Modeling the Basic Meanings of Path Relations

In the field of spatial reasoning, point-to-point relations have been thoroughly examined, but only l i t t le attention has been payed to the modeling of path relations. We propose a computational model that extends the existing referential semantics for point-to-point relations to path relations. On the linguistic side, we present some research on German path prepositions as well as results on their English counterparts. This analysis of path prepositions is used to extract a semantic model for path relations. On the geometric side, we examine the characteristics of trajectories and propose a computational method to find an appropriate path relation for a given si tuat ion. Finally, we show how our findings on the linguistic and the geometric sides can be brought together to form a consistent model. 1 I n t r o d u c t i o n Space plays a central role in human cognit ion, and has therefore been a research focus in different disciplines like (computational) linguistics [Lakoff, 1987], cognitive sciences [Kosslyn, 1994], psychology [Landau and JackendofF, 1993], and artif icial intelligence [MaaB et a/., 1993]. Sophisticated conceptual [Egenhofer, 1991] and computat ional models [Gapp, 1994] have been developed that made it possible to compute the appropriateness of spat ia l relations in specific situations, thereby providing a better understanding of what is meant by certain spatial expressions. These results paved the way for intelligent systems that are able to analyze and generate natural language descriptions of space [Wahlster et al, 1998]. Wi th in the field of spatial relations, so-called topological (e.g. near or a t ) and projective relations (like r i g h t o f or above) have been most thoroughly examined. Both groups are point-to-point relations as they establish a spatial relation between two objects of arbitrary shape. (Between is an exception from this rule as it requires at least three objects to be computed correc ; t ly[Habel, 1989].) A different kind of spatial relations are the so-called path relations (e.g. a long , around, or pas t ) . Much less attention has been payed to their conceptualization [Kriiger and Maafi, 1997] and computat ion than in the case of point-to-point relations. This may be due to the greater complexity: the problem of computing the most appropriate path relation can only be solved if there is a path which consists at least of a simple line with a start ing point and an ending point. As we wi l l show in section 5, topological, projective, and path relations share nevertheless several (geometric) concepts. In section 2, we present the basic ideas and concepts we wil l use throughout the paper. Based on an analysis of the linguistic side of the problem (section 3), we propose a semantic model which is described in section 4. In the fol lowing section, we turn to the geometric side, and establish a basic framework for geometric path relations. These two sides are integrated to form a consistent model which then is applied to various examples (section 6). Finally, we summarize our findings and give an outlook on future directions. 2 K e y C o n c e p t s a n d R e l a t e d W o r k Following [Herskovits, 1986] we distinguish between the basic meaning of a spatial relation and its instantiation in a concrete si tuat ion: An object to be localized (LO) is set, in relation to a reference object (RO). Furthermore, a frame of reference has to be established in order to distinguish different spatial relations. Determining the origin and the orientation of the reference frame depends on a mul t i tude of factors such as the point of view of the observer/addressee, or the intrinsic orientation of the RO [Maafi, 1993]. The computat ion of topological 384 COGNITIVE MODELING Table 1: Several German preposit ions used for path descript ion prototypical meaning, on a scale f rom 'zero' (not appl icable) to 'one' ( ful ly appl icable). For a detailed descript ion of the factors and a lgor i thms used to determine the DA of pointto-point relations, refer to [Gapp, 1997]. Path relations differ f rom their topogical and projective counterparts in two ways. On one side, the LO is expected to be pathl ike: either its shape has to be pathlike, or it can be abstracted to a pathl ike shape. In some cases, this also holds for the RO. ( In the computa t ion of the appl icabi l i ty of po in t to -po in t relat ions, the shape of the LO is of lesser importance.) On the other side, the computa t ion of path relat ions cannot be reduced to a simple two point p rob lem. Figure 1 i l lustrates this fact: Trajectory (a) is certainly a better match for a relat ion a l o n g i (describing a path tha t fol lows the form of the RO) than is t ra jectory (b) . B u t there is no single point on either t ra jec tory tha t can be used to determine the appl icabi l i ty of this re la t ion. (Wh i le one may argue about the meaning of u a long " , a l o n g i might actual ly capture a meaning facet.) 3 Analysis of German Path Preposit ions From a computa t iona l perspective, one of the main problems in understanding natura l language is its inherent ambigui ty . Th is is also t rue for path preposit ions: The German path preposi t ion u z u " (roughly " t o " , " to wards" ) , for example, is used to describe trajectories tha t lead towards the reference object. The descript ion "der VVeg zu dem Park" ( u the way to the pa rk " ) can mean different th ings. It is not obvious where the traWe use the term '(path) preposition' although, from a strictly linguistic perspective, not all of them are prepositions per se. and project ive relat ions relies on a frame of reference. It is used to extract, the two essential parameters that are needed to evaluate the app l icab i l i t y of po in t to -po in t relations: the distance of the LO f rom the RO (topological relations), and the angle d ispar i ty f rom a prototypica l direct ion (project ive relat ions). Figure 1: T w o trajectories [Gapp, 1997] proposed a model for spatial relations that is based on a three level referential semantics. On the lowest level, only purely visual in fo rmat ion is avail able. Th is i n fo rma t ion is abstracted on the semantic level to a geometr ical representation which the referent ia l semantic i tself relies on. Idealized meanings of spat ia l relat ions are compared to the actual s i tuat ion tak ing into account contextual factors tha t are modeled on the conceptual layer. A degree of applicability (DA) is computed tha t rates how well a re lat ion corresponds to the KRAY AND BLOCHER 385 jectory starts, nor where it ends: Does it end outside the park, just at the border, or inside? The verbalization process is affected by ambiguity, too: To describe a path that starts wi th in the park and leads outside of i t , one could, for example, use "aus" (approx. "out o f " ) or "von" (approx. "from'). Table 1 lists some of the most commonly used German path prepositions. We tried to express the basic meanings of the main uses in natural language and at a finer level of detail in a more formal syntax. It should be stated that the table does not contain a complete description of all possible meanings of path prepositions but only meaning facets. The formal syntax is based on a subset of Egenhofers semantics for topological relations [Egenhofer, 1991]. " I " stands for containment of the LO wi th in the HO (inside), "M for contact of the boundaries of LO and RO (meet), and ''D" for the disjoints ss of them (disjoint). So, a formula like