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<Paper uid="C88-2156">
  <Title>Figuring out Most Plausible Interpretation from Spatial Descriptions</Title>
  <Section position="3" start_page="764" end_page="766" type="metho">
    <SectionTitle>
2 The Potential Model
</SectionTitle>
    <Paragraph position="0"> At the center of potential model is a potential function, which gives a value indicating the cost for accepting the relation to hold ainong a given set of arguments. The lower is the value provided by a potential function, the more plausible is the corresponding relation. We allow the value of potential functions to range from 0 to potential trace of gradual approximation</Paragraph>
    <Paragraph position="2"> +co. A potential function may give a minimal value for more than one combination of arguments. S~dt case may be taken as an existence of ambiguity.</Paragraph>
    <Paragraph position="3"> A primitive potential function is defined for each spatial relation. A potential function for overall situation is constructed by adding primitive potential functions for spatial relations involved. null When a potential function is formulated from a given set of information, the system will seek iora combination of arguments which may minhnize the value of potential function. We use a gTadual approximation method to obtain an approximate solution. Starting from an appropriate combination of arguments, the system changes the current set of values by a small amount proportional to a virtual force obtained by differentiating the potential function. This process will be repeated until the magnitude of virtual fo:cce becomes less than a certain threshold. Figure 1 illustrates those idea.</Paragraph>
    <Paragraph position="4"> Unfortunately, using the gradual approximation ,nay not find a combination which makes a given potential function minimum. When there are some locally minimal solutions, this method will terminate with a combination appropriately near one of them. Which nfinimal nohtion is chosen depends on the initial set of argmnents. We assume there exists some heuristic which predicts a suttldently good i,fitim values and the above approximation process works rather as an adjustment than as a means for finding solution.</Paragraph>
    <Paragraph position="6"/>
    <Section position="1" start_page="764" end_page="764" type="sub_section">
      <SectionTitle>
2.1 The Spring Model
</SectionTitle>
      <Paragraph position="0"> We use an imaginary, virtual mechanical spring between constrained objects to represent constraint on distance. If the distance between the two objects is equal to the natural length of the spring, the relative position is most plausible.</Paragraph>
      <Paragraph position="1"> The more extended or compressed the spring, the more (virtual) force is required to maintain the position, corresponding to the interpretation being less plausible.</Paragraph>
      <Paragraph position="2"> An integration of the force needed either to extend or compress the spring is called an elastic potential. TILe spring model, subclass of the potential model, takes an elastic potential as a potential function. Let the positions of two objects connected by a spring of natural length L and elastic constant It&amp;quot; be (x0, y0) and (xl,yl), respectively. Then the potential is given by the following formula: P(xo, Yo, xx, Y*) = I((v~l: xdeg)2 + (y' -- ydeg)2 - L)~ 2 See figure 2 for the shape of this function.</Paragraph>
    </Section>
    <Section position="2" start_page="764" end_page="765" type="sub_section">
      <SectionTitle>
2.2 Inhibited Region and Inhib-
</SectionTitle>
      <Paragraph position="0"> ited Half Plane Unlike other primitive potential functions introduced so far, inhibited region and half plane pose a discontinuous constraint on the possible region of position. By inhibited region and half plane we mean a certain region and half plane is inhibited for an object to enter,.respectively. Inhibited regions and half planes are not global in the sense that each is defined only for some particular object. Inhibited region is less basic concept because it can be represented by a logical combination of inhibited half plane.</Paragraph>
      <Paragraph position="2"> An inhibited half-plane is chaxacterized by its directed boundary line. A directed boundary line in turn is characterized by the orientation 0 (measured counter-clockwise from the orientation of x-axis) and a location (X, Y) of a point (referred to as a characteristic point) on it. The inhibited half plane is the right hand side of the directed boundaa'y.</Paragraph>
    </Section>
    <Section position="3" start_page="765" end_page="766" type="sub_section">
      <SectionTitle>
2.3 Directional Potential
</SectionTitle>
      <Paragraph position="0"> Suppose we want to represent a constraint that an object B is to the direction 0 of another object A (measured counter-clockwise). Let the position of A and B be (x0, y0) and (Xl, yl), respectively. We use the following potential function to represent the constraint: P(xo, Yo, xl,y,) .-'. I'\[1(-(xl - xo) sin 0 + (Yl - Yo) cos ~) 2 + 1(2 When viewed horizontally from A, this function represents a hyperbola. If this function is cut vertically to the intended direction, this represent a parabola (upside down). See figure 3 for the shape of this function. Note that the notion of direction defined here denotes that in everyday life, which is not very rigid.</Paragraph>
      <Paragraph position="1"> Since the value of the potential function defined above P jumps from +oo to -co if one proceeds for the -0 direction.</Paragraph>
      <Paragraph position="2"> We add inhibited half planes in the - 0 direction, so that it is impossible to put the object in this region.</Paragraph>
    </Section>
  </Section>
  <Section position="4" start_page="766" end_page="766" type="metho">
    <SectionTitle>
3 A Method of Gradual
Approximation
</SectionTitle>
    <Paragraph position="0"> A maximally plausible position is obtained by revising a tentative solution repeatedly.</Paragraph>
    <Paragraph position="1"> The move/~ = (Ax, A~) at each step is given as follows:</Paragraph>
    <Paragraph position="3"> where K is a positive constant.</Paragraph>
    <Paragraph position="4"> This basic move may be complicated by taldl~g inhibited regions into account. The following subsection explain how it is done.</Paragraph>
    <Section position="1" start_page="766" end_page="766" type="sub_section">
      <SectionTitle>
3.1 Avoiding to Place Ob-
</SectionTitle>
      <Paragraph position="0"> jects within its Inhibited Haft Plane An algoritlu~n ior escaping from inhibited halt plane is applied when an object is placed within its inhibited half plane. If such a situation is detected, the algoritl~un defined below will push the object out of an inhibited half plane in n steps. At this time, any influences from other constraints axe taken into account. Thus, the move d = (d~, dr) of the object at each step is the sum of dr) I = (dv~,dv,) (a component vero tical to the boundary) and d~o = (dp.,dp,) (a component in par,'flld to the boundary). Supdeg pose the initial position of an object is (x0, y0), then each of which is defined as follows:</Paragraph>
      <Paragraph position="2"> resents the distance from the initial position to the boundary of the inhibited half planedeg Note that the inhibited half plane is characterized by its directed boundary with a characteristic point (X, Y) and the orientation 0.</Paragraph>
      <Paragraph position="3"> dvo = V(I~ co~ 20 + 1~ sh 0 cos 0) up, = c(1: sin o cos o + 1~ sin ~o) where, C is a positive co.rant, and / = (f., \],) is a virtual force from other constraints. Figure</Paragraph>
    </Section>
    <Section position="2" start_page="766" end_page="766" type="sub_section">
      <SectionTitle>
Inhibited Region
</SectionTitle>
      <Paragraph position="0"> Once ~m object has been put out of an in-Mbited \]taft plane, one must want it not to ihave it re-enter the same inhibited half plane.</Paragraph>
      <Paragraph position="1"> However~ the gradual approximation algorithm may try to push the object there again. An algorithm for avoiding to push objects into it watches out for such situation. If it detects, it will recourse the gradual move.</Paragraph>
      <Paragraph position="2"> Suppose an inhibited half plane is characterized by 0 and (X, Y) on the boundary. Suppose aho that the next position suggested by the gradual approximation algorithm is (x, y).</Paragraph>
      <Paragraph position="4"> then, the next position will be forced into the h ddbited half plane; In, such a case, the move is x aoditled and the new destination becomes: (:d, y') =: (x - (1 + e)LsinO, y + (1 + e)Lcos O) where, e is a positive infinitesimal.</Paragraph>
      <Paragraph position="5"> See figure 5.</Paragraph>
    </Section>
  </Section>
  <Section position="5" start_page="766" end_page="768" type="metho">
    <SectionTitle>
302 Dependency
</SectionTitle>
    <Paragraph position="0"> It would require a great amount of comput&amp;.</Paragraph>
    <Paragraph position="1"> tion, if the position of all objects have to be deter~fined at once. Fortunately, human-human commmtication is not so nasty as this is the case; natural language sentences contain many cues which help the hearer understand the inpttt. ~br example, in normal conversations, the utter~uce Kyoto University is to the north of</Paragraph>
    <Section position="1" start_page="766" end_page="766" type="sub_section">
      <SectionTitle>
Kyoto Station
</SectionTitle>
      <Paragraph position="0"> is given in the context in which the speaker has already given the position of Kyoto Station, or s/he can safely assume the hearer knows that fact. If such a cue is carefully recognized, the amount of computation must be significantly reduced.</Paragraph>
      <Paragraph position="1"> Dependency is one such cue. By dependency we mean a partial order according to which position of objects are determined. SPRINT is designed so that it can take advantage of it.</Paragraph>
      <Paragraph position="2"> Instead of computing everything at once, the spatial reasoner can determine the position of objects one by one. An object whose position does not depend on any other objects is chosen as the origin of local coordinate. SPRINT determines the temporary position of objects from the root of the dependency network. The position of an object will be determined if the position of all of its predecessors is determined. -Figure 6 shows how SPRINT does this.</Paragraph>
      <Paragraph position="3"> This algorithm has three problems:  1. the total plausibility may no~ be maximal. 2. in the worst case, the above may result in contradiction.</Paragraph>
      <Paragraph position="4"> 3. objects may be underconstrained.</Paragraph>
      <Paragraph position="5">  Currently, we compromise with the first problem. More adequate solution may be to have an adjustment stage after initial contlgnlation of objects are obtained. The second problem will be addressed in the next subsection. The third remains as a future problem.</Paragraph>
    </Section>
    <Section position="2" start_page="766" end_page="768" type="sub_section">
      <SectionTitle>
3.3 Resolving Contradiction
</SectionTitle>
      <Paragraph position="0"> Adding new information may result in inconsistency. In order to focus an attention to this  GIVEN TEXT: (1) Kyoto University is to the north of Kyoto Station. (2) Ginkakuji(temple) is to the northeast of Kyoto Station. (3) Kyoto University is to the west of Ginkakuji(temple).  GIVEN TEXT: (1) Mt.Hiel is to the north of Kyot,o Station. (2) Kyoto University is to the north of Kyoto Station. (3) Shugakuin is to the north of Kyoto University. (4) Shug~kuin is to the south of Mt.Hiei.  problem, let us temporalily restrict the spatial coordinate as one-dimensional. Suppose an object is given a maximally plausible position x0. Suppose also that a new inhibited region (interval I in a one dimensional world) is given as a new constraint. Then the position of the object is recomputed so as to take this new constraint into account. If the interval I accidentally involves x0, then the object may be moved out of interval I. This is the situation in which the object tends to move to the position x0 but cannot due to the inhibited half space. In this case, the parent node in the dependency is tried to move in the reverse direction to resolve this situation. A situation is worse than the above ff the inhibited region (or interval) is too wide to fit in a space. This problem rises especially when we  take size into account. Suppose the position of two objects A and B are already given maximally plausible positions x0 and xl(xo &lt; xl), respectively. Suppose now the third object C with width being wider than Xl -x0 is declared to exist between A and B. This causes a failure because there is no space to place C.</Paragraph>
      <Paragraph position="1"> The solution to this problem comprises in two stages. First, the reason of the failure is analyzed. Then, parents of the current objects are moved gradually so that the inconsistency can be removed. Figure 7 illustrates how this works.</Paragraph>
    </Section>
  </Section>
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