ALGEBRAIC PROPERTIES OF KNOWLEDGE
REPRESENTATION SYSTEMS
Jerzy W. Grzymala-Busse
Department of Computer Science
University of Kansas
Lawrence, KS 66045
ABSTRACT.
New concepts of knowledge representation systems, like object and attribute factors, connectedness relations, and seven kinds of
homomorphisms of knowledge representation systems are introduced.
Some
properties of these homomorphisms, related to factors and connectedness,
are shown.
The theory presented here may be used to aggregate sets of
objects, attributes, and descriptors of the original system in order to
produce a simpler system, which preserves the description function of
the original system.
In some applications, the new system is sufficient
representation of the original one.
KEYWORDS:
knowledge representation systems, object factors, attribute
factors, connectedness relations, homomorphisms.
Introduction
The concept of a knowledge representation system was introduced by Z.
Pawlak [2,4,5].
The model arose from the problems of learning theory,
inductive reasoning, expert systems and others.
This approach makes
dealing with two basic problems possible - such as reducing a knowledge
base and deriving rules out of a knowledge base.
Several systems to support medical diagnoses have been implemented on
the basis of this approach[5]. An example of synthesis of a control algorithm, based also on the concept of the knowledge representation system, has been developed and implemented in a cement plant, to control a
rotary kiln [3].
The main aim of this paper is to study transformations of knowledge representation systems while preserving their basic functions. These transformations, called homomorphisms, are applicable in simulation of big
systems by their smaller homomorphic images. In[l] it was shown that homomorphic images may be constructed in such a way that values of some
attributes may still be determined on the basis of other attributes,
while not losing any accuracy.
In this paper basic properties of homomorphisms of knowledge representation systems, with respect to connectedness relations, are given.
Knowledge Representation Systems
In this paper we will adopt the concept of a knowledge representation
system as introduced by Z. Pawlak.
According to this approach, a know-
432
ledge representation system has three components: the set of objects,
the set of attributes, and the set of descriptors.
For each pair (object, attribute), a descriptor is assigned by a function, called a description function.
E.g., if our system is a school, then its objects
are students.
Objects are characterized by attributes, and in our example, they are: the students name, age, gender, test names,etc.
For
example, a student's name is John Smith, and an attribute is Gender.
A
description function assigns descriptor Male to the pair (John Smith,
Gender).
Formally, a knowledge representation syste m S (or briefly,
an ordered quadruple (U, Q, v, p), where
U is a nonempty finite
Q is a nonempty finite
V is a nonempty sinite
and O is a function of
set, and its elements
set, and its elements
set, and its elements
U x Q onto V, called
system S) is
are called objects of S,
are called attributes of S,
are called descriptors of S,
a description function of S,
Let x E U. Let S x denote the following knowledge representation system
({x}, Q, Vx, 0a), where V x = {0(x,a) la E Q}, and 0x: {x}xQ ~ v x is a
restriction of O to {x} x Q.
S x will be called an object factor Ùf S ,
associated with x
(or, briefly, an object factor of S).
Let a E Q. Let S a denote the following knowledge representation system
0 , {a}, Va, pa),where V a = {p(x,a) Ix E U}, and Pa: Ux{a} ~ V a is a restriction of p to U x {a}.
S a will be called an attribute factor of S,
associated with a
(or, briefly, an attribute factor of S).
Example.
Let S = (U, Q, v, 0) be a knowledge representation system,
where U = {x 1,x 2,x 3,x 4,x s,x 6,x 7,x 8,x 9}, Q = {a l,a 2,a a,a 4}, V = {d l,d2,
d a,d4,d s,d6}, and O is given by the Table i.
Table i.
xI
x2
x3
x4
x5
x6
x7
x8
x9
System
a1
a2
a3
a4
d4
ds
ds
d4
ds
ds
d4
ds
ds
d1
d2
ds
dl
d2
d3
d1
d2
d2
d4
d2
d6
d4
d2
d1
d2
d3
dl
d2
d3
d1
d2
d2
d6
d4
d2
d2
S.
Connectedness
Relations
Let S = (U, Q, v, p) be a knowledge representation
two binary relations, a and ~, as follows
= {(aid) ~ Q x V I 0(x,a) = d for some x E U},
and
= {(x,d) E U x V I 0(x,a) = d for some a E Q}.
433
system.
We define
By the A-D-connectedness relation ~* on S and 0-D-connectedness relation
~* on S we mean the smallest equivalence relations on Q U V and U U V,
containing ~ and m, respectively.
For any r E Q U V, [r]~* means the
equivalence class of ~* containing r.
Similarly, for any t E U U V,
[t]~, means the equivalence class of ~* containing t. The equivalence
classes of ~* and ~* will be called A-D-components of S and O-Dcomponents of S, respectively.
For d E V, the set {ai(a,d ) 6 e} will be denoted by ad, and the set
{xI(x,d) E ~} will be denoted by ~d.
For x E U
For a E Q
and
and
d E V, the set {a]0(x,a ) = d} will be denoted by c&xd.
d E V, the set {x[0(x,a) = d} will be denoted by Wad.
In the example, ~ = {(ai,d4),(al,ds),(a2,dx),(a2,d2),(a2,d3),(a3,d2)
,
(a3,d4),(a3,d6),(a4,dl),(a4,d2),(a4,d3)}.'
There is only one A-Dcomponent of S - the entire set U U V.
O-D-components of S are {xl,x4,
xT,dl,d4} and {x2,x3,xs,x6,xs,xg,d2,d3,ds,d6}.
Moreover, ~d 2 = ~x2d 2 =
{a2 ,a3 ,a4 }.
Homomorphisms
of Knowledge R e p r e s e n t a t i o n
Let S I = (UI,QI,Vl,pl)
tion systems.
Sjstems
and $2 = (U2,Q2,V2,02)
be knowledge
representa-
Let h 0 be a function of U I into U2, let h A be a function of QI into Q2,
and let h D be a function of U I into U 2.
The triple (ho, hA, hD) is called an OTA-D-homomorphism of S I into S~
(or briefly a homomorphism of S, into $2) iff for all x E U I and a E QI"
hD
(01 (x,a)) = 02 (h0(x), hA(a)).
The set of all 0-A-D-homomorphisms
Homo_A-D(SI ~ $2).
of S I into S2 will be denoted by
A h o m o m o r p h i s m (ho, hA, hD) of S, into S2 is called a h o m o m o r p h i s m
$I onto $2 iff all three functions h0, hA, h D are onto.
of
Let QI=Q2 and h A transform each element into itself, i.e. h A = id. Then
a pair (ho, hD) is called an O - D - h o m o m o r p h i s m of S, into $2 iff a triple
(ho, id., hD) is a h o m o m o r p h i s m of $I into $2. For U, = U2 and ho= id.,
we define an A - D - h o m o m o r p h i s m of S, into $2 as a pair (hA, hD) iff (id.,
hA, hD) is a homomorphism of S, into $2.
For V, = V2 and h D = id., a
pair (ho, hA) is said to be an O - A - h o m o m o r p h i s m of $I into $2 iff (h 0,
hA, id.) is a h o m o m o r p h i s m of $I into $2.
The set of all O-D-homomorphisms (A-D-homomorphisms, O-A-homomorphisms)
of S, into S2 will be denoted by H o m 0 _ D ( S , ~ $2) (HOmA-D(S, ~ S2),
Hom0_A(SI ~ S2 ), respectively).
Let QI = Q2, vl = v2, hA = id., and h D = id..
Then a function ho is
called an O - h o m o m o r p h i s m of S, into $2 iff (h O, id., id.) is a homom o r p h i s m of $I into $2.
Similarly, we define an A - h o m o m o r p h i s m of $I
into $2 and a D / h o m o m o r p h i s m of $I into $2.
The set of all 0-homomorphisms (A-homomorphisms, D-homomorphisms) of S I
into S2 will be denoted by H o m 0 ( S 1 ~ $2) (HOmA(SI~ $2), H o m D ( S I ~ $2),
respectively).
434
All six of the a b o v e m e n t i o n e d s p e c i a l h o m o m o r p h i s m s
onto iff c o r r e s p o n d i n g 0 - A - D - h o m o m o r p h i s m s
are onto.
of $I into
$2 are
In T a b l e s 2-8 p r e s e n t e d are k n o w l e d g e r e p r e s e n t a t i o n s y s t e m s w h i c h are
h o m o m o r p h i c images of the k n o w l e d g e r e p r e s e n t a t i o n s y s t e m f r o m Table i.
Table
2.
h0 :
Yl
Y2
Y3
Y4
Ys
S y s t e m $I.
(h O, h A , hD)
Table
bl
b2
e2
e2
e4
el
e2
ea
hA:
e4
e2
el
e2
hD :
[ xI
x2
x3
x4
xs
x6
X7
X8
X9
$I
[ Yl
Y2
Y3
Yl
Y2
Y3
Y4
Ys
Ys
Q
Q1
bl
.a2
b2
•a3
bl
a4
b2
I dl
el
d2
e2
d3
d4
e3
e4
ds
e2
d6
e2
I al
V
Vl
is an O - A - D - h o m o m o r p h i s m
of S onto
SI .
3.
Yl
Y2
Y3
Y4
Ys
al
el
e4
eI
a2
e,
e2
e3
eI
a3
el
e2
e3
el
a4
el
e2
e3
el
e4
e2
e2
e2
e4
h0:
hD:
S y s t e m S2 .
(ho, hD) is an O - D - h o m o m o r p h i s m
Table
S
S
xI
x2
x3
x4
x5
x6
x7
. X8 .x9
$2
Yl
Y2
Y3
Yl
Y2
Y3
Y4
Ys
V
dI
d2
d3
d4
ds
d6
V2
el
e2
e3
el
e4
e3
"
of S onto
S2 .
4.
xl
x2
x3
x4
x5
x6
x7
x8
x9
bl
b2
b3
el
el
e2
e3
el
e2
e3
h A:
e4
e4
el
e2
e3
el
e2
e3
hD :
el
e4
e4
el
e2
e2
el
e2
e2
e4
e4
el
S y s t e m S3 .
(hA, hD) is an A - D - h o m o m o r p h i s m
Q
a!
a2
a3
a4
Q3
bl
b2
b3
b2
v
d1
d2
d3
d4
ds
d6
V3
eI
e2
e3
el
e4
e3
of S onto
435
S3 .
Ys
Table
5.
Yl
Y2
Ys
Y4
Ys
bl
b2
b3
d4
di
d4
ds
ds
d4
ds
d2
d2
d3
dl
d6
d4
d2
d2
x2
x3
x4
x$
x6
x7
x8
x9
U4
Yl
Y2
Y3
Yl
Y2
Y3
Y4
Ys
Y~
aI
a2
a3
a4
b1
b2
bs
b2
Q
Q4
of
S onto
S4 .
6.
Yl
Y2
Y3
Y4
Ys
aI
a2
a3
a4
d4
ds
ds
d4
ds
dl
d2
d3
dl
d2
d4
d2
d6
d4
d2
d,
System S5 .
h 0 is a n O - h o m o m o r p h i s m
Table
xI
hD:
S y s t e m S4 .
(h0, hA) is a n 0 - A - h o m o m o r p h i s m
Table
U
hA:
d2
d3
d,
d2
of
U
xI
x2
x3
x4
x5
x6
x7
x8
x9
Us
Yl
Y2
Y3
Yt
Y2
Y3
Y4
Ys
Ys
Q
[
aI
a2
a3
a4
Q6
I
bx
b2
bs
b2
ho:
S onto
S5 .
7.
x1
x2
x3
x4
xS
x6
x7
x8
x9
bl
b2
b3
d4
ds
ds
d4
ds
ds
d4
ds
ds
di
d2
d3
dt
d2
d3
dt
d2
d2
d4
d2
d6
hA:
d4
d2
d6
d4
d2
d2
S y s t e m S6 .
h A is an A - h o m o m o r p h i s m
of
S onto
S6 .
436
Table
8.
x1
X2
X3
X4
XS
X6
X7
X8
X9
as
a=
aa
a4
eI
e4
e4
el
e4
e4
eI
e4
e4
el
e2
%
el
e=
ea
el
e2
ea
el
e2
ea
eI
e2
e2
el
e2
ea
el
e2
ea
eI
ee
ee
System S7 .
h D is a D - h o m o m o r p h i s m
V
d1
d2
da
d4
ds
d6
V7
eI
e2
e3
eI
e4
e3
hD :
el
e2
e2
.
of S onto
S7 .
By the d u a l of a c o n c e p t or p r o p o s i t i o n , c o n t a i n i n g a r e f e r e n c e to one
of U or Q or to e l e m e n t s of that set, we m e a n the c o n c e p t or p r o p o s i t i o n
formed f r o m the first by r e p l a c i n g each such r e f e r e n c e by a r e f e r e n c e to
the other U or Q.
Thus " o b j e c t f a c t o r of S" and " a t t r i b u t e f a c t o r of S"
are dual concepts.
Similarly, " O - D - h o m o m o r p h i s m of S I into $2" and
" A - D - h o m o m o r p h i s m of S I into S=" are also d u a l concepts.
The c o n c e p t of
" 0 - A - h o m o m o r p h i s m of S I into $2" is self-dual.
F r o m this p o i n t we w i l l
take duals of n o n - s e l f - d u a l c o n c e p t s or p r o p o s i t i o n s for granted.
A l t h o u g h U and Q are i n d i s t i n g u i s h a b l e in the a b o v e sense in the abs t r a c t system, the real w o r l d i n t e r p r e t a t i o n of a c o n c e p t or p r o p o s i t i o n
m a y be d i f f e r e n t f r o m that of its dual, since the m e a n i n g s g i v e n to
" o b j e c t " and " a t t r i b u t e " are quite d i f f e r e n t .
Let S = (U, Q, v, p) be a k n o w l e d g e r e p r e s e n t a t i o n system.
An O - D - c h a i n
in S w i t h an i n i t i a l e l e m e n t x E U and a t e r m i n a l e l e m e n t x n ~ U (or
n ~ - E V) is a s e q u e n c e
Xl,
all
,
dl,
a12 ,
x2,
a22 ,
a s e q u e n c e Xl, a11 , dl,
an.l, ~, x n, ann, an),
w h e r e all e l e m e n t s
..., dn_1, (or dl,
d2,...,
a12 , x2,
x I, x 2,...,
d 2,..., dn)
Xn_l, an_l,~_l, an_l,
a22,
d2,...,
an_l,n,
xn
(or
Xn_ I , a n - l , n - 1 , dn-l,
x n are m e m b e r s of U, all e l e m e n t s d I, d 2,
are m e m b e r s of V, all e l e m e n t s a 11, a~2,
a22 , an_l~n_l, an_l, n (or all , a12 , a22,... , an_l,n_l, an_1,n, ann) are
m e m b e r s of Q,
0 (xi, aii) = di for all i = i, 2,..., n-l, and
0 (Xi+l, ai,i+l) = di for all i = 0, i,..., n-2 (or i = 0, i,..., n-l).
In the example,
x 2, a 2, d 2, a a, x 5, a I, ds,
al,
x3
is an O - D - c h a i n
of S.
O b v i o u s l y , for any two e l e m e n t s x, y E U we have (x, y) E ~* iff there
e x i s t s an O - D - c h a i n in S w i t h its initial e l e m e n t x and its t e r m i n a l
e l e m e n t y.
Properties
of H o m o m o r p h i s m s
B e l o w are s t a t e d
basic
structural
properties
437
of h o m o m o r p h i s m s
of k n o w l e d g e
representation
systems.
Proposition 4. i. Let S I = (U I , Qx , Vl , 01 ) and S 2 = (U 1 , Qt , Vl , 01 ) be
knowledge representation systems with Q1 = Q2Let a E Q1.
Let (S 1)a
and (S2)a be attribute factors of S t and S 2, respectively.
Then
Horn0_ D (S t -+ S 2) = N Hom0-D ((St)a -+ (S2)a).
aEQ 1
The proof of this result is straightforward.
For the sake of simplicity, for any function of a set X into a set Y,
and for any subset X' of X, the set {f(x)[x E X'} will be denoted by
f(X').
Similarly, for a function of XxY into Z, for X'E X, and for
y E Y, the set {f(x,y) Ix E X'} will be denoted by f(X',y).
Corollary 4.2.
Let S I = (U t, QI, vt, 01) and S 2 = (U 2, Q2, V2, 02) be
knowledge representation systems with Qt = Q2 and VI = V2.
Let a ~ QI"
Let (S 1)a and (S 2)a be attribute factors of St and $2, respectively.
Then
Hom0(SI-+ $2) = N Hom0((St)a -+ (S2)a)aeQ t
Proposition 4.3.
Let S = (Ut, Q1, Vl, 01) and S 2 =
knowledge representation systems with Ut = U2 and QI
function of V I into V 2 . Then h D is a D-homomorphism
the relation 6 is a function, where
6 = {(Ot (x,a), 02(x,a)) I x E U I , a E Q1}Moreover, if 0 is a function, then h D is equal to ~
unique.
The next result relates the connectedness
of knowledge representation systems.
relations
(U2, Q2, v2, 02) be
= Q2.
Let h D be a
of S I i n t o $2 iff
and therefore
with homomorphisms
Proposition 4.4.
Let S I = (UI, QI, QI, 01) and S~ = (U2, Q2, V2, 02)be
knowledge representation systems.
Let (h0, hA, h D) be a homomorphism of
$I into $2. Let (~i)* and (~2)* be A-D-connectedness relations on $I
and S 2 , and let (601)* and (~h)* be 0-D-connectedness relations on S I and
S 2 , respectively.
Then
(i)
(ii)
V x e U I, h 0 ([x] (601)*) £ [h0(x)](602)*
,
Vd E V 1
,
hD
([d] (601),) c_ [hD(d)](602),
(iii)
Va
E Q1, hA ([a](~l)*)
_c [hA(a)](~2),
(iv)
gd
E VI, h D ([d](~1)*)
_c [hD(d)](~2),
,
Proof.
We show (i) only, since remaining proofs are analogous.
Suppose
that (i) is false, i.e. there exist x,y E U I such that (x,y) E (601)* and
(h0(x), h0(Y))~ (0~)*.
For x and y there exists an 0-D-chain in S I with
its initial element x and its terminal element y, since (x,y) E (601)*.
Let this 0-D-chain in S I be
xl,a~t,dl,a12,x2,a22,d2,...,Xn_t,an_1,n_
I , dn_t,an t,n,Xn, where xt= x
and Xn = Y. For S 2 we may construct a sequence h0(x 1), hA(at t), hD(d t),
hA(at2), h0(x2), hA(a22), hD(d2),..., hO(Xn-1), hA(an-l,n-t), hD(dn-t),
hA(an-l,n), h0(xn)"
As follows
from the definition
of homomorphism
438
of S I into S 2 , the above
sequence is an 0-D-chain in S 2 with its initial element ho(x ) and its
terminal element ho(Y), therefore (ho(x) , ho(Y)) E (~2)*, a contradiction.
Proposition 4.5. Let $I = (UI, QI, V1, 01) and $2 = (U2, Q2, V2, 02) be
knowledge representation systems.
Let h 0 be a function of U, into U2,
h A be a function of QI into Q2, and h D be a function of VI into V2. Then
(ho, h A , hD) is a homomorphism of S x into $2 iff
Va E QI Vd E V I , h0(Ooad)_c ~hA(a)hD(d ).
Proof.
Let
i.e. 01(x,a)
(hO, hA, hD)
since {ho(x)
Let V a
(ho, hA, hD) be a homomorphism of S I into S2 . Let x E ~a(d),
: d. Then hD(01 (x,a)) = hD(d ) = 02 (ho(x), hA(a)), since
is a homomorphism of S I into $2. Thus ho(x ) ~ OOhA(a)hD(d),
IP2 (ho(x), hA(a)) = hD(d)} = ~OhA(a)hD(d ).
E Q I V d E V I,
ho(~oad) _c COhA(a)hD(d).
Let x be an arbitrary member of U I and let a be an arbitrary member of
QI. Let Pl (x,a) be denoted by d. Then ho(x ) E ~OhA(a)hD(d), since
ho(x) E ho(~Oad). Hence 02(ho(x), hA(a)) = hD(d)- "~But hD(d)=hD(01 (x,a)),
and therefore (ho, hA, hD) is a homomorphism of S I into S2 .
Corollary 4.6.
Let $I = (UI, QI, V1, 01) and $2 = (U2, Q2, V2, ~ ) be
knowledge representation systems.
Let h 0 be a function of UI into U2,
h A be a function of QI into Q2, and h D be a function of VI into V2. Then
(ho, hA, hD) is a homomorphism of Sl into $2 iff
V Q S QI v V ~ V1, hO(a~Q~ad ) ~ n ~hA(a)hD(d).
deV
de
Conclusions
We defined seven kinds of homomorphisms of knowledge representation
systems.
The interpretation of each of them is clear from the definitions and examples.
There is a great variety of applications of homomorphisms of knowledge representation systems. E.g., let our system S =
(U, Q, V, 0) be a hospital, objects be patients, and attributes be
tests.
Results of the patients' tests are arranged in a table, defining
the description function.
We want to research some disease.
For the
needs of the research, we may classify all patients in blocks, according
to the degree of disease progression.
In a similar way, we may classify
attributes in blocks, so that all tests in a block give - from the point
of view of the disease researcher - the same information.
We may also
classify results of tests for patients (descriptors) into blocks, using
many possible criteria, such as reducing the accuracy of the tests. Now,
we may try to produce a table, in which blocks of objects correspond to
lines and blocks of attributes correspond to columns.
For any block B 0
of objects and any block BA of attributes, if the set {0(x,a) ix E BO,
a E BA} is equal to one of the blocks of descriptors, we may fill in the
blocks of descriptors, and thus what we have produced is a homomorphic
image S' of the knowledge representation system S. System S' may be
much simpler than system S, yet we may do the same research using S'
instead of S.
439
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440