Approximation of Relations. Andrzej Skowron. Warsaw University. Banacha 2, Warsaw, Poland. Jaroslaw Stepaniuk

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1 Approximation of Relations 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 [3]. In generalized approximation spaces we dene the lower and upper set approximations. We illustrate the introduced notions with dierent types of relation approximation. 1 Introduction Investigations on relation approximation are well motivated both from theoretical and practical points of view. The equality approximation is fundamental for a generalization of the rough set approach [3] 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 good tools for function approximation. Finally, let us also mention some applications of relation approximation to discrete optimization problems [7] where approximations of input-output relations of programs are investigated. The relation approximation based on the rough set approach is formulated in [3] and investigated in several papers (see e.g.[7]). In this paper we introduce a generalization of approximation spaces formulated in [3] and we also present a generalization of set approximation in these spaces. Our intention is to give a general tool for the investigation of relation approximations. 2 Generalized approximation spaces In this section we present a generalization of the lower and upper approximations of sets, introduced in [3] and [9]. First we recall the denitions from [3], [9]. 1

2 An approximation space is an ordered pair R= (U; IN D), where U is a nonempty set and IN D UU is an equivalence relation called the indiscernibility relation. By [x] IND we denote an equivalence class of the relation IN D dened by the object x. The lower and upper approximations of a set X are dened as follows: 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 [4] as follows: (x; X) = jx \ [x] INDj j[x] IND j where j:j denotes set cardinality. Hence we have Moreover X = fx 2 U : (x; X) = 1g and X = fx 2 U : (x; X) > 0g: (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 [9] 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 = jx \ [x] INDj j[x] IND j where x 2 X and X U. By putting = 0 and f = id we obtain the case considered in [3]. 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; 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. 2

3 Hence we have: L((id); X) L((f ); X) and U((id); X) U((f ); X) for any set X and function f satisfying the conditions formulated above. 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 jy j for any 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 inclusion vagueness : 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. One can observe an analogy with fuzzy set theory [1],[8], 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 some partial information about any considered object accessible. In general it is not possible to identify an object having that partial information only. In rough set theory [3] 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 respect to a given set of attributes. Thus we obtain an uncertainty function I : U! P(U) dened by I(x) = Inf?1 (Inf(x)) for any x 2 U [4],[5]. In general, an uncertainty function I on U is any function from U into P(U) satisfying the condition x 2 I(x) for any x 2 U. The inclusion vagueness 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 [6]. This is the notion of a structural element. Let I be a given uncertainty function and let P : I(U)! f0; 1g. 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. An approximation space is a system R = (U; I; ; P ) where U is a non-empty set of objects, : P(U) P(U)! [0; 1], I : U! P(U) P : I(U)! f0; 1g are inclusion vagueness, uncertainty and structurality functions, respectively. The lower and upper approximations of a set X U in R are dened by: L(R; X) = fx 2 U : P (I(x)) = 1 and (I; )(y; X) = 1 for any y 2 T fi(z) : x 2 I(z)g 3

4 and U(R; X) = fx 2 U : P (I(x)) = 1 and (I; )(y; X) > 0 for any y 2 I(x)g; respectively. In the case when I is dened by the indiscernibility relation the above denitions can be written in a simpler form, namely L(R; X) = fx 2 U : P (I(x)) = 1 and (I; )(x; X) = 1g and U(R; X) = fx 2 U : P (I(x)) = 1 and (I; )(x; X) > 0g, respectively. This holds because in the considered case the following implication is true: if I(x) \ I(y) 6= ; then I(x) = I(y). In rough set theory I(x) = [x] IND for any x 2 U, (X; Y ) = jx \Y j=jxj for any X; Y U and P is the identity on I(U). For variable precision rough set model approach only the inclusion vagueness is dened dierently, namely (f )(X; Y ) = f (jx \ Y j=jxj) where f is specied as above. There is one more aspect of set approximation which we would like to discuss. It concerns information availability for the approximated sets. In [3] sets are represented by listing their elements in an information system. Sets are approximated on the basis of available information about objects. In this case the approximated sets are taken to be exact, which does not hold in general, e.g. we can only have some information about relations which we would like to approximate. We consider the case when information about approximated sets in a given approximation space R = (U; I; ; P ) is specied by a projection function (from Rinto (R 1 ;... ;R k )) J : P(U)! P(U 1 [...[U k ), where R i = (U) i ; I i ; i ; P i ) for any i = 1;...; k are approximation spaces and U i \ U j 6= ; for i 6= j. If X U, then the set J(X) \ U i is called the (J; i)- projection of X. 3 Approximation of relations In this section we show examples of structurality and inclusion vagueness functions. We also present some relationships between dierent approximation spaces related by projections. Finally we formulate an optimization problem. Let R = (U; I; ; P ) and R i = (U i ; I i ; i ; P i ) for i = 1;... ; k be approximation spaces, where U = U 1... U k. If R U 1... U k, then by i (R) we denote the projection of the k-ary relation R onto the i-th axis i.e. i (R) = fx : 9x x i?1 9x i x k R(x 1 ;... ; x i?1 ; x; x i+1 ;...; x k )g: Let us consider examples of properties P (Q 1 ;... ; Q k ; G) dening structurality functions. They are described by: P (Q 1 ;... ; Q k ; G)(I(x)) = 1 i Q 1 y (I(x))... Q k y k 2 k (I(x))(y 1 ;...; y k ) 2 G where Q i 2 f8; 9g for i = 1;... ; k; G U 1... U k and x 2 U. Applying in the construction of relation approximation the structural elements dened above we choose to consider only those I(x) which have these additional properties P (Q 1 ;... ; Q k ; G). 4

5 The denition of the vagueness function for relations can be based on dierent idea than the cardinality of the intersection with a structural element, e.g. for k = 2 and a threshold p(0 < p < 1) let us assume (X; Y ) = c i w=j 1 (X)j = c, where w = jfy (X) : jy (Y ) : (y 1 ; y 2 ) 2 Y j=j 2 (Y )j pgj. In [3] it is assumed that structural elements are of the form I(x) = I 1 (x 1 )... I k (x k ) for x = (x 1 ;... ; x k ) 2 U and I i (x i ) = [x i ] IND i, i.e. the i-th information function is dened by the indiscernibility relation IN D i U i U i. In this case I is also dened by the indiscernibility relation. Assuming = (f ) and P (x) = 1 for all x 2 U we obtain the following equalities [9] for the approximation of a relation R in R o = (U; I; ; P ): L(R o ; R) = f(x 1 ;... ; x k ) 2 U : jr \ [x 1] IND1... [x k ] IND kj j[x 1 ] IND1... [x k ] IND kj U(R o ; R) = f(x 1 ;... ; x k ) 2 U : jr \ [x 1] IND1... [x k ] IND kj j[x 1 ] IND1... [x k ] IND kj 1? g > g: Proposition. Let R = (U; I; ; P ), R 0 = (U; I; 0 ; P ), R i = (U i ; I i ; i ; P i ) for i = 1;... ; k be approximation spaces, where U = U 1... U k ; I i is dened by an indiscernibility relation IN D i U i U i, i.e. I i (x i ) = [x i ] IND i for x i 2 U i ; I(x) = I 1 (x 1 )... I k (x k ) for x = (x 1 ;... ; x k ) 2 U. We assume also that P (I(x)) = P i (I i (x i )) = 1 for x = (x 1 ;... ; x k ) 2 U, i = (f i) for some 0 i < 0:5. If J and J X (where ; 6= X U) are projections of R into (R 1 ;... ;R k ) dened by J(R) \ U i = i (R) and J X (R) \ U i = i (R \ X) for R U and then (X; Y ) = minf i (J(X) \ U i ; J(Y ) \ U i ) : 1 i kg 0 (X; Y ) = minf i (J X (X) \ U i ; J X (Y ) \ U i ) : 1 i kg 1., 0 are monotonic with respect to the second argument; 2. L(R; R) = fx 2 U : j[x i ] IND i \ i (R)j=j[x i ] IND ij 1? i for i = 1;... ; kg; 3. L(R 0 ; R) = fx 2 U : j i (R \ I(x))j=j[x i ] IND ij 1? i for i = 1;...; kg; 4. U(R; R) = fx 2 U : j[x i ] IND i \ i (R)j=j[x i ] IND ij > i for i = 1;...; kg; 5. U(R 0 ; R) = fx 2 U : j i (R \ I(x))j=j[x i ] IND ij > i for i = 1;...; kg; 6. L(R; R) = k i=1l(r i ; i (R)) and U(R; R) = k i=1u(r i ; i (R)); 7. L(R 0 ; R) L(R; R) and U(R 0 ; R) U(R; R). 2 Finally let us formulate an important problem related to relation approximation. 5

6 Optimization Problem: INPUT: approximation spaces R = (U; I; ; P );R i = (U i ; I i ; i ; P i ), for i = 1;... ; k, where U = U 1...U k ; projection J from R into (R 1 ;... ;R k ); family F of functions from [0; 1] k into [0; 1]. OUTPUT: f opt 2 F such that S(f opt ) = inffs(f) : f 2 F g, where S(f) = supfj(x; Y )? f opt ( 1 (X 1 ; Y 1 );... ; k (X k ; Y k ))j : X; Y Ug, = (I; ), i = (I i ; i ), X i = J(X) \ U i, Y i = J(Y ) \ U i for i = 1;...; k. Strategies for nding optimal (sub-optimal) solutions for this problem are important for applications of rough set methods in process control area. Results related to this problem will be presented in our next papers. 4 Conclusions We have proposed the generalizations of approximation space and set approximation as a tool for the investigation of relation approximation. These denitions allow to omit some drawbacks of the classical denitions. We hope that they can be considered as a good starting point for investigations of important problems related to relation approximation, e.g. the optimization problem for controllers design. References [1] Dubois D., Prade H. and Yager R.: Fuzzy Sets for Intelligent Systems, Morgan Kaufmann, San Mateo [2] Pawlak Z. : Rough Relations, Bull. Acad. Polon. Sci, Ser. Tech. Sci. vol. 34 (9-10), 1986, [3] Pawlak Z. : Rough Sets. Theoretical Aspects of Reasoning about Data, Kluwer Academic Publishers, Dordrecht [4] Pawlak Z., Skowron A. : Rough Membership Functions In: M.Federizzi, J.Kacprzyk and R.R.Yager (eds.): Advances in the Dempster-Shafer Theory of Evidence. John Wiley and Sons, New York 1994, [5] Pawlak Z.: Hard and soft sets, Proceedings of the International Workshop on Rough Sets and Knowledge Discovery, Ban, Alberta, Canada, October 12-15, 1993, [6] Serra, J.: Image Analysis and Mathematical Morphology, Academic Press, New York - London [7] Skowron A., Stepaniuk J. : Towards an Approximation Theory of Discrete Problems, Fundamenta Informatice 15(2), 1991, [8] Zadeh L: Fuzzy sets, Information and Control 8, 1965, [9] Ziarko W. : Variable Precision Rough Set Model, Journal of Computer and System Sciences, vol. 46 (1), 1993,