1. No category

THE SUPERIORITY OF RELATIVE CRITERIA IN
PARTIAL MATCHING AND GENERALIZATION
Paul J. Kline
Center f o r C o g n i t i v e Science
The U n i v e r s i t y o f Texas a t A u s t i n
ABSTRACT
r e p r e s e n t a t i o n could only account f o r a s t u d e n t ' s
a b i l i t y t o remember t h e s o l u t i o n s t o p r o b l e m s t h a t
have been s o l v e d p r e v i o u s l y .
However, i f i n s t e a d
o f m e r e l y l o o k i n g f o r a f u l l match w e a r e a b l e t o
find
a
rule
whose
left-hand
side
"closely
approximates"
the
description
of
the
exercise
problem,
then
we
may
be
able
to
provide
a
computational
account
of
what
happens
when
a
s t u d e n t recognizes the s i m i l a r i t y of an e x e r c i s e
p r o b l e m to a p r e v i o u s e x a m p l e .
Some
familiar
justifications
for concluding
t h a t a d a t a e l e m e n t D is an u n a c c e p t a b l e match to a
pattern
element
P
are
examined
for
their
s u i t a b i l i t y for partial-matching applications.
All
of
these
absolute
criteria
are
found
to
be
u n a c c e p t a b l e i n t h a t t h e y r u l e o u t some p l a u s i b l e
partial
matches.
A
p a r t i a l matching program,
REIAX, i s d e s c r i b e d w h i c h j u s t i f i e s t h e c o n c l u s i o n
t h a t a data element D
i s a n u n a c c e p t a b l e match t o
a p a t t e r n e l e m e n t P on t h e g r o u n d s t h a t some o t h e r
d a t a e l e m e n t D b is a b e t t e r match to P. The use of
s u c h r e l a t i v e c r i t e r i a makes i t p o s s i b l e t o compute
a l l o f t h e k i n d s o f mappings t h a t K l i n g
[ 9 ] has
claimed
are
required
to account
for intuitions
about
similarity.
The
ability
to
form
u n r e s t r i c t i v e mappings a l l o w s RELAX t o c o n s t r u c t
g e n e r a l i z a t i o n s t h a t have e l u d e d o t h e r a p p r o a c h e s
Even i f t h i s can b e d o n e , h o w e v e r , t h e s u c c e s s
is
achieved
at
the
cost
of
a
considerable
complication
in
the
notion
of
executing
the
right-hand side of a r u l e .
I n t h o s e cases where i t
t u r n s o u t t h a t a f u l l match i s a c h i e v e d t o a r u l e ' s
left-hand
side,
executing
the
right-hand
side
amounts o n l y t o a s t r a i g h t f o r w a r d e x e c u t i o n o f a
f u l l y i n s t a n t i a t e d procedure.
However, i n cases
where o n l y a p a r t i a l match has been a c h i e v e d , t h e
r i g h t - h a n d s i d e a c t i o n s w i l l need t o b e m o d i f i e d t o
r e f l e c t the d e p a r t u r e s from a f u l l match.
This
paper w i l l o n l y b e concerned w i t h the q u e s t i o n o f
how t o p a r t i a l - m a t c h p r o b l e m d e s c r i p t i o n s i n a way
t h a t p r o p e r l y models s t u d e n t s ' a b i l i t i e s t o d e t e c t
s i m i l a r i t i e s between p r o b l e m s .
[5, 6 ] .
I
INTRODUCTION
The work d e s c r i b e d i n t h i s p a p e r grows o u t o f
an attempt to s i m u l a t e the b e h a v i o r of j u n i o r - h i g h
school students l e a r n i n g geometry.
An examination
o f t h e g e o m e t r y t e x t b o o k t h e y were u s i n g
[ 8 ] made
i t c l e a r t h a t s t u d e n t s were e x p e c t e d t o a p p r e c i a t e
the s i m i l a r i t y o f t h e i r e x e r c i s e problems t o the
examples p r o v i d e d i n t h e t e x t , and t o use t h o s e
examples a s a g u i d e i n f i n d i n g s o l u t i o n s .
Protocol
analyses o f students using t h i s t e x t suggests t h a t
s t u d e n t s do o f t e n ( b u t not always) a p p r e c i a t e the
s i m i l a r i t i e s t h a t are intended
[3].
II SOME ABSOLUTE CRITERIA
An a p p r o a c h w h i c h m i g h t be e x p e c t e d to y i e l d a
s a t i s f a c t o r y p a r t i a l m a t c h e r would b e t o examine
the o p e r a t i n g p r i n c i p l e s of a f u l l matcher f o r a
p r o d u c t i o n system and t o d e t e r m i n e w h i c h p r i n c i p l e s
c o u l d b e r e t a i n e d and w h i c h p r i n c i p l e s w o u l d have
to be discarded.
Each c o n d i t i o n i n t h e l e f t - h a n d
s i d e of a p r o d u c t i o n r u l e is a p r o p o s i t i o n which
can c o n s i s t o f c o n s t a n t s , v a r i a b l e s ( e x i s t e n t i a l l y
q u a n t i f i e d ) , or other p r o p o s i t i o n s .
A s a n example
w e have ( b e l i e v e s V s u b j e c t ( f l a t e a r t h ) ) , where
( f l a t e a r t h ) i s a n embedded p r o p o s i t i o n . The V
p r e f i x i n d i c a t e s t h a t Vsubject is a v a r i a b l e .
We
m i g h t choose t o i n t e r p r e t t h e c o n s t a n t s b e l i e v e and
f l a t a s p r e d i c a t e s and t h e c o n s t a n t e a r t h a s a n
argument o f t h e p r e d i c a t e f l a t , b u t t h e r e i s n o
need f o r t h e m a t c h e r t o b e c o n c e r n e d w i t h t h e s e
semantic i s s u e s .
As the term p r o p o s i t i o n s u g g e s t s ,
( b e l i e v e s V s u b j e c t ( f l a t e a r t h ) ; can b e t r u e o r
f a l s e , b u t what i s more i m p o r t a n t f r o m t h e p o i n t o f
view of the matcher
is
whether
there
is
some
p r o p o s i t i o n i n t h e d a t a base l i k e ( b e l i e v e s B i l l
( f l a t e a r t h ) ) w h i c h can s e r v e a s i t s i n s t a n t i a t i o n .
One way t o model such b e h a v i o r i s t o r e p r e s e n t
the worked-out
examples as
<condition 1 ,.........>
=>
<action1,...>
rules,
where
the
left-hand
side
d e s c r i b e s t h e example p r o b l e m and t h e r i g h t - h a n d
side i n d i c a t e s the steps required f o r i t s s o l u t i o n .
The d e s c r i p t i o n o f a n e x e r c i s e p r o b l e m can t h e n b e
compared
against
the
left-hand
sides
of
these
rules.
I n t e r p r e t i n g these r u l e s a s p r o d u c t i o n s
would i m p l y t h a t t h e d e s c r i p t i o n o f t h e e x e r c i s e
problem
would
have
to
completely
satisfy
all
c o n d i t i o n s o f one o f t h e r u l e s b e f o r e any a c t i o n
could be taken towards f o r m u l a t i n g i t s s o l u t i o n .
In o t h e r words, a f u l l - m a t c h comparison using t h i s
This
work
was
sponsored
by
NSF
grant
no.
IST-7918474 to Jane P e r l r a u t t e r and by ONR c o n t r a c t
n o . N00014-78-C-0725 t o John R . A n d e r s o n .
A f u l l m a t c h e r uses t h e f o l l o w i n g c r i t e r i a t o
decide whether a set of i n s t a n t i a t i o n s c o n s t i t u t e s
296
a s a t i s f a c t o r y match
production r u l e :
to
the
left-hand
side
of a
2. Constants
Students f i n d problems i n v o l v i n g the a l g e b r a i c
r e l a t i o n " < " very r e m i n i s c e n t o f the " > " v e r s i o n s
of those problems.
In such cases p a r t i a l matches
between these problems would have to v i o l a t e the
f u l l - m a t c h c r i t e r i o n t h a t a c o n d i t i o n and
its
i n s t a n t i a t i o n c o n t a i n the same c o n s t a n t s .
The
u n s u i t a b i l i t y of f u l l - m a t c h c r i t e r i a (1 ) and (2)
f o r p a r t i a l matching i s not c o n t r o v e r s i a l .
All
f o u r a l g o r i t h i m s reviewed i n
[ 4 ] f o r inducing a
general d e s c r i p t i o n of a concept from examples can
deal w i t h these kinds of d e p a r t u r e s from a f u l l
match.
1. There must be an i n s t a n t i a t i o n of each
condition proposition.
2. The constants contained in a c o n d i t i o n
p r o p o s i t i o n must a l s o be contained in
its instantiation.
3. There should be a one-one f u n c t i o n t h a t
a s s o c i a t e s each v a r i a b l e w i t h a unique
binding
so
that
all
condition
propositions
containing
that variable
have
instantiations
containing
its
binding.
3. V a r i a b l e Binding Requires 1-1
Functions
Imagine a person who has seen many N a t i o n a l
League b a s e b a l l games, but is watching American
League b a s e b a l l f o r the f i r s t t i m e .
In
the
N a t i o n a l League, besides p l a y i n g in the f i e l d , a
p i t c h e r takes h i s t u r n a t b a t ; i n the American
League, however, the p i t c h e r is replaced in the
b a t t i n g l i n e u p b y the designated h i t t e r .
I f the
b a s e b a l l fan t r i e s t o i n s t a n t i a t e h i s p a t t e r n f o r
N a t i o n a l League b a s e b a l l using the l i n e u p s f o r t h i s
game should he bind the v a r i a b l e V - N L p l t c h e r to the
p i t c h e r or to the designated h i t t e r ?
The c o r r e c t
answer seems to be b o t h ; to the p i t c h e r if he is
i n s t a n t i a t i n g p r o p o s i t i o n s d e s c r i b i n g the N a t i o n a l
League p i t c h e r ' s r o l e i n the f i e l d , but t o the
designated h i t t e r when i n s t a n t i a t i n g p r o p o s i t i o n s
about the N a t i o n a l League p i t c h e r ' s r o l e as a
batter.
On the o t h e r hand, an American League fan
watching N a t i o n a l League b a s e b a l l f o r the f i r s t
time has the problem of too few p l a y e r s r a t h e r than
too many. He is in the p o s i t i o n of having the same
constant, NLpitcher,
as
the
binding
of
two
different
variables,
V-ALpitcher
and
V-ALdesighitter.
4. Each c o n d i t i o n p r o p o s i t i o n should have
the
"same"
structure
as
its
instantiation.
In the simplest case
they should have the same number of
terms and corresponding terms should be
in the same o r d e r .
In some p r o d u c t i o n
systems allowances are made f o r unequal
numbers
of
arguments
or
symmetric
p r e d i c a t e s ; however, the important p o i n t
i s t h a t t h e r e i s never any ambiguity
about whether an i n s t a n t i a t i o n has the
proper s t r u c t u r a l requirements and those
t h a t d o n ' t are simply unacceptable.
A . F a i l u r e o f the Absolute C r i t e r i a
We now examine each of these c r i t e r i a in t u r n
to
determine
their
suitability
for
partial
m a t c h i n g . We w i l l f i n d t h a t w h i l e each c o n s t i t u t e s
a w o r t h w h i l e goal f o r a p a r t i a l matcher to attempt
t o achieve (so t h a t a f u l l match w i l l b e found i f
one is a v a i l a b l e ) , none of these c r i t e r i a can be an
a b s o l u t e requirement if we want to be able to
d e t e c t the same s i m i l a r i t i e s t h a t students can
detect.
1 . Every c o n d i t i o n has an i n s t a n t i a t i o n
Textbooks sometimes provide more i n f o r m a t i o n
in the statement of example problems than is
s t r i c t l y required f o r t h e i r s o l u t i o n .
This e x t r a
i n f o r m a t i o n would prevent the examples from being
seen as r e l e v a n t to the s o l u t i o n of e x e r c i s e
problems not s h a r i n g those i n e s s e n t i a l f e a t u r e s i f
w e were t o r e t a i n the f u l l - m a t c h c r i t e r i o n t h a t a l l
p r o p o s i t i o n s in a problem d e s c r i p t i o n must r e c e i v e
an i n s t a n t i a t i o n .
1
_b
F i g . 1: &) Prove RN * OY. b) Prove R'XN' - O'XY'.
Marked o b j e c t s are known to be congruent.
F i g . 1 shows two problems t h a t some students
can see as q u i t e s i m i l a r — F i g . 1b being j u s t an
angle v e r s i o n o f the problem i n F i g . 1a.
This
a p p a r e n t l y r e q u i r e s t h a t a p a r t i a l matcher accept
the
proposition
(angle
VR'
VX
VO')
as
an
i n s t a n t i a t i o n of the p r o p o s i t i o n (segment VR VO).
That segment can be matched by angle is j u s t
another
example
of matching
one
constant
to
another.
What is more i n t e r e s t i n g is t h a t the
second argument, VO, of segment must be bound to
the t h i r d argument, VO/, of angle r a t h e r than to
the second argument, VX.
Many f u l l matchers w i l l a l l o w two v a r i a b l e s t o
be bound to the same c o n s t a n t ; however, a d d i t i o n a l
mechanisms are then r e q u i r e d to deal w i t h the
inevitable
cases
where
this
is
inappropriate.
These a d d i t i o n a l mechanisms play no r o l e in p a r t i a l
m a t c h i n g , so it s i m p l i f i e s the d i s c u s s i o n to make
the one-one assumption.
297
DP p r e d i c a t e s .
A l s o , when a v a r i a b l e such as
V - N L p l t c h e r r e c e i v e s a second b i n d i n g , a l l of the
p r o p o s i t i o n s mentioning t h a t v a r i a b l e , whether they
a l r e a d y have a n i n s t a n t i a t i o n using the f i r s t
b i n d i n g or n o t , are checked to see if they have an
i n s t a n t i a t i o n using t h i s new b i n d i n g .
Thus RELAX
computes a l l o f the k i n d s o f mappings t h a t K l i n g
claims
are
necessary
for
capturing
the
correspondences between s i m i l a r problems and does
so
using
"best
use"
criteria
which
express
p r e f e r e n c e s t h a t are meaningful only to a program
t h a t i s aware t h a t i t has these o p t i o n s r e g a r d i n g
the form o f i t s mappings.
A. Best Use of the C o n d i t i o n
When p a r t i a l matching the d e s c r i p t i o n of Example 1
u s i n g the d e s c r i p t i o n of Example 2 as d a t a , the
v a r i a b l e V s q r 1 r e q u i r e s two b i n d i n g s so t h a t we can
have both of the i n s t a n t i a t i o n s
One use f o r the output from p a r t i a l matching a
s e t o f c o n d i t i o n s t o a set o f data i s i n computing
a g e n e r a l i z a t i o n , i . e . , a new s e t of c o n d i t i o n s
which r e c e i v e s a f u l l match when i n s t a n t i a t e d by
e i t h e r the o r i g i n a l c o n d i t i o n s o r the o r i g i n a l
data*
Since the o r i g i n a l c o n d i t i o n s and the
o r i g i n a l data p l a y e n t i r e l y p a r a l l e l r o l e s i n the
d e f i n i t i o n o f g e n e r a l i z a t i o n , one i s led r a t h e r
n a t u r a l l y (though not Inescapably) t o the idea t h a t
the output of the p a r t i a l match should not depend
in any e s s e n t i a l way on which set of p r o p o s i t i o n s
we c a l l data and which set we c a l l c o n d i t i o n s .
In
f a c t , in implementing RELAX we found t h a t t h e r e are
c o m p u t a t i o n a l advantages to v i e w i n g the matching
process a s searching f o r c o n d i t i o n s t o i n s t a n t i a t e
data at the same time as it is searching f o r data
to i n s t a n t i a t e conditions.
For example,
this
viewpoint
suggests
that
in
addition
to
the
requirement t h a t each i n s t a n t i a t i o n be the best
a v a i l a b l e use o f i t s data p r o p o s i t i o n , t h e r e should
a l s o be a complementary requirement t h a t each
i n s t a n t i a t i o n b e the best a v a i l a b l e use o f i t s
condition proposition.
I n t h i s t h i r d (and l a s t ) o f
RELAX's r e l a t i v e c r i t e r i a f o r p a r t i a l m a t c h i n g ,
cost
is
again measured
i n terms o f m u l t i p l e
assignments; but since i n t h i s c o n t e x t c o n d i t i o n s
are
thought
of
as
instantiations
of
data
p r o p o s i t i o n s , a m u l t i p l e assignment means t h a t a
constant i n the data i s the b i n d i n g f o r more than
one c o n d i t i o n v a r i a b l e .
The two i n s t a n t i a t i o n s
mentioned above f o r (male V - N L p l t c h e r ) are e q u a l l y
good in t h i s respect s i n c e both A L p i t c h e r and
A L d e s i g h i t t e r are b i n d i n g s of o n l y one v a r i a b l e ;
thus both i n s t a n t i a t i o n s are p e r m i t t e d
i n the
p a r t i a l match.
However, any attempt to get a
second i n s t a n t i a t i o n f o r (male V-KLcatcher) would
be ruled out by t h i s new c r i t e r i o n .
These two i n s t a n t i a t i o n s are taken from Table 1
which shows the f u l l set found by RELAX.
The 15
i n s t a n t i a t i o n s in Table 1 are the s u r v i v o r s of the
185
instantiations
that
RELAX
considered
in
computing t h i s match.
Of these 185 p a i r i n g s , 100,
or 54% were r e j e c t e d by the "best use" c r i t e r i a
w i t h o u t the need f o r any search at a l l .
In 21
cases, p a i r i n g s which s a t i s f i e d these c r i t e r i a when
f i r s t made subsequently were r e j e c t e d by them as
the match proceeded and b e t t e r uses were found f o r
t h e i r conditions or t h e i r data.
Table 1: The set of i n s t a n t i a t i o n s found as the
p a r t i a l match o f the examples i n F i g . 2 .
IV SOME EXAMPLES
A. A G e n e r a l i z a t i o n Example
F i g . 2 shows the g e n e r a l i z a t i o n problem t h a t
l e d Hayes-Roth & McDermott to be concerned w i t h
multiple
assignments
for
variables.
The
g e n e r a l i z a t i o n t h a t Hayes-Roth A McDerraott wanted
but were unable to get SPROUTER to produce was:
The demonstration t h a t t h i s p a r t i a l match w i l l
lead to the c o r r e c t g e n e r a l i z a t i o n w i l l have to be
postponed u n t i l a l a t e r s e c t i o n which g i v e s the
d e t a i l s o f the approach taken toward d i s j u n c t i o n s .
However, t h e r e are a number of t h i n g s we can say at
t h i s p o i n t t o defend the view t h a t t h i s i s the
There is a small square above a small
c i r c l e and one of these s m a l l f i g u r e s is
inside a large t r i a n g l e .
300
c o r r e c t p a r t i a l match f o r these examples.
The f a c t
t h a t e x a c t l y the same correspondences are obtained
when Example 2 is t r e a t e d as the set of c o n d i t i o n s
and Example 1 is t r e a t e d as the data provides a
demonstration t h a t RELAX at l e a s t f u n c t i o n s as
I n t e n d e d . However, how can correspondences such as
be defended?
The answer to t h i s is t h a t there is
an obvious a l t e r n a t i v e to viewing F i g . 2 as two
examples
in need
of a g e n e r a l i z a t i o n .
That
a l t e r n a t i v e is to see Example 1 as the s i t u a t i o n at
time _t1 and Example 2 as the s i t u a t i o n at a l a t e r
time t2, where the task is to c h a r a c t e r i z e the
transformation
that
has
occurred.
One
c h a r a c t e r i z a t i o n of the t r a n s f o r m a t i o n t h a t changes
Example 1 i n t o Example 2 i s :
The t r i a n g l e t h a t o r i g i n a l l y c o n t a i n s the
square moves down to c o n t a i n the c i r c l e .
The q u e s t i o n a b l e correspondences mentioned above
have an obvious relevance f o r c h a r a c t e r i z i n g t h i s
transformation.
F i g . 3 : The p a r t i a l match obtained f o r the l i s t s
in Table 2.
Lowercase e n t r i e s are from
POWERSET (COME BACK); uppercase from
POWERSET (YALL COME BACK).
B. A T r a n s f o r m a t i o n Example
The next example comes from observations of
two s u b j e c t s l e a r n i n g to program in LISP made by
John R. Anderson and h i s collegues at CMU.
As an
e x e r c i s e in t h e i r textbook [10] these s u b j e c t s had
to w r i t e a LISP f u n c t i o n to compute the powerset
( i . e . , the set o f a l l subsets) o f the set (YALL
COME BACK).
This problem is q u i t e d i f f i c u l t f o r
novices and both s u b j e c t s had l i t t l e success u n t i l
they h i t upon the r e p r e s e n t a t i o n f o r the problem
shown in Table 2.
Once in t h i s form, it appeared
t h a t both s u b j e c t s simply d i d some p a t t e r n matching
to a r r i v e at the f o l l o w i n g s o l u t i o n :
POWERSET(YALL
COME BACK) r e q u i r e s two copies of P0WERSET(C0ME
BACK) and the second copy must have YALL CONSed
onto the f r o n t o f a l l o f i t s s u b l i s t s .
was given to RELAX the r e s u l t s were as shown in
F i g . 3.
Here each s u b l i s t was s u c c e s s f u l l y matched
s t a r t i n g from the l e f t end up u n t i l the p o i n t where
the YALL was encountered and also was s u c c e s s f u l l y
matched s t a r t i n g from the r i g h t end u n t i l the YALL
was encountered.
Extending the match from the l e f t
end
any
farther
was
ruled
out
by
the
b e s t - u s e - o f - t h e - d a t a and b e s t - u s e - o f - t h e - c o n d i t i o n
criteria.
I f the r e s u l t s o f t h i s p a r t i a l match could b e
used as data by o t h e r r u l e s , then we could
c o n s t r u c t a r u l e t h a t would recognize t h a t an atom
had been CONSed onto each s u b l i s t because the
c h a r a c t e r i s t i c t r a n s f o r m a t i o n produced by CONS is
found in each case.
In g e n e r a l , however, w i t h o u t
some n o t i o n of a c o n s t i t u e n t , REIAX would not be
able to d e t e c t o t h e r cases where a l i s t ( r a t h e r
than an atom) has been CONSed onto each s u b l i s t .
Thus, w h i l e REIAX f a l l s s h o r t o f accounting f o r a l l
o f the p a t t e r n matching t h a t s u b j e c t s can d i s p l a y
in an e x e r c i s e l i k e t h i s , it does appear to be a
promising b e g i n n i n g .
Table 2: The r e s u l t s of successive c a l l s
to the f u n c t i o n POWERSET(L).
E x c l u s i v e d i s j u n c t i o n s a r i s e i n two d i f f e r e n t
ways in the process of forming g e n e r a l i z a t i o n s .
What we might c a l l e x t e r n a l d i s j u n c t i o n s have t h e i r
source in the f a c t t h a t f o r any set of examples e^ ,
. . . , _e_
there
i s the
logically correct,
but
unillumlnating, generalization
or
or . • .
The goal of g e n e r a l i z a t i o n programs t h a t attempt to
account f o r e x t e r n a l d i s j u n c t i o n s
[ 7 , 11 ] is to
merge as many
of
the
as
possible
into
c o n j u n c t i v e g e n e r a l i z a t i o n s so as to minimize the
number o f d i s j u n c t s i n the r e s u l t i n g d i s j u n c t i v e
normal form g e n e r a l i z a t i o n .
In p a t t e r n matching terms, the need f o r one
copy i s t r i v i a l — i t r e s u l t s from a f u l l match o f
the sets in P0WERSET(C0ME BACK) using the sets in
POWERSET(YALL COME BACK) as d a t a . The need f o r the
second copy, however, r e q u i r e s a p a r t i a l match of
P0WERSET(C0ME BACK)
to
the
remaining sets in
POVERSET(YALL COME BACK).
When the problem of f i n d i n g t h i s p a r t i a l match
301
I n t e r n a l d i s j u n c t i o n s , by c o n t r a s t , do not
emerge u n t i l a p a r t i a l match has been computed
between two examples e i and
ej.
Then each
i n s t a n t i a t i o n i n the p a r t i a l match
yields
the
again
logically
unillumlnating, generalization
correct,
Example 1 from F i g . 2 producea a f u l l match to
t h i s g e n e r a l i z a t i o n by matching the f l r a t member of
every d i s j u n c t i o n , w h i l e Example 2 produces a f u l l
match by matching the second member of every
disjunction.
In f a c t , t h e r e is a new f u l l - m a t c h
c r i t e r i o n r e q u i r i n g t h a t the same p o s i t i o n be
matched i n a l l d i s j u n c t i o n s i n a r u l e .
Now i t
should come as no s u p r i s e
t h a t when p a r t i a l
matching we w i l l want to v i o l a t e t h i s c r i t e r i o n and
match d i f f e r e n t p o s i t i o n s or even both p o s i t i o n s of
some d i s j u n c t i o n s .
However, t h i s a n t i c i p a t e s an
approach t o p a r t i a l matching d i s j u n c t i o n s t h a t
there is i n s u f f i c i e n t space to e l a b o r a t e on h e r e .
but
Just as in the case of e x t e r n a l d i s j u n c t i o n s , the
way to produce p l e a s i n g g e n e r a l i z a t i o n s seems to be
to remove as many d i s j u n c t i o n s as p o s s i b l e .
The
s i m p l e s t case o f t h i s i s the r e d u c t i o n o f a l l
d i s j u n c t i o n s of the form (c or c) to the s i n g l e
constant £.
Less obvious cases where d i s j u n c t i o n s
can be removed are best understood by reference to
F i g . 4a which shows a p o r t i o n of the f u l l network
of b i n d i n g s t h a t can be e x t r a c t e d from Table 1.
Although
the
natural
way of viewing the
p a r t i a l match
found
in
the
POWERSET
example
presented above is as a c h a r a c t e r i z a t i o n of the
t r a n a f o r m a t i o n s r e q u i r e d to go from POWERSET(COME
BACK) to POWERSET(YALL COME BACK), it is also
i n f o r m a t i v e to look at the g e n e r a l i z a t i o n t h a t
would r e s u l t from t h i s p a r t i a l match.
F i g . 4b
shows the p a t t e r n of b i n d i n g s found f o r the p a r t i a l
match of the s u b l i s t (COME) to the s u b l i s t (YALL
COME) and i n d i c a t e d convergences t h a t should lead
to d i s j u n c t i o n s .
The g e n e r a l i z a t i o n obtained f o r
t h i s s u b l i s t ( i g n o r i n g the need f o r t y p e - t o k e n
distinctions) is:
( b e f o r e Lparen (Come or Y a l l ) )
( b e f o r e (Lparen or Y a l l ) Come)
( b e f o r e Come Rparen)
F i g . 4: A summary of s e l e c t e d v a r i a b l e b i n d i n g s
ja) from Table 1 and b) from F i g . 3.
This
generalization
provides
a
disjunctive
r e p r e s e n t a t i o n of the f a c t t h a t YALL ia an o p t i o n a l
f i r a t element o f these l i s t s .
I n the case where
the YALL's are p r e s e n t , the second p o s i t i o n of
every d i s j u n c t i o n is matched and the f i r a t two
propositions
of
this
generalization
are
i n a t a n t i a t e d by ( b e f o r e Lparen Y a l l ) and ( b e f o r e
Y a l l Come), r e s p e c t i v e l y .
On the o t h e r hand, when
the YALL's are absent, the f i r s t p o s i t i o n o f every
d i s j u n c t i o n is matched so in t h i s case these two
p r o p o s i t i o n s are both i n s t a n t i a t e d by the s i n g l e
p r o p o s i t i o n ( b e f o r e Lparen Come).
REIAX operates
on
this
network
in
order
to
determine a way of r e a l i z i n g each of these b i n d i n g s
in a g e n e r a l i s a t i o n .
The goal is to choose a
d i f f e r e n t l a b e l f o r each l i n k i n t h i s network,
where p o s s i b l e l a b e l s are the terms at e i t h e r end
of a l i n k or the d i s j u n c t i o n of those terms.
Link
x in F i g . 4a receives the l a b e l Vsqr 1 because three
"conditions are s a t i s f i e d : 1) Vaq2 has only one
connection to the network, 2) Vaqr 1 can have the
b i n d i n g Vsqr 2 in a f u l l match, and 3) the l a b e l
Vsqr 1 is not a l r e a d y used as the l a b e l f o r some
other l i n k .
Link z r e c e i v e s the l a b e l V e r l g
because these c o n d i t i o n s are a l s o s a t i s f i e d t h e r e .
Link y is then forced to have the l a b e l (Vsqr 1 or
V c j J 2 ) because both of i t s endpoints have a l r e a d y
been used as l a b e l s .
A• I m p l i c a t i o n s f o r Semantic Approaches
,f
Obtaining a sensible generalization f o r t h i s
problem i s important because i t helps b o l s t e r our
confidence in the unorthodox correspondences made
b y REIAX i n p a r t i a l matching these l i s t s .
If a
l e f t p a r e n t h e s i s must be matched to the atom YALL
to get the c o r r e c t g e n e r a l i z a t i o n , then what about
p a r t i a l - m a t c h i n g programs t h a t compare the semantic
c a t e g o r i z a t i o n of terms before making assignments?
Surely these two terms are s u f f i c i e n t l y d i f f e r e n t
s e m a n t i c a l l y t h a t such programs should be r e l u c t a n t
to see them aa c o r r e s p o n d i n g .
Table 3: The g e n e r a l i z a t i o n r e s u l t i n g from
the p a r t i a l match i n Table 1 .
How s e r i o u s a problem t h i s poses depends
l a r g e l y on whether these p a r t i a l matchers use
semantic relatedness as an abeolute c r i t e r i o n or aa
a r e l a t i v e one.
Programs such as K l i n g ' s ZORBA
[ 9 ] which make semantic relatedness an absolute
c r i t e r i o n w i l l e i t h e r r e j e c t the assignement o f
LPAREN to YALL or w i l l have to decrease the amount
of semantic relatedneas r e q u i r e d to the p o i n t where
this
criterion
has
little
ability
left
to
d i s c r i m i n a t e u s e f u l from useless matches.
On the
o t h e r hand, in Winston's program
[ 1 2 ] the c l o s e r
REIAX produces the g e n e r a l i z a t i o n in Table 3
from the i n s t a n t i a t i o n s in Table 1. It can be seen
t h a t Table 3 c o n t a i n s p r o p o s i t i o n s expressing the
g e n e r a l i z a t i o n t h a t Hayes-Roth & NcDermott wanted
f o r the examples o f F i g . 2 .
302
two terms are s e m a n t i c a l l y the more " p o i n t s " t h a t
assignment c o n t r i b u t e s to an o v e r a l l match score
which determines the best p a r t i a l match. By making
semantic relatedness a r e l a t i v e c r i t e r i o n i n t h i s
manner, Winston allows f o r the p o s s i b i l i t y t h a t
o t h e r f a c t o r s can compensate f o r the semantic
divergence of two
terms.
(Winston's
partial
matcher r e l i e s t o t a l l y on one-one mappings, so
presumably there is no way f o r h i s program to
produce
the
correct
generalization
for
this
p a r t i c u l a r example, however.)
ACKNOWLEDGEMENTS
The author would l i k e to acknowledge the
hundreds of hours of d i s c u s s i o n t h a t he has had
w i t h John R. Anderson about these t o p i c s . The l a c k
of a shared consensus on these issues has, if
anything,
only
made
these
discussions
more
v a l u a b l e . H e l p f u l comments on a previous d r a f t of
this
paper
were
also
provided
by
G. I b a ,
J. P e r l m u t t e r ,
M. Schustack,
and
the
IJCAI
reviewers.
VI THE LAST ABSOLUTE CRITERION
REFERENCES
Now RELAX does obey one absolute c r i t e r i o n ,
namely, t h a t a p r o p o s i t i o n and i t s i n s t a n t i a t i o n
must have the same s t r u c t u r e .
However, as was
discussed above, the problem in F i g .
1]3 may be
seen as a v e r s i o n of the problem in F i g . 1a using
angles
instead
of
segments.
Detecting
the
s i m i l a r i t y of these problems r e q u i r e s t h a t the
c o n d i t i o n (segment VR VO) have the i n s t a n t i a t i o n
(angle VR' VX V O ' ) .
To enable RELAX to f i n d the
c o r r e c t p a r t i a l match f o r t h i s example we followed
the lead of Hayes-Roth & McDermott ( c f .
t h e i r use
of SCR's) and switched to a more f i n e - g r a i n e d
representation.
Finer g r a i n was obtained by
" e x p l o d i n g " each p r o p o s i t i o n in the d e s c r i p t i o n of
these problems in such a way t h a t every l i n k in the
semantic network r e p r e s e n t a t i o n of t h a t p r o p o s i t i o n
i t s e l f becomes a f u l l p r o p o s i t i o n i n the exploded
version.
Thus (segment VR VO) becomes the set of
propositions
[2]
[3]
[4]
[5]
(Rel segment VsegRO)
(Arg1 VR VsegRO)
(Arg2 VO VsegRO).
For these c o n d i t i o n p r o p o s i t i o n s ,
instantiations
RELAX f i n d s
the
[6]
(Rel segment VsegRO) -> (Rel angle VangR'XO')
(Arg1 VR VsegRO) -> (Arg1 VR' VangR'XO')
(Arg2 VO VsegRO) -> (Arg3 VO' VangR'XO').
No use is made of the data
VangR'XO').
[I]
[7]
p r o p o s i t i o n (Arg2 VX
[8]
RELAX was able
to c a l c u l a t e
the c o r r e c t
p a r t i a l match f o r the two problems in F i g . 1 when
they were described i n t h i s d e t a i l .
In
the
process, Arg2 was matched to Arg3 f i v e times and to
Arg2 t h r e e t i m e s .
This success suggests t h a t the
r e l a t i v e c r i t e r i a o u t l i n e d i n t h i s paper are also
adequate,
at least in p r i n c i p l e ,
f o r matching
different structures.
However, f i n d i n g the 36
i n s t a n t i a t i o n s i n the f i n a l match required searches
to evaluate 230 candidate i n s t a n t i a t i o n s .
Thus it
was c l e a r t h a t by recoding whole problems i n t o t h i s
f i n e - g r a i n e d r e p r e s e n t a t i o n we had placed a l a r g e
computational burden on REIAX. Work is in progress
on d e f i n i n g p r i n c i p l e s
t h a t would permit
the
program
to
switch
dynamically
to
the
more
f i n e - g r a i n e d r e p r e s e n t a t i o n f o r j u s t those p o r t i o n s
o f the problem d e s c r i p t i o n where i t i s r e q u i r e d .
This c a p a b i l i t y would e f f e c t i v e l y f r e e RELAX from
the
need
to
rely
on
any
absolute
criteria
whatsoever.
[9]
[10]
[II]
[12]
303
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