Tolerance Approximation Spaces. Andrzej Skowron. Institute of Mathematics. Warsaw University. Banacha 2, Warsaw, Poland

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1 Tolerance Approximation Spaces Andrzej Skowron Institute of Mathematics Warsaw University Banacha 2, Warsaw, Poland Jaroslaw Stepaniuk Institute of Computer Science Technical University of Bialystok Wiejska 45A, Bialystok, Poland Abstract We generalize the notion of an approximation space introduced in [8]. In tolerance approximation spaces we dene the lower and upper set approximations. We investigate some attribute reduction problems for tolerance approximation spaces determined by tolerance information systems. The tolerance relation dened by the so called uncertainty function or the positive region of a given partition of objects have been chosen as invariants in the attribute reduction process. We obtain the solutions of the reduction problems by applying boolean reasoning [1]. The solutions are represented by tolerance reducts and relative tolerance reducts. 1 Introduction We discuss a generalization of the approximation space denition introduced in [8]. Our investigations are motivated by the results of [3], [7], [18] and [20] concerning sets with the boundary regions less crisp than in the case presented in [8] as well as by papers [6], [13], [17] on relation approximation. Investigations on relation approximation are well motivated both from theoretical and practical points of view. The equality approximation [4], [5] is fundamental for a generalization of the rough set approach [8] to the case of an indiscernibility relation being based on an approximation of the equality relations in the value sets of attributes rather than on the exact equality relations in these sets. Applications of rough set methods in process control require some new tools for investigation of function approximation. Finally, let us also mention some applications of relation approximation to discrete optimization problems (see [13] where approximations of input-output relations of programs are investigated). We propose several modications of the approximation space denition [8]. The rst one concerns the so called uncertainty function. We assume that objects are perceived by information about them. Information Inf (x) about the object x is represented by its attribute value vector. The set of all objects with similar value vectors creates the tolerance set I(x). In [8] all objects with the same value vector create the indiscernibility class. The relation y 2 I(x) is in this case an equivalence relation. We consider a more general case when it can be a tolerance relation [11]. The second one is related to structural elements introduced by analogy with mathematical morphology [12]. In the construction of the lower and upper set approximations only tolerance sets being structural elements are considered. The structural elements are also used in construction of approximation of relations [17]. The third one introduces a generalization of a rough membership function [9]. We assume that only incomplete information about perceived objects is available. In consequence to answer a question whether an object x belongs to an object set X we have to

2 answer a question whether its tolerance set I(x) is included in X. Hence we take as a primitive notion a vague inclusion function rather than the fuzzy or rough membership function. Our approach allows to unify dierent cases considered in [3], [8], [20]. One can dene a variety of the lower and upper set approximations in the case when the relation y 2 I(x) is a tolerance (and not equivalence) relation. In general, experiments related to a particular approximation space can show which type of approximation is the best e.g. from the point of view of classication of new objects. We also consider generalized approximation spaces determined by tolerance information systems (A; D) where A is an information system and D is a family of discernibility relations determining the similarity (tolerance) relation between objects. The main two problems investigated in the paper are related to attribute reduction. The solution for the rst problem consists of the family of minimal attribute sets, called tolerance reducts, dening the same tolerance relation as the initial set of attributes. The solution for the second problem consists of the family of minimal attribute sets, called relative tolerance reducts, dening the same positive region [8] as the initial set of attributes. 2 Generalized approximation spaces In this section we present a generalization of the lower and upper approximations of sets. First we recall the denitions from [8], [9]. An approximation space is an ordered pair R = (U; IND), where U is a non-empty set and IND U U is an equivalence relation called the indiscernibility relation. By [x] IND we denote an equivalence class of the relation IND dened by the object x. The lower and upper approximations of a set X are dened by: X = fx 2 U : [x] IND Xg and X = fx 2 U : [x] IND \ X 6= ;g respectively. The rough membership function of a set X U (in a given approximation space R) is dened [9] by: (x; X) = j [x] IND \ X j j [x] IND j where j:j denotes set cardinality. Hence we have X = x 2 U : (x; X) = 1 and X = x 2 U : (x; X) > 0 : Moreover (x; X) = 1 i [x] IND X; (x; X) > 0 i [x] IND \ X 6= ;; (x; X) = 0 i [x] IND \ X = ;. The denition of lower and upper approximations of sets has been generalized in [20] by introducing the so called variable precision rough set model. Let be a real number within the range 0 < 0:5 and let f : [0; 1]! [0; 1] be a non-decreasing function such that f (t) = 0 i 0 t and f (t) = 1 i 1? t 1. The f -membership function (f ) can now be dened by (f )(x; X) = f (t); where t = j [x] IND \ X j j [x] IND j where x 2 X and X U. Assuming = 0 and f equal to the identity id on [0; 1] we obtain the case considered in [8]. The lower and upper approximations of a set X U with respect to the membership function (f ) can be presented in the following form: L((f ); X) = fx 2 U : (f )(x; X) = 1g and U((f ); X) = fx 2 U : (f )(x; X) > 0g, respectively. Let us observe that the following facts hold: if (f )(x; X) = 1, then not necessarily [x] IND X; if [x] IND X, then (f )(x; X) = 1; 2

3 if (f )(x; X) > 0, then [x] IND \ X 6= ;; if [x] IND \ X 6= ;, then not necessarily (f )(x; X) > 0; if (f )(x; X) = 0, then not necessarily [x] IND \ X = ;; if [x] IND \ X = ;, then (f )(x; X) = 0. Hence we have: L? (id); X L? (f ); X U? (f ); X U? (id); X for any set X and function f satisfying the conditions formulated above. The variable precision rough set model can be treated as a method of boundary region thinning. The membership functions introduced above can be extended to functions from P(U) P(U) into the interval [0; 1] of reals, namely (f )(X; Y ) = f (t), where t = jx \ Y j=jxj for any ; 6= X; Y U and 0 < 0:5. The extension can be treated as a measure of inclusion vagueness. We must decide what conditions should a function, called vague inclusion : P(U) P(U)! [0; 1] satisfy to be an appropriate measure for the degree of inclusion of sets. Here we only assume monotonicity with respect to the second argument, i.e. (X; Y ) (X; Z) for any Y Z; where X; Y; Z U: We also assume (;; Y ) = 1 for any Y U. One can observe an analogy with fuzzy set theory [2], [19], i.e. that is a fuzzy inclusion function. The dierence is that we are taking as a primitive notion a function of the above form measuring the degree of set inclusion rather than the degree membership function for objects. The reason is that in general one can only expect to have only partial information about any considered object accessible. In general it is not possible to identify an object having that partial information only. For example, in rough set theory the information Inf (x) about an object x is specied by the vector of attribute values on that object x. This information Inf (x) denes the set [x] of all objects indiscernible with x i.e. the set fy 2 U : Inf (x) = Inf (y)g. Thus we obtain an uncertainty function I : U! P(U) dened by I(x) = Inf?1 (Inf (x)) for any x 2 U [11]. In general, an uncertainty function I on U is any function from U into P(U) satisfying the condition x 2 I(x) and y 2 I(x) i x 2 I(y) for any x; y 2 U. It means that we assume the relation xiy i y 2 I(x) is a tolerance relation. The vague inclusion function and the uncertainty function I dene the membership function (I; )(x; X) = (I(x); X), where x 2 U, X U. We would like to add one more condition to the denition of set approximation which arises by analogy with mathematical morphology [12]. This is the notion of a structural element. Let I be a given uncertainty function and let P : I(U)! f0; 1g (where I(U) = fi(x) : x 2 Ug). Any set X 2 I(U) satisfying P (X) = 1 is called a P -structural element (in I(U)). The function P is called the structurality function. The structurality functions allow us to construct set approximations by introducing some global conditions on sets I(x) out of which the set approximations are constructed. Let us consider two simple examples of structurality functions. Let P t (X) = 1 i jxj=juj > t where t is a given threshold and X U. In this case the set approximations built with a help of P t will be constructed out of sets I(x) with cardinality greater than tju j. The second example of structurality function is related to a xed information system A = (U; A). Let b 2 A and v 1 ;... ; v k be a given sequence of values of b, called the b-pattern, where fv 1 ;... ; v k g V b. We assume P b (X) = 1 i there exists an injection f : f1;... ; kg! X such that b(f(i)) = v i for i = 1;... ; k where X U. Hence the set approximations built with a help of P b will be constructed only out of sets I(x) having the b-pattern v 1 ;... ; v k. An approximation space is a system R = (U; I; ; P ) where U is a non-empty set of objects, I : U! P(U), : P(U) P(U)! [0; 1], P : I(U)! f0; 1g are uncertainty, vague inclusion and structurality functions, respectively. The classical denition of approximation space [8] corresponds to the approximation spaces R = (U; I; ; P ), where P (I(x)) = 1 for any x 2 U, fi(x) : x 2 Ug creates a 3

4 partition of U; (X; Y ) = jx \ Y j=jxj for any X; Y U, X 6= ;. The denition of the lower and the upper set approximations can be written as follows: and L(R; X) = fx 2 U : P (I(x)) = 1 & (I(x); X) = 1g U(R; X) = fx 2 U : P (I(x)) = 1 & (I(x); X) > 0g A modication of this approach is presented in [20] for the so called variable precision rough model. The approximation spaces R = (U; I; ; P ), where 0 < 0:5 are dened in the same way as before with only one exception, namely the vague inclusion is dened by f (t) = ( 0 if 0 t (t? )=(1? 2) if < t < 1? 1 if t 1? The set approximations are dened by L(R; X) = fx 2 U : P (I(x)) = 1 & (I(x); X) = 1g and U(R; X) = fx 2 U : P (I(x)) = 1 & (I(x); X) > 0g: If the uncertainty function I denes a tolerance relation not being an equivalence relation then there is a variety of possibilities to dene the lower and upper set approximation. Let R be an approximation space and let r U U. A lower and an upper approximations of X U in R (relatively to r) are dened by and L r (R; X) = fx 2 U : P (I(x)) = 1 & 8y(xry?! (I(y); X) = 1)g U r (R; X) = fx 2 U : P (I(x)) = 1 & 8y(xry?! (I(y); X) > 0)g; respectively. There are dierent possible choices for r, e.g. (i) r 1 is the identity relation; (ii) xr 2 y i y 2 T fi(z) : x 2 I(z)g; and (iii) xr 3 y i y 2 I(x). In this case we have L r3 (R; X) L r2 (R; X) L r1 (R; X) and U r3 (R; X) U r2 (R; X) U r1 (R; X). It depends on particular application which type of set approximation to choose. One can also show that the Variable Precision Rough Sets Model with Asymmetric Bounds [3] can be described in the introduced model of generalized approximation spaces. 3 Tolerance information systems and tolerance reducts We present in this section a generalization of a reduct notion to the case when indiscernibility of objects is dened by a tolerance relation. If A = (U; A) is an information system [8] and B A then INF(B) = finf B (x) : x 2 Ug is the set of information vectors Inf B (x) = f(a; a(x)) : a 2 Bg. If u 2 INF(C) and B C A then ujb = f(a; w) 2 u : a 2 Bg i.e. ujb is the restriction of u to B. A tolerance information system is a pair (A; D) where A = (U; A) is an information system, D = (D B ) BA and D B INF(B) INF(B) is a relation, called the discernibility relation, satisfying the following conditions: (i) INF(B) INF(B)? D B is a tolerance (indiscernibility) relation; (ii) ((u? v) [ (v? u) (u 0? v 0 ) [ (v 0? u 0 )) & ud B v?! u 0 D B v 0 for any u; v; u 0 ; v 0 2 INF(B) i.e. D B is monotonic with respect to the discernibility property; (iii) non(ud C v) implies non(ujbd B vjb) for any B C and u; v 2 INF(C). A (B; D B )-tolerance function I[B; D B ] : U?! P(U) is dened by y 2 I[B; D B ](x) i non (Inf B(x)D B Inf B(y)) for any x; y 2 U. The set I[B; D B ](x) is called the tolerance set of x. The relation INF(B) INF(B)? D B expresses similarity of objects in terms of accessible information about them. 4

5 The set RED(A; D) is dened by fb A : I[A; D A ] = I[B; D B ] and I[A; D A ] 6= I[C; D C ] for any C Bg: Elements of RED(A; D) are called tolerance reducts of (A; D) (or, tolerance reducts, in short). It follows from the denition that the tolerance reducts are minimal attribute sets preserving (A; D A ) - tolerance function (or, equivalently, the tolerance relation A dened by x A y i non(inf A (x)d A Inf A (y))). The tolerance reducts of (A; D) can be constructed in an analogous way as reducts [16] of information systems. More precisely, we show that tolerance reducts of (A; D) are computable relatively to the family f(a; D)[u; v] : ud A vg where (A; D)[u; v] = fb A : ujb D B vjb & (non(ujc D C vjc) for any C B)g. One can check (applying the assumption (ii)) that if B 2 (A; D)[u; v] then B is a subset of the set of attributes on which u and v are dierent. Let F(A; D) = fc A : 8u; v 2 INF(A) (ud A v?! 9B 2 (A; D)[u; v](b C)g, so any set from F(A; D) consists of a minimal set of attributes sucient to discern between any discernible (by attributes from A) information vectors u and v. We have the following: Proposition 3.1. C 2 F(A; D) i I[A; D A ] = I[C; D C ] for any C A. Proof.? Suppose I[A; D A ] = I[C; D C ] and C 62 F(A; D) for some C A. Hence for some u 0 ; v 0 2 INF(A), discernible by A i.e. u 0 D A v 0, the set C does not contain any set from (A; D)[u 0 ; v 0 ]. In consequence non(u 0 jc D C v 0 jc). We also have (u 0 jc D C v 0 jc) (from the assumptions I[A; D A ] = I[C; D C ] and u 0 D A v 0 ), a contradiction.?! Suppose now I[A; D A ] 6= I[C; D C ] and C 2 F(A; D). Then either y 0 2 (I [A, D A ](x 0 )? I[C; D C ](x 0 )) or y 0 2 (I[C; D C ](x 0 )? I[A; D A ](x 0 )) for some x 0 ; y 0 2 U. Let u 0 = Inf A(x 0 ) and v 0 = Inf A(y 0 ). In the former case non(u 0 D A v 0 ) and u 0 jc D C v 0 jc. The last condition implies u 0 D A v 0 (from (iii)) but this contradicts the rst condition. In the latter case we have non(u 0 jc D C v 0 jc). From the assumption C 2 F(A; D) it follows that there exists B 0 C satisfying B 0 2 (A; D)[u 0 ; v 0 ]. Hence u 0 jb 0 D B0 v 0 jb 0 0j, so by (iii) u 0 jc D C v 0 jc, a contradiction. 2 We have the following theorem: Theorem 3.2. Let (A; D) be a tolerance information system, where A = (U; A) and A = fa 1 ;... ; a m g. Let ga;d be a boolean function of m boolean variables a 1;... ; a m corresponding to attributes a 1 ;... ; a m and dened by ga;d? a 1;... ; a m = ^_ f^b : B 2 (A; D)[u; v]g : ud A v where B = fa : a 2 Bg. We have the following equivalence: a i 1 ^... ^ a i k is a prime implicant of ga;d i fa i1 ;... ; a ik g 2 RED(A; D). Sketch of the proof. From the condition (ii) of the denition of tolerance information system it follows that for any x i ; x j 2 U it is enough to consider only attributes discerning between objects x i and x j, so one can apply the discernibility matrix M(A) = (c ij ) where c ij = fa 2 A : a(x i ) 6= a(x j )g, i; j = 1;... ; n, and n = juj (see [16]). For any non-empty entry c ij of the discernibility matrix M(A) we consider all minimal subsets B of attributes sucient to discern (by D A ) between information vectors u = Inf A (x i ), v = Inf A (x j ). Proposition 3.1 characterizes the attribute sets discerning (by D A ) between given information vectors u; v. One of these sets must be taken to keep the discernibility (by D A ) between u and v. Hence we obtain the expression _n^ B : B 2 (A; D)[u; v]o 5

6 which should be satised with respect to the discernible (by D A ) information vectors u; v (where B is the set of boolean variables corresponding to attributes from B). The conjunction ga;d of the formulae of the above form described in the theorem expresses a global constraint. Its satisability is necessary and sucient for keeping as an invariant the discernibility between any pair of discernible (by D A ) information vectors in a given information system A, and in the consequence for keeping as an invariant the tolerance relation A (it follows from the condition (ii) of the denition of the tolerance information system that any two information vectors indiscernible by D A will be indiscernible by D B for any subset B of A). The constructed boolean formula ga;d has the following property: ga;d(val) = 1 i A = B where B = fa : val(a ) = 1g for any valuation val of boolean variables a 1;... ; a m. Hence we have that a i 1 ^... ^ a i k is a prime implicant of ga;d i B = fa ik ;... ; a ik g is a minimal set of attributes such that A = B (or equivalently I[A; D A ] = I[B; D B ]). 2 We also obtain Corollary 3.3. NP-hard. The problem of computing minimal tolerance reduct of (A; D) is Proof. Analogous to presented in [16]. 2 We say that an approximation space R = (U; I; ; P ) is determined by a tolerance information system (A; D) where A = (U; A) if I = I[A; D A ]. If R is determined by (A; D) then the set RED(A; D) is called the tolerance reduct set of R. Example 1. Let us consider an information system A presented in Table 1 and the discernibility relations D B for B A dened by ud B v i j(u? v) [ (v? u)j > 1 for any u; v 2 INF(B). (i.e. information vectors v and u discern among two attributes at least). Height Weight Hair Eyes 1 Short Light Dark Blue 2 Tall Heavy Dark Blue 3 Tall Heavy Dark Brown 4 Tall Heavy Red Blue 5 Short Light Blond Blue 6 Tall Heavy Blond Brown 7 Tall Heavy Blond Blue 8 Short Light Blond Brown Table 1. We have U = f1; 2; 3; 4; 5; 6; 7; 8g, A = fh; W; R; Eg where H; W; R; E are abbreviations for Height, Weight, hair, Eyes. One can calculate the value of I = I[A; D] applying the discernibility matrix. The rows and columns are labelled by information vectors u i from INF(A). On the position corresponding to the row labelled by u and the column labelled by v there is the set of all attributes discerning between objects with information u and v. All elements of (A; D)[u; v] are some subsets of this set. u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 1 - H,W H,W,E H,W,R R H,W,R,E H,W,R R,E u 2 H,W - E R H,R R,E R H,W,R,E u 3 H,W,E E - R,E H,W,R,E R R,E H,W,R u 4 H,W,R R R,E - H,W,R R,E R H,R,E u 5 R H,W,R H,W,R,E H,W,R - H,W,E H,W E u 6 H,W,R,E R,E R R,E H,W,E - E H,W u 7 H,W,R R R,E R H,W E - H,W,E u 8 R,E H,W,R,E H,W,R H,R,E E H,W H,W,E - Table 2 6

7 We have I(1) = f1; 5g, I(2) = f2; 3; 4; 7g, I(3) = f2; 3; 6g, I(4) = f2; 4; 7g, I(5) = f1; 5; 8g, I(6) = f3; 6; 7g, I(7) = f2; 4; 6; 7g, I(8) = f5; 8g. We have ga;d(h; W; R; E) = H ^ W ^ R ^ E (to simplify notation we omit star superscripts). Hence RED(A; D) = fh; W; R; Eg. 2 4 Tolerance decision tables and relative tolerance reducts In this section we assume that A = (U; A[fdg) is a xed decision table (with the decision d 62 A) and (A; D) is a tolerance information system where A = (U; A). If u 2 INF(A) then u A = fx 2 U : Inf A (x) = ug. By [A ] we denote a binary relation in INF(A) INF(A) dened by u[a ]v i 9i(either ua Y i or va Y i ) where Y i = fx 2 U : d(x) = ig. We denote by R[A; D] the generalized approximation space R = (U; I; ; P ) where the vague inclusion function is dened by (X; Y ) = jx \ Y j=jxj for X; Y U ((;; Y ) = 0 for any Y ), P (I(x)) = 1 for any x 2 U and the uncertainty function I is equal to I[A; D A ]. We assume in this section the lower approximation of X U is dened by L(R[A; D]; X) = fx 2 U : (I(x); X) = 1g. The positive region of R[A; D] with respect to d, denoted by POS[A; D A ; d], is dened by POS[A; D A ; d] = S L(R[A; D]; Y i ) : i 2 V d & Y i = fx 2 U : d(x) = ig. The set of relative tolerance reducts of (A; D) with respect to d is dened by B 2 REL RED(A; D A ; d) i B is a minimal (with respect to inclusion) subset of A satisfying the following condition: POS[A; D A ; d] = POS[B; D B ; d] where B = (U; B): The elements of REL RED(A; D A ; d) are called relative tolerance reducts of (A; D A ; d). One can show that the relative tolerance reducts can be computed in an analogous way as the tolerance reducts. In fact, we have the following: Theorem 4.1. Let (A; D) be a tolerance information system, where A = (U; A), A = fa 1 ;... ; a m g and let A = (U; A [ fdg) be a decision table. Let ha ;D be a boolean function of variables a 1;... ; a m corresponding to attributes a 1 ;... ; a m dened by ha ;D(a 1;... ; a m) = ^_ f^b : B 2 (A; D A )[u; v]g : ud A v & u[a ]v : Then we have the following equivalence: fa i1 ;... ; a ik g 2 REL RED(A ; D) i a i 1 ^... ^ a i k is a prime implicant of ha ;D: The proof is analogous as in the case of Theorem The above theorem gives us a method for computing the relative tolerance reducts. Example 2. Let us consider a decision table A = (U; A [ fdg) presented in Table 3 and D dened as in Example 1. We have IND(A) = fu 1 ;... ; u 8 g. For simplicity of notation we omit star superscripts in the discernibility matrix presented in Table 4. 7

8 We have Height Weight Hair Eyes d 1 Short Light Dark Blue 1 2 Tall Heavy Dark Blue 1 3 Tall Heavy Dark Brown 1 4 Tall Heavy Red Blue 2 5 Short Light Blond Blue 2 6 Tall Heavy Blond Brown 1 7 Tall Heavy Blond Blue 2 8 Short Light Blond Brown 1 Table 3. u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 1 - H,W H,W,E H,W,R,d R,d H,W,R,E H,W,R,d R,E u 2 H,W - E R,d H,R,d R,E R,d H,W,R,E u 3 H,W,E E - R,E,d H,W,R,E,d R R,E,d H,W,R u 4 H,W,R,d R,d R,E,d - H,W,R R,E,d R H,R,E,d u 5 R,d H,W,R,d H,W,R,E,d H,W,R - H,W,E,d H,W E,d u 6 H,W,R,E R,E R R,E,d H,W,E,d - E,d H,W u 7 H,W,R,d R,d R,E,d R H,W E,d - H,W,E,d u 8 R,E H,W,R,E H,W,R H,R,E,d E,d H,W H,W,E,d - Table 4. ha ;D (H,W,R,E) = (H^W_H^R_ W^R) ^ (R^E) ^ (H^W_H^E_W^E) = (H^R^E)_(W^R^E). Hence REL RED(A ; D) = fb 1 ; B 2 g where B 2 = fh; R; Eg, B 2 = fw; R; Eg. One can check that I[B 2 ; D B2 ] = I[B 1 ; D B1 ] 6= I[A; D A ]. For example I[B 1 ; D B1 ](1) = f1; 2; 5g and I[A; D A ](1) = f1; 5g. 2 In applications it can also be important to check if the distances between the lower and upper approximation of decision classes determined by relative tolerance reducts are suciently large. This observation suggests that in the process of attribute reduction some lower bounds on distances between decision classes should be preserved [15]. Conclusions We have presented a generalization of the approximation space notion [8] to cover some of its modications considered e.g. in [9] and [3], [20]. Two problems of attribute reduction with the tolerance relation I and the positive region as invariants in the reduction have been solved. We have showed that boolean reasoning [1] can be applied to obtain the solutions in the form of tolerance reducts and relative tolerance reducts, respectively. This work has been supported by the grant # 8-S from the State Committee for Scientic Research (Komitet Badan Naukowych) References [1] Brown, F.N.: Boolean reasoning, Kluwer, Dordrecht [2] Dubois D., Prade H. and Yager R.: Fuzzy Sets for Intelligent Systems, Morgan Kaufmann, San Mateo [3] Katzberg J.D., Ziarko W.: Variable Precision Rough Sets with Asymmetric Bounds, Proceedings of the International Workshop on Rough Sets and Knowledge Discovery, Ban, Alberta, Canada, October 12-15, 1993, pp [4] Marcus, S.: Tolerance relations, Cech topologies, learning process, Bull. Acad. Polon. Sci., Ser. Math., in print. 8

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