Fuzzy Logic

FUZZY LOGIC AND APPROXIMATE

9.2 Fuzzy Logic

9.2. 1 Classical Logics Revisited

Logics as bases for reasoning can be distinguished essentially by their three topic- neutral (context-independent) items: truth values, vocabulary (operators), and reasoning procedure (tautologies, syllogisms).

In Boolean logic, truth values can be 0 (false) or 1 (true), and by means of these truth values, the vocabulary (operators) is defined via truth tables.

Let us consider two statements, A and E, either of which can be true or false, that is, have the truth value 1 or O. We can construct the following truth tables:

A B /\ V xv => <=> ?

1 1 1 1 0 1 1 1

1 0 0 1 1 0 0 1

0 1 0 1 1 1 0 0

0 0 0 0 0 1 1 0

There are 2²²= 16 truth tables, each defining an operator. Assigning meanings (words) to these operators is not difficult for the first 4 or 5 columns: the first obviously characterizes the "and," the second the "inclusive or," the third the

"exclusive or," and the fourth and fifth the implication and the equivalence. We will have difficulties, however, interpreting the remaining nine columns in terms of our language.Ifwe have three statements rather than two, this task of assign- ing meanings to truth tables becomes even more difficult.

So far it has been assumed that each statement,AandB,could clearly be clas- sified as true or false. If this is no longer true, then additional truth values, such as "undecided" or a similar description, can and have to be introduced, which leads to the many existing systems of multivalued logic.Itis not difficult to see how the above-mentioned problems of two-valued logic in "calling" truth tables or operators increase as we move to multivalued logic. For only two statements and three possible truth values, there are already 3³²= 729 truth tables! The uniqueness of interpretation of truth tables, which is so convenient in Boolean logic, disappears immediately because many truth tables in three-valued logic look very much alike.

The third topic-neutral item of logical systems is the reasoning procedure itself, which is generally based on tautologies such as

modus ponens:

modus tollens:

syllogism : contraposition :

(A /\ (A ^==}B» ^==}B

«A ^==}B) /\ -, B) ^==}-,A

«A ==}B) /\ (B ==}C» ^==}(A==} C) (A^==}B)^==}(-,B ==}-,A)

Let us consider the modus ponens, which could be interpreted as:"IfAis true and if the statement 'If A is true then B is true' is also true, then B is true."

The term true is used at different places and in two different senses: All but the last "trues" are material trues, that is, they are taken as a matter of fact, while the last 'true" is a topic-neutral logical "true ." In Boolean logic, however, these

"trues" are all treated the same way [see Mamdani and Gaines 1981, p. xv]. A distinction between material and logical (necessary) truth is made in so-called extended logics: Modal logic [Hughes and Cresswell 1968] distinguishes between necessary and possible truth, and tense logic between statements that were true

in the past and those that will be true in the future. Epistemic logic deals with knowledge and belief and deontic logic with what ought to be done and what is permitted to be true. Modal logic, in particular, might be a very good basis for applying different measures and theories of uncertainty, as indicated in chapter 4.

Another extension of Boolean logic is predicate calculus, which is a set theo- retic logic using quantifiers (all, etc.) and predicates in addition to the operators of Boolean logic.

Fuzzy logic [Zadeh 1973a, p. 101] is an extension of set-theoretic multival- ued logic in which the truth values are linguistic variables (or terms of the lin- guistic variable truth).

Since operators, like v, /-; -',=>in fuzzy logic are also defined by using truth tables, the extension principle can be applied to derive definitions of the opera- tors. So far, possibility theory (see section 8.1) has primarily been used in order to define operators in fuzzy logic, even though other operators have also been investigated (see, for instance, Mizumoto and Zimmermann [1982]), and could also be used. In this book, we will limit considerations to possibilistic interpre- tations of linguistic variables, and we will also stick to the original proposals of Zadeh [1973a]. To the interested reader, however, we suggest supplemental study of alternative approaches such as those by Baldwin [1979], Baldwin and Pilsworth [1980], Giles [1979, 1980], and others.

Ifv(A) is a point inV=[0, I],representing the truth value of the proposition

"u isA"orsimplyA, then the truth value of notA is given by v(notA)=1-v(A)

Definition 9-5

Ifv(A) is a normalized fuzzy set, v(A) = {(Vi,

IlJli

= 1, . .. , n, Vi E [0, I]), then by applying the extension principle, the truth value of v(not A) is defined as

v(notA)= {(l-Vi,IlJi=1,... , n,Vi E[0,

In

In particular, "false" is interpreted as "not true," that is, v(false)= {(l-Vi'IlJi=1, . . . ,n,Vi E[0,I])

Example 9-4

Let us consider the terms true and false , respectively, defined as the following possibility distributions :

v(true)

=

{(.5, .6), (.6, .7), (.7, .8), (.8, .9), (.9, 1),(1, 1)}

v(false)

=

v(not true)

=

{(.5,.6), (.4, .7), (.3, .8), (.2, .9), (.1, 1), (0, I)}

Then

v(very true)

=

{(.5, .36), (.6, .49), (.7, .64), (.8, .81), (.9, 1), (1, I)}

v(very false)

=

{(.5, .36), (.4, .49), (.3, .64), (.2, .81),(.1, 1), (0, I)}

Ithas already been mentioned that fuzzy logic is essentially considered as an application of possibility theory to logic. Hence the logical operators "and," "or,"

and "not" are defined accordingly.

Definition 9-6

For numerical truth valuesv(A)andv(B),the logical operationsand, or, not,and impliedare defined as

v(A)^1\ v(E)= v(A^1\ E)=min{v(A), v(B)}

v(A)v v(B)= v(Av B)

=

^max{yeA), v(B)}

....,v(A)= 1- yeA)}

YeA)=>v(E)=v(A=>B)=....,v(A)v v(E)

= max{I - yeA), v(B)}

If

yeA)

=

{(Vi, Ui)}, U i E[0,1],v,E[0,1]

v(B)

=

{(Wj,P j)},

Pi

^E[O,I],ffij E[O,I]

i=1,. . . ,n;j =1,. .. ,m then

v(A andB)=v(A)1\v(B)=

{(u

^=min{

v.,

^Wj},

i

=

1,.. . ,

n;

^j

⁼

^{1,... ,}

m}

(This is equivalent to the intersection of two type 2 fuzzy sets.) The other oper- ators are defined accordingly.

Example 9-5

Let v(A)=true= {(.5, .6), (.6, .7), (.7, .8), (.8, .9), (.9, 1),(1, I)}.

Then

--.,ii(A)= {(O, 1), (.1, 1), (.2, 1), (.3, 1),(.4, 1), (.5,.4), (.6, .3), (.7, .2), (.8, .1) }

9.2.2 Linguistic Truth Tables

As mentioned at the beginning of this section, binary connectives (operators) in classical two- and many-valued logics are normally defined by the tabulation of truth values in truth tables. In fuzzy logic, the number of truth values is, in general, infinite. Hence tabulation of the truth values for operators is not possi- ble. We can, however, tabulate truth values, that is, terms of the linguistic vari- able "Truth," for a finite number of terms, such as true, not true, very true, false, more or less true, and so on.

Zadeh [1973a, p. 109] suggests truth tables for the determination of truth values for operators using a four-valued logic including the truth values true, false, undecided, and unknown. "Unknown" is then interpreted as "true or false"

(T+F), and "undecided" is denoted bye.

Extending the normal Boolean logic with truth values true (I) and false (0) to a (fuzzy) three-valued logic (true= T,false= F, unknown= T+F), with a uni- verse of truth values being two-valued (true and false), we obtain the following truth tables, in which the first column contains the truth values for a statement A and the first row those for a statementB [Zadeh 1973a, p. 116]:

1\ T F T+F

T T F T+F

F F F F

T+F T+F F T+F Truth table for "and"

v T F T+F

T T F T

F T F T+F

T+F T T+F T+F Truth table for "or"

1t

^{T+F T+F}Truth table for "not"^{TF FT}

Ifthe number of truth values (terms of the linguistic variable truth) increases, one can still "tabulate" the truth table for operators by using definition 9-6 as follows: Let us assume that thez<hrow of the table represents "not true" and the

l'

column "more or less true." The (i,jyhentry in the truth table for "and" would then contain the entry for "not true /\ more or less true." The resulting fuzzy set would, however, most likely not correspond to any fuzzy set assigned to the terms of the term set of "truth ." In this case, one could try to find the fuzzy set of the term that is most similar to the fuzzy set resulting from the computations. Such a term would then be called linguistic approximation. This is an analogy to sta- tistics, where empirical distribution functions are often approximated by well- known standard distribution functions.

Example 9-6

LetV= {O, .1, .2, .. . , I } be the universe, true= {(.8, .9), (.9,1),(1, I)},

more or less true

=

{(.6, .2), (.7,.4), (.8, .7), (.9, I), (1, I)}, and almost true= {(.8, .9), (.9, 1), (1, .8)}.

Let "more or less true" be the jlhrow and "almost true" thelhcolumn of the truth table for "or."

Then "more or less truev almost true" is the(i, j)thentry in the table:

more or less true v almost true

= {(.6, .2), (.7,.4), (.8, .7), (.9,1), (1, I)} v {(.8, .9), (.9,1),(1,.8)}

= {(.6, .2), (.7,.4) ,(.8, .9), (.9, 1),(1, I)}

Now we can approximate the right-hand side of this equation by true= {(.8, .9), (.9, 1),(1, I)}

This yields

"more or less true v almost true" '" "true."

Baldwin [1979] suggests another version of fuzzy logic-fuzzy truth tables, and their determination: The truth values on which he bases his suggestions were shown graphically in figure9-3. They were defined as

true={(v,).ltrue(v)=v)lvE[0, I]}

false={(v,).lfalse(V)=1- ).ltrue(v»!vE[0, I]}

very true={(v, ().ltrue(v»2)lv^E[0,

Il}

fairly true={(v, ().ltrue(v» !/2)lvE[0,

l]}

undecided = {(v, l)lvE[0,lj]

Very false and fairly false were defined correspondingly, and absolutely true ={(v,11at(v))lvE[0, I])

{I for v=1 withIlat(v) -

- 0 otherwise absolutely false={(v, Ila/(v))/vE[0,l])

{I for v=0 withIla/(v) =

o

^otherwise

Hence

(very) truek ~ absolutely true ask~⁰⁰

(very) falsek _~ absolutely false ask_~⁰⁰ (fairlyl'true~ undecided ask~⁰⁰ (fairlyl'false~undecided ask ~⁰⁰

Using figure 9-3 and the interpretations of "and" and "or" as minimum and maximum, respectively, the following truth table results [Baldwin 1979, p. 318]:

yep) v(Q) v(P and Q)

«r

or Q)

false false false false

true false false true

true true true true

undecided false false undecided

undecided true undecided true

undecided undecided undecided undecided

true very true true very true

true fairly true fairly true true

Some more considerations and assumptions are needed to derive the truth table for the implication. Baldwin considers his fuzzy logic to rest on two pillars : the denumberably infinite multivalued logic system of Lukasiewicz logic and fuzzy set theory :

Implication statements are treated by a composition of fuzzy truth value restrictions with a Lukasiewicz logic implication relation on a fuzzy truth space. Set theoretic con- siderations are used to obtain fuzzy truth value restrictions from conditional fuzzy lin- guistic statements using an inverse truth functional modification procedure. Finally true functions modification is used to obtain the final conclusion [Baldwin 1979, p. 309].