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CM3112 Artificial Intelligence
Fuzzy logic: Connectives
Steven Schockaert
SchockaertS1@cardiff.ac.uk
School of Computer Science & Informatics Cardiff University

Fuzzy logic connectives
x: truth degree of “the glass is filled with wine” y: truth degree of “the wine is excellent”
f(x,y): truth degree of “the glass is filled with wine AND the wine is excellent”
Fuzzy logic is truth-functional: the truth degree of a conjunction is a matter of convention

Fuzzy logic connectives: conjunction
A t-norm (triangular norm) T is a [0,1]2-[0,1] mapping satisfying
1.boundary condition: 2.symmetry: 3.associativity: 4.monotonicity:
T(1,x) = x
T(x,y) = T(y,x)
T(x,T(y,z)) = T(T(x,y),z)
x1 ≤ x2 ⇒ T(x1,y) ≤ T(x2,y)
Intuitively, fuzzy conjunction should at least satisfy the properties of a t-norm

Fuzzy logic connectives: conjunction
Minimum TM(x,y) = min(x,y)
idempotence: min(x,x) = x
Product
TP(x,y) = x.y
Łukasiewicz TW(x,y) = max(0,x+y-1)
law of contradiction: TW(x,1-x) = 0

Fuzzy logic connectives: conjunction
Property: For every t-norm T and x in [0,1], it holds that T(0,x) = 0

Fuzzy logic connectives: conjunction
Property: For every t-norm T and x in [0,1], it holds that T(0,x) = 0
Proof: Using the monotonicity of T and the boundary condition on T, we find
T(0,x) ≤ T(0,1) = 0

Fuzzy logic connectives: conjunction
Property: The minimum is the greatest t-norm, i.e. for every t-norm T, it holds that T(x,y) ⩽ min(x,y) for all x and y in [0,1]

S(x, S(y, z)) = 1 T (1 x, 1 S(y, z))
= 1 T (1 x, 1 1 T (1 y, 1 z))
= 1 T (1 x, T (1 y, 1 z))
= 1 T (1 1 T (1 x, 1 y), 1 z)
Proof: Assume without loss of generality (due to the
Fuzzy logic connectives: conjunction
Property: The minimum is the greatest t-norm, i.e. for every = 1 T (T (1 x, 1 y), 1 z)
t-norm T, it holds that T(x,y) ⩽ min(x,y) for all x and y in [0,1]
= S(S(x, y), z) symmetry of T) that x ⩽ y. Then we have
monotonicity
assumption
T(x,y) ⇥ T(x,1) = x = min(x,y) boundary condition

Fuzzy logic connectives: conjunction
Property: The minimum is the only idempotent t-norm, i.e. the only t-norm T such that T(x,x)=x for all x in [0,1]

Fuzzy logic connectives: conjunction
Property: The minimum is the only idempotent t-norm, i.e. the only t-norm T such that T(x,x)=x for all x in [0,1]
Proof: suppose that T is idempotent and there is an (x,y) in [0,1]2 such that T(x,y) ≠ min(x,y), or equivalently since min is the greatest t-norm, T(x,y) < min(x,y). Fuzzy logic connectives: conjunction Property: The minimum is the only idempotent t-norm, i.e. the only t-norm T such that T(x,x)=x for all x in [0,1] Proof: suppose that T is idempotent and there is an (x,y) in [0,1]2 such that T(x,y) ≠ min(x,y), or equivalently since min is the greatest t-norm, T(x,y) < min(x,y). Suppose without loss of generality that x ≤ y (since T is symmetric). Then we have T(x,y) < x. Fuzzy logic connectives: conjunction Property: The minimum is the only idempotent t-norm, i.e. the only t-norm T such that T(x,x)=x for all x in [0,1] Proof: suppose that T is idempotent and there is an (x,y) in [0,1]2 such that T(x,y) ≠ min(x,y), or equivalently since min is the greatest t-norm, T(x,y) < min(x,y). Suppose without loss of generality that x ≤ y (since T is symmetric). Then we have T(x,y) < x. Using the monotonicity of T, we also find T(x,x) ≤ T(x,y) < x A contradiction since T was assumed to be idempotent. Fuzzy logic connectives: conjunction Fuzzy logic connectives: disjunction A t-conorm S is a [0,1]2-[0,1] mapping satisfying 1.boundary condition: 2.symmetry: 3.associativity: 4.monitonicity: S(0,x) = x S(x,y) = S(y,x) S(x,S(y,z)) = S(S(x,y),z) x1 ≤ x2 ⇒ S(x1,y) ≤ S(x2,y) Intuitively, fuzzy disjunction should at least satisfy the properties of a t-conorm Fuzzy logic connectives: disjunction Property: for every t-norm T, the mapping S defined by S(x,y) = 1-T(1-x,1-y) is a t-conorm, called the dual of T Fuzzy logic connectives: disjunction Maximum SM(x,y) = max(x,y) idempotence: max(x,x) = x distributivity: max(min(x,y),z) = min(max(x,z),max(y,z)) distributivity: min(max(x,y),z) = max(min(x,z),min(y,z)) Probabilistic sum SP(x,y) = x+y-x.y Łukasiewicz SW(x,y) = min(1,x+y) excluded middle: SW(x,1-x) = 1 xABxAxB x⇤A⇧B⇥x⇤A⌃x⇤B Fuzzy set operations x⇤A⌅B⇥x⇤A⌥x⇤B (A (A ( (A ⇤S B)(x) = S(A(x), B(x)) ⇤⇧⇥⇤⌃⇤ x⇤A⌅B⇥x⇤A⌥x⇤B x⇥A⌅Bx⇥A⇧x⇥B x⇥A⌅Bx⇥A⇧x⇥B (A B)(x) = T (A(x), B(x)) ( Crisp ⇧T B)(x)=T(A(x),B(x)) x⇥A⌅Bx⇥A⇧x⇥B x⇥A⌅Bx⇥A⇧x⇥B ⌅S B)(x) = S(A(x), B(x)) x⇥A⇤Bx⇥A⌃x⇥B x⇥A⇤Bx⇥A⌃x⇥B x ⇤ coA ⇥ ¬(x ⇤ A) A⌅T B)(x)=T(A(x),B(x)) )( )= ( ( ) ( ) ⇧T x⇥A⇤Bx⇥A⌃x⇥B (A⌅T B)(x)=T(A(x),B(x)) (A⌅T B)(x)=T(A(x),B(x)) Fuzzy x⇥A⇤Bx⇥A⌃x⇥B A ⌅S B)(x) = S(A(x), B(x)) x ⇤ coA ⇥ ¬(x ⇤ A) (A ⇤S B)(x) = S(A(x), B(x)) (A ⇤S B)(x) = S(A(x), B(x)) (coA)(x) = 1 A(x) (A⌅TB x TAx,Bx) (coA)(x) = 1 A(x) (A ⇤ B)(x) = S(A(x), B(x)) S Operations on fuzzy sets can straightforwardly be defined, using their logical definitions Fuzzy set operations Cheap hotel-price Around € 75 Fuzzy set operations Cheap hotel prices which are around € 75: intersection using TM Fuzzy set operations Hotel prices which are cheap or around € 75: union using SM Fuzzy set operations Hotel prices which are not around € 75 Fuzzy logic connectives: implication A implicator I is a [0,1]2-[0,1] mapping satisfying boundary conditions: decreasing 1st argument: increasing 2nd argument: I(0,1)=I(1,1)=I(0,0) =1, I(1,0)=0 x1 ≤ x2 ⇒ I(x1,y) ⩾ I(x2,y) y1 ≤ y2 ⇒ I(x,y1) ≤ I(x,y2) Which condition is redundant? Fuzzy logic connectives: implication IT (x, y) = sup{| ⇤ [0, 1] ⌅ T (x, ) ⇥ y} Property: Let S be a t-conorm, and let IS be the [0,1]2-[0,1] mapping defined by IS (x, y) = S(1 x, y) then IS is an implicator, which we call the S-implicator generated by S Fuzzy logic connectives: implication Property: Let T be a continuous t-norm, and let IT be the [0,1]2-[0,1] mapping defined by IT (x, y) = sup{| ⇥ [0, 1] ⇤ T (x, ) y} then IT is an implicator, which we call the residual implicator or R-implicator generated by T Fuzzy logic connectives: implication CHAPTER 2. PRELIMINARIES FROM FUZZY SE Kleene-Dienes implication Gödel implication Table 2.2: Popular choices of implicators. S–implicator ISM (a, b) = max(1 a, b) R–implicator 1 ifa⇤b ISP (a, b) = 1 a + ab ISW(a,b)=min(1,1a+b) ITW(a,b)=min(1,1a+b) infimum and supremum that for every implicator I ITP (a, b) = tors and residual implicators satisfy these requirements [57]. We fi ITM (a, b) = b otherwise 1 ifa⇤b b otherwise a Łukasiewicz implication an f (sup ) inf ( ) T o [ ) T(a,b) c a I (b,c) (2.2 ra tt aa y dd u ) ) r) 2e we C3 y ) 2 ) s rg properties arTe(wa,eIll–(kan,obw))n⇤. Fbor example, (2.22), (2.23), (2.26) and (2(.2 ) 2 2 e 2 r 2 al ⇤⌃⇤ T s are the more intuitive of the two, as they straightforwardly gener n of implication: p ⇧ q ⇥ ¬p⌥q. On the other hand, residual implica I (a,I (b,c)) = I (b,I (a,c)) (2.27 0, 1]. ThisT propTerty is somTetimTes also called Galois correspondence er to the notion of a generalized implication, preserving more impor Fuzzy logic connectives: implication The following lemma reveals a number of additional logical propert T(IT(a,b),IT(b,c)) ⇤ IT(a,c) (2.28 l implication such as transitivity and shunting. Of central import residual implicators. T(IT(a,b),IT(c,d)) ⇤ IT(T(a,c),T(b,d)) (2.29 uation principle, which holds for any left–continuous t–norm T an ( ( )) ( ( ) ) (2.30 op.27 a Ta,I b,c ⇤I I a,b,c left–continuouTs t–norm,Tit hTolds that implicator I [135]: T Property: For any continuous t-norm T it holds that: erties are well–known. For example, (2.22), (2.23), (2.26) and (2 I (1,a)=a ( T (a,Tb) ⇤ c ⌃ a ⇤ IT (b, c) (residuation property) is shown in [181]; (2.28) is shown in [90]; and (2.24) and (2.30) a⇤b⌃I (a,b)=1 ( T only need to show (2.29). By applying (2.24) (twice) and (2.26), 1]. This property is sometimes also called Galois corresponde T(IT(a,b),c) ⇤ IT(a,T(b,c)) ( he following lemma reveals a number of additional logical pro T(a,I (a,b))⇤b ( ONNECTIVES (modus ponens) esidual implicators. I (a,T(b,I (c,d))) ⇤ I (a,I (c,T(b,d))) = I (T(a,c),T(b,d)) TTTTTT IT(T(a,b),c) = IT(a,IT(b,c)) ( ( left–continuous t–norm, it holds that I (a,I (b,c)) = I (b,I (a,c)) T T T T (exchange principle) is a continuous t–norm, it holds that T(I (a,b),I (b,c)) ⇤ I (a,c) I (1,a)=a ( TTTT T(I (a,b),I (c,d)) ⇤ I (T(a,c),T(b,d)) ( T(a,I (a,b))=min(a,b) (divisibility()2. TTT T a⇤b⌃I (a,b)=1 T (transitivity) 5) ar ew [0, nc 2. Tpe 2. 2. C ⇤ a T 2.2 2.2 r T(a,I (b,c)) ⇤ I (I (a,b),c) (2.3 TT TT T T(I (a,b),c) ⇤ I (a,T(b,c)) (2 elates the ordering of t-norms to an ordering of their correspondin 2. (2 2. (2 2.2 31 (2