REIHE COMPUTATIONAL INTELLIGENCE

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U N I V E R S I T Y OF D O R T M U N D

REIHE COMPUTATIONAL INTELLIGENCE

C O L L A B O R A T I V E R E S E A R C H C E N T E R 5 3 1

Design and Management of Complex Technical Processes and Systems by means of Computational Intelligence Methods

Axiomatic Considerations of the Concepts of R-Implication and T-Norm

Helmut Thiele

No. CI-35/98

Technical Report ISSN 1433-3325 April 1998

Secretary of the SFB 531 ⋅ University of Dortmund ⋅ Dept. of Computer Science/XI 44221 Dortmund ⋅ Germany

This work is a product of the Collaborative Research Center 531, “Computational

Intelligence”, at the University of Dortmund and was printed with financial support

of the Deutsche Forschungsgemeinschaft.

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Axiomatic Considerations of the Concepts of R-Implication and T-Norm

^∗

Helmut Thiele

Abstract In the paper presented we investigate under which conditions there exists a one-to- one correspondance between a class of generalized R-implications and a class of generalized T-norms based on the mutual definability of these classes. Furthermore, we study which properties of functions of the one class will be translated into properties of functions of the other class by the bijection mentioned above. This paper can be considered as a continuation of author’s paper dealing with the same problematics for S-implications on the one hand and S-norms (T-conorms) and negations on the other hand.

Keywords R-implications, T-norms, mutual definability, bijections between classes of gen- eralized R-implications and classes of generalized T-norms, translating properties of R- implications and T-norms by this bijection.

1 Basic Definitions and Fundamental Results

By 〈0, 1〉 we denote the set of all real numbers r with 0 ≦ r ≦ 1.

We define

FUNCT(2) =

_def

l Φ Φ : 〈 0, 1 〉× 〈 0, 1 〉 → 〈 0, 1 〉 q ^.

In [26] we have defined the functional operators RIMP and TNOR with RIMP, TNOR : FUNCT(2) → FUNCT(2) as follows where τ,π ∈ FUNCT(2) and r, s ∈ 〈0, 1〉

Definition 1

1. RIMP(τ)(r, s) =

def

sup m ^{t t} ∈ 〈0, 1〉∧ τ(r,t) ≦ s r

2. TNOR( π)(r, s) =

def

inf m ^{t t} ∈ 〈0, 1〉∧ π(r,t) ≧ s r ^.

We underline that definition 1 generalizes the well-known residuation operation and the gen- eration of a T-norm by a given implication, respectively.

The following six theorems and corollaries one can find already in [26], but without proof.

Theorem 1

For every r, s ∈ 〈0, 1〉 ,

TNOR a RIMP(τ) f (r, s) ≦ τ(r, s) . Proof

Assume r, s ∈ 〈0, 1〉. By definition of the operator TNOR we have to prove inf m ^{t t} ∈ 〈 0, 1 〉∧ RIMP( τ )(r,t) ≧ s r

^≦

τ (r, s) . (1)

By definition of inf and RIMP it is sufficient to show

∃t d ^t ^{∈ 〈0, 1} 〉∧ sup m t′ t′ ∈ 〈0, 1 〉∧ τ (r,t′) ≦ t r

^{≧ s}

∧ t ≦ τ(r, s) i

(2)

∗Revised version of a paper originally published in International Conference on Onformation Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU ’98), Paris, France, July 6–10, 1998

1

(3)

hence by definition of sup it is sufficient to show

∃t∃t′ c ^t,t ′ ∈ 〈0, 1〉∧ τ(r,t′) ≦ t ∧ t′

≧ s

∧ t ≦ τ(r, s) h ^.

(3) We put

t =

def

τ(r, s) t′ =

_def

s . (4)

Obviously, (3) holds.

Theorem 2

If the function τ is monotone and left-hand continuous with respect to 〈 0, 1 〉 and its second argument, then for every r, s ∈ 〈 0, 1 〉 ,

τ(r, s) ≦ TNOR a RIMP(τ) f (r, s) . Proof

Assume r, s ∈ 〈0, 1 〉 . By definition of the operator TNOR we have to prove τ(r, s) ≦ inf m ^{t t} ∈ 〈0, 1 〉∧ RIMP(τ)(r,t) ≧ s r ^.

(1)

By definition of inf it is sufficient to show

∀t c ^t ∈ 〈0, 1〉∧ RIMP(τ)(r,t) ≧ s →τ(r, s) ≦ t h ^,

(2)

hence by definition of RIMP it is sufficient to show

∀t d ^t ∈ 〈0, 1〉∧ sup m t′ t′ ∈ 〈0, 1〉∧ τ(r,t′) ≦ t r

^{≧ s}

→ τ(r, s) ≦ t i ^.

(3)

By definition of sup we have

sup m τ(r,t′ t′ ∈ 〈0, 1〉∧ τ(r,t′) ≦ t r

^{≦ t .}

(4)

Furthermore, as for every fixed r ∈ 〈0, 1 〉 the function τ(r, s) is left-hand continuous with respect to s ∈ 〈0, 1 〉 , we obtain

τ d ^{r, sup} m t′ t ′ ∈ 〈0, 1 〉∧ τ (r,t′) ≦ t ri

^{≦ sup}

m τ(r,t′) t′ ∈ 〈0, 1 〉∧ τ (r,t′) ≦ t r ^.

(5)

Hence from (4) and (5) we get

τ d ^{r, sup} m t′ t′ ∈ 〈0, 1〉∧ τ(r,t′) ≦ t ri

^{≦ t .}

(6)

In order to prove (3) we assume

sup m ^t ′ t′ ∈ 〈0, 1〉∧ τ(r,t′) ≦ t r

^{≧ s .}

(7)

Because for every fixed r ∈ 〈0, 1〉 the function τ(r, s) is monotone with respect to s ∈ 〈0, 1〉, we obtain

τ (r, s) ≦ τ d ^{r, sup} m ^t ′ t ′ ∈ 〈0, 1 〉∧ τ (r,t′) ≦ t ri ^,

(8)

hence from (8) and (6) we get

τ(r, s) ≦ t . (9)

2

(4)

Corollary 3

If the function τ is monotone and left-hand continuous with respect to 〈0, 1〉 and its second argument, then for every r, s ∈ 〈0, 1〉,

TNOR a RIMP(τ) f (r, s) = τ(r, s) . Proof

By theorems 1 and 2.

Definition 2

1. FUNCT(2, M2, LHC2)

=

_def

R S T ^{ϕ ϕ}

: 〈 0, 1 〉× 〈 0, 1 〉 → 〈 0, 1 〉 and ϕ is monotone and left-hand continuous with respect to 〈 0, 1 〉 and its second argument

U V W

2. FUNCT(2, M2, RHC2)

=

def

R S T ^{ϕ ϕ}

: 〈0, 1〉× 〈0, 1〉 → 〈0, 1〉 and ϕ is monotone and right-hand continuous with respect to 〈0, 1〉 and its second argument

U V W

Corollary 4

The operator RIMP is an injection from FUNCT(2, M2, LHC2) into FUNCT(2).

Proof

By corollary 3.

Now, we are faced with the problem to characterize the image of the class FUNCT(2, M2, LHC2) generated by the operator RIMP. The following theorems and corrollaries solve this problem.

Assume π : 〈 0, 1 〉× 〈 0, 1 〉 → 〈 0, 1 〉 . Theorem 5

If the function π is monotone and right-hand continuous with respect to 〈0, 1〉 and its second argument, then for every r, s ∈ 〈0, 1〉,

RIMP a TNOR(π) f (r, s) ≦ π(r, s) . Proof

Like theorem 2.

Theorem 6

For every r, s ∈ 〈0, 1 〉 ,

π(r, s) ≦ RIMP a TNOR(π) f (r, s) Proof

Like theorem 1.

Corollary 7

If the function π is monotone and right-hand continuous with respect to 〈 0, 1 〉 and its second argument, then for every r, s ∈ 〈0, 1〉 :

RIMP a TNOR(π) f (r, s) = π(r, s) . Proof

By theorems 5 and 6.

3

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Corollary 8

1. TNOR is a bijection from FUNCT(2, M2, RHC2) onto FUNCT(2, M2, LHC2) 2. RIMP is the inversion of TNOR.

Proof ad 1.

By thorems 5 and 6 we get

FUNCT(2, M2, RHC2) j RIMP a FUNCT(2, M2, LHC2) f , (1)

hence by monotonicity of TNOR

TNOR a FUNCT(2, M2, RHC2) f

j TNOR

b RIMP a FUNCT(2, M2, LHC2) fg . (2)

By corollary 3 we have

TNOR b RIMP a FUNCT(2, M2, LHC2) fg

j FUNCT(2, M2, LHC2),

(3)

hence by (2)

TNOR a FUNCT(2, M2, RHC2) f

j FUNCT(2, M2, LHC2) .

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ad 2.

In analogy to 1.

2 Axioms for Characterizing T-Norms and R-Implications

Definition 3 (Axioms for Characterizing T-Norms) Assume τ : 〈0, 1〉× 〈0, 1〉 → 〈0, 1〉.

TN1.

∀s b s ∈ 〈0, 1〉→τ(0, s) = 0 g

TN2.

∀r b r ∈ 〈0, 1〉→ τ(r, 0) = 0 g

TN3.

∀s b s ∈ 〈0, 1〉→τ(1, s) = s g

TN4.

∀r b ^r ∈ 〈0, 1 〉→ τ (r, 1) = r g

TN5.

∀r∀s∀t c ^{r, s,t} ∈ 〈0, 1 〉∧ r ≦ s →τ(r,t) ≦ τ(s,t) h

TN6.

∀r∀s∀t c ^{r, s,t} ∈ 〈0, 1 〉∧ s ≦ t →τ(r, s) ≦ τ(r,t) h

TN7.

∀r∀s b r, s ∈ 〈0, 1〉→τ(r, s) = τ(s, r) g

TN8.

∀r∀s∀t b r, s,t ∈ 〈0, 1〉→ τ a r, τ(s,t) f = τ a τ(r, s),t fg

Definition 4 (Axioms for Characterizing R-Implications) Assume π : 〈0, 1〉× 〈0, 1〉 → 〈0, 1〉.

RIM1.

∀s b ^s ∈ 〈0, 1〉→ π(0, s) = 1 g

RIM2.

∀s b s ∈ 〈0, 1〉→ π(1, s) = s g

RIM3.

∀r b r ∈ 〈0, 1〉→π(r, 1) = 1 g

RIM4.

∀r∀s c ^{r, s} ∈ 〈0, 1〉∧ r ≦ s →π(r, s) = 1 h

RIM5.

∀ r ∀ s c ^{r, s} ∈ 〈 0, 1 〉∧ π (r, s) = 1 → r ≦ s h

4

(6)

RIM6.

∀r∀s∀t c ^{r, s,t} ∈ 〈0, 1〉∧ r ≦ s →π(r,t) ≧ π(s,t) h

RIM7.

∀r∀s∀t c ^{r, s,t} ∈ 〈0, 1〉∧ s ≦ t → π(r, s) ≦ π(r,t) h

RIM8.

∀r∀s∀t d ^{r, s,t} ∈ 〈0, 1〉→ c π(r,t) ≧ s ↔ π(s,t) ≧ r hi

RIM9.

∀ r ∀ s ∀ t b ^{r, s,t} ∈ 〈 0, 1 〉→π a ^r, π (s,t) f ⁼ π a ^s, π (r,t) fg

3 On Translating Properties of Functions by Applying the Functional Operator TNOR

Theorem 9

Assume π : 〈 0, 1 〉× 〈 0, 1 〉 → 〈 0, 1 〉 . 1. TNOR(π) fulfils TN1 if π(0, 0) = 1

2. TNOR(π) fulfils TN2 without any assumption for π 3. TNOR(π) fulfils TN3 if π fulfils RIM2

4. TNOR(π) fulfils TN4 if π fulfils RIM4 and RIM5 5. TNOR(π) fulfils TN5 if π fulfils RIM6

6. TNOR(π) fulfils TN6 without any assumptions for π 7. TNOR(π) fulfils TN7 if π fulfils RIM8

8. TNOR( π ) fulfils TN8 if π fulfils RIM5, RIM6, RIM7, RIM9 and π is right-hand con- tinuous with respect to 〈 0, 1 〉 and its second argument.

Proof ad 1. TN1

Assume s ∈ 〈0, 1 〉 . We have to prove

TNOR(π)(0, s) = 0 . (1)

By definition of TNOR it is sufficient to show

inf m ^{t t} ∈ 〈0, 1〉∧ π(0,t) ≧ s r ^{= 0 .}

(2)

In order to prove (2), by definition of inf it is sufficient to show

∃t c ^t ∈ 〈0, 1〉∧ π(0,t) ≧ s ∧ t = 0 h ^.

(3)

But (3) holds because of π(0, 0) = 1.

ad 2. TN2

Assume r ∈ 〈0, 1〉. We have to prove

TNOR(π)(r, 0) = 0 . (1)

By definition of TNOR it is sufficient to show

inf m ^{t t} ∈ 〈0, 1 〉∧ π (r,t) ≧ 0 r ^{= 0 .}

(2)

Because π(r,t) ≧ 0 holds for every r,t ∈ 〈0, 1 〉 , we get π(r, 0) ≧ 0, hence (2) holds.

ad 3. TN3

Assume s ∈ 〈0, 1〉. We have to prove

TNOR(π)(1, s) = s . (1)

5

(7)

By definition of TNOR it is sufficient to show

inf m ^{t t} ∈ 〈0, 1 〉∧ π (1,t) ≧ s r ^{= s .}

(2)

Because of RIM2 we have

∀t b ^t ∈ 〈0, 1〉→ π(1,t) = t g ^,

(3)

hence we have

inf m ^{t t} ∈ 〈0, 1〉∧ π(1,t) ≧ s r ^{= inf} m ^{t t} ∈ 〈0, 1〉∧ t ≧ s r ^.

(4)

Obviously, we have

inf m ^{t t} ∈ 〈0, 1〉∧ t ≧ s r ^{= s,}

(5)

hence from (4) and (5) we get (2).

ad 4. TN4

Assume r ∈ 〈0, 1〉. We have to prove

TNOR( π )(r, 1) = r . (1)

By definition of TNOR it is sufficient to show

inf m ^{t t} ∈ 〈0, 1 〉∧ π (r,t) ≧ 1 r ^{= r .}

(2)

In order to show (2), it is sufficient to prove

∀r b ^r ∈ 〈0, 1 〉→ π (r, r) = 1 g

(3) and

∀r∀t c ^r,t ∈ 〈0, 1〉∧ π(r,t) = 1 → t ≧ r h ^.

(4)

But, RIM4 implies (3) and RIM5 implies (4).

ad 5. TN5

Assume for r, s,t ∈ 〈0, 1 〉

r ≦ s . (1)

We have to prove

TNOR(π)(r,t) ≦ TNOR(π)(s,t) . (2)

By definition of TNOR it is sufficient to show

inf m ^t ′ t ′ ∈ 〈 0, 1 〉∧ π (r,t ′

≧ t

r

^{≦ inf}

m ^t ′ t ′ ∈ 〈 0, 1 〉∧ π (s,t ′ ) ≧ t r ^.

(3)

By definition of inf it is sufficient to prove

∀t′ c t′ ∈ 〈0, 1 〉∧ π (s,t′) ≧ t → π(r,t′) ≧ t h ^.

(4)

Assume π(s,t′) ≧ t. From r ≦ s and RIM6 we get π(r,t′) ≧ π(s,t′), hence (4) holds.

ad 6. TN6

Assume for r, s,t ∈ 〈 0, 1 〉

s ≦ t . (1)

6

(8)

We have to prove

TNOR(π)(r, s) ≦ TNOR(π)(r,t) . (2)

By definition of TNOR it is sufficient to show

inf m t′ t ′ ∈ 〈0, 1〉∧ π(r,t′) ≧ s r

^{≦ inf}

m t′ t′ ∈ 〈0, 1〉∧ π(r,t′) ≧ t r ^.

(3)

By definition of inf it is sufficient to show

∀ t ′ c ^t ′ ∈ 〈 0, 1 〉∧ π (r,t ′ ) ≧ t →π (r,t ′ ) ≧ s h ^.

(4)

But, (4) holds trivially because of s ≦ t.

ad 7. TN7

Assume r, s ∈ 〈0, 1〉. We have to prove

TNOR(π)(r, s) = TNOR(π)(s, r) . (1)

By definition of TNOR it is sufficient to show

inf m ^{t t} ∈ 〈0, 1〉∧ π(r,t) ≧ s r ^{= inf} m ^{t t} ∈ 〈0, 1〉∧ π(s,t) ≧ r r ^.

(2)

By definition of inf it is sufficient to show

∃ t c ^t ∈ 〈 0, 1 〉∧ π (r,t) ≧ s h if and only if ∃ t c ^t ∈ 〈 0, 1 〉∧ π (s,t) ≧ r h ^.

(3)

But, (3) holds because of assumption RIM8.

ad 8. TN8

Assume r, s,t ∈ 〈0, 1 〉 . We have to prove

TNOR(π) a r, TNOR(π)(s,t) f = TNOR(π) a TNOR(π)(r, s),t f . (1)

By definition of TNOR it is sufficient to show (2) inf m ^{u u} ∈ 〈0, 1〉∧ π(r, u) ≧ TNOR(π)(s,t) r ⁼

inf m ^{v v} ∈ 〈0, 1〉∧ π a TNOR(π)(r, s), v f

≧ t

r ^.

By definition of inf it is sufficient to prove

∃u c ^u ∈ 〈0, 1〉∧ π(r, u) ≧ TNOR(π)(s,t) h

(3)

if and only if

∃v c ^v ∈ 〈0, 1〉∧ π a TNOR(π)(r, s), v f

≧ t

h ^.

(4) I ( B ).

Assume for u ∈ 〈 0, 1 〉

π(r, u) ≧ TNOR(π)(s,t), i. e.

(5)

π(r, u) ≧ inf m ^{w w} ∈ 〈0, 1〉∧ π(s, w) ≧ t r ^.

(6)

Because π is right-hand continuous with respect to 〈 0, 1 〉 and its second argument, there ex- ists a w ∈ 〈0, 1 〉 such that

π(r, u) ≧ w (7)

7

(9)

and

π(s, w) ≧ t . (8)

We have to prove (4), i. e.

∃v e ^v ^{∈ 〈0, 1} 〉∧ π d ^inf m ^{x x} ∈ 〈0, 1 〉∧ π (r, x) ≧ s r ^{, v} i

^{≧ t}

j ^.

(9)

Because of RIM6 we get

π d ^inf m ^{x x} ∈ 〈 0, 1 〉∧ π (r, x) ≧ s r ^{, v} i

^{≧ sup}

m π (x, v) x ∈ 〈 0, 1 〉∧ π (r, x) ≧ s r ^,

(10)

hence it is sufficient to show

∃v d ^v ∈ 〈0, 1〉∧ sup m π(x, v) x ∈ 〈0, 1〉∧ π(r, x) ≧ s r

^{≧ t}

i ^.

(11)

By definition of sup it is sufficient to show

∃v∃x c ^{v, x} ∈ 〈0, 1 〉∧ π (x, v) ≧ t ∧ π(r, x) ≧ s h ^.

(12) Put

v = π(s, u) (13)

and

x =

def

r . (14)

Then it is sufficient to prove

π a r,π(s, u) f

≧ t

(15)

and

π(r, r) ≧ s . (16)

We prove (15).

By (7) and RIM7 we get

π a s,π(r, u) f

≧

π(s, w), (17)

hence by (8)

π a s,π(r, u) f

≧ t,

(18)

hence by RIM9, (15) holds.

Finally, (16) holds because of RIM5.

II ( A ).

In analogy to I.

4 On Translating Properties of Functions by Applying the Functional Operator RIMP

Theorem 10

Assume τ : 〈 0, 1 〉× 〈 0, 1 〉 → 〈 0, 1 〉 . 1. RIMP(τ) fulfils RIM1 if τ(0, 1) = 0

8

(10)

2. RIMP(τ) fulfils RIM2 if τ fulfils TN3

3. RIMP(τ) fulfils RIM3 without any assumption 4. RIMP(τ) fulfils RIM4 if τ fulfils TN4

5. RIMP( τ ) fulfils RIM5 if τ fulfils TN4 and τ is left-hand continuous with respect to

〈 0, 1 〉 and its second argument 6. RIMP(τ) fulfils RIM6 if τ fulfils TN5

7. RIMP(τ) fulfils RIM7 without any assumption 8. RIMP(τ) fulfils RIM8 if τ fulfils TN6 and TN7

9. RIMP(τ) fulfils RIM9 if τ fulfils TN3, TN6, TN8, and τ is left-hand continuous with respect to 〈0, 1〉 and its second argument.

Proof

Like theorem 9.

5 Conclusions

The theorems and corollaries above give the possibility to derive numerous “translating”

and “bijection” theorems for classes of functions. In the following we discuss only one ex- ample.

To this end we define Definition 5

1. FUNCT(2, C2) =

_def

R S T ^{ϕ ϕ}

: 〈 0, 1 〉× 〈 0, 1 〉 → 〈 0, 1 〉 and ϕis continuous with respect to 〈 0, 1 〉 and its second argument

U V W

2. FUNCT(2, M2, C2) =

def

R S T ^{ϕ ϕ ∈}

FUNCT(2,C2) and ϕ is monotone with respect to 〈0, 1〉 and its second argument

U V W

Lemma 11

For every ϕ , if ϕ ∈ FUNCT(2, M2, C2), then TNOR( ϕ) and RIMP( ϕ) belong to FUNCT(2, M2, C2).

Proof

Like corollary 8.

Definition 6

Assume π,τ : 〈0, 1〉× 〈0, 1〉 → 〈0, 1〉.

π is said to be an R-implication

=

def

π fulfils

RIM2.

∀s b s ∈ 〈0, 1〉→ π(1, s) = s g

RIM4.

∀r∀s c ^{r, s} ∈ 〈0, 1〉∧ r ≦ s →π(r, s) = 1 h

RIM5.

∀r∀s c ^{r, s} ∈ 〈0, 1〉∧ π(r, s) = 1 → r ≦ s h

RIM6.

∀ r ∀ s ∀ t c ^{r, s,t} ∈ 〈 0, 1 〉∧ r ≦ s →π (r,t) ≧ π (s,t) h

RIM8.

∀ r ∀ s ∀ t d ^{r, s,t} ∈ 〈 0, 1 〉→ c π (r,t) ≧ s ↔ π (s,t) ≧ r hi

RIM9.

∀r∀s∀t b ^{r, s,t} ∈ 〈0, 1 〉→π a r, π(s,t) f = π a s, π(r,t) fg

9

(11)

Remember the concept of T-norm which can be defined as follows:

τ is said to be a T-norm

=

def

τ fulfils the axioms TN4, TN5, TN7, and TN8.

Then we get the following theorem:

Theorem 12

1. If τ is a T-norm with τ ∈ FUNCT(2, C2),

then RIMP(τ) is an R-implication with RIMP(τ) ∈ FUNCT(2, C2) 2. If π is an R-implication with π ∈ FUNCT(2, C2),

then TNOR( π) is a T-norm with TNOR( π) ∈ FUNCT(2, C2)

3. RIMP and TNOR are bijections between the class of T-norms and the class of R- implications (restricted in both cases to the class FUNCT(2, C2)).

Proof

By theorems 9, 10, corollaries 3, 7, 8, and lemma 11.

Remark

Because T-norms and R-implications are monotone with respect to 〈0, 1〉 and their second argument, in the theorem above FUNCT(2, C2) can be replaced by FUNCT(2, M2, C2).

In a forthcoming paper we will publish a proof of theorem 10 and discuss further examples.

Acknowledgement

The author likes to thank Mario Thalheim for his help in preparing the manuscript.

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