Preferred and Alternative Mental Models in Spatial Reasoning

Spatial Inference Tasks: Intervals In One Dimension Artificial Intelligence (AI) research on qualitative spatial reasoning has developed several sets of spatial relations (e.g., Cohn & Hazarika, 2001), and we chose the set of thirteen interval relations by Allen (1983) for the following reasons: The calculus is well investigated from a formal point of view (Ligozat, 1990) and has some interesting computational properties (Nebel & Bürckert, 1994). It is used in applications dealing with small-scale spaces like webpage design (Borning, Lin, & Marriott, 1997, 2000), as well as in applications with large-scale spaces like Geographic Information Systems (GIS, e.g., Longley, Goodchild, Maguire, & Rhind, 2001). 244 RAUH, HAGEN, KNAUFF, KUSS, SCHLIEDER, STRUBE The set of interval relations was originally introduced by James Allen (1983) for temporal reasoning with intervals, but was soon transferred to the domain of one-dimensional spatial reasoning (e.g., Freksa, 1991). In Table 1, the 13 interval relations are listed, together with the verbalizations that we used in our experiments. For convenience, a diagram is provided displaying the relationship between the two intervals X and Y, whereas the geometric semantics are given in the last column, where the ordering of start points and endpoints is given. Learning Interval Relations The aforementioned interpretation problem is resolved as follows: Before our participants solve reasoning problems, they have to read descriptions of the spatial relationship of a red and a blue interval using the 13 qualitative relations (in German) in the first phase of all our experiments—the definition phase. Each verbal description is presented with a short commentary about the location of the start point and endpoint of the two intervals, together with a diagram with a red and blue interval that match the description. In a subsequent phase—the learning phase—we test to see if the participants understand the relations: The learning phase consists of trial blocks, during which participants are presented with the one-sentence description of the red and blue interval (e.g., “The red Table 1 The 13 qualitative interval relations, associated natural language verbalizations, a graphical realization, and ordering of start points and endpoints (adapted and augmented according to Allen, 1983). Allen’s temporal term Relation symbol Natural language verbalization Graph. ex. Point ordering s=start point e=endpoint before X < Y X lies to the left of Y sX Y X lies to the right of Y sY – . For the above example, the short-hand notation would be “o – di.” To evaluate participants’ inferences, it is necessary to know which relationships can hold between the end terms of an interval-based three-term series problem. These possible relationships can also be regarded as models in the usual logical sense. Therefore, we will use the terms solution and models in the following synonymously. In Table 2, the complete list of solutions/models for all three-term series problems that can be constructed with the 12 interval relations (omitting the trivial “=” relation) is given. In this composition table, one can read off the possible relationships of two intervals X and Z in the corresponding cell of the table, given the composition of interval relation of X and Y (rows) and the interval relation of Y and Z (column). The example above also shows that there are many three-term series problems that have more than one solution: For the three-term series problem “o di,” there are five possible relationships between X and Z, namely “X < Z,” “X m Z,” “X o Z,” “X fi Z,” or “X di Z.” In total, there are 42 three-term series problems that have three solutions, 24 that have five solutions, 3 that have nine, and another 3 that have thirteen solutions. These indeterminate problems play a major role in the subsequent experiments. Note that the number of models can be unequivocally counted. Experiment 1: Existence of PMMs In this experiment, we explore whether people come up with the same spatial configuration to inference problems that have multiple solutions. If the construction of initial solutions in spatial configuration problems could be accomplished solely by an idiosyncratic procedure then there would be no reason to prefer one configuration over the other. As a consequence, for all indeterminate tasks, one should observe that the frequencies of generated solutions should follow an equal distribution. On the other hand, if a general 246 RAUH, HAGEN, KNAUFF, KUSS, SCHLIEDER, STRUBE model construction process exists, people should come up with the same mental model. One should observe preferences for certain solutions for each task.