Plausible Inferencing Using Extended Composition
Michael N. Huhns
MCC
3500 West Balcones Center Drive
Austin, Texas 78759
Abstract
T h i s paper considers the c o m p o s i t i o n of tuples
f r o m t w o relations in order to derive a d d i t i o n a l
tuples of one of these relations. O u r purpose
is to determine when the c o m p o s i t i o n is p l a u sible and for which relation the new tuples are
derived. We first present a f o r m a l definition of
c o m p o s i t i o n and our extension to i t . We next
define conditions on the domains and ranges of
the relations t h a t are necessary for extended
c o m p o s i t i o n to occur. We then show how a set
of u n d e r l y i n g a t t r i b u t e s , independently specified for each r e l a t i o n , is sufficient for d e t e r m i n i n g plausible c o m p o s i t i o n , when the p r i m i t i v e s
are combined according to an algebra. F i n a l l y ,
we apply our m e t h o d for extended c o m p o s i t i o n
to a representative g r o u p of semantic relations
and evaluate the results.
1
as C Y C [Lenat et a/., 1986, Lenat and G u h a , 1988], the
m a p p i n g for a relation is only p a r t i a l l y specified; other
tuples for the relation are added as knowledge is entered.
T h e procedure for composing relations o u t l i n e d in this
paper provides a means of i n f e r r i n g a d d i t i o n a l tuples belonging to an i m p l i c i t l y defined r e l a t i o n .
A composite relation results f r o m a p p l y i n g the binary operation of composition to two binary relations.
T h i s operation has the f o l l o w i n g definition
[Stanat and M c A l l i s t e r , 1977]:
D e f i n i t i o n 1 Let Ri be a relation from set A to set B
and Rj be a relation from set B to set C. The composite
relation f r o m A to C, denoted Ri ■ Rj, is
Introduction
The construction of a large knowledge base is difficult
and requires techniques t h a t can facilitate knowledge acquisition. Rather t h a n requiring that all knowledge in
the base be entered explicitly, a system could be provided
w i t h a basic set of facts and an inference mechanism for
inferring additional facts from these [Baker et al, 1987].
An ideal system would be able to generate all valid and
no invalid inferences. One way to approach this ideal is
to provide a set of specialized inference procedures t h a t
collectively generate a valid set of inferences. In this paper we develop one such procedure, based on an extended
composition of semantic relations f r o m a knowledge base.
Figure 1 contains examples of this type of composition.
The procedure has the effect of constructing new inference rules, which, when executed, generate extensions to
the knowledge base.
2
L a r r y M . Stephens
Center for Machine Intelligence
University of South Carolina
Columbia, South Carolina 29208
Extended Composition
A binary relation R consists of a set A (the domain),
a set B (the range), and a mapping that specifies the
set of tuples (a, b) belonging to R, where
and
The mapping may be explicit by listing all the
tuples in R or i m p l i c i t by providing rules for selecting the
tuples. In a large frame-based knowledge system, such
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Knowledge Representation
Extended composition can also be shown to be associative and not commutative.
We would like to have an algorithmic way of determ i n i n g when
is nonempty and whether it is a
subset of Ri or Rj or neither. Our method for making
this determination is based on two premises:
• the domains and ranges of the two relations must
be type-compatible, and
• the primitives (defined below) of the relations must
combine compatibly.
If the first premise is satisfied by relations Ri and Rj,
then the primitives of the two relations can be combined
to yield the primitives of the composed relation,
The primitives of
can then be compared to those
of Ri and Rj to determine if
is a subset of R i ,
Rj, b o t h , or neither.
their fundamental characteristics. For example, attributes are not generally transferable t h r o u g h other
relations.
F u n c t i o n a l : T h e d o m a i n of a Functional relation is
in a specific spatial or t e m p o r a l position w i t h respect to the range of the r e l a t i o n . For example, in
an instance of the componentOf r e l a t i o n , such as
Wheel.component Of.Car, the Wheel is in a specific
spatial position w i t h respect to the Car. T h i s property does not hold for J u r o r . m e m b e r O f . J u r y .
H o m e o m e r o u s : In each instance of a Homeomerous rel a t i o n , the element of the d o m a i n must be the same
k i n d of t h i n g as the element of the range, e.g., in
P i e S l i c e . p i e c e o f . P i e , the slice is the same stuff as
the pie.
T h e type c o m p a t i b i l i t y specified by the first premise
results in the f o l l o w i n g necessary conditions for the extended c o m p o s i t i o n of relations:
1. T h e intersection of sets B and C m u s t be n o n e m p t y ;
otherwise, the relation Ri © Rj w i l l be empty.
2. For the derived tuples to be elements of R i , the
intersection of sets B and V m u s t be nonempty.
3. For the derived tuples to be elements of Rj, the
intersection of sets A and C m u s t be nonempty.
These conditions, represented using Venn diagrams in
Figure 2, e l i m i n a t e m a n y of the possibilities for extended
c o m p o s i t i o n . An algebra based on p r i m i t i v e s of the relations eliminates a d d i t i o n a l implausible compositions.
3
P r i m i t i v e s for Semantic Relations
T h e second premise above requires a set of p r i m i t i v e s
t h a t describe each relation and a set of rules for comb i n i n g p r i m i t i v e s . We have postulated a group of ten
p r i m i t i v e s , based on a l i t e r a t u r e survey [Chaffin and
H e r r m a n n , 1987, Cohen and Loiselle, 1988, Wierzbicka, 1984, W i n s t o n et a/., 1987] and an analysis of numerous semantic relations in the C Y C knowledge base
[Lenat and G u h a , 1988]. These p r i m i t i v e s are independently d e t e r m i n a b l e for each relation and relatively selfe x p l a n a t o r y . T h e y specify a relationship between an element of the d o m a i n and an element of the range of the
semantic r e l a t i o n being described. T h e p r i m i t i v e s , described n e x t , have values f r o m the set X = {+, 0, — } ,
where + indicates t h a t the relationship holds, — t h a t it
does n o t , and 0 t h a t it is n o t applicable.
C o m p o s a b l e : Some semantic relations can never be
m e a n i n g f u l l y composed w i t h other relations due to
S e p a r a b l e : T h e d o m a i n of a Separable relation can be
t e m p o r a l l y or spatially separated f r o m the range,
and can thus exist independently of the range. For
the above componentOf example, the Wheel can be
separated f r o m the Car and can exist independently.
For Wheel.made Of Aluminum, the Aluminum cannot
be separated f r o m the Wheel if the Wheel is still to
exist.
S t r u c t u r a l : T h e d o m a i n and range of a S t r u c t u r a l rel a t i o n have a hierarchical relationship in terms of a
physical structure. For example, in Wheel.compo­
nentOf Car, the hierarchical structure is f r o m part
to whole and the S t r u c t u r a l property of compo­
nentOf has a — value.
T e m p o r a l : T h e d o m a i n and range of a T e m p o r a l relat i o n are ordered in regard to a t e m p o r a l structure.
For example, there is no notion of t i m e in the relat i o n pieceOf indicated by a value of 0 for T e m p o r a l ;
in causedBy, a value of — indicates t h a t the range
element precedes the d o m a i n element.
I n t a n g i b l e : T h e d o m a i n and range of an Intangible rel a t i o n have a hierarchical relationship in terms of
ownership or m e n t a l inclusion. As an example, the
relarion ownedBy has a value of — for Intangible,
because the element owned is i n t a n g i b l y included in
the owner's sphere of influence.
(Note: values of the last three p r i m i t i v e s for the
converse of a relation are opposite to those for the
forward relation.)
N e a r : T h e d o m a i n of a relation w i t h property Near is
physically or t e m p o r a l l y close to the range.
C o n n e c t e d : T h e d o m a i n of a relation w i t h property
Connected is physically or t e m p o r a l l y connected to
Huhns and Stephens
1421
• In order to compose, two relations must have the
same hierarchical direction for their Structural,
Temporal, and Intangible primitives.
• If Ri has the property Connected and Rj does not,
then
(and
cannot have the property
Connected. Therefore,
is not
a subset of Ri.
• If Ri has the property Separable and Rj does not,
then
has the property Separable. Therefore,
may be a
subset of Ri.
the range. A connection, which may be indirect, is
indicated by +; no connection is denoted by —.
I n t r i n s i c : A semantic relation has the property I n t r i n sic if the relation is an a t t r i b u t e of the stufflike nature of its domain or range. For example, the relation hasDensity is an intrinsic property of its dom a i n , so t h a t if Aluminum.has Density.5, then every
piece of Aluminum inherits this value for its density.
To test our hypotheses, we have selected a representative set of relations (Table 1), including part-whole,
subclass, ownership, causal, and a t t r i b u t i o n relations.
For each of these relations, Table 2 shows the values we
have assigned to the above primitives. The domains and
ranges of the relations, shown in Table 1, are also needed
to determine plausibility.
4
Algebra of Relation Primitives
We assume t h a t the results of composing two semantic
relations can be determined f r o m the results of combining their ten relation primitives (the accuracy of this
assumption is evaluated below) as follows:
(1)
where
and o is the combination operator. T h a t is, for the purposes of relation
composition, each relation can be represented solely by
a vector of values for its ten relation primitives. It thus
becomes necessary to define precisely how two of these
vectors combine.
We assume t h a t the primitives are orthogonal and
f o r m a linear basis for the set of relations. The combination operator o can thus be defined in terms of a separate
operation table for each p r i m i t i v e , as shown in Table 3.
Each operation table is symmetric and has been derived
f r o m empirically determined rules for relation composit i o n , such as the following:
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Knowledge Representation
The resultant algebra enables the primitives of the
composed relation to be derived. If these derived p r i m itives match the primitives of one (or b o t h ) of the composing relations, then a tuple of one (or both) of these
can be instantiated; else, the knowledge base can be
searched to find all relations t h a t match the resultant
primitives, and, if not already instantiated, these can
be presented to a user as potential new tuples for the
knowledge base.
As an example of this inference procedure, assume
t h a t a user has entered the assertions Wheel.componentOf.C&r and Car.ownedBy.Grover.
Combining the
primitives f r o m Table 1 for componentOf and ownedBy
according to the combining rules in Table 3 yields the
following vector of primitives for the resultant relation:
Because
this vector matches the primitives of ownedBy and does
not match those of componentOf, the inference is that
Wheel, own edBy.Grover.
The plausibility of this result is checked by comparing the types of the domain and range of this relation instance w i t h the types specified for ownedBy in
Table 2. To do this, a taxonomy of types is needed
that enables the intersection of domains and ranges
to be determined. Such a taxonomy is typically part
of frame-based knowledge-representation systems. The
types used for our examples are f r o m the C Y C ontology [Lenat and Guha, 1988].
Using this ontology
and Table 2, we find t h a t Wheel is an instance of
I n d i v i d u a l O b j e c t , Grover is an instance of T a n g i b l e & I n t a n g i b l e O b j e c t , and these match the domain
and range of ownedBy. The resultant inference is thus
deemed plausible.
5
Results
The above
of relations
the form of
Each entry
inference procedure was applied
shown in Tables 1 and 2. The
a composition m a t r i x , are shown
in Table 4 is equivalent to a rule
to the set
results, in
in Table 4.
of the form
(2)
The results reflect the order of composition, e.g.,
as well as
which was not addressed in either
[Cohen and Loiselle, 1988] or [Winston et a/., 1987]. Because each of the operators for combining primitives is
symmetric, the composition m a t r i x is nearly symmetric. The only exceptions result f r o m type compatibility,
Huhns and Stephens
1423
w h i c h sometimes excludes a c o m p o s i t i o n f r o m o c c u r r i n g .
For e x a m p l e ,
because the intersection of the range of / w i t h the d o m a i n of / is e m p t y .
T h e f o l l o w i n g are specific examples of plausible inferences predicted by the extended c o m p o s i t i o n of relations
(where —> denotes logical i m p l i c a t i o n ) :
6
Discussion and Conclusions
T h e inference procedure and results presented in this
paper extend the work of previous researchers. Chaffin
and H e r r m a n n [1987] identify a set of relation elements
(relation p r i m i t i v e s ) t h a t can be used to describe and
classify relations. Each relation element is a f u n d a m e n t a l
p r o p e r t y t h a t holds between the d o m a i n and range of the
relation.
W i n s t o n et al. [1987] define three independent relation
elements, inclusion, connection, and similarity; these are
used to describe spatial i n c l u s i o n , m e r o n y i n i c inclusion,
and class inclusion. W h e n any inclusion relation is combined w i t h another, they f i n d t h a t a v a l i d inference can
be made and t h a t the resultant relation is the one havi n g the fewest relation elements. In a d d i t i o n , W i n s t o n et
al. identify three dependent elements of connection t h a t
explain the t r a n s i t i v i t y , b u t not the composability, of six
m e r o n y m i c relations.
Cohen and Loiselle [1988] i d e n t i f y t w o deep structures
for relations: hierarchical and temporal, each h a v i n g a d i rection. Each r e l a t i o n is hierarchical, t e m p o r a l , or b o t h .
W h e n t w o relations are composed, the resultant relat i o n may have any of several possible deep structures,
depending on the properties of the composing relations.
T h e y f o u n d t h a t inferences are most plausible when either the hierarchical or t e m p o r a l directions of the t w o
composing relations are the same as t h a t in the composed r e l a t i o n . Like W i n s t o n et al., they do not consider
type consistency in composing relations.
We extend the research efforts cited above by basing
relation c o m p o s i t i o n on set theory. On this basis, we
conclude t h a t t y p i n g of the d o m a i n and range elements
may restrict c o m p o s i t i o n , independently of any relation
a t t r i b u t e restrictions. In a d d i t i o n , we extend the work
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Knowledge Representation
of [ W i n s t o n et al., 1987] by e x p l i c i t l y considering the h i erarchical nature of the inclusion relations, as suggested
by [Cohen and Loiselle, 1988]. T h i s leads to a means of
defining the p r i m i t i v e a t t r i b u t e s of the converse of a rel a t i o n a n d , consequently, of composing a converse w i t h
other relations.
We provide a vector of ten p r i m i t i v e s for each of 21
t y p i c a l relations. T h i s vector representation provides a
more powerful basis for r a n k i n g and classifying relations
t h a n does the linear o r d e r i n g in [ W i n s t o n et al., 1987].
Since there are three possible values for each of the ten
p r i m i t i v e s , our representation provides for 3 1 0 = 59,049
different basis vectors t h a t can be used to represent relations. T h e number of relations t h a t could be represented
is a c t u a l l y m u c h greater because of the large number of
types t h a t could be chosen for the domains and ranges.
T h e inference procedure we developed for relation
c o m p o s i t i o n is based on several assumptions. T h e forem o s t of these is t h a t relation c o m p o s i t i o n is equivalent to
a c o m b i n a t i o n of the corresponding vector of p r i m i t i v e s .
T h e correctness of this assumption is borne o u t by the
p l a u s i b i l i t y of the predicted inferences, shown in Table
4. A second assumption is t h a t each relation p r i m i t i v e
is o r t h o g o n a l to the others. T h i s s i m p l i f y i n g assumption
greatly increases the efficiency of the inference procedure by y i e l d i n g o p e r a t i o n tables (see Table 3) t h a t are
independent of each other. A l t h o u g h the v a l i d i t y of the
results supports this assumption also, there is some evidence t h a t the chosen p r i m i t i v e s are N O T o r t h o g o n a l .
For example, the p r i m i t i v e s Connected, Homeomerous,
and I n t r i n s i c combine dependently according to the foll o w i n g rule t o y i e l d compositions w i t h a t t r i b u t e relations
not predicted by our algebra:
Such
a rule
w o u l d yield the valid inference densidensityOf, w h i c h does not result f r o m
our relation algebra. It could be applied after extended
c o m p o s i t i o n and viewed as an a d d i t i o n a l inference mechanism.
Other v a l i d inferences are missing f r o m Table 4, i n cluding
memberOf
isA
memberOf and
componentOf
attribute Of
attribute Of.
However, we feel
t h a t these omissions do not d i m i n i s h the u t i l i t y of our
results, in t h a t our procedure is designed for correctness
instead of completeness. In a d d i t i o n , m a n y knowledgebased systems have other inference mechanisms t h a t
could generate these missing inferences. For example,
an a u t o m a t i c classifier [ L i p k i s , 1981] w o u l d generate the
inference
memberOf
memberOf
T h e p o t e n t i a l for generating new inferences in a large
knowledge base, such as the one in C Y C , is enormous.
C Y C , currently w i t h
relations, could have approxi m a t e l y sixteen m i l l i o n possible compositions. Of these,
2 0 % are predicted to be plausible, based on the percentage of v a l i d entries in T a b l e 4. For a l l possible values
o f relation p r i m i t i v e s , n o more t h a n 3 1 % could b e composed validly due to prohibited entries in the operation
tables for c o m b i n i n g p r i m i t i v e s . T h e one m i l l i o n assertions now in the C Y C knowledge base can be combined
using the predicted compositions to yield many new inferences.
However, there are t w o m a j o r problems w i t h extended
c o m p o s i t i o n . F i r s t , reason maintenance for the resultant
inferences is c o m p u t a t i o n a l l y p r o b l e m a t i c , because the
inferences depend not only on the relations being composed, b u t also on the relation p r i m i t i v e s for all of the
relations involved. Second, assigning values for the relat i o n p r i m i t i v e s is conceptually p r o b l e m a t i c . T h e values
are subjective and must be entered m a n u a l l y for each
relation in a knowledge base. T h e v a l i d i t y of the i n ferences generated by extended composition are directly
dependent on these values.
Nevertheless, we expect t h a t the relation p r i m i t i v e s
can be used for classifying relations, as well as generating new inferences, and for suggesting plausible analogies. T h e procedure for extended composition appears
to be a viable technique for increasing the i n f o r m a t i o n
in an existing knowledge base. Because the procedure
has the effect of generating new inference rules and then
a p p l y i n g t h e m , it yields plausible inferences t h a t are not
w i t h i n the deductive closure of the original knowledge
base.
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