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Many-valued Similarity- Theory and Applications

of Fuzzy Reasoning

- May 2007, Prague
- Esko Turunen, Ph.D
- Tampere University of Technology Finland

Motivation

Lecture I

Aristotelian logic

All human beings are mortal Sokrates is a human

being

Sokrates is mortal

It took thousands of years before Aristotelian

informal logic was expressed in a formal way,

today known as First-order Boolean Logic

In 1960s Zadeh introdused Fuzzy Logic

More generally (fuzzy rule systems)

Red apples are ripe This apple is more or less red

IF x is in A1 and y is in B1 THEN z is in C1

IF x is in An and y is in Bn THEN z

is in Cn

This apple is almost ripe

What is the mathematical formalism of Zadehs

Fuzzy Logic? We claim it is Pavelka - Lukasiewicz

many-valued logic, in particular, many-valued

similarity.

In science, we always want to minimize the set of

axioms and maximize the set of the consequences

of these axioms. Thus, consider the following

Definition 1. Let L be a non-void set, 1 an

element of L and ?, a binary and

unary operation, respectively, defined on L such

that, for all x, y, z in L, we have

Then the system L ?L, ?,,1? is called

Wajsberg algebra.

Now, define on a Wajsberg algebra L a binary

relation by (5) x y iff x?y 1.

Then, by (2) we have

Hence, by (1) we have

which implies (6) x?x 1, i.e. x x.

Let x?y 1 and y?z 1, that is, let x y

and y z. By (2),

thus

(7) if x y, y z then x z.

Let x?y 1 and y?x 1, that is, let x y

and y x. By (3), 1?y 1?x, thus, x y. We

conclude

(8) if x y, y x then x y.

Equations (6) (8) show that (5) defines an

order on L.

Next we show that the element 1 is the greatest

element with respect to this order, in other

words

(9) for all x in L, x ?1 1, i.e. x 1.

To this end, we first reason, by (3), (1) and

(6), (x?1)?1 (1?x)?x x?x 1, that is

(10) (x?1)?1 1

On the other hand, by (1), (10), (1) and (2),

in other words,

Thus,

so by (8)

First exercise. Show that the arrow operation is

antitone in the first variable, that is, if x y

then y?z x ?z. Hint use equation (2)

Proposition 1. In a Wajsberg algebra L, for any

x, y, z in L,

Proof. Since y 1 and ? is antitone in the first

variable we reason that 1?x y?x, therefore x

y?x, thus (11) holds.

To establish (12) we first verify

Indeed, if x y?z then, as the arrow operation

is antitone in the first variable, we have

By (3),

and, by (11),

so that altogether we have

By definition (5),

Thus, (14) holds.

Applying (14) to (z?x)?(x?y)?(z?y) 1, which

holds by (2), we conclude (x?y)?(z?x)?(z?y)

1, i.e. (12) holds. Finally, by (11) and (3),

and by (12),

Therefore

which, by (14), implies x?(y?z) y?(x?z).

By a similar argument y?(x?z) x?(y?z).

We conclude that (13) holds.

Second exercise. Show that the arrow operation is

isotone in the second variable, that is, if x

y then z?x x?y. Hint use equation (12)

Proposition 2. In a Wajsberg algebra L, for any x

in L,

Proof. By (11), x (1)?x, by (4), (1)?x

x?1, hence

On the other hand, by (4) and (1),

and, as the arrow operation is

antitone in the first variable,

which, by (3), implies that

Next we reason,

and by (13),

by (4) and (1),

and, by (11),

thus

Hence,

and, by (18),

The (in-)equalities (17) and (19) now imply

equation (15).

The (in-)equality (16) follows by (11), (4) and

(1), indeed,

Remark. Condition (16) implies that 1 is the

least element in the corresponding Wajsberg

algebra L and will therefore denoted by 0. We

write x instead of (x).

Third exercise. Show that, for all elements x, y

in a Wajsberg algebra L, hold

Hint Apply (13) and (4) for (20), moreover

(4), (20) and (4) for (21) and finally, (20),

(4) and (20) for (22).

Till now we have seen that Wajsberg algebra

axioms generate an order relation on L. Our aim

is to show that, after a suitable stipulation, L

becomes a lattice, that is, all pairs x, y of

elements of L have the greatest lower bound in L

and the least upper bound in L with respect to

the order relation given by (5). For l.u.bx,y

we set

First we realize, by (11), that

and then, by (11) and (3),

Let now z be such an element of L that x, y z.

Then x?z 1 thus, by (1), (x?z)?z z.

Since the arrow operation is antitone on the

first variable, we first reason z?x y?x,

and then (y?x) ?x (z?x) ?x z.

We conclude that (y?x) ?x coincide

with l.u.bx,y, i.e. that (23) is a correct

definition.

For g.l.bx,y we set

First we realize that

which is the case.

On the other hand, if z is such an element of L

that

Then

therefore

and so

We conclude that (24) is a correct definition.

Fourth exercise. Show that, for all elements x, y

in a Wajsberg algebra L, hold de Morgan laws

Hint Apply (24) and (20).

Define on a Wajsberg algebra L a binary operation

for each x, y, z in L via

Then we have

Proposition 3. In a Wajsberg algebra L, for any

x,y,z in L,

Lecture II

commutativity

associativity

isotonity

Proof. By (26), (20), (21) and (26),

respectively, we have

Thus, (27) holds.

therefore

We have established (29). For (28) we reason in

the following way

Fifth exercise. Show that, for all elements x,

y, z in a Wajsberg algebra L, hold

Galois connection

Remark. Equations (27) (31) mean that lattice L

generated by Wajsberg algebra axioms is a

residuated lattice. Thus, all properties valid

in a residuated lattice hold in Wajsberg

algebras, too. For example, the meet and

join operations are associative, commutative and

absorption holds. For all x, y, z in L,

By (24), (23), (21) and (26) we reason that

and, by commutativity of ?, we have

Proposition 4. In a Wajsberg algebra L, for any

x, y in L,

Proof (of prelinearity). By (32) and (6),

and, similarly,

therefore

Thus,

Therefore (34b) holds.

Sixth exercise. Show that, for all elements x, y,

z in a Wajsberg algebra L, holds

Hint (21), (25), (23), (3), (13), (21)

Remark. Residuated lattices such that (34a) and

(34b) hold are called BL-algebras (Basic

Logic algebras by P. Hajek 1997), moreover,

BL-algebras such that a double negation low x

x holds are known as MV-algebras (Multi Valued

algebras by C.C. Chang 1957). Hence, Wajsberg

algebras are MV-algebras. Even more is true,

these two structures coincide each MV-algebra

generates a Wajsberg algebras and vice versa.

In an MV-algebra, there is a binary operation

In a Wajsberg algebra,

a sum operation is introduced by a formula

Seventh exercise. Show that, for all elements x,

y, z in a Wajsberg algebra L, hold

For the sake of completeness, we present the

original MV-algebra axioms of C.C. Chang. It will

be an extra exercise to show that they hold in

Wajsberg-algebras!

We needed only four equational axioms to

establish a rich structure. However, to be

able to introduce fuzzy inference in an axiomatic

way, we will still need two more

axioms. Unfortunately, they are not equational.

First consider a completeness axiom

where L is an MV-algebra, called complete

MV-algebra. For such algebras we have e.g.

Proposition 5. In a complete MV-algebra L, for

any x, y in L,

Proof. Since the operation

is isotone, we have, for each i in ?,

therefore

Conversaly,

for each i,

by the Galois connection, equivalent to

Therefore

Again by the Galois connection we conclude

We have demonstrated equation (49). Equation (50)

can be

shown in a quite similar manner. Indeed, since

for each i,

we have

Conversaly, trivially

Thus, by the Galois connection,

We shall conclude

which is equivalent to

This completes the proof of equation (50).

To establish (51) we first realize, as the arrow

operation is antitone in the first variable,

Therefore

Conversaly,

therefore

thus

hence

whence

which is equivalent to

The proof is complete.

Eighth exercise. Prove in a complete MV-algebra

Ninth exercise. Prove in a complete MV-algebra

An element b of an MV-algebra L is called an

n-divisor of an element a of L, if

If all elements have n-divisors for all natural

n, then L is called divisible. An

MV-algebra L is called injective if it is

complete and divisible. We will see that

the six axioms of an injective MV-algebra are

sufficient to construct fuzzy IF-THEN

inrefence systems.

A canonical example of an injective MV-algebra is

the Lukasiewicz algebra defined on the real unit

interval 0,1 1 1, x 1 x, x?y min1, 1

x y.

Di Nola and Sessa proved in 1995 that an

MV-algebra L is injective if, and only if L is

isomorphic to F(L), where F(L) is the MV-algebara

of all continous 0,1-valued functions on the

set of all maximal ideals of L, and

1(M) 1, (f?g)(M) min1, 1 f (M) g(M),

f(M) 1 f(M), for any maximal ideal M of L.

Tenth exercise. Write the MV-operations on the

Lukasiewicz structure, that is

Proposition 5. In an injective MV-algebra L, any

n-divisor is unique.

Lecture III

Proof. It is enough to show that the statement

holds in any injective MV-algebra F(L).

To this end, let

n a natural number and g, h two n-divisors of f.

Let M be

a maximal ideal of L.

If f(M) a lt 1, then ng(M) (ng)(M) a

(nh)(M) nh(M).

Thus, g(M) h(M).

Now assume f(M) 1. Then

If (n-1)g(M) would be equal to 1, then g(M)

should be equal to 0, which it is clearly not.

Let a counter assumption

Therefore (n-1)g(M) lt 1.

Similarly (n-1)h(M) 1-h(M) lt 1.

g(M) lt h(M) hold. Then

which implies a contradiction h(M) lt g(M).

An assumption h(M) lt g(M) leads to a similar

contradiction, too.Therefore h(M) g(M).

We conclude h g and the proof is complete.

By Proposition 5, we may denote the unique

n-divisor of an element a by a/n.

For any maximal ideal M of an injective

MV-algebra L, it holds that n(f(M)/n) f(M),

moreover,

We therefore conclude (f(M)/n) f/n(M), that is,

in F(L),

map first, then divide equals to divide first,

then map.

Eleventh exercise. Prove that in the Lukasiewicz

structure,

Clearly, in the Lukasiewicz structure, we have

Thus, in F(L),

Summarizing

Proposition 6. In any injective MV-algebra L,

- Definition 2. Let L be an injective MV-algebra

and let A be a non-void set. A fuzzy - similarity S on A is such a binary fuzzy relation

that, for each x, y, and z in A, - S(x,x) 1 (everything is similar to itself),
- S(x,y) S(y,x) (fuzzy similarity is symmetric),
- (iii) S(x,y)?S(y,z) S(x,z) (fuzzy similarity is

weakly transitive).

Recall an L-valued fuzzy subset X of A is an

ordered couple (A,µX), where the member- ship

function µXA?L tells the degree to which an

element a in A belongs to the fuzzy subset X.

Given a fuzzy subset (A,µX), define a fuzzy

relation S on A by (54) S(x,y) µX(x)?µX(y).

This fuzzy relation is trivially symmetric, by

(6) it is reflexive and, by (2), transitive.

So, Any fuzzy set generates a fuzzy similarity

this is true for L being any residuated lattice

Proposition 7. Consider n injective MV-algebra L

valued fuzzy similariteis Si, i 1,...,n on a

set A. Then a fuzzy binary relation S on A

defined by

is an L valued fuzzy similarity on A.

More generally, any weighted mean SIM is

an L valued fuzzy similarity, where

Proof. (i) (ii) obvious, (iii) by Proposition 6.

Example 1 Countriess, Example 2 Functionality

The idea of partial similarity is not new.

Indeed, in 1988 Niiniluoto quoted from Mill

(1843) by defining If two objects A and B agree

on k attributes and disagree on m attributes,

then the number

can be taken to

measure the degree of similarity or partial

identity between A and B.

Obviously,

sim is a reflexive and symmetric fuzzy relation.

It is weakly transitive with respect to the

Lukasiewicz t-norm (and, therefore, can be

considered as an injective MV-algebra valued

similarity).

To see this,

assuming there are N attributes,

study the following Venn-diagram

B

It is easy to see that k t r N, 0 s. Then

we have

A

m

p

k

t

s

r

q

C

which holds true.

Is worth noticing that, among all BL-algebras (in

particular, among continuous t-norms) injective

MV-algebras are the only structures where the

average of similarities is a similarity.

Therefore the following consideration can be done

only in such a structure.

An Algorithm to Construct Fuzzy IF-THEN Inference

Systems

Let us now return to our starting point, a fuzzy

rule based system

Rule 1 IF x1 is in A11 and x2 is in A12 and

.... and xm is in A1m THEN y is in B1 Rule 2 IF

x1 is in A21 and x2 is in A22 and .... and xm is

in A2m THEN y is in B2 Rule n IF

x1 is in An1 and x2 is in An2 and .... and xm is

in Anm THEN y is in Bn

Here all Aij.s and Bj are fuzzy but can be crips

actions, too. As usual, it is not necessary that

the rule base is complete, some rule combinations

can be missing without causing any difficulties.

It is also possible that different IF-part causes

equal THEN-part, but it is not possible that a

fixed IF-part causes two different THEN-parts. We

will not need any kind of defuzzification

methods, everything is based on an experts

knowledge and properties of injective MV-algebra

valued similarity.

Step 1. Create the dynamics of the inference

system, i.e. define the IF-THEN rules and give

shapes to the corresponding fuzzy sets.

Step 2. If necassary, give weights to various

IF-parts to emphasize their importance.

Step 3. List the rules with respect to the mutual

importance of their IF-parts.

Step 4. For each THEN-part, give a criteria on

how to distinguish outputs with equal degree of

membership.

- A general framework for the inference system is

now ready. - Asssume then that we have an actual input Actual

(X1,...,Xm). A corresponding output - Y is counted in the following way.
- Consider each IF-part of each rule as a crisp

case, that is µAij(xj) 1 holds. - Count the degree of similarity between Actual and

the IF-part of Rule i, i 1,...,n. - Since µAij(Xj)?µAij(xj) µAij(Xj)?1

µAij(Xj), we only need to calculate averages - or weighted averages of membership degrees!
- (3) Fire an output Y such that µBk(Y)

Similarity(Actual, Rule k) corresponding to the - greatest similarity degree between the input

Actual and the IF-part of a Rule k. If such - a maximal rule is not unique, then use the

preference list given in Step (3), and if there - are several such outputs Y, use a creteria given

in Step (4).

Note that counting the actual output can be

viewed as an instance of Generalized Modus

Ponens in the sense of (injective MV-algebra

valued) Lukasiewicz-Pavelka logic

where ? corresponds to the IF-part of a Rule, ß

corresponds to the THEN-part of the Rule a is

the value Similarity(Actual, Rule k) and b 1.

This gives a many-valued logic based theoretical

justification to fuzzy inference.

In the rest part of the lecture deals with real

world case studies where we have applied the

above metodology and algorithm.