Difference Functions of Dependence Spaces Jouni J¨arvinen
Abstract Here the reduction problem is studied in an algebraic structure called dependence space. We characterize the reducts by the means of dense families of dependence spaces. Dependence spaces defined by indiscernibility relations are also considered. We show how we can determine dense families of dependence spaces induced by indiscernibility relations by applying indiscernibility matrices. We also study difference functions which connect the reduction problem to the general problem of identifying the set of all minimal Boolean vectors satisfying an isotone Boolean function.
1 Introduction Z. Pawlak introduced his notion of information systems in the early 1980’s [11]. Information concerning properties of objects is the basic knowledge included in information systems, and it is given in terms of attributes and values of attributes. For example, we may express statements concerning the color of objects if the information system includes the attribute “color” and a set of values of this attribute consisting of “green”, “yellow”, etc. It should be noted that relational databases can be viewed as information systems in the sense of Pawlak. In an information system each subset of attributes defines an indiscernibility relation, which is an equivalence on the object set such that two objects are equivalent when their values of all attributes in the set are the same. It may turn out that a proper subset of a set of attributes classifies the objects with the same accuracy as the original set, which means that some attributes may be omitted. An attribute set is a reduct of an attribute set , if is a minimal subset of which defines the same indiscernibility relation as . The reduction problem means that we want to enumerate all reducts of a given subset of attributes. This work is devoted to the reduction problem in a dependence space. It is based on some papers of the same author, in particular on [5]. The fundamental notion appearing in the present paper is the concept of a dense family of a dependence space. We prove that our definition of dense families agrees with the definition presented earlier in the literature [10]; this result appeared also in [7]. Proposition 4.1
Turku Centre for Computer Science TUCS, Lemmink¨aisenkatu 14, FIN-20520 Turku, Finland. Email:
[email protected]
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characterizes reducts in dependence spaces by the means of dense families. Also difference functions are defined by using dense families (cf. [5]). Proposition 5.2 characterizes reducts by the means of difference functions and Proposition 6.1 contains a construction of a dense family in the dependence space of an information system starting with its indiscernibility matrix; this result appeared also in [6]. As stated above, this paper gives a survey of some results concerning reducts and their construction, but the presented formulations are simpler than the formulations published earlier. It also completes proofs of some theorems published without proofs in the quoted papers.
2 Preliminaries All general lattice theoretical and algebraic notions used in this paper can be found in [2, 4], for example. An oredered set (many authors use the shorthand poset) is a join-semilattice if the join
exists for all . An equivalence
relation on the semilattice if and relation on is a congruence imply for all ' . We denote by the & congruence class of , that is, !#"$ % ( . ) * + and *,+ exist for all * An ordered set is a lattice if . Let us consider a lattice . An element - is meet-irreducible if ./ ,10 implies 2 3 or 2#0 . We denote the set of all meet-irreducible elements 5'6 4 78 * (in case has a unit) of by 9 . The following lemma can be found in [2], for example. Lemma 2.1. If
is a finite lattice, then : &
Hence :1C
=@(
P
. This means that C is meet-dense.
=@(
B
Let be an ordered set and * . We say that is covered by , and , if Y and 0 Y implies 0 . It is known (see e.g. [2]) that write in a finite lattice the set of the elements of covered by exactly one * 7 element of is 9 . Thus, by Lemma 2.2, a subset of a finite lattice is meet-dense if and only if it contains all elements of which are covered by exactly one element of . A family of subsets of a set is said to be a closure system on if is I , we have . closed under intersections, which means that for all We denote by the power set of , i.e., the set of all subsets of . A closure operator on a set I is an extensive, idempotent and isotone map ; I *I , , and implies that is to say,I for all . A subset of is closed (with respect to ) if . A closure system on defines a closure operator on by the rule
Conversely, if
"D
&
I
DW(
P
is a closure operator on , then the family
H#"
I
&
(
of closed subsets of is a closure system. The relationship between closure systems and closure operators is bijective; the closure operator induced by the closure system is itself, and the closure system induced by the closure operator Iis . It is well-known that if is a closure system on , then the ordered set is a lattice in which
$
!
D/,
D
and
" !#
D/
D
for all D . Next we consider meet-dense subsets of the lattice system on a finite set.
I
%
$% $% $% '&
, where
I
is a closure
Proposition 2.3. Let be a meet-dense subset of a lattice , where closure system onI a finite set . & I P (a) For all , 2"D DW( I (b) For all the following three conditions are equivalent: LI (i) ;I I (ii) for all D , D implies D ; (iii) for all D , 5DN4 implies 5D 4 .
621
'&
is a
I
Proof. (a) Because , and D , we obtain by Lemmas 2.1 and 2.2 that
D
'"D
?&
9
$% $%
"D
&
I
I
I
DW(
DW(
&
$
"D
D
"D
&
&
I
D
DW(
I
DW(
DW( . . If D and . I Conversely, if for all D I & I "D I
%$
if and only if
"D
9
&
?I
I
?&
I
DW(
DW(
for all D
P
implies , then %, "% "% . Hence by (a), % % . Thus, (i) and (ii) are equivalent. Also (ii) and (iii) are equivalent since for all if and only if , '& . Hence, (b) Let
"D
&
"DO
I
I
I
LI
DW(*
D
I
I
I
D
DW(
, then I
D
I
D
A"D
D
5D
B
3 Dense families of dependence spaces We recall Novotn´y’s and Pawlak’s [9] definition of dependence spaces. We note that in [7] J¨arvinen studied infinite dependence spaces.
#
Definition. If is a finite nonempty set and is a congruence on the semilattice , then the ordered pair is said to be a dependence space.
beI a dependence space. Recalling the finiteness of , it is Let clear that for every , the congruence class has a greatest element I : . It was noted in [8] that for all ,
if and only if
$
P
In [8] it was also observed that is a closure operator on . We denote by the closure system corresponding to the closure operator . Hence, the family consists of the greatest elements of the classes.
&
#
( and be the congruence relation on Example 3.1. Let and let be the subset of which satisfies By Lemma 5.1 contains an element from each nonempty difference D , where D . Assume that is not minimal set with respect to that property, that is, there exist a which also contains an element from each nonempty difference -D , where DO . By Lemma 5.1 this implies that . But implies and hence , a contradiction! Therefore, is minimal set with respect to the property of containing an element from each nonempty difference XD , where D . This implies that is a reduct of by Proposition 4.1. On the other hand, suppose that is a reduct of . Then contains an element from each nonempty difference XD , where D , and thus . Suppose that . This means that there exists a vector 5
%$
!
'&
%
%
%
626
"%
such that . Let be the subset of which satisfies . Then obviously is a set which contains an element from each nonempty difference . Since , is not a reduct of , a contradiction! D , where D Hence, . if and only if there exist (b) By Lemma 5.1 it is obvious that an D such that D 4 and D . This is equivalent to the condition thatI# such that if and only if there exist an D 5DN4 5D and . I/ D Suppose that . This implies that for some , and hence . Assume that D D 4 D D D . Since , this implies ,a & Y" D D D 4 ( . Assume contradiction! Hence, ?& that there exists a ." D D D 4 ( such that . Clearly, this implies that and hence ,a ?& " 5D D 5DN4 ( . contradiction! Thus, %& Conversely, suppose that " 5D D 5D 4 ( . . Assume that there exists a such that Then obviously I3 . This implies that there exists an D such that WD and WD 4 . We obtain that for some D WD such that WDN4 , a contradiction! Hence, . B
"%
% '&
'%
&
&
&
%
& $% '& $% '&
% & % &
% '& are the Hence, the minimal true vectors of the difference function of
%
I
characteristic vectors of the reducts of
.
Example 5.3. Let us consider the dependence space defined in Ex ( " 6 ( " ( ( is known to be dense in . ample 3.1. The family / " " 6 The differences WD are all nonempty for all D . Hence, X"6 ( , " 6 ( , X" (
%
where 6
" "
(
6
"
,
"
,
6
%
"
stands for . The function has the minimal true vectors and 6 6 6 , which implies by Proposition 5.2 that 6 ( ( .
= = , The dual of a Boolean function , denoted by , is defined by where and = denote the complements of and = , respectively. It is well-known that and that the DNF expression of is obtained from that of by exchanging and , as well as constants
and 6 , and then expanding
the resulting
6 ,
, , 6
, is formula. the dual For example,
R of : .
,
6 , , It is known (see e.g. [1]) that for any isotone Boolean function , & " A ( . Let us denote simply by . By Proposition 5.2:
A
A
627
A
A
%
" "
5D
5D
$% '& $% '&
& ?& D D
.D
4
5DN4
(
P
(
" "6 ( "6 ( " ( ( is known to be dense in the depenThe family dence space of Example 3.1. Let us denote by the difference function of the set . Then
"
"
6
WD $
?&
6
D
$% "
6
$
6
'&
5DN4 ( P 6 (
R This means that and .
6Z,
, ,
6 , , $ #" 6 6 6 6 6 ( , as stated in Example 5.3. Hence, PPP PPP = =SQ = Q Remark. Let and ! = be a pair of isotone Boolean functions given by their minimal true vectors and , respectively. Let us consider the following problem; test whether and are mutually dual. In [3] Fredman and& Khachiyan showed that& this problem can be & !&
, where solved in time . This implies that for an isotone Boolean function given by its minimal true vectors and for a subset , a new& vector U & /& &
can be computed in time , where (see [1], for
example). This means also that for any isotone Boolean function given by
can be computed in time , where its minimal true vectors,& & & !& .
6 An application to information systems
L
, where is a set of objects, An information system is a triple X " ( is a set of attributes, and "R( is an indexed set of value sets of attributes. All these sets are assumed to be finite and nonempty. Each attribute is a function which assigns a value of the attribute to objects (see e.g. [9, 10, 11]). I For any , the indiscernibility relation of is defined by
'
#"
=
&
=
for all A
(
P
It is known that is an equivalence relation on such that its equivalence classes consist of objects which are indiscernible with respect to all attributes in . Let us define the following binary relation on the set :
1
P
So, two subsets of attributes are in the relation if and only if they define the same indiscernibility relation. It is known (see e.g. [8, 9]) that is a congru ence on the semilattice . Hence, the pair is a dependence
#
628
I
I
space. It can be easily seen that is a reduct of in the dependence space if and only if is a minimal subset of which defines the same indiscernibility relation as P.PP Assume that "= =*( . Then the indiscernibility matrix of is an -matrix 0 such that
$
0
&
" A
=
(
. Thus, the entry 0" consists of the attributes which do not for all 6 discern objects = and = (cf. discernibility matrices defined in [12]). It is now trivial that I P = = 0
Next we show how matrices of preimage relations induce dense families. L
is an information system and " ( Proposition 6.1. If 0 is the indiscernibility matrix of , then the family & 6 ( #"0
%
is dense in the dependence space
.
%
. If Proof. By Proposition 3.3 it suffices to show that
I =% = , then for all 6 , 0 iff = = iff I I 0 , which implies . Hence, .
I If , = = iff , then for all 6 I 0 iff = = , which implies . Thus, also and hence ! .
%
%
%
%
We conclude this paper by an example. L
%
I
0
iff
iff B
( , Example 6.2. Let us consider an information system " where the object set "6 ( consists of five persons, the attribute set consists of the attributes Age, Eyes, and Height, and the corresponding value sets are " Young, Middle, Old ( , " Blue, Brown, Green ( , and ! #" Short, Normal, Tall ( . Let the values of the attributes be defined as in the following table.
1 2 3 4 5
Age Young Young Middle Old Young
Eyes Blue Brown Brown Green Brown
Height Short Normal Tall Normal Normal
For example, the indiscernibility relation of the "attribute
set is an equivalence on which has the equivalence classes "6$( , " ( , " ( , and " ( . If we denote 5$#&% ' , )(+*'-, , and 0A)./' 0%2143 , then the indiscernibility matrix of is the following: 629
&
"$ (
" S(
&
&
" S(
%
&
"$0$(
&
"0 (
" (
&
" S(
" (
" (
&
"$0$(
" (
"$0$(
" , "$ ( , " ( , "$0$( , *( consisting of the By Proposition 6.1, the family . Let us denote entries of is dense in the dependence space .& by the difference function of the set . Then " D $ D D 4 ( " 6 6 6 6 6 6 ( . This means that R,80
@,?0
Z,? and R 80 , Z 80 , @ @,
@,?0
R,80 . Obviously, and 6 6 are the minimal true vectors of . Thus, 6 6 , 6 6 "$ ( , " 0 ( , and "$ 0$( are the reducts of in .
%
&
Acknowledgement The author thanks the referee for his valuable comments and suggestions.
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