introduce strong and weak preimage relations, which are suitable for the investigation ... we introduce matrices of preimage relations and show how we can ...
Preimage Relations and Their Matrices Jouni J¨arvinen University of Turku, Department of Mathematics, FIN-20014 TURKU
1 Introduction Information systems (in the sense of Pawlak) are used for representing properties of objects by the means of attributes and their values. However, sometimes there are situations in which we cannot give an exact value of an attribute for an object; we can only approximate the incompletely known value by a subset of values, in which the actual value is expected to be. In these kind of information systems we can define several information relations on the object set. It seems that these relations have many common properties. Here we introduce strong and weak preimage relations, which are suitable for the investigation of those common features. Dependence spaces are general settings for the study of reducts of attribute sets, for example. In here we consider especially dense families of dependence spaces, and we give a characterization of reducts by the means of dense families. Dependence spaces defined by strong and weak preimage relations are also studied. In addition to this, we introduce matrices of preimage relations and show how we can determine dense families of dependence spaces defined by strong and weak preimage relations by using these matrices. This paper is structured as follows. In the next section we define information systems and some information relations. Section 3 contains the definition of strong and weak preimage relations. In Section 4 we give some basic concepts concerning dependence spaces and show how strong and weak preimage relations define dependence spaces. Finally, in Section 5 dense families of dependence spaces and matrices of preimage relations are studied.
2 Information Systems and Information Relations An information system is a triple S = (U, A, {Va }a∈A ), where U is a set of objects, A is a set of attributes, and {Va }a∈A is an indexed set of value sets of attributes. All these sets are assumed to be finite and nonempty. Each attribute is a function a: U → ℘(Va ) which assigns subsets of values to objects such that a(x) 6= ∅, for all a ∈ A and x ∈ U (see e.g. [5,6]). If |a(x)| = 1, then the information of the attribute a for the object x is complete (or deterministic), and we usually write a(x) = {v} simply by a(x) = v. If |a(x)| > 1, then the information of the attribute a for the object x is incomplete (or nondeterministic). For example, if a is “age”, and x is 25 years old, then a(x) = 25. It is also possible L. Polkowski and A. Skowron (Eds.): RSCTC' 98, LNAI 1424, pp. 139–146, 1998. c Springer-Verlag Berlin Heidelberg 1998
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that we know the age of x only approximately, say between 20 and 28. In this case a(x) = {20, . . . , 28}. In the following we shall present 16 information relations between objects, which are based on their values of attributes. These relations can be found in [5,6], for example. Suppose S = (U, A, {Va }a∈A ) is an information system and let B(⊆ A) be a subset of attributes. The first eight relations reflect indistinguishability between objects: • strong (weak) indiscernibility: (x, y) ∈ ind(B) (wind(B)) iff a(x) = a(y) for all (some) a ∈ B; • strong (weak) similarity: (x, y) ∈ sim(B) (wsim(B)) iff a(x) ∩ a(y) 6= ∅ for all (some) a ∈ B; • strong (weak) forward inclusion: (x, y) ∈ fin(B) (wfin(B)) iff a(x) ⊆ a(y) for all (some) a ∈ B; • strong (weak) backward inclusion: (x, y) ∈ bin(B) (wbin(B)) iff a(y) ⊆ a(x) for all (some) a ∈ B, where −a(x) denotes the complement of a(x) in Va . For example, two objects are strongly B-indiscernible if they have the same values for all attributes in B, and they are weakly B-similar if there is an attribute in B such that these objects have at least one common value for this attribute. The following eight relations reflect distinguishability between objects: • strong (weak) diversity: (x, y) ∈ div(B) (wdiv(B)) iff a(x) 6= a(y) for all (some) a ∈ B; • strong (weak) right orthogonality: (x, y) ∈ rort(B) (wrort(B)) iff a(x) ⊆ −a(y) for all (some) a ∈ B; • strong (weak) right negative similarity: (x, y) ∈ rnim(B) (wrnim(B)) iff a(x) ∩ −a(y) 6= ∅ for all (some) a ∈ B; • strong (weak) left negative similarity: (x, y) ∈ lnim(B) (wlnim(B)) iff −a(x) ∩ a(y) 6= ∅ for all (some) a ∈ B. For example, two objects are weakly B-diverse if their values for all attributes in B are not the same, and two objects are strongly right B-orthogonal if they have no common value for any attribute in B.
3 Preimage Relations All information relations defined in the end of the previous section are similar in the following sense. Two objects belong to a certain strong (resp. weak) information relation with respect to an attribute set B if and only if their all (resp. some) value sets of B-attributes are in a specified relation. For example, objects x and y are in the relation sim(B) if and only if a(x) ∩ a(y) 6= ∅ for all attributes a in B. In this section we introduce preimage relations. This notion allows us to study properties of information relations in a more abstract setting. We denote by Rel(A) the set of all binary relations on a set A. The complement of any relation R(∈ Rel(A)) is −R = {(x, y) ∈ A2 | (x, y) 6∈ R}. The set of all maps from A to B is denoted by B A . Moreover, we assume that U and Y are nonempty sets, and R ∈ Rel(Y ).
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For any map f ∈ Y U , the preimage relation of R is f −1 (R) = {(x, y) ∈ U 2 | f(x)Rf(y)}. Thus, two elements x and y are in the relation f −1 (R) if and only if their images f(x) and f(y) are in the relation R. For example, if R is the equality relation, =, then f −1 (R) is the kernel of the map f, ker f = {(x, y) | f(x) = f(y)}. The following obvious lemma shows that f −1 (R) inherits many properties from R. [ Lemma 1.]If R is reflexive, then f −1 (R) is reflexive, and similar conditions hold when R is irreflexive, symmetric, or transitive It is also true that
f −1 (−R) = −f −1 (R).
[ Example 1.]Suppose S = (U, A, {Va}a∈A ) is an information system, which is described in the following table.
We denote V = on Y by setting
S
Age (years) Weight (kg) Height (cm) P1 {22, . . . , 26} {48, . . . , 54} {154, . . . , 157} P2 {26, . . . , 33} {73, . . . , 78} {170, . . . , 175} P3 {24, . . . , 29} {51, . . . , 58} {159, . . . , 162} P4 {31, . . . , 37} {75, . . . , 82} {157, . . . , 165} a∈A
Va and Y = ℘(V ) − {∅}. Let us define the binary relation SIM (W1 , W2 ) ∈ SIM ⇐⇒ W1 ∩ W2 6= ∅.
The preimage relations of SIM with respect to the attributes Age, Weight, and Height are the following. Age−1 (SIM ) = {(x, y) ∈ U 2 | Age(x) ∩ Age(y) 6= ∅} = ∆ ∪ {(P1, P2), (P2, P1), (P1, P3), (P3, P1), (P2, P3), (P3, P2), (P2, P4), (P4, P2)} Weight−1 (SIM ) = {(x, y) ∈ U 2 | Weight(x) ∩ Weight(y) 6= ∅} = ∆ ∪ {(P1, P3), (P3, P1), (P2, P4), (P4, P2)} Height−1 (SIM ) = {(x, y) ∈ U 2 | Height(x) ∩ Height(y) 6= ∅} = ∆ ∪ {(P1, P4), (P4, P1), (P3, P4), (P4, P3)} Here ∆ is the diagonal relation of U ; that is, ∆ = {(x, x) | x ∈ U }. Two objects are in the relation Age−1 (SIM ) if and only if their ages are possibly the same, for example. Next we shall extent the notion of preimage relations in a natural way. Let A(⊆ Y U ) be a nonempty set of functions. The strong and weak preimage relations of a subset B(⊆ A) are defined by SR (B) = {(x, y) ∈ U 2 | (∀f ∈ B)f(x)Rf(y)}; WR (B) = {(x, y) ∈ U 2 | (∃f ∈ B)f(x)Rf(y)},
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respectively. The following properties are clear by the definition of strong and weak preimage relations. [ Lemma 2.]If B, C ⊆ A and f ∈ A, then (a) SR ({f}) =TWR ({f}) = f −1 (R); S (b) SR (B) = {f −1 (R) | f ∈ B} and WR (B) = {f −1 (R) | f ∈ B}; (c) SR (∅) = U × U and WR (∅) = ∅; (d) SR (B ∪ C) = SR (B) ∩ SR (C) and WR (B ∪ C) = WR (B) ∪ WR (C); (e) If B ⊆ C, then SR (C) ⊆ SR (B) and WR (B) ⊆ WR (C); (f) If B 6= ∅, then SR (B) ⊆ WR (B); (g) −SR (B) = W(−R) (B) and −WR (B) = S(−R) (B).
The following obvious proposition shows that also strong and weak preimage relations inherit many properties from the original relation. [ Proposition 1.]Let ∅ 6= B ⊆ A. If R is reflexive, then SR (B) and WR (B) are reflexive, and similar conditions apply when R is irreflexive or symmetric. Moreover, if R is transitive, then SR (B) is transitive. Information relations are preimage relations, as we see in the following example. [ Example S 2.]Assume S = (U, A, {Va }a∈A ) is an information system. As before, we set V = a∈A Va and Y = ℘(V ) − {∅}. Now we can define the following relations on the set Y . (W1 , W2 ) ∈ IN D (W1 , W2 ) ∈ SIM (W1 , W2 ) ∈ F IN (W1 , W2 ) ∈ BIN (W1 , W2 ) ∈ DIV (W1 , W2 ) ∈ RORT (W1 , W2 ) ∈ RN IM (W1 , W2 ) ∈ LN IM
⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒
W1 = W2 ; W1 ∩ W2 6= ∅; W1 ⊆ W2 ; W1 ⊇ W2 ; W1 6= W2 ; W1 ⊆ −W2 ; W1 ∩ −W2 6= ∅; −W1 ∩ W2 6= ∅.
It can be easily seen that −IN D = DIV , −SIM = RORT , −F IN = RN IM , and −BIN = LN IM . For any subset B(⊆ A) of attributes, ind(B) = SIND (B) sim(B) = SSIM (B) fin(B) = SF IN (B) bin(B) = SBIN (B) div(B) = SDIV (B) rort(B) = SRORT (B) rnim(B) = SRNIM (B) lnim(B) = SLNIM (B)
and and and and and and and and
wind(B) = WIND (B); wsim(B) = WSIM (B); wfin(B) = WF IN (B); wbin(B) = WBIN (B); wdiv(B) = WDIV (B); wrort(B) = WRORT (B); wrnim(B) = WRNIM (B); wlnim(B) = WLNIM (B).
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Because the relation IN D is trivially reflexive, symmetric, and transitive, then by Proposition 1, the relation ind(B) = SIND (B) is reflexive, symmetric, and transitive. Moreover, by Lemma 2(g), −IN D = DIV implies that −ind(B) = −SIND (B) = W(−IND) (B) = WDIV (B) = wdiv(B), for example. Preimage relations allow us to define also different kind of information relations, as we see in the following example. [ Example 3.]Suppose S = (U, A, {Va }a∈A ) is an information system, which is given by the following table. P1 P2 P3 P4
Height (cm) Weight (kg) 186 80 157 59 172 64 166 52
Let us now consider the usual order relation > on N. The preimage relations of > with respect to the attributes Height and Weight are Height−1 (>) = {(P1, P2), (P1, P3), (P1, P4), (P3, P2), (P3, P4), (P4, P2)} Weight−1 (>) = {(P1, P2), (P1, P3), (P1, P4), (P3, P2), (P3, P4), (P2, P4)}. For all B ⊆ A, S> (B) = {(x, y) ∈ U 2 | (∀a ∈ B)a(x) > a(y)}; W> (B) = {(x, y) ∈ U 2 | (∃a ∈ B)a(x) > a(y)}. For example, S> (A) = {(P1, P2), (P1, P3), (P1, P4), (P3, P2), (P3, P4)} and W> (A) = S> (A) ∪ {(P2, P4), (P4, P2)}.
4 Dependence Spaces of Preimage Relations Dependence spaces are algebraic structures which are suitable for the study of reducts of attribute sets, for example. An equivalence relation Θ on ℘(A) is a congruence on the semilattice (℘(A), ∪) if for all X1 , X2 , Y1 , Y2 ⊆ A, X1 ΘX2 and Y1 ΘY2 implies X1 ∪ Y1 ΘX2 ∪ Y2 . The congruence class of a subset B(⊆ A) is B/Θ = {C ⊆ A | BΘC}. If A is a finite nonempty set and Θ is a congruence on the semilattice, (℘(A), ∪) then, by the definition of Novotn´y and Pawlak, the pair D = (A, Θ) is called a dependence space (see e.g. [1,2,3,4]). Assume U and Y are nonempty sets, let R ∈ Rel(Y ), and let A(⊆ Y U ) be a finite S W subset of functions. Let us now define the following two binary relations ΘR and ΘR on the set ℘(A): S (B, C) ∈ ΘR ⇐⇒ SR (B) = SR (C); W (B, C) ∈ ΘR ⇐⇒ WR (B) = WR (C).
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S W So, two subsets of functions are in the relation ΘR (resp. ΘR ) iff they define the same strong (resp. weak) preimage relation. By Lemma 2(d) it is easy to see that the following proposition holds. S W [ Proposition 2.]The pairs (A, ΘR ) and (A, ΘR ) are dependence spaces.
By the previous proposition we can now define in an information system S = (U, A, {Va}a∈A ) a dependence space with respect to any information relation presented in Section 2. For example, the dependence space defined by strong diversity is Ddiv = (A, Θdiv ), where Θdiv = {(B, C) ∈ ℘(A)2 | div(B) = div(C)}. In the theory of information systems the notion of reducts is defined usually only with respect to the strong indiscernibility relations. In such cases a reduct of a set B of attributes is a minimal subset C of B, which defines the same strong indiscernibility relation as B. In here we define reducts in a more abstract setting of dependence spaces. Let D = (A, Θ) be a dependence space and B ⊆ A. We say that a subset C (⊆ A) is a reduct of B, if C ⊆ B and C is minimal in B/Θ. In the framework of dependence spaces we can study reducts of subsets of attributes in information systems with respect to any type information relation, as we see in the next example. [ Example 4.]Let S = (U, A, {Va }a∈A ) be an information system. Let us consider the S dependence space Dsim = (A, ΘSIM ). Two subsets B and C of attributes are now in S the relation ΘSIM if and only if they define the same strong similarity relation; that is, sim(B) = sim(C). A reduct of a subset B(⊆ A) of attributes is a minimal subset C of B, which defines the same strong similarity relation as B.
5 Dense Families and Matrices of Preimage Relations Dense families of dependence spaces are families of subsets which contain enough information about the structure of dependence spaces. In this section we shall show how we can find reducts of subsets by applying dense families. Moreover, we will study how in dependence spaces defined by preimage relations, dense families can be determined by using matrices of preimage relations. Suppose A is a set. Then each family H (⊆ ℘(A)) defines a binary relation Γ (H) on ℘(A) as follows. (B, C) ∈ Γ (H) iff for all X ∈ H, B ⊆ X ⇐⇒ C ⊆ X. It is easy to see that Γ (H) is a congruence on the semilattice (℘(A), ∪). Let D = (A, Θ) be a dependence space. We say that a family H (⊆ ℘(A)) is dense in D if Γ (H) = Θ [4]. Next we shall show how we can find dense families of dependence spaces which are defined by strong and weak preimage relations. Assume Y is a nonempty set and let U = {x1 , . . . , xn } be finite. If R is a binary relation on Y and A (⊆ Y U ) is a finite subset of functions, then the matrix of preimage relations of R is an n × n-matrix M(R) = (cij )n×n such that cij = {f ∈ A | f(xi )Rf(xj )}
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for all 1 ≤ i, j ≤ n. Thus, the entry cij consists of functions f ∈ A such that xi and xj are in the preimage relation f −1 (R) (cf., discernibility matrices defined in [7]). The following lemma is trivial. [ Lemma 3.]If A (⊆ Y U ) is a finite nonempty set of functions and M(R) = (cij )n×n is the matrix of preimage relations of R (∈ Rel(Y )), then for all B ⊆ A, (a) (xi , xj ) ∈ SR (B) iff B ⊆ cij ; (b) (xi, xj ) ∈ WR (B) iff B ∩ cij 6= ∅. Our next proposition shows how matrices of preimage relations define dense families of dependence spaces. [ Proposition 3.]Assume A (⊆ Y U ) is a finite nonempty set of functions and M(R) = (cij )n×n is the matrix of preimage relations of R (∈ Rel(Y )). Then S (a) {cij | 1 ≤ i, j ≤ n} is dense in the dependence space (A, ΘR ); W (b) {−cij | 1 ≤ i, j ≤ n} is dense in the dependence space (A, ΘR ). [ Proof.](a) Let us denote H = {cij | 1 ≤ i, j ≤ n}. We have to show that Γ (H) = S S ΘR . If (B, C) ∈ ΘR , then for all 1 ≤ i, j ≤ n, B ⊆ cij iff (xi, xj ) ∈ SR (B) iff S ⊆ Γ (H). (xi, xj ) ∈ SR (C) iff C ⊆ cij , which implies (B, C) ∈ Γ (H). Hence, ΘR If (B, C) ∈ Γ (H), then for all 1 ≤ i, j ≤ n, (xi , xj ) ∈ SR (B) iff B ⊆ cij iff C ⊆ S cij iff (xi , xj ) ∈ SR (C), which implies SR (B) = SR (C). Thus, also Γ (H) ⊆ ΘR and S so Γ (H) = ΘR . W (b) Let us denote K = {−cij | 1 ≤ i, j ≤ n}. If (B, C) ∈ ΘR , then for all 1 ≤ i, j ≤ n, B ⊆ −cij iff B ∩ cij = ∅ iff (xi, xj ) 6∈ WR (B) iff (xi , xj ) 6∈ WR (C) iff W C ∩ cij = ∅ iff C ⊆ −cij , which implies (B, C) ∈ Γ (K). Hence, ΘR ⊆ Γ (K). If (B, C) ∈ Γ (K), then for all 1 ≤ i, j ≤ n, (xi, xj ) ∈ WR (B) iff B ∩ cij 6= ∅ iff B 6⊆ −cij iff C 6⊆ −cij iff C ∩ cij 6= ∅ iff (xi , xj ) ∈ WR (C), which implies W W WR (B) = WR (C). So, also Γ (K) ⊆ ΘR and hence Γ (K) = ΘR . The next proposition, which can be found in [2], characterizes the reducts of given subset of attributes by the means of dense sets. [ Proposition 4.]Let D = (A, Θ) be a dependence space and let H (⊆ ℘(A)) be dense in D. If B ⊆ A, then C is a reduct of B iff C is a minimal set which contains an element from each nonempty differences B − X, where X ∈ H. [ Example 5.]Suppose S = (U, A, {Va}a∈A ) is an information system, which is described in the following table. Age (years) Weight (kg) Height (cm) P1 {22, . . . , 26} {48, . . . , 54} {154, . . . , 157} P2 {26, . . . , 33} {73, . . . , 78} {170, . . . , 175} P3 {24, . . . , 29} {51, . . . , 58} {159, . . . , 162} P4 {31, . . . , 37} {75, . . . , 82} {157, . . . , 165} Let us denote a = Age, b = Weight, and c = Height. If R = SIM , then the preimage matrix of R is the following.
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P1 P2 P3 P4 P1 A {a} {a, b} {c} P2 {a} A {a} {a, b} P3 {a, b} {a} A {c} P4 {c} {a, b} {c} A By Proposition 3, the family H = {{a}, {c}, {a, b}, A} is dense in the dependence S space Dsim = (A, ΘSIM ). Next we determine the reducts of the set A in this dependence space. The differences A − X, where X ∈ H, are A − {a} = {b, c}, A − {c} = {a, b}, A − {a, b} = {c}, and A − A = ∅. The first three of them are nonempty, and clearly {a, c} and {b, c} are minimal sets which contain an element from these differences. Then by Proposition 4, the set A has the reducts {a, c} and {b, c} in the dependence space Dsim . Thus, {a, c} and {b, c} are minimal sets which define the same strong similarity relation as A. Similarly, the family K = {−{a}, −{c}, −{a, b}, −A} = {∅, {c}, {a, b}, {b, c}} W is dense in the dependence space Dwsim = (A, ΘSIM ). The differences A − X, where X ∈ K, are A − ∅ = A, A − {c} = {a, b}, A − {a, b} = {c}, and A − {b, c} = {a}. They all are nonempty and obviously {a, c} is the only minimal set which contains an element from these differences. This means that {a, c} is the only reduct of A in the dependence space Dwsim . So, {a, c} is the unique minimal set which defines the same weak similarity relation as A.
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