Many-valued Similarity - Theory and Applications of Fuzzy Reasoning

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Many-valued Similarity- Theory and Applications
of Fuzzy Reasoning

• Esko Turunen
• Tampere University of Technology Finland

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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.
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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.
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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.
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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).
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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.
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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
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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
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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.
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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!
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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? L, yi i?G Í L.
Proof. Since the operation
is isotone, we have, for each i in ?,
therefore
Conversely,
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
Conversely, trivially
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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
Conversely,
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
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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
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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.
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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,
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• 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 Countries, Example 2 Functionality
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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.
It is worth noting 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.
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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.
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• 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 a 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.