Figuring out Most Plausible Interpretation from Spatial Descriptions 
Atsushi Yamada,Toyoaki Nishida and Shuji Doshita 
Department of Information Science 
Kyoto University 
Sakyo-ku, Kyoto 606, Japan 
phone: 81-75-751-2111 ext. 5396 
emaih yamada or nishida%doshita.kuisokyoto-u.junet%japan~relay.cs.net 
Abstract 
The problem we want to handle in this paper 
is vagueness. A notion of space, which we basi- 
cally have, plays an important part in the fac- 
ulty of thinking and speech. In this paper, we 
concentrate on a particular class of spatial de- 
scriptions, namely descriptions about positional 
relations on a two-dimensional space. A theo- 
retical device we present in this paper is called 
the potential model The potential model pro- 
vides a means for accumulating from fragmen- 
tary information. It is possible to derive max- 
imally plausible interpretation from a chunk of 
information accumulated in the model. When 
new information is given, the potential model 
is modefied so that that new information is 
taken into account. As a result, the interpre- 
tations with maximal plausibility may change. 
A program called SPRINT(SPatial Relation IN- 
Terpreter) reflecting our theory is in the way of 
construction. 
i Introduction 
Natural language description is vague in many 
ways. The real world described with natural 
language has continuous expanse and transi- 
tion, although the natural language itself is a 
descrete symbolic system. Vagueness plays an 
important role in our communication in that 
it allows us to transfer partial information. 
Suppose a situation in which a boy is looking 
around for his toy. Even if we cannot tell ex- 
764 
actly where it was if we know it was somewhere 
around my desk~ we can transfer him this par~ 
tim information by telling that his toy is around 
my desk. It would be nice if we can communio 
cute with our robot in the same way. We also 
use vague expression in the case of thinking it 
useless to give more detailed information to th6 
hearer. 
A theoretical device we present in this pa- 
per for the interpretation of such wague infor- 
mation is called potential model. The potem 
tial model employs both continuous and clis~ 
continuous functions to represent spatial rela- 
tions, so that the probability changes either 
continuously or discontinously, depending on 
the nature of a given description. Currently, we 
are concentrating on a particular class of spa- 
tial relations, namely positional relations on a 
two-dimensional space, although the potential 
model is more general. We assume objects to 
be sizeless. 
A program called SPRINT (SPatial Relation 
INTerpreter) reflecting our theory is in the way 
of construction. 
2 The Potential Model 
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 pro- 
vided by a potential function, the more plausi- 
ble is the corresponding relation. We allow the 
value of potential functions to range from 0 to 
potential 
trace of gradual approximation 
X 
Figure 1: Potential Model and Gradual Ap- 
proximation 
+co. A potential function may give a minimal 
value for more than one combination of argu- 
ments. S~dt case may be taken as an existence 
of ambiguity. 
A primitive potential function is defined for 
each spatial relation. A potential function for 
overall situation is constructed by adding prim- 
itive potential functions for spatial relations in- 
volved. 
When a potential function is formulated from 
a given set of information, the system will seek 
iora combination of arguments which may min- 
hnize the value of potential function. We use 
a gTadual approximation method to obtain an 
approximate solution. Starting from an appro- 
priate 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 thresh- 
old. Figure 1 illustrates those idea. 
Unfortunately, using the gradual approxima- 
tion ,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 ap- 
propriately 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,fi- 
tim values and the above approximation process 
works rather as an adjustment than as a means 
for finding solution. 
1 / ~/ X=Xe KL'/ t~,, / 
• Y,,- ( m, y,B 
Figure 2: Distance Potential 
2.1 The Spring Model 
We use an imaginary, virtual mechanical spring 
between constrained objects to represent con- 
straint on distance. If the distance between the 
two objects is equal to the natural length of the 
spring, the relative position is most plausible. 
The more extended or compressed the spring, 
the more (virtual) force is required to maintain 
the position, corresponding to the interpreta- 
tion being less plausible. 
An integration of the force needed either to 
extend or compress the spring is called an elas- 
tic potential. TILe spring model, subclass of the 
potential model, takes an elastic potential as a 
potential function. Let the positions of two ob- 
jects connected by a spring of natural length L 
and elastic constant It" be (x0, y0) and (xl,yl), 
respectively. Then the potential is given by the 
following formula: 
P(xo, Yo, xx, Y*) = I((v~l: x°)2 + (y' -- y°)2 - L)~ 2 
See figure 2 for the shape of this function. 
2.2 Inhibited Region and Inhib- 
ited Half Plane 
Unlike other primitive potential functions in- 
troduced 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 ba- 
sic concept because it can be represented by a 
logical combination of inhibited half plane. 
765 
I(x=, y,) 
j 0 
Figure 3: Directional Potential 
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 orien- 
tation 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. 
2.3 Directional Potential 
Suppose we want to represent a constraint that 
an object B is to the direction 0 of another ob- 
ject A (measured counter-clockwise). Let the 
position of A and B be (x0, y0) and (Xl, yl), re- 
spectively. We use the following potential func- 
tion to represent the constraint: 
P(xo, Yo, xl,y,) .-'. I'\[1(-(xl - xo) sin 0 + (Yl - Yo) cos ~) 2 + 1(2 
When viewed horizontally from A, this func- 
tion represents a hyperbola. If this function 
is cut vertically to the intended direction, this 
represent a parabola (upside down). See figure 
3 for the shape of this function. Note that the 
notion of direction defined here denotes that in 
everyday life, which is not very rigid. 
Since the value of the potential function de- 
fined above P jumps from +oo to -co if one 
proceeds for the -0 direction. 
We add inhibited half planes in the - 0 direc- 
tion, so that it is impossible to put the object 
in this region. 
766 
3 A Method of Gradual 
Approximation 
A maximally plausible position is obtained by 
revising a tentative solution repeatedly. 
The move/~ = (Ax, A~) at each step is given 
as follows: 
• ~ = (Az, A~) = K. (OP\]Ox, OP/Oy)~ 
where K is a positive constant. 
This basic move may be complicated by taldl~g 
inhibited regions into account. The following 
subsection explain how it is done. 
3.1 Avoiding to Place Ob- 
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). Sup° 
pose the initial position of an object is (x0, y0), 
then each of which is defined as follows: 
dv~ = -L sin 0In 
dv~ = L cos 0/u 
where, L : I(xo- Y)~in 0. (~0 - Y)cos 01 rep 
resents the distance from the initial position to 
the boundary of the inhibited half plane° Note 
that the inhibited half plane is characterized 
by its directed boundary with a characteristic 
point (X, Y) and the orientation 0. 
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 
4 illustrates how this works. 
next move ~ ) 
I inhibited half plane 
position ' ~'" ~.' ..\ k N~/CYx'~,"N 
of the object 
effects from other constraint 
a component in parallel to 
the directed boundary 
previous position ? 
~~kkl inhibited 
~ \ ~ -(x,y) 
next position 
(generated by gradual 
approximation algorithm) 
half plane 
Figure 4: Pushing an Object Out of an Ixthib- 
ited Region 
Figure 5: Avoiding to Push an Object into an 
Inhibited Region 
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. 
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. 
Suppose an inhibited half plane is character- 
ized by 0 and (X, Y) on the boundary. Sup- 
pose aho that the next position suggested by 
the gradual approximation algorithm is (x, y). 
~f 
L = x,,;inO- ycosO- XsinO + YcosO > 0 
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. 
See figure 5. 
302 Dependency 
It would require a great amount of comput&. 
tion, if the position of all objects have to be de- 
ter~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 in- 
pttt. ~br example, in normal conversations, the 
utter~uce 
Kyoto University is to the north of 
Kyoto Station 
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. 
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. 
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 cho- 
sen 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. 
This algorithm has three problems: 
1. the total plausibility may no~ be maximal. 
2. in the worst case, the above may result in 
contradiction. 
3. objects may be underconstrained. 
Currently, we compromise with the first prob- 
lem. 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. 
3.3 Resolving Contradiction 
Adding new information may result in incon- 
sistency. In order to focus an attention to this 
767 
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). 
DEPENDENC~: 
Kyoto Station 
Ginkakuji ~Kyoto University (a) 
RESULT(PRODUCED BY SPRINT): 
\]-I~ITERPRETQT IOH J J " 
• ~ " llJ, l @ 
%% # J I 
__~) Kyoto University 
initial )lacement / (after Interpretation of (1)) 
• t ¢ ) 
----k-. • 2 .~" I I 
. Kyoto 
University'~ ~. ~ Ginka.kujl. - " 
(after Interpretation of (3)) l '-.. ,, ~ . "..~ 
~C'" , t / ~" trace of gradual moves 
~-~-\ ~ ~ .~ I ",, . I 
"'Q~yoto StOlon \ hfitial placement 
Figure 6: Positioning using Dependency 
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. 
BEFORE INTERPRETATION OF (4): 
I hi. TEIRpRE ~ ," ";" 'i ~.~Sh.gak.in' 
"~, ttl i$ J 11 ¢ 
" Mt.Hiei 
t t J 
............ __~d~(yot o Station 
AFTER INTERPRETATION OF (4): 
~ • s" r 
t t.Iliei 
t ¢ 
....... ...... 
\] , ~ }Kyoto University 
J "L L $ t L¶ 
........ ;._.2~,~.y°2~st±9°_ ~. ...... 
Figure 7: Resolving contradiction. 
problem, let us temporalily restrict the spatial 
coordinate as one-dimensional. Suppose an ob- 
ject is given a maximally plausible position x0. 
Suppose also that a new inhibited region (inter- 
val 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 in- 
volves x0, then the object may be moved out of 
interval I. This is the situation in which the ob- 
ject 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 in- 
hibited region (or interval) is too wide to fit in 
a space. This problem rises especially when we 
768 
take size into account. Suppose the position of 
two objects A and B are already given maxi- 
mally plausible positions x0 and xl(xo < 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. 
The solution to this problem comprises in two 
stages. First, the reason of the failure is ana- 
lyzed. Then, parents of the current objects are 
moved gradually so that the inconsistency can 
be removed. Figure 7 illustrates how this works. 
4 Related Work 
The problem of vagueness has not been stud- 
ied widely in spatial reasoning\[Kui78,Lav77, 
+~++ !i7?+W~:(++~:+ij. W~:,.& by Drew McDermott nard. 
0.1!,,::..++:..~:d+ i;i~.~,..+~ v~,~g~v.:++e.~ of spatial concept and 
i;+:+<;,/,:+~;:~+i>&~.0.:ed ,~+ '~.~++,,~:~c*,icM device called fuzz 
~,<@,<#,e;:{:!ii,, A+ .~a>:,~ bo:~c denotes a region in 
~,:"~0~,~:~, .+,,+ ~;:i~c++ <+b~e,::,a; ).m~,y 1mssib!y exist° Possi- 
i::)~g~'a;y ,~>.~" ¢;L~.c ~:x~aeea~:e ia u:nitOrm\]y positive in a 
~"e++;'.~ bo::~', ~_.~d '.+ero outside the box. 
'! :Sa=.i;~+ +~',,:~-:,.+~da+,,+,:¢<,:+~. has a cottple of drawbacks. 
'i++'2~.+~;~,;, ~;.g+,+ ~,&:+~;pv, ~ff &e reglon+ must be rectan- 
~,:u~iia~ + ~>:~: .r:.~:+~;?+c., Se.e(>~d, Davis had to have the 
'.ai'ih~+~; i{~.eir ~:pp~:oa~x:h ha+s a sig~fificant ditfi- 
+:'~i~,:V :i~+ m(+dc?~u.~: v~trious spatiM c~atcepts. For 
,~:~.~vtu~)i~e+ eh(: ~ne+~n~,tg of a'roundneas is hard to 
:rcpre~+e.~.d, w~¢~.~ f~za box, sin.ca it is still hard to 
dra~.w a~._,:~ e'~acg b~m+c+.dm.+y ~;o distinguish the re+. 
gi~,>~, wh;c_k .~ ~u'o,~_ad something from that that 
)+ ~:+.O~;o 
::.H   .dmg Remarks 
'+J;;!~+,t,.':~.(::~-: of ~+,ia~rM ia~guage description con° 
~;~h~.~ IU;:+ o~; hard issues, mdy a fragment of 
wi6.(:.i~ h;.:r~; boca addressed in this paper. Re- 
g+~ ..... ~!"':,!.:~,q~,+ ~,he i.a.~g~age understanding process as 
~+ re~c~-~-~::,,'~.,r:&e:~. (R' ~;he described worM, we have 
.:i;,~,~c.+"i o~i~ ~+~::,a~.ia~ descriptions only declaring 
A c,'~<i~.~- ,!d :m~jor problems related to this 
, k ~:;:./+~{;e:~:,~,~.~c me~tod should be developed 
~,,~ d~_:~;c_~,_+~/hae ac~,u.a:i values of the model. 
C¢~r:~'e:a~ly+ we shodd coMess that param- 
e~;<~: v+&~e~,; ~++rc detcnnhted rather subjec- 
~;~ve:~ ~:~ ~;h++t m~mple prob!e~as may be 
:::o!,v(,,d+ :!1~:~. fi~.t.:~'e, we waa~t to apply adap- 
t+ A~thougTh the c~rre, nt program is forced 
;~:~ :L.garc o~+~ ~ost plausible co~fllgura~ 
~io~ from giee.a hffo~matio~l, there do ex- 
is~ +t;~+.u+.,+:~ i'~++ width ~;hh~gs are so undercon- 
~;~>a~.~ed_ ~:b+~,~ figuri~g out temporary con- 
~i_g0~+:~+~ti(>:,_~. ia "+.adess or rather harr~dul. 
¢?~i c,i;.+i:r~;% og~e~! problems are remain ~m 
~>!,,+,+~d+ +~'iI~+e m~del should be extended as that 
~d@+.:% ca~'~ h~ve ~ize aa~d shape; initial place- 
~u,:~.e>.~+~ he,~_~.ri~ic ~tto~M be i~morporated; etc. 
Those seem to be less hm'd. ih~ f,z~ ~}~.e bd~,:az;~ 
implementation of SP.~INT :is be~g ext;e:~,ded° 

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B. Klfipeac:.~° iVtode~Jng spati~d k~m~vJ:+ 
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M.A+ Lavin. Computer Analyd,~ 0j: 
Scenes from a Moving Viewing .Point. 
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1977. 

G.S. Novak Jr. Represent;sCions of 
knowledge in a prograra ibr .~:oiw. 
htg physics problems, -.hL Proceeding~ 
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