Some examples of generated fuzzy implicators

Transcription

1 Some examples of generated fuzzy impliators Dana Hliněná and Vladislav Biba Department of Mathematis, FEEC, Brno University of Tehnology Tehniká 8, Brno, Czeh Rep. v Abstrat Conjuntors in MV-logi with truth values range [0, 1] are monotone extensions of the lassial onjuntion. Let f : [0, 1] [0, ] be a stritly dereasing funtion, suh that f(1) = 0, then we an define onjuntor C : [0, 1] 2 [0, 1] by C(x, y) = f ( 1) (f(x) + f(y)), where the pseudo-inverse f ( 1) is given by f ( 1) (x) = sup{t [0, 1]; f(t) > x}, f is alled an additive generator of C. A funtion I : [0, 1] 2 [0, 1] is said to an impliator if and only if I(1, 0) = 0 and I(0, 0) = I(0, 1) = I(1, 1) = 1 and I is non-inreasing in its first omponent and non-dereasing in its seond omponent. The unary operator n : [0, 1] [0, 1] is alled negator if for any a, b [0, 1] holds a b n(b) n(a), n(0) = 1, n(1) = 0. Starting with the onjuntor C and standard negation N s (x) = 1 x, we an introdue the impliation operator in [0, 1] valued logi as follows: I C (x, y) = 1 C(x, 1 y). Another way of extending the lassial binary impliation operator to the unit interval [0, 1] uses the residuation R C with respet to the left-ontinuous onjuntor C R C (x, y) = sup{z [0, 1]; C(x, z) y}. There exists several onstrutions of impliators. We will ompare these impliators and some their properties will be given. 1 Introdution We an reall definitions of the most important onnetives in MV-logi. Definition 1 An unary operator n : [0,1] [0,1] is alled negator if for any a,b [0,1] it holds (i) a < b n(b) n(a), (ii) n(0) = 1,n(1) = 0. The negator n is alled strong negator if and only if the mapping n is one to one. Evidently, strong negator is ontinuous and its inverse n 1 is strong negator too. The negator n is alled involutive negator if and only if for all a [0,1], n(n(a)) = a. It an be easily proved that involutive negator is strong and n 1 = n. Example 1 (1) n(a) = 1 a involutive negator,

2 (2) n(a) = 1 a 2 (3) n(a) = 1 a 2 (4) n(0) = 1,n(a) = 0 if a > 0 strong, non-involutive negator, involutive negator, non-strong negator. Definition 2 A non-dereasing mapping C : [0,1] 2 [0,1] is alled onjuntor if for any a,b [0,1] it holds (i) C(a,b) = 0 whenever a = 0, or b = 0 (ii) C(1,1) = 1. Commonly used onjuntors in MV-logi are the triangular norms. Definition 3 A triangular norm (t-norm for short) is a binary operation on the unit interval [0,1], i.e., a funtion T : [0,1] 2 [0,1] suh that for all x,y,z [0,1] the following four axioms are satisfied: (T1) Commutativity T(x, y) = T(y, x), (T2) Assoiativity T(x,T(y,z)) = T(T(x,y),z), (T3) Monotoniity T(x,y) T(x,z) whenever y z, (T4) Boundary Condition T(x, 1) = x. Example 2 The following are the four basi t-norms: Minimum T M given by Produt T P given by T M (x,y) = min(x,y), T P (x,y) = x y, Lukasiewiz t-norm T L given by T L (x,y) = max(0,x + y 1), Drasti produt T D given by T D (x,y) = { min(x, y) if max(x, y) = 1, 0 otherwise. Remark 1 Note, that the dual operator to the onjuntor C defined by a non-dereasing mapping D : [0,1] 2 [0,1], suh that D(a,b) = 1 whenever a = 1 or b = 1 and D(0,0) = 0 is alled the disjuntor D. Commonly used disjuntors in MV-logi are the triangular onorms. Triangular onorms (also alled S norms) are dual to t norms under the order reversing operation whih assigns 1 x to x on [0,1]. Definition 4 A funtion I : [0,1] 2 [0,1] is said to be an impliator if I(1,0) = 0, I(0,0) = I(0, 1) = I(1, 1) = 1, I is non-inreasing in its first omponent and non-dereasing in its seond omponent.

3 Starting with the onjuntor C and the negation n, we an introdue the impliation operator in [0, 1]-valued logi as follows: I C (x,y) = n(c(x,n(y))). Another way of extending the lassial binary impliation operator to the unit interval [0, 1] uses the residuation R C with respet to a left-ontinuous onjuntor C R C (x,y) = sup{z [0,1];C(x,z) y}. When starting with the impliator I we an define a onjuntor C I as follows C I (x,y) = inf{z [0,1];I(x,z) y}. Remark 2 Note that in the lassial logi it is C = C IC but in MV-logi C is not equal to C IC in general. Our onstrutions of impliators will make use of extending the lassial inverse of funtion. One way of extending is desribed in next definitions. Definition 5 Let ϕ : [0,1] [0,1] be a non-dereasing funtion. The funtion ϕ ( 1) whih is defined by ϕ ( 1) (x) = sup{z [0,1];ϕ(z) < x}, is alled the pseudo-inverse of the funtion ϕ, with the onvention sup = 0. Definition 6 Let f : [0,1] [0,1] be a non-inreasing funtion. The funtion f ( 1) whih is defined by f ( 1) (x) = sup{z [0,1];f(z) > x}, is alled the pseudo-inverse of the funtion f, with the onvention sup = 0. 2 The onstrution of impliators based on generators There exist several onstrutions of impliators via generalized onjuntors and disjuntors. Detailed desribtion of these onstrutions are in [4]. Main ontributions of this paper are new properties of these generalized impliators whih are desribed in next propositions. Let f be a stritly dereasing funtion suh that f(1) = 0 and g be a stritly inreasing funtion suh that g(0) = 0. Then we an define the impliators I f,i g as follows: I f (x,y) = f ( 1) (f(y + ) f(x)), I g (x,y) = g ( 1) (g(1 x) + g(y)), where f(y + ) = lim f(x). Now, we reall definitions of some important properties of impliators x y + whih we will investigate. Definition 7 An impliator I is alled border impliator if for all b [0,1] it holds I(1,b) = b.

4 Definition 8 An impliator I is said to satisfy the exhange priniple, if I(x,I(y,z)) = I(y,I(x,z) for all x,y,z [0,1]. Definition 9 A border impliator is alled ontrapositive impliator with respet to a given negator n if for all a,b [0,1] it holds I(a,b) = I(n(b),n(a)). The main ontributions of our paper are, infat orrollaries of the following tehnial result, whih has not been published to our knowledge yet. Proposition 1 Let be a positive real number, then for pseudo-inverse of positive multiple of any left-ontinuous funtion f we get ( ( f(x)) ( 1) = f ( 1) x ). Proof. Let f be a non-dereasing funtion, then and then f ( 1) (x) = sup{z [0,1];f(z) < x} ( f) ( 1) (x) = sup {z [0,1]; f(z) < x} = sup { z [0,1];f(z) < x } Now, the proof for the ase of non-inreasing funtion is analogous. First, we will investigate the properties of I f impliators: ( = f ( 1) x ). Proposition 2 Let f : [0, 1] [0, ] be a left-ontinuous, stritly dereasing funtion suh that f(1) = 0. Then I f is border impliator and moreover I f = R C, where C is the onjuntor generated by additive generator f. It is well known that generators of ontinuous Arhimedean t-norms are unique up to a positive multipliative onstant, and this is also true for the f generators of I f impliators. The next theorem is a orrollary of Proposition 1. Theorem 1 The f generator of an I f impliator is uniquely determined up to a positive multipliative Seond, we turn our attention to I g impliators and their properties. Proposition 3 Let g : [0, 1] [0, ] be a left-ontinuous, stritly inreasing funtion suh that g(0) = 0. Then I g is border and ontrapositive impliator and moreover I g = R C, where C is the onjuntor generated by additive generator f, f (x) = g(1 x). Remark 3 Note, that if f(x) = g(1 x), then impliators I f and I g are idential. Proposition 4 Let g be a left-ontinuous, stritly inreasing funtion suh that g(0) = 0. Then impliator I g satisfies the exhange priniple. Theorem 2 The g generator of an I g impliator is uniquely determined up to a positive multipliative Generalization of I g impliators is given in next proposition.

5 Proposition 5 Let n be a negator, g be a left-ontinuous, stritly inreasing funtion suh that g(0) = 0. Then the funtion I g n : [0,1] 2 [0,1] whih is defined by is border impliator. I g n (x,y) = g( 1) (g(n(x)) + g(y)), Proposition 6 Let n be an involutive negator, g be a left-ontinuous, stritly inreasing funtion suh that g(0) = 0. Then impliator I g n is ontrapositive impliator with respet to the negator n. Proposition 7 Let g be a left-ontinuous, stritly inreasing funtion suh that g(0) = 0. Then impliator I g n holds the exhange priniple. Theorem 3 The g generator of an I g n impliator is uniquely determined up to a positive multipliative Aknowledgement. Supported by Projet 1ET of the Program Information Soiety and by Projet MSM of the Ministry of Eduation. Referenes [1] B. De Baets, R. Mesiar. Residual impliators of ontinuous t-norms Pro. EUFIT 96, Aahen, 27-31, 1996 [2] S. Gottwald. Fuzzy Sets and Fuzzy Logi Vieweg, Braunshweig, 1993 [3] R. Mesiar. Generated onjuntors and related operators in MV-logi as a basis for AI appliations ECAI 98 Workshop 17, Brighton, 1 5, 1998 [4] D. Smutná. On many valued onjuntions and impliations Journal of Eletrial Engineering 10/s vol. 50, 8 10, 1999