, match more closely with human thinking. In order to construct more intuitive and easier-to-use spatial query languages, this paper proposes a top-down approach to spatial reasoning. We share this goal with the recent work by Albrecht (1996), but pursue a significantly different approach. In lieu of analyzing users behavior in using a given set of operations implemented in a particular geographic information system (GIS), we start with a small set of operations that have been found to be fundamental to much of human thought and language. These operations characterize the behavior of spatial configurations called image schemata (Johnson 1987, Lakoff and Johnson 1980). Image schemata have recently been considered as an important basis for GIS query languages and user interfaces (Mark 1989, Mark and Frank 1996). We select a particular small-scale space a room space to study the properties of its objects such as a ball, a box, a, pen, a sheet of paper, a table, and the room itself with respect to two major image schemata: the container and surface. This small-scale space retains the bodily experiences of how people assimilate image schemata. This concrete study reveals what kinds of inferences are possible or impossible. Since we start with the spatial objects, operations upon them, and spatial relations among them, we axiomatize the spatial inferences rather than attempt to derive them from a Cartesian coordinate space. We share the interest of Hayes (1990) for analyzing the states and changes of states of our scenario; however, we focus on the description of the behavior of solid objects rather than liquid substances. The remainder of this paper summarizes the essence of image schemata (Section 2) and provides a detailed analysis of the ontology of the test case, a room space (Section 3). Section 4 formalizes these natural-language descriptions through algebraic specifications and generalizes the observations from the room space to an algebra for objects that act as containers and surfaces. Section 5 demonstrates how this algebra is applicable to objects in geographic space. Conclusions and future work are addressed in Section Image Schemata Image schemata are recurrent patterns that people learn through physical (bodily) and repetitive experiences (Johnson and Lakoff 1980). For example, all infants experience the image schemata of a container by putting food and other objects into their mouths. People perceive and understand situations as recurrent patterns that are extended, transformed, and metaphorically projected to produce meaningful situations. Johnson (1987) pointed out that image schemata are pervasive, well defined, and full of sufficient internal structure to constrain our understanding and reasoning. Image schemata imply that experiences are organized into meaningful structures before and independently of concepts; however, concepts can impose more constraints to already existing structures. Image schemata are composed of parts and relations that allow for an organization of many different perceptions or events. Contrary to concepts, image schemata are dynamic structures operating at a higher level of abstraction and generality than concrete images. Thus, they are adaptable structures according to the context for organizing situations; however, they become relatively stable by being located in our network of meaning. Johnson (1987) presented a partial list of image schemata (Table 1), which covers only what he considers to be the most important image schemata. Many of these image schemata are fundamentally spatial in nature. Moreover, some of them are related to spatial orientations. For example, given the image schema of a container, whose structural elements are the interior, boundary, and exterior, the in-out orientation becomes the differentiation, separation, and enclosure between interior and exterior.

3 Container Balance Full-Empty Iteration Compulsion Blockage Counterforce Process Surface Restraint Removal Enablement Attraction Matching Part-Whole Mass-Count Path Link Collection Contact Center-Periphery Cycle Splitting Merging Object Scale Superimposition Table 1: The partial list of image schemata from Johnson (1987) Since image schemata are conceptual models of human perception and cognition and they explain how different spatial relations are used in language through prepositions, they form an excellent basis for describing geographic scenes and for designing GIS query languages. Mark and Frank (1996) used image schemata to describe the experiential and formal models of geographic scene. Mark (1989) went further in the use of image schemata by suggesting that image schemata should be used to define good user interfaces. Thus, user interfaces including command and query languages would be compatible with the views that users have of the system. Similarly, Kuhn and Frank (1991), and again Kuhn in a later work (1993), supported the idea that image schemata and their metaphorical mappings are the fundamental theories to build efficient user interfaces. Freundschuh and Sharma (1996) linked image schemata with the spatial concepts that children learn through story books and suggested that there is a progressive process of spatial knowledge understanding and that some image schemata seem to be more fundamental or basic than others. In the complexity ranking of image schemata, container and surface appear to be the two most basic image schemata. Freundschuh and Sharma also presented how locatives prepositions relate to image schemata. They stated that while in and out correspond to locative prepositions associated with a container, on and off correspond to locative prepositions associated with a surface. Despite the interest of the GIS research community in image schemata, approaches to formalize them have been limited. Kuhn and Frank (1991) used algebraic specification to formalize interface metaphors that are instances of the image schemata container and surface. Although their formalization reflects the basic behavior of surface and container, it does not capture the difference between these two schemata, because only the use of different names for the constructors (insert vs. put) and the observers (in vs. on) makes the surface and container specifications different. Since image schemata are structures of high level of abstraction that are often used in combinations or in metaphorical projections, we avoid ambiguities by starting with the ontology of a concrete scenario, a room space. 3. Room Space A room space is an ubiquitous example of a small-scale space that contains manipulable objects. As such, the room space constitutes a representative scenario where people can experience recurrent manifestations of image schemata. We assume a neatly organized room space with six major objects: a box, a ball, a table, a sheet of paper, a pen, and a room. All these items are solid objects that people manipulate by changing their locations. We assume, for the time being, that all objects may only be completely on or off a surface, i.e., no part of, say, the paper may extend beyond the tabletop. Similarly, all objects can be only completely in or out of a container. The discussion of partially on and off and partially inside and outside relates to the part-whole image schema, which would introduce more complex variations of the simpler cases. Subsequently, we analyze the room from simple to more complex configurations by describing the operations upon the objects and their interrelationships. In this ontology, rather than defining the objects themselves, we are interested in describing the behavior of the objects in relation to changes of the spatial relations in the configurations, called scenes.

4 3.1 Ball in a Box The box serves as a container so that it can contain other objects. In our scenario the box is in an upright position and objects can be moved into it. The box is a container by creation, i.e., it does not need to have another object to be a container. There is some logic in saying that the box always contains something, be it pens, papers, or air. In our scenario, however, we assume that the box is empty if there is only air in it. In that case, the ball is outside the box. By moving the ball into the box, the ball is in the box (Figure 1a). The box can be moved and the ball moves with it (Figure 1b). The ball can be removed and the ball would be again outside the box (Figure 1c). (a) (b) (c) Figure 1: Ball and box: (a) moving the ball into the box; (b) moving the box with the ball in it; and (c) removing the ball from the box. 3.2 Ball in a Box in a Room The box with the ball can be moved into a room. The room acts as another container and the same operations and properties apply to box and room, and ball and room, as did apply to ball and box. The interesting aspect here is, however, the interplay between the ball (in the box) and the room. Moving the box (with the ball) into the room means that the ball is in the room as well (Figure 2a). Moving the box that contains a ball around the room keeps the same properties between ball and box, and ball and room (Figure 2b). However, if the box is moved out of the room and the box contains the ball, then the ball goes with it (Figure 2c). Moving the ball into the box that is in the room means to move the ball into the room and then into the box (Figure 2d). Reversibly, if the ball is removed from the box while the box is in the room, the ball is still in the room (at least for some non-zero time) (Figure 2e). Finally, removing the ball that is in the box from the room implies to remove the ball from the box and then from the room (Figure 2f). We assume, for the time being, that the box is small enough with respect to the room to have a free space between box and room when the box is in the room.

5 (a) (b) (c) (d) (e) (f) Figure 2: Ball, box, and room: (a) moving the box with the ball into the room; (b) moving box in room with ball in it; (c) removing the box with the ball from the room; (d) moving the ball into the box in the room; (e) removing the ball from the box in the room; and (f) removing the ball from the box out of the room. 3.3 Paper on a Table The table acts as a surface. Similar to the box, which is a container by creation, the table is a surface by creation and in its horizontal position, one can put objects onto it. If the sheet of paper is put on the table, the paper and the table have contact (Figure 3a). If the paper is moved around on the table, it stays in contact with the table (Figure 3b). If it is removed from the table, the paper loses its contact with the table. When the paper is removed from the table, the paper is off the table (Figure 3c). The table may be moved quickly called jerked so that the paper is also removed from the table (Figure 3d).

6 (a) (b) (c) (d) Figure 3: Paper and table: (a) putting the paper onto the table; (b) moving the paper around the table; (c) removing the paper from the table; and (d) jerking the table with the paper. 3.4 A Pen on a Paper on a Table The pen can be put onto the paper that lays on the table. Although the same operations and properties apply to pen and paper as did apply to the paper and table, the interesting aspect of this configuration is the relationship between the pen (on the paper) and the table. While the pen is now on the table, it does not have contact with the table (Figure 4a). The same situation happens if the pen was moved onto the paper while the paper was already on the table (Figure 4b). Being on the surface is transitive, contact is not. If the paper is moved around on the table the pen will either move with it or end up being in contact with the table. The paper may be removed from the table such that it keeps the pen on it (Figure 4c). The paper on the table may also be jerked so that the pen on the paper lies now on the table and the paper is removed from the table (Figure 4d). The pen on a paper alone may be moved such that it remains on the paper as long as it has contact with the paper. A pen that is moved horizontally and loses the contact with the paper may remain on the table as long as it has contact with the table; however, it might have been moved off the table as it was moved off the paper, because the paper and table were aligned along a common edge and the pen was moved across that edge. (a) (b) (c) (d) Figure 4: Pen, paper, and table: (a) putting the paper with the pen onto the table; (b) putting the pen onto the paper on the table; (c) removing the paper with the pen from the table; and (d) jerking the paper with the pen on the table.

7 3.5 Ball in a Box on a Table A box with a ball in it can be moved onto a table such that the ball is on the table, but it does not have contact with the table (Figure 5a). While the box is moved onto the table, the ball remains in it (Figure 5b). If the box is removed from the table, the ball will move with it (Figure 5c). The ball may be removed from the box that is on the table so that the ball is not anymore on the table (Figure 5d). There are different judgments, however, about the relation between the ball and the table. While some people might argue the ball is on the table, others might argue the ball is not on the table because it does not have direct contact with the table. In this work, we assume the ball is on the table and we keep the resolution of the argument for future human-subjects testing. (a) (b) (c) (d) Figure 5: Ball, box, and table: (a) putting the box with the box onto the table; (b) moving the box with the ball around the table; (c) removing the box with the ball from the table; and (d) removing the ball from the box on the table. 3.6 Paper on a Table in a Room A sheet of paper on a table can be moved into a room (Figure 6a). Because the table is in the room, the paper on the table is also in the room. The paper can be moved on the table and will keep the same operations and properties as the paper on the table (Section 3.3). If the paper is removed from the table, the paper remains in the room, at least for a non-zero time (Figure 6b). If the table is moved around the room, either the paper is moved with the table and remains on it (Figure 6c) or the paper is removed from the table with a fast movement (Figure 6d). If the table is moved out of the room keeping the paper on it, the paper will be out of the room as well (Figure 6e).

8 (a) (b) (c) (d) (e) Figure 6: Paper, table, and room: (a) moving the table with the paper into the room; (b) removing the paper from the table that is in the room; (c) moving the table with the paper around the room; (d) jerking the table with the paper in the room; and (e) removing the table with the paper from the room. 4. The Container-Surface Algebra 4.1 Formalization of Spatial Relations in the Room Space Each particular configuration in the study case was formally specified by using algebraic specification (Liskov and Guttang 1986) and implemented with the functional programming language Gofer v (Bailey 1990, Jones 1993). Rather than defining particular objects, we define for each configuration a main data type Scene. A Scene allows us to describe all the objects within a configuration and the spatial relations between any two objects. In particular, the use of a Scene permits to make references to objects that are not involved in, but affected by an operation. Moreover, a Scene can reflect one or more than one change product of an operation. In isolation (i.e., without a Scene) these important effects would be invisible. A sort identifier introduces the data type to be defined, a Scene, and the data types used in the specification. The specification identifies the possible operations between any two objects within a Scene. We call basic constructors those operations that modify the state of a Scene and are not defined in terms of any other constructor. On the other hand, we call derived constructors those operations that modify the state of the Scene and are derived from basic constructors. From the basic constructors, spatial relations are derived as observers, operations that do not modify the state of the Scene. Finally, axioms of the configuration constrain the behavior of the derived constructors and observers. For each Scene, its objects are pre-defined. We assume that a new Scene involves only objects that are empty, i.e., they do not have any object inside or over them. Therefore, if an object A has not been moved into or onto an object B, A is outside or off B, respectively. The specifications reveal that the operation move around does not modify the properties that define a scene. Likewise,

9 move around does not affect the observers, i.e., spatial relations within a scene. Pseudo code of the specification for a simple configuration is presented in table 2. The complete specification written using Gofer is available in [ Sort: Scene, Ball, and Box Operations: Basic constructors: CreateScene :: MoveBallIntoBox :: Non-Basic constructors: removeballfrombox :: moveboxaround :: Observers: isballinbox :: isballoutsidebox :: Box x Ball -> Scene Scene -> Scene Scene -> Scene Scene -> Scene Scene -> Bool Scene -> Bool Variables: Box box Ball bal Scene sce Axioms: removeballfrombox(createscene(box,bal)) = CreateScene(box,bal) removeballfrombox(moveballintobox(sce)) = sce moveboxaround(createscene(box,bal)) = CreateScene(box,bal) moveboxaround(moveballintobox(sce)) = MoveBallIntoBox(sce) isballinbox(createscene(box,bal)) = False isballinbox(moveballintobox(sce)) = True isballoutsidebox(sce) = not (isballinbox(sce)) Table 2: Pseudo code of the specification for a ball and a box. 4.2 Generalization to a Container-Surface Algebra From the six configurations we obtained two groups of objects with different behaviors. In one group, possible operations are move into and remove from. This group involves objects that act as containers such as box and room. In the other group, operations are move onto, remove from, and jerk (a fast movement). Surfaces, such as paper and table, belong to this group. The six configurations with balls, boxes, pens, paper, tables, and rooms provide a comprehensive treatment of situations between containers and surfaces, that is, they cover all possible combinations of objects with respect to containers and surfaces (Table 4). Using these configurations we analyze transitive properties of the spatial relations in and on with respect to the repetition and composition of operations move into and move onto. Room configurations Image schemata combinations Ball in box Object in container Ball in box in room Object in container in container (container-container nesting) Paper on table Object on surface Pen on paper on table Object on surface on surface (surface-surface nesting)

10 Pall in box on table Paper on table in room Object in container on surface (container-surface composition) Object on surface in container (surface-container composition) Table 4: Mapping the room configurations onto combinations of image schemata. The container-surface algebra considers a formalization based only on the possible operations that involve containers and surfaces. In no case do we define the structural elements of a container or surface. We expect to capture the difference between surface and container in terms of their behaviors within a spatial configuration. Like in the specifications for each configuration, we use a type Scene. In this case however, we define Scene in terms of a set of Objects so that different configurations may be specified. A generic type Object is defined only with a numeric identifier. The generic Object permits the use of the same object not only as a surface or container, but also as the object being moved into or onto another object through multiple operations. The data type Scene has three possible states: (1) it is a new Scene; (2) it is a Scene where the last performed operation was to move an object into another object that acts as a container; and (3) it is a Scene where the last performed operation was to move an object onto another object that acts as a surface (Figure 7). The constructors MoveInto(o1, o2, s) and MoveOnto(o1, o2, s) are read as o2 is moved into and onto o1, respectively, within the scene s. data Scene = CreateScene(Set Objects) MoveInto(Object, Object, Scene) MoveOnto(Object, Object, Scene) Figure 7: The three constructors to make a change a scene. Although in our study case an object is either in or on another object, the specification does not constrain that move into or onto be mutually exclusive operations. This approach agrees with Mark s (1989) claim that in natural languages there might not exist a one-to-one mapping between possible relations and two objects. For example, people may use the sentences the paper is in the bookshelf and the paper is on the bookshelf" interchangeably to refer to the same configuration. The image schemata adopted will determinate the appropriate spatial relation. Changes of states are also possible by means of the derived constructors removefrom and jerk (Figure 8). The operation removefrom has particular characteristics with the repetition or composition of the constructors MoveInto and MoveOnto. Likewise, the operation jerk results interesting when applied over an object placed on a surface. These operations reveal differences in the behavior of the surface and container schemata. removefrom :: (Object, Object, Scene) -> Scene jerk :: (Object, Object, Scene) -> Scene Figure 8: Changing a scene through derived constructors Difference between Surface and Container Specifications Since the basic constructors of our specification are MoveInto and MoveOnto, these operations are not subject of constraints. Thus, differences between surface and container are specified within the axioms that constrain the behavior of derived constructors and observers. Figure 8 illustrates one of these differences. An operation to remove an object A from an object B, either a surface or a container, when B is inside of a container C, results in the object A remaining in the container C, at least for some non-zero time (Figure 9a). On the other hand, an operation to remove an object A from an object B, either a surface or a container, while B is on a surface C, results in the object A

11 being also off the surface C (Figure 9b). While in the first case the spatial relation is hold, in the second one it does not longer exist. C C A B A B (a) A B C A B C (b) Figure 9: Difference in the operation removefrom. The operation jerk has an additional effect to the normal operation remove when a surface is involved (Figure 10). If an object A is on a surface B, and B is on another surface C, the jerk operation over B removes B from C and moves A onto C. b a c a b c Figure 10: Operation jerk over a surface Spatial Relation Axioms Basically, the specification reveal three axioms of spatial relations. However, the transitive property reflected in these axioms should be verified with human-subjects testing. Pseudo code of the axioms presented in Tables 5, 6, and 7 is based on the algebraic specification written using the functional programming language Gofer. The spatial relation in results from an operation move into and is transitive for subsequent move into or move onto operations (Table 5). isin(o1, o2, CreateScene(set)) = False isin(o1, o2, MoveInto(o3, o4, sce)) = True if (isin(o1, o3, sce) or (o1 == o3)) and (isin(o3, o2, sce) or ison(o3, o2, sce) or (o4 == o2)) otherwise isin(o1, o2, sce) isin(o1, o2, MoveOnto(o3, o4, sce)) = False if ((o1 == o3) and (o2 == o4)) otherwise True if (isin(o1, o3, sce) and (ison(o3, o2, sce) or isin(o3, o2, sce) or (o4 == o2)) otherwise isin(o1, o2, sce) Table 5: Axioms for object o2 is in object o1

12 The spatial relation on results from an operation move onto and is transitive for subsequent move onto or move into operations (Table 6). ison(o1, o2, CreateScene(set)) = False ison(o1, o2, MoveInto(o3, o4, sce)) = False if (o1 == o3) and (o2 == o4) otherwise True if ison(o1, o3, sce) and (isin(o3, o2, sce) or ison(o3, o2, sce) or (o4 == o2)) otherwise ison(o1, o2, sce) ison(o1, o2, MoveOnto(o3, o4, sce)) = True if (ison(o1, o3, sce) or (o1 == o3)) and (ison(o3, o2, sce) or isin(o3, o2, sce) or (o4 == o2)) otherwise ison(o1, o2, sce) Table 6: Axioms for object o2 is in object o1 The spatial relation contact is result of an operation move onto and is not transitive for subsequent move onto or move into operations (Table 7). isincontact(o1, o2, CreateScene(set)) = False isincontact(o1, o2, MoveInto(o3, o4, sce)) = isincontact(o1, o2, sce) isincontact(o1, o2, MoveOnto(o3, o4, sce)) = True if ((o1 == o3) and (o2 == o4)) otherwise isincontact(o1, o2, sce) Table 7: Axioms for object o2 in contact with object o1 5. The Application of the Container-Surface Algebra to a Larger-Scale Space Exploring the idea that scale may affect the coherence of our container-surface algebra, we analyzed sentences of the natural-language that comprise large geographic configurations and use spatial relations in and on. Consequently, we assume that these configurations involve container and surface image schemata. Like in the room space, we selected configurations that represent a complete treatment of situations between surfaces and containers. Once spatial configurations were selected, we applied the constructors, as we did apply in the room space, and we obtained the spatial relations from the container-surface algebra. The result of this exercise is presented in Table 8. A basic problem with this analysis is that the behavior of objects in a large-scale space differs from the behavior of objects in a small-scale space. A large-scale space contains non-manipulable objects that are larger than the human body and require motion to experience them (Kuipers 1978, Zubin 1989, Freundschuh and Egenhofer 1997). Geographic objects might be considered static objects and therefore, they might not be moved into or onto other objects. Since the operations move into and move onto are the basic constructors of the container-surface algebra, this algebra would not be applicable to geographic objects. However, the notion of geographic objects in a map space (Freundschuh and Egenhofer 1997) allows to metaphorically project the operations of a small-scale space to those operations in a large-scale space. Accepting the sensibility of the container-surface algebra for a large-scale space, the derived spatial relations in and on seem to be consistent with a commonsense judgment. The relation in contact, however, presents some discrepancy. For example, consider the configuration of a mountain is in a national park on a peninsula. Based on the container-surface algebra, the mountain is on but not in contact with the peninsula. However, someone might argue that the mountain is in contact with the peninsula since there is no physical boundary that separates these two geographic objects. Unlike this configuration, all the configurations in the room space include containers with physical boundaries that separate the exterior from the interior. Assuming containers with physical boundaries, the container-surface algebra seems to be valid for configurations that involve objects of the similar scale. For being applicable to geographic objects in a non-map space, it would be necessary to define constructors representative of the

13 behavior of these geographic objects. The observers (spatial relations) defined in terms of these new constructors should be equivalent to the observers defined with the constructors for the smallscale space. No satisfied with the analysis of a larger-scale space, we explored configurations where there is a combination of objects belonging clearly to different scales. Following the same approach to the large-scale space, we applied the operations and we obtained the spatial relations (Table 9). In this case, the results showed a similar problem with respect to the sensible application of the constructors as they did show in a large-scale space. In addition to this problem, spatial relations present discrepancies that reflect a scale dependency of the container-surface algebra. Putting a sheet of paper on a table that is on a peninsula does not imply that the paper is on the peninsula. This examples breaks the transitive property of the algebra for the spatial relation on. Even more, it seems that for combinations of objects far a part in a ranking of scales, the spatial relation in might fail. For instance, consider a paper in a room in a house, and continue throughout even larger objects such as a peninsula in a national park. It might be arguable to say that the paper is in the national park. 6. Conclusions and Future Work In this paper we presented spatial relation algebra with a small set of cognitive plausible operators. These operators define the spatial relations in, on, and in contact. The innovative approach of this work is the formalization of these operations based on the behavior of two basic image schemata, surface and container. By applying the container-surface algebra within different scales, the transitive properties of the operators in and on were found to differ from how people would reason about them. Therefore, it might be interesting to explore whether this discrepancies reveal a change in the scale and types of space. An area for further investigation is to explore how the spatial relation algebra is affected by incorporating the part-whole image schemata. Part-whole schema appears to be an important factor for discriminating spatial relations. In addition, by modifying the algebra to incorporate liquids and their behavior, these operations might be used in broader range of real world situations. Finally, like the work by Mark and Egenhofer (1994), we believed that a human-subjects testing should be apply to verify if the container-surface algebra corresponds to how people reason about the spatial relations. In particular, human testing can help to determine when the transitive properties of these relations is not longer sensible. 7. Acknowledgments We want to thank Kathleen Hornsby and Doug Flewelling for their contributions on this paper. We are also grateful to Christian Sanhueza who helped us with the figures. 8. References J. Albrecht (1996) Universal Analytical GIS Operations. Ph.D. Thesis, University of Vechta, Germany. R. Bailey (1990) Functional Programming with Hope. Ellis Horwood, London. M. Egenhofer (1994) Deriving the Composition of Binary Topological Relations. Journal of Visual Languages and Computing 5(2): M. Egenhofer and D. Mark (1995) Naive Geography. in: A. Frank and W. Kuhn (Eds.), Spatial Information Theory A Theoretical Basis for GIS, International Conference COSIT 95, Semmering, Austria. Lecture Notes in Computer Science 988, pp. 1-15, Springer-Verlag, Berlin. A. Frank (1991) Qualitative Spatial Reasoning about Cardinal Directions. in: D. Mark and D. White (Eds.), Autocarto 10, Baltimore, MD, pp

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