A new structure for fuzzy systems: Fuzzy propositional logic and multi-universe representation of fuzzy decision processes

F!OZ2Y

sets and systems ELSEVIER

Fuzzy Sets and Systems 85 (1997) 325-354

A new structure for fuzzy systems: Fuzzy propositional logic and multi-universe representation of fuzzy decision processes Turhan

(~iftqibasl

Electrical and Electronics Engineering Department, Hacettepe University, Ankara, Turkey Received May 1995; revised August 1995

Abstract

A new structure has been developed for fuzzy logical algebra. This is called as "Fuzzy propositional logic structure" in which "Proposition" is defined as a more generalized form of fuzzy proposition defined by Zadeh. With this structure, the fuzzy reasoning mathematics becomes very simple, and very easy to understand. The fuzzy sets and fuzzy propositions are represented in the extended universe and all logical operations can be performed by the known logical formulas already developed for nonfuzzy logic. Mutual exclusive and complementary properties of propositions have been defined. Notion of a fuzzy basis for the universe is developed. The decision formula is transformed into the output universe, hence all calculations become one-universe operations. In short, this new structure brings about unification and simplification in all stages of fuzzy decision process.

Keywords: Fuzzy propositional logic; Truth function; Membership function; Approximate reasoning; Fuzzy relations; Fuzzy control

1. Introduction

There has been much interest in recent years in the fuzzy set theory, fuzzy reasoning and decision theory, and fuzzy control systems. After Zadeh's pioneering work on fuzzy set theory [17], interesting developments have been achieved towards system theory applications using fuzzy logic approach. Following Zadeh's tutorial papers [18, 19] on fuzzy reasoning theory, Mamdani and Asilian [12], Mamdani [11], Zadeh [20], Lee [9], Larsen [8], Dubois and Prade [3] published important contributions on mathematical representation and applications of fuzzy logic. Some new contributions can be found in [1, 4, 6, 7, 10, 16].

In the last few years, the literature on this subject has been growing so rapidly that it is difficult to give a complete list of mathematical developments. However, the mathematical tools that are being used depend on the above literature. In this paper a new approach is proposed as an extension to Zadeh's works. This approach can be summarized as following: (I) Zadeh introduced both "set theory algebra" and "propositional logic algebra", but the developments of the theory have been built on a set theoretic approach. In fact, Zadeh's definition of proposition was only limited to single predicates. In this paper the structure of fuzzy algebra has been built upon "fuzzy propositional logic", in which a

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Turban ~ifigiba~l / Fuzz3, Sets and Systems 85 (1997) 325 354

proposition is not a single predicate but "the set of predicates about the subject matter". With this definition, propositional logic algebra has been used as the mathematical tool in the structure of fuzzy logic. This is more convenient for the decision processes, since decision process is in fact a propositional operation. (II) The definition of "cylindrical extension" of a fuzzy set defined by Zadeh [,19], is applied to fuzzy propositions; and the "extension fuzzy proposition" is considered as another representation of the same proposition, which we shall call "multiuniverse extension" or "multi-universe extension of the representation" of the same fuzzy proposition. The fuzzy proposition in the original universe and in the extended universe are two different representations in different universes of the same physical/linguistic system. This is analogous to higher dimensional representation of a point, defined in a lower dimensional space. More important than that, it is possible to define AND, OR operations between fuzzy propositions in different universes by extending the fuzzy propositions into a common higher dimensional universe, since in this universe these operations become valid. "Cartesian product" operation of fuzzy propositions in different universes is nothing but AND operation in this extended universe. Similarly "Composition" operation of relations in different universes, defined by Zadeh [,19] as max rain product, is equivalent to "AND" operation between fuzzy relations in the extended universe. By this representation the fuzzy logical algebra can be developed by using the known techniques of nonfuzzy propositions, hence confusions of fuzzy implication algebra are removed, and important results in fuzzy conditional statements are obtained. (lII) Semi-exclusive, exclusive, semi-complementary, complementary propositions have been defined and the notion of "fuzzy basis" is developed, it has been proved that the defining propositions of fuzzy control systems should form a semi-complementary and semi-exclusive set. (IV) The combined implication rules of a fuzzy control system have been obtained in Boolean algebraic formulas, they have been transformed into the output universe by projection/shadow formulas,

hence the status of the resultant control can be calculated directly using fuzzy propositional operations in the input and in the output universes separately as one-universe calculations. This method is, in fact, the known method that was developed by Mamdani [,11] and Larsen [-8], and the propositional logic structure developed in this paper gives the mathematical base in a clear presentation, away from the complicated presentation of the previous literature.

2. Theory and notation 2.1. Preliminary definitions on .fizzzy sets 2.1.1. Fuzzy set and membership fimction Definition (Zadeh [-53). Consider a set U in the classical (non-fuzzy) sense which will be called as the "universe" set. A fuzzy set is defined in U, by a membership function which assigns a membership value in [-0, 1] for each element of U. The membership function is shown as #A(u) where A is the name of the set, and the letter u will be used as a generic element of the universe U. 2.1.2. Fuzzy proposition and truth function Definition. Consider a set U of "statements about a subject matter" that will be called as the universe set. A fuzzy proposition is defined in U, by a "truth function", which assigns a "truth value" in E0 1] for each element of U (for each statement). The truth function is shown as #A(U) where A is the "name of the proposition", and the letter u will be used as a generic element of the universe U (hence is a statement about the subject matter). 2.1.3. Notation The universes (sets of statements) are represented by capital letters U, V, W, U1, U2, etc. The statements (predicates) (elements of universes) will be represented as lowercase letters u, v, etc. of the related universes. Fuzzy sets and propositions will be represented in capital letters as A, B, C, R, ... or in

Turhan ~iftqiba~t / Fuzzy Sets and Systems 85 (1997) 325-354

lowercase bold letters u, ux, u 2, ,V, ¥1, V2, of the related universes. Note that the above definition of proposition is different from the definition of Zadeh [9], since Zadeh defines the single predicates (what we called statements in this paper) as proposition. Our definition is closer to the "term set" definition of Zadeh [193. .

.

.

-

-

-

Example 2.1 (Finite discrete case). Consider the subject matter "Age of the man". Assume that the set of possible statements (predicates) about the age of the man is limited to the following finite discrete universe set: Universe = {"Over or around 80", "around 70", "around 60", "around 50", "around 40", "around 30", "around 20", "less than or around 10"}. To obtain a proposition about the age of a man, a truth value should be assigned for each statement in the universe set. ul = { 1

327

0.6 0.1 0 0 0 0 0}

is a proposition that can be named in short as "very old". u2 = {0 0 0.1 0.6 1 0.6 0.1 0}

is a proposition that can be named in short as "middle aged". Example 2.2 (Continuous case).

'

A1

A2

A3

40

70

1

lO

t

Fig. 1. Truth functions for three propositions.

assigning a counterpart in the physical world. Mathematically, only the truth function should be considered.

2.1.4. Possibility function Definition. Truth function is also called [5] as the "possibility function" or as the "possibility distribution function" by considering statements with high truth values as "more possible", but statements with low truth values as "less possible". The name "possibility function" can be considered as the backward interpretation of the name "truth function". When the real world is given, one can assign a truth function that will mathematically describe its status. When a truth function is given one can use the possibility notion to find a status in the real world, specifically, a state having possibility 0.8 is more possibly the real status than the state having possibility 0.6.

Subject matter of U = "Age of the man".

2.1.5. "OR" Operation of fuzzy propositions Universe (statement set) = {statements " T h e man is around t years old." l0 < t < 100} Propositions A1, A2, A3 are defined by the truth functions given in Fig. 1. Proposition A1 can be named in short as "young boy", A2 can be named as "around 40", A3 can be named as "around 65-75". Note that giving a short name is only for naming this proposition and for intuitive interpretation for

Definition. Given any two fuzzy propositions A, B in the same universe (statement set) U by their truth functions ~tA(U) and #~(u), the fuzzy proposition "A O R B", shown as (A V B) or as (A + B) is defined in the same universe by its truth function: #a v B(U) = max{ t~a(U), #B(U)}.

(2.1)

2.1.6. "'AND" Operation of fuzzy propositions Definition. For any two propositions A, B in the same universe (statement set) U with truth

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Turhan ~ifigibaq~ / Fuzzy Sets and Systems 85 (1997) 325 354

functions [~A(U) and p~(u), the fuzzy proposition "A AND B" shown as (A A B) or as (A" B) is defined in the same universe by its truth function:

PA A~(U) = min { #A(U), #B(U)}.

(2.2)

2.1.7. "Negation" operation of fuzzy propositions Definition. Given a fuzzy proposition A in the universe U, fuzzy proposition "Negation of A" { 7 A } also called as " N O T A" is defined in the same universe by its truth function given as:

Other definitions than (2.1) and (2.2) for union and intersection have also been given in the literature [17-20], and as long as the definitions satisfy the above basic properties of nonfuzzy logical systems, they are accepted as the "fuzzy generalization of nonfuzzy logical structure". Some of the alternative definitions that have been used in the literature are given below: ]Xa V B(u) = IRA(U) -}- ]XB(U) -- [dA(U)" JXB(U)

(2.1')

(Algebraic sum union)

PA v B(U) = Min{pa(U) + #B(U), 1}

(2.1")

t.t~a(U) = 1 -- pA(U).

(Bounded sum union) Note that the negation operation should not be named as "complement", as it is commonly used in set theory, since in fuzzy propositional logic this word will be used for a different meaning as will be defined in Section 6 below.

PA A B(U) = pA(U)" pB(U)

(2.2')

(Algebraic product intersection)

PA AB(U) = Max{0, pA(U) + pB(U) -- 1 }

(2.2")

(Bounded product intersection)

2.1.8. Fuzzy sets described as fuzzy proposition For each fuzzy set in the universe U having its membership function IrA(U), there is a "corresponding fuzzy proposition" with the universe (statement set) defined as: ith statement of the fuzzy proposition = "ui is an element of U" having truth value equal to membership value. Therefore: truth function of the fuzzy proposition = membership function of the related fuzzy set. Therefore, for each fuzzy set, a related fuzzy proposition can be defined so that fuzzy set operations and fuzzy propositional operations will be equivalent. However, the converse is not so obvious. It is not always intuitive to find a fuzzy set counterpart of the proposition on the subject matter "temperature of the process" or "beauty of a flower". Although Zadeh solved this problem by definitions of "term sets" and "compatibility functions", defining the systems upon the propositional logic instead of the set theory logic supplies the mapping between the physical world and mathematics in a more intuitive way. It has been shown by Zadeh [17], that the basic properties of nonfuzzy logical system (associative, commutative laws, distributive law and De-Morgan theorem) are applicable between these operations.

2.1.9. Note for continuous universe case In the definitions of AND, OR operations above, usage of Inf and Sup is applicable instead of Min and Max operations for the continuous universe case.

2.1.10. Sub-proposition Definition. Given two fuzzy propositions A and B in the same universe U, the proposition "A" is said to be a sub-proposition of the proposition "B" if

pA(U) <~#B(U) for all u in U.

(2.3)

This is shown in short as A -G>. A.

2.2. Special propositions 2.2.1. Identity proposition Definition. The fuzzy proposition defined by the truth function which assigns truth value '1' for all statements in the universe is called the identity proposition. Note that "giving T R U E value for all statements" can be interpreted as "any", "any one

Turhan ~iftfiba~t / Fuzzy Sets and Systems 85 (1997) 325-354

accepted", "all", "all of them possible". The identity proposition is represented as "(1)v", or as "(1)" when the universe is obvious. Hence in mathematical terms: /~(~)(u)= 1 for a l l u i n U. 2.2.2. Null proposition

329

truth value "l", while all others have truth value "0", is called "crisp proposition". The interpretation of a crisp proposition can be made as "only and definitely U is true (possible, accepted)" where u is the statement with truth value "1". For the finite universe U = {ul, u2, -.. , U n } , the crisp propositions will be represented by fig, i.e.,

1={1,0,0 .... ,0},

.,0}

. . . . .

Definition. The fuzzy proposition defined by the

truth function which assigns truth value "0" for all statements in the universe is called null proposition. Note that giving FALSE value to all statements can be interpreted as "none", "none of them true", "no one accepted"; "none of them possible". The Null proposition is represented as "(0)v', or as "(0)" when the universe is obvious. Hence in mathematical terms: /~o)(U)=0

for a l l u i n U.

Similarly, for the continuous case the crisp proposition having the element u such that 1 = true, and all others 0 = false will be represented as ft. 2.2.5. Semi-true proposition Definition. A fuzzy proposition is called "semi-true fuzzy proposition" if all truth values are greater than or equal to 0.5 i.e. u is a semi-true proposition iff u i> (0.5).

2.2.3. Constant proposition 2.2.6. Semi-false proposition Definition. The fuzzy proposition in the universe

U, having all truth values to be a constant "e" is called a constant proposition and represented as (~)v or as (~) when the universe is obvious. The following lemma, giving interesting properties of constant fuzzy propositions, is important in the sequel. Lemma 2.1. (a) "AND" operation of a fuzzy proposition u in U via a constant proposition (~)v, changes the truth values of u as "decreasing the values more than ct to value ~". (Limiting truth values from above at value co) (b) "OR" operation of a fuzzy proposition u in U, via a constant proposition (:t)v changes the truth values of u as "increasing the values less than to value c~". (Limiting truth values from below at value ~.)

Definition. A fuzzy proposition is called as "semifalse fuzzy proposition" if all truth values are less than or equal to 0.5 i.e. u is a semi-false proposition iff u ~< (0.5).

It is trivial by definition that for any fuzzy proposition u in the universe U, n+(1)v :(1)v,

n'(O)u:(O)v.

(2.4a, b)

Two other trivial but important facts are: u'(1)v = u,

u + (0)v = u,

(2,5a, b)

which shows that (1) is the ineffective element of " A N D " operation, and "(0)" is the ineffective element of "OR" operation. The following lemmas, generalize the concept of ineffective element of operations to fuzzy propositions.

Proofs are obvious from the definitions. 2.2.4. Crisp proposition Definition. The special case of a fuzzy proposition,

in which only one statement of the universe has

Lemma 2.2. (a) " A N D " operation of a fuzzy proposition via a semi-true proposition changes the large truth values only. (b) "OR" operation of a fuzzy proposition via a semi-false proposition changes the small truth

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Turhan ~iftfiba~l / Fuzzy Sets" and Systems 85 (1997) 325 354

values only. Thus semi-false propositions can be considered as "small term" for OR operation. Proofs are obvious from the definitions.

Let the fuzzy propositions be defined as: A1 = {1 0.6 0.4 0}

in U

A 2 = {0 0.6 1 0.6}

in U

A3={0

Lemma 2.3. I f u, ul, u2, u 3,

0 0 1}

inU

114. are fuzzy proposiB 1 = {0 0.2 0.6 0.9 1}

tions in the same universe U, then

in V

(a) ul ~
Ul'U~
(2.6a)

B 2 = {0 0.6 1 0.6 0}

(b) Ul ~
Ul + u ~ < u 2 + u ,

(2.6b)

Interpretations of the a b o v e propositions can be m a d e as:

(C) I11 ~ U2 a n d u 2 ~ u 3

implies ul ~< u3, (2.6c)

al

=

in V

t e m p e r a t u r e cold

(d) ul ~< U2

implies (ul A N D u2) = Ul,

(2.6d)

A2 = t e m p e r a t u r e w a r m

(e) ul ~< u2

implies (ul O R u2) = u2,

(2.6e)

A3=

t e m p e r a t u r e definitely hot

(f) ul~
~

(g) ul ~< U2 and u3 ~< u4 ~

BI=

pressure high

Ul+U3~
B2=

pressure m e d i u m

ul" u3 ~< U2" U4..

Then we have:

(2.6g) Proof. Assume that the result is not obtained by one statement in the statement set. Then the initial a s s u m p t i o n is not satisfied for this statement. Hence the proof. Unlike nonfuzzy systems, A N D and O R operations of a fuzzy proposition via its negation have the property:

( = temp not cold) = {0 0.4 0.6 1} ( = pressure not high) = {1 0.8 0.4 0.1 0}

~B1

- q A 1 /~ A 2 ( = t e m p not cold and warm)

{0 0.4 0.6 0.6} V A2 ( = temp not cold or warm) {0 0.6 1 1} Note: A1 A ~ A 1 ( = t e m p cold and not cold)

u'(-qu) ~< (0.5) u + ( ~ u ) ~> (0.5)

(a semi-false proposition) (a semi-true proposition)

E x a m p l e 2.3. Let two different universes U and V be defined as below:

= {0 0.4 0.4 0} (a semi-false proposition) B 2 ~ / ~ I B 2 ( = pres. med. or not med.)

= {1 0.6 1 0.6 1} (a semi-true proposition) Also note:

Subject m a t t e r of U = t e m p e r a t u r e of the process U = {low, below normal, a b o v e normal, high}

(1)v = any t e m p level possible (accepted)

Subject m a t t e r of V = pressure of the process V = {low, below medium, medium, a b o v e medium, high}

(1)v = any pressure level possible = {1 1 1 1 1}

= .{111

1}

(0)v = no t e m p level possible = {0 0 0 0}

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Turban ~iftfibaqt /Fuzzv Sets and Systems 85 (1997) 325 354

(0)v = no pressure level possible = {0 0 0 0 0} (0.4)v = all pressure levels have 0.4 possibility (acceptability) = {0.4 0.4 0.4 0.4 0.4}

2.4. Cross product of two fuzzy propositions of different universes Let A and B be two fuzzy propositions in different universes U and V, respectively. The "Cross Product" A x B of these propositions is defined by Zadeh [18, 19] as the two-universe proposition

(0.6)v = all temp. levels have 0.6 possibility WAx B( u, V) =

= {0.6 0.6 0.6 0.6}

min { ]/A(U), ~B(U) }

(2.7)

with the interpretation "both u and v".

B1 A (0.8)v = {0 0.2 0.6 0.8 0.8} (Bi limited from above at 0.8 truth value) B 1 V (0.3)v ~--{0.3 0.3 0.6 0.9 1}

(Bt limited from below at 0.3 truth value)

2.3. Multi-universe fuzzy proposition (fuzzy relation) Definition. Let U and V be two universes (sets of statements about their subject matters). The "twouniverse fuzzy proposition" (fuzzy relation) is defined in the universe U x V by assigning a truth value for each pair of statements (Zadeh [19]). Note: 1. The truth function is defined from U x V ( = set of pairs of statements) into [0, 1]. 2. Truth values of the pairs need not be dependent on the truth values of each statement, if defined. The truth function of a two-universe fuzzy proposition is represented as fiR(U, V). A multi-universefi~zzy relation is similarly defined as the fuzzy proposition in the universe U~ x U2 x •.. x U, by the truth function I~R(Ul, U2. . . . . U,). A finite, discrete two-universe fuzzy proposition can be shown in matrix form as:

A =

0.2

0.3

1

0.4

0.1

0.6

0

0.3

0.7

0

0.9

0

where the (i, j ) t h element shows the truth value of the (i, j ) t h statement pair.

2.5. Representation of a fuzzy proposition in a higher universe (Extension of a fuzzy proposition into a higher universe fuzzy proposition) (called as Cylindrical extension by Zadeh [19].) Definition. Consider a fuzzy proposition A defined in the universe U by the truth function #A(U). The fuzzy proposition defined in the Cartesian product universe U x V by the truth function /;/Ext[A](U, V) = ~/A(U)

(2.8)

is called "Extension of the fuzzy proposition A into the higher universe U x V" or "Representation of the fuzzy proposition in the higher universe U x V" and is shown in short as E x t v x v { A } v or as Ext {A} when the universes are obvious. For generalization of the above discussions, consider a fuzzy proposition A in the n-universe U1 x -.. x U,. It is possible to represent this fuzzy proposition in the m-universe U1 x ... x Urn (m > n) with the truth function ]dExt 'IA~(Ul, "'" , Un, Un + 1 . . . . .

Urn) = ]'IA(Ul . . . . .

Un)"

(2.9) In matrix representation (of the finite, discrete case), the known values of the n-dimensional relation matrix are copied as the values of the newly added dimensions. Example 2.4. F o r the universes in Example 2.3 consider the fuzzy propositions: A1 = temperature cold = {1 0.6 0.4 0}

in U

Turban (iftfiba~l / Fuzzy Sets and Systems 85 (1997) 325-354

332

B1 = pressure high = {0 0.2 0.6 0.9 1}

Extv×v{A,} =

Extv×v{Bl} =

1

1

1

1

1

0.6 0.4

0.6

0.6

0.6

0.6

0.4

0.4

0.4

0.4

0

0

0

0

0

0

0.2

0.6

0.9

1

0

0.2

0.6

0.9

1

0

0.2

0.6

0.9

1

0

0.2

0.6

0.9

1

in V

~PROJ(R)(/,/il,

... ,

Uim)

u,)},

(2.12a)

= Minq,{~R(ull . . . . ,ui,,, ... ,u,)},

(2.12b)

= Maxq,{l~,(uil . . . . . ui . . . . . . ~SHAD(R)(Ul 1 .....

2.6. Projection and shadow o f a f u z z y proposition 2. 6.1 Projection

Definition. Given a fuzzy proposition "R" on U x V defined by the truth function pR(U, V), the

fuzzy proposition "A" defined on U by the truth function ]JA(U) = max {#R(U, V)}

truth function:

(2.10)

Uim)

where the maximum and minimum are taken on the complement q' of the index set q = {i1,i2 . . . . . i,,} of the m-dimensional universe (Zadeh [,19]). In words it can be stated as: Projection (Shadow) is obtained by taking maximum (minimum) of all truth values for all combinations of discarded indices, keeping the "standing" indices fixed. Note that Projection operation places emphasis on high truth values (most possible ones), while Shadow operation emphasizes on the low truth values (least possible ones). In the definitions of Projection and Shadow operations in continuous universes, Inf and Sup operations should be used instead of Min and Max operations.

v~V

is called Projection of R on U (Zadeh [-19]), and also shown as A = Pr~u{R}u×v 2.6.2. Shadow

Definition. Given a fuzzy proposition "R" on U x V defined by the truth function pR(u, v), the fuzzy proposition "B" defined on U by the truth function /~8(u) = min {l~g(U, v)}

(2.1 1)

2. 7. ' "AND" operation o f two f u z z y propositions in different universes

"A and B" (A /~ B or shortly A' B) of the fuzzy proposition A (defined by ~ A ( U ) in the universe U) and the fuzzy proposition B (defined by #s(v) in the universe V) is defined by AND operation in their extended universe representation as: {A}u'{B}v = { A } u × v ' { B } v × v

(2.13a)

which is equivalent to PA. B(u, v) = min{#a(u), pB(v)}.

(2.13b)

L'~V

is called the Shadow of R on U, and also shown as B = Shadv{R}v×v. 2.6. 3. Generalization

Given a multi (n) universe relation R, Projection and Shadow of R on a lower (m < n) dimensional universe are similarly defined by the m-dimensional

Note that the above definition of applying " A N D " operation to two fuzzy propositions A and B in different universes U and V is equivalent to Zadeh's definition of cross product A × B, also given in Section 2.4. Mamdani [,11] recognized the fact that cross product operation can be thought of as " A N D " operation in different universes. Expressing all fuzzy propositions in the higher dimensional representation shows this fact more clearly.

Turhan ~iftfiba~l / Fuzzy Sets and Systems 85 (1997) 325-354

333

Example 2.5. Consider the same universes in Example 2.3, with the fuzzy propositions:

Example 2.6. For Example 2.5:

A1 = temperature cold = {1 0.6 0 0}

Az V B1 = {temperature hot} or {pressure low}

A 2 =

0

pressure high = {0 0 0.1 0.8 1}

Then from the definition of " A N D " operation

0

0

0

0.2

0.2

0.2

0.2

0.2

0.8

0.8

0.8

0.8

0.8

1

1

1

1

1

V = {0 0.2 0.8 1}u/k {1 0.8 0.1 0 0 } v 0

0

0

0

0.2

0.2

0.2

0.2

0.2

0.8

0.8

0.8

0.8

0.8

1

1

1

1

1

1

0.8

0.1

0

0

1

0.8

0.1

0

0

1

0.8

0.1

0

0

1

0.8

0.1

0

0

0

0

0

0

0

0.2

0.2

0.1

0

0

0.8

0.8

0.1

0

0

1

0.8

0.1

0

0

propositions

0

A2 'B1 -- {temperature hot} and {pressure low}

0

same

of

= {0, 0.2, 0.8, 1}v V {1, 0.8,0.1, 0, 0Iv

temperature hot = {0 0.2 0.8 1}

B~ = pressure low = {1 0.8 0.1 0 0} B 2 =

the

1

0.8

0.1

0

0

1

0.8

0.1

0

0

1

0.8

0.1

0

0

1

0.8

0.1

0

0

1

0.8

0.1

0

0

1

0.8

0.2

0.2

0.2

1

0.8

0.8

0.8

0.8

1

1

1

1

1

2.9. Representation o f a physical system by a fuzzy proposition - size o f the universe

A physical system can be represented by several models in different universes. Although the system is a one-universe system in principle, it can be represented by a higher universe model. More specifically, suppose that in a physical system, a fuzzy proposition has been modelled as: Subject matter: Temperature of the process

2.8. " O R " operation of two fuzzy propositions in different universes

"A or B" (A V B or shortly A + B) of the fuzzy proposition A in the universe U and the fuzzy proposition B in the universe V is defined as the "OR" operation of fuzzy propositions in the extended universe U x V. This can also be expressed

Universe: {Low, Below medium, Above medium, High} Proposition: u = A = {1 0.6 0.4 0} Interpretation: Temperature cold in one-universe representation. The same physical system can also be modelled in two-universe representation as:

as~

Subject: (Temperature, Pressure) of the process {A}u + {B}v = {A}v×v + {B}l:×v

(2.14a)

which is equivalent to: UA+B(U, V) = max { PA(U), #B(V)}.

(2.14b)

Universe: {Low, Below medium, Above medium, High} x {Low, Below medium, Medium, Above medium, High}

Turhan (2iftciba.~t / Fuzzv Sets and Systems 85 (1997) 325 354

334

Proposition:

The truth value of the (i,j)th element in the universe U x V is only dependent on the truth value of the ith element in the universe U. This could be expressed in a different notation as

proJection/shadow is then accepted as the inverse procedure of the extension operation. Both one-universe and two-universe models can represent the same system equivalently. Even after the one-universe model has been chosen and calculations (by using AND, OR, N O T operations and their combinations) have been made on this system, whenever this model becomes insufficient, one can just add a new universe and continue. Since all calculations that were made up to that time are valid in the higher universe with the new variable equal to 1 = ANY, after this step calculations can continue in the higher universe.

(u, v) = {temp is cold, pressure is any}

2.9.1. Notation

Extu×v{U} =

1

1

1

1

1

0.6

0.6

0.6

0.6

0.6

0.4

0.4

0.4

0.4

0.4

0

0

0

0

0

Interpretation: {temperature is cold, pressure is any}.

L e m m a 2.4. For any fuzzy proposition AI, A 2 in the

The representation of a proposition "A" after extension into the higher universe U x V will also be called with the same name "A" assuming that all definitions and calculations that had been made up to that time were made in this higher universe. The original proposition in the original universe U will then be represented as {A}v. If representations, of both one-universe, and two-universe fuzzy propositions should be shown in one equation, as in "extension", "projection", "shadow'" operations, the following notation will be used:

same universe U,

Extv × v {A }v

extension into U × V of the fuzzy proposition A in U

Projv{A}v×v

projection into U of the fuzzy proposition A in U x V

Shadv{A}v×v

shadow into U of the fuzzy proposition A in U x V

= (A, 1v)

={{1 0.6 0.4 0},{1 1 1 1 1}}. This is analogous to "representing the point x = 5 in the real axis as ( x , y ) = (5,0) in the complex plane". The following lemma will constitute a base for the equivalence between a fuzzy proposition and its representations in the higher universes:

Extv×v{Al" A2}c, = Extc,×v{A, }v'Extv×v{A2}v, Extv×v{A~ + Az}v

= Extv×v{A1}v + Extv×v{Az}u, Extc,× v{~A~ }c, = -nExtv ×v{A1 }v.

In words, "making boolean operations and taking extension into the higher universe" is equivalent to "taking extension and makin,o the boolean operation in the extended universe". Proof. Obvious by definition. Equivalence axioms can be proved between a fuzzy proposition in the universe U and its extensions in the higher universes. It should be noted that if the projection and the shadow of a fuzzy proposition into a lower universe are equal, they are accepted in the same equivalence class, and

2.10. "AND" operation of relations in d~fferent universes (Composition) Definition. Consider the universes (statement sets) U (with n elements), V (with m elements), and W (with p elements). Again consider the relation R in the universe U x V and the relation S in the universe V x W. The composition of the relation R and S (R ~ S) is defined by " A N D " operation in the extended universe U x V × W. The composition of R (n x m relation matrix) with S (m × p relation matrix) was defined by Zadeh

Turhan ~iftfiba6z / Fuzzy Sets and Systems 85 (1997) 325 354

in [19] as the max min matrix product of the matrices R and S with the result in universe U x W. The definition of this paper ("AND" operation in extended universe representation) gives the result in the extended universe U x V x W. If the result is requested in U x W universe, the projection can be taken and the same result of max min product is obtained. The definition of this paper is also consistent with logical reasoning, i.e. R o S means that both relations hold. (The fact that "composition is nothing but A N D operation" was also stated by M a m d a n i in [11].) In a more general terminology, the composition of R1 in the universe (U1 x ... x U,) and R2 (in the universe (U1 x ... x Urn) can be obtained by representing both relations in the extended universe (by including unpresent dimensions so that both relations will have a c o m m o n universe). Then composition can be performed by " A N D " ing the relations as fuzzy propositions in this extended universe. 2.11. "'OR" operation of relations in different universes Performing " O R " operation between the relation R1 in the universe (U1 x .-. x U,) and Rz (in the universe U1 x ... x Urn) can be obtained by representing both relations in the extended universe (by including unpresent dimensions so that both relations will have a c o m m o n universe). Then " O R " operation can be performed as a fuzzy proposition in this extended universe. 2.11.1. Unification o f the notation The definitions in this section showed that only two operations " A N D " and " O R " are sufficient for all fuzzy proposition operations. Cartesian product and Composition that are used in the literature are replaced by " A N D " operation. This will clear all confusions that were present in the developing phases of the fuzzy reasoning logic.

335

valid by considering the propositions in the extended universe, fuzzy implications can be defined by the known formulas as in non-fuzzy Boolean propositional algebra. Defining the implication A ~ B in terms of propositions A and B, and composing via any other fuzzy proposition one obtains the resultant fuzzy proposition in the extended universe. If the result is requested in the universe U or in the universe V, projection or shadow should be taken. Several different implication definitions that are used in the literature will be investigated below by the new notation.

3. ".4 ~ B "

(IF A T H E N B)

Consider the fuzzy propositions A in the universe U and B in the universe V. By representing the same fuzzy propositions in the extended (U x V) universe, the implication statement can be defined in two interpretations that are direct generalizations of the nonfuzzy version. Implication version-l: A ~ B = -7 A + A- B

(3.1)

Implication version-2: A ~ B -- -7 A + B

(3.2)

These definitions are the same as the definitions of Zadeh [18], expressed in the extended universe terminology. Other definitions are also possible which can be obtained by equivalent expressions in Boolean algebra. One important definition is "full conjunctive normal form" as in Boolean algebra, which has all variables in each term as given below: Implication version-3: A ~B = A'B + ~A'B

+ ~A'-nB

(3.3)

3.0. Verification of the implication definitions

2.12. Fuzzy conditional statements

The verification of these definitions will be made by showing that

Remembering that all operations AND, OR, N O T between propositions of different universe are

Composition (A ~ B). A is equivalent to "A and B"

Turhan ~iftfibaql / Fuzzy Sets and Systems 85 (1997) 325 354

336

and

Version-3 for implication gives: (A ~ B ) . A

Composition (A ~ B)- (~B) is equivalent to "(~A) and (~B)"

=- ( ~ A . B

+ ~A.-nB

= A.~A.B

+ A.B).A

+ A.-nA.~B

+ A.B

(3.8)

Note that in the daily usage o f logic, the result of "(A ~ B) and A" is B because A is already known, hence is not mentioned. Similarly, it is used in the daily usage of logic that: The result of "(A ~ B ) and ~ B " is ~ A since ---nB is given and therefore not mentioned. But in the (nonfuzzy) Boolean algebra,

In view of the previous discussions, "A. B" is the correct result, and the remaining part is the error term which is zero for the nonfuzzy case, and "semifalse proposition" (small term) for the fuzzy case. By taking the projection of "(A ~ B).A" into universe V, the result will be close to B, the difference from B coming from: (I) the error terms given above; (II) the error coming from the projection of the correct result A-B into universe V, which will be discussed in Section 7 below.

"(A ~ B) and A" is equivalent to "A and B" (3.4) and

Example 3.1. Consider the universes in Example 2.3 above, and the fuzzy propositions: A1 ( = temperature cold) = {1, 0.6, 0, 0}

"(A ~ B) AND -nB" is equivalent to " ~ A AND ~ B "

(3.5)

A2 ( = temperature hot) = {0, 0.2, 0.8, 1}

by the truth table in Boolean algebra and logically more reasonable.

BI ( = pressure low) = {1, 0.8, 0.1, 0, 0}

A

B

A=~B

(A=*B)'A

A'B

(A=>B)'~B

~A'~B

Using version-1 for implication,

0 0

0 1

1 1

0 0

0 0

1 0

1 0

AliBI

l l

0 1

0 l

0 l

0 1

0 0

0 0

B2 ( = pressure high) = {0, 0, 0.1, 0.8, 1}

= ( ~ A 1 ) + (AI"B1)

= {0 0.4 1 1} + {1 0.6 0 0}'{1 0.8 0.1 0 O}

The same is applicable in fuzzy logic by considering both A and B in the extended universe U x V. If the result is requested in the universe U or V, projection can be taken and the same result in the literature is obtained. 3.1. The composition "A ~ B A N D A "

0

0

0

0

0

0.4

0.4

0.4

0.4

0.4

1

1

1

1

1

1

1

1

1

1

+

Version-1 for implication gives: (A~B).A

= (~A + A'B)'A

=~A'A

+ A'B

(3.6) Version-2 for implication gives: (A ~ B). A - (-nA + B). A = A . ~ A

+ A. B

(3.7)

1

1

1

1

1

0.6

0.6

0.6

0.6

0.6

0

0

0

0

0

0

0

0

0

0

1

0.8

0.1

0

0

1

0.8

0.1

0

0

1

0.8

0.1

0

0

1

0.8

0.1

0

0

Turhan ~iftfiba~ / Fuzzy Sets and Systems 85 (1997) 325 354

0

0

0

0

0

1

1

1

1

1

0.4

0.4

0.4

0.4

0.4

0.6

0.6

0.6

0.6

0.6

1

1

1

1

1

0

0

0

0

0

1

1

1

1

1

0

0

0

0

0

+

1

0.8

0.1

0

0

1

0.8

0.1

0

0

0.6

0.6

0.1

0

0

0.6

0.6

0.4

0.4

0.4

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Hence,

is the calculated result in U x V. (Version-1 of implication is used. The same result is obtained for version-2 and version-3 of implications.) Note that the correct reply AI'B~ is

A1 :=>B1 1

O.8

0.1

0

0

0.6

0.6

0.4

0.4

0.4

1

1

1

1

1

1

1

1

1

1

by version-1 for implication. A1 - B1

Similarly using version-2 and version-3 of implication one can obtain:

1

0.8

0.1

0

0

0.6

0.6

0.1

0

0

0

0

0

0

0

0

0

0

0

0

and the error terms for version-1 and version-2 implications are A l" (-hA1) error matrix:

A1 ~ B1 1

0.8

0.1

0

0

1

0.8

0.4

0.4

0.4

1

1

1

1

1

1

1

1

1

1

by version-2 for implication.

A1 ~ B1 1

0.8

0.1

0

0

0.6

0.6

0.4

0.4

1

0.8

0.9

1

0.4 1

1

0.8

0.9

1

1

by version-3 for implication.

To verify the above results, the fuzzy counterpart of (A1 ~ B1)' (A1) will be calculated. Since the result will be in U x V, the projection into the universe V will be found for comparison with B1. By using version-1 for implication {(A 1 :=~ B1).A1}u× v

z

337

1

0.8

0.1

0

0

0.6

0.6

0.4

0.4

0.4

1

1

1

1

1

1

1

1

1

1

A 1 . - - 1 A 1 ~__

0

0

0

0

0

0.4

0.4

0.4

0.4

0.4

0

0

0

0

0

0

0

0

0

0

which is a semi-false term (small term), and increases "the values less than 0.4", to "0.4" in the second row by " O R operation" via the correct result A1 "B1. The same error term is obtained for version-3 implication. Projection of the resultant matrix into the universe V, gives B'I = Projv{(A1 ~ B1)" A1} = {1 0.8 0.4 0.4 0.4} 7~ B1, the difference coming from the error term. An important point that must be mentioned is that the projection of the "correct result" AI"B1 into V gives the original fuzzy proposition Ba in this example. However, (as also mentioned above) there is an additional source of error so that if

338

Turban ~iftqibaqt / Fuzzy Sets and Systems 85 (1997) 325 354

maximum truth value of A 1 is not "1", the projection of A1 "B~ into V does not give B1, the original proposition in V. This will be analyzed in Section 7 below. (See Lemma 7.7). 3.2. The composition "A ~ B AND N O T B "

By using version-1 for implication, {(A1 => BI )' (-7B1)} v ×v

1

0.8

0.1

0

0

0.6

0.6

0.4

0.4

0.4

1

1

1

1

1

1

1

1

1

1

0

0.2

0.9

1

1

0

0.2

0.9

l

1

0

0.2

0.9

l

1

0

0.2

0.9

1

1

0

0.2

0.1

0

0

0

0.2

0.4

0.4

0.4

0

0.2

0.9

1

1

0

0.2

0.9

1

1

Version-1 for implication gives: (A ~ B)" ( 7 B ) --- ( 7 A + A' B)' ( 7 B) =-TA.~B

(3.9)

+ A'B'~B

Version-2 for implication gives: (A ~ B). ~ B =- (-TA + B ) . - 7 B = 7A'-TB

(3.10)

+ B'TB

Version-3 for implication gives: (A ~ B ) . ~ B

~_ ( - 7 A . ~ B = 7A.~B

+ ---7A.B + A .B).---7B +TA.B.TB

(A1 ::~"B1)' ("-~B1) =

+ A.B.--TB

(3.11) According to logical reasoning, the correct reply is " T A .-7B"

is the result in U x V by version-1 implication. (Version-2 and version-3 implications also give the same result.) Note that the correct reply --TA~ .TB~ is

and the error terms for each version are: "A- B ' 7 B "

by using version-1 implication

"B.TB"

by using version-2 implication

"~A.B.TB

+ A.B.TB"

by using version-3 implication The error terms are again semi-false propositions (small terms), and only affect small truth values of the result.

-qA1 "-7B1 =

A1.B1.TB

1 =

0

0

0

0

0.2

0.4

0.4

0.4

0

0.2

0.9

1

1

0

0.2

0.9

1

1

0

0.2

0.1

0

0

0

0.2

0.1

0

0

0

0

0

0

0

0

0

0

0

0

for version-1 implication,

A~ ( = temp cold) = {1 0.6 0 0}

Implication A~ ~ B1 was obtained in the previous example for the three versions of the implication. To verify these results, the fuzzy counterpart of (A1 = ~ B 1 ) ' ~ B 1 will be calculated and the difference from ~ A I '-7B1 will be found. Since this result is in U x V, the projection into U will be found for comparison with -7A1.

0

and the error terms are:

Example 3.2. Consider the same universes and fuzzy propositions in Example 3.1 above.

B1 ( = pressure low) = {0 0.8 0.1 0 0}

0

B1 "7B1 = 7 A i "B1 "-"7B1 + A1 "B1 ' ~ B 1 0

0.2

0.1

0

0

0

0.2

0.1

0

0

0

0.2

0.1

0

0

0

0.2

0.1

0

0

for version-2 and version-3 implications,

339

Turhan (2iftfiba~l / Fuzzy Sets and Systems 85 (1997) 325-354

all of which are small terms (semi-false propositions) giving the error between the correct reply and the obtained matrix. Projection of the resultant matrix to the universe U, is different from ---TA 1 due to the error term. Ai = Projo{(A1 ~ B 1 ) ' ( ~ B 1 ) } = {0.2 0.4 1 1}

sions are also possible one of which is "full conjunctive normal form" given below: --TA. B. C + ---7A . - T B . C + --hA. B . ~ C +--nA.~B.~C + A.~B.-TC

+ A.TB.C

(4.5)

+ A.B.C

--7A1 = {0 0.4 1 1}

5. The fuzzy proposition "IF A THEN B ELSE C" 4. The fuzzy proposition "IF A THEN (IF B THEN C)" Consider the fuzzy propositions A, B, C in different universes U, V and W, respectively. By representing the same fuzzy propositions in the extended universe U × V × W, implication can be defined in the Boolean algebra terminology. The definition of Zadeh [-18] for this statement is

The fuzzy propositions A in the universe U, and B, C in the universe V can be represented in the universe U × V and the definition of this implication can be given in this extended universe. The mathematical interpretation for this verbal expression by Zadeh [18] is: IF A T H E N B E L S E C -

A.B +-TA.C

(5.1)

"A ~ (B ~ C)"

Another mathematical interpretation for this relation by Zadeh [18] is:

which will be equivalent to

IF A T H E N B ELSE C = (A ~ B ) . ( ~ A ~ C) (5.2)

7A+A.(-TB+B.C)=TA+A.TB+A'B.C

(4.1)

which can be simplified by version-1 implication as: = ( A . B + ~ A ) . ( - - n A . C + A)

by using version-1 of implication, or = A.B +~A.C

-TA + ( ~ B + C) = 7 A + - T B + C

+ A.-7A + A.-TA.B.C

(4.2) (5.3)

by using version-2 of implication. The definition of Mamdani [11] for this statement is:

This is equal to Eq. (5.1) plus small error terms. Verification by composing via A:

(A. B) ~ C

"IF A T H E N B ELSE C" AND "A" = {A.B +--TA.C}.A

which can be expressed in the open form as ~ ( A - B ) + A ' B ' C = ~ A +'--7B + A . B ' C

(4.3)

by using version-1 of implication, and ~ ( A - B) + C = -TA + ~ B + C

(4.4)

by using version-2 of implication. Note that they all have the same truth table for the nonfuzzy case. In the fuzzy case, all three forms are approximate to each other. Different other ver-

= A.B + A.~A.C

which gives the correct result A-B plus a small error term. Verification by composing via -TA: "IF A T H E N B ELSE C " AND " ~ A " = {A'B +-TA'C}'~A =~A'A'B

+~A'C

Turban Ciftqiba~t /Fuzzv Sets and Systems 85 (1997) 325 354

340

which gives the correct result " - l A • C" plus a small error term.

The measurement Ao which can be a fuzzy or crisp proposition, will be composed via R, hence Ro = Ao' R Ro = A o ' { A I " B 1 + - q A l " {

6. Applications to fuzzy controllers

The propositional logic summarized above will be applied in this section to the fuzzy controllers that were developed in the previous literature.

A 2 • B2 + ~ A 2 • { A3 'B3 + ~ A 3 ' {

+ A k ' B k +--lAg}} 6.1. Decision rule f o r 1-input, 1-output system

The common usage of fuzzy set theory/propositional logic in fuzzy controllers starts by preparing implication statements as the following:

... }}

is the output in the extended universe. Projection of Ro into V gives the output of the controller, which will be defuzzified to obtain the control output of the system: Bo = Projv{Ro}.

IF A 1 T H E N B1

(6.3)

(6.4)

The open form of Eqs. (6.3) and (6.4) for a typical 1-input, 1-output 4-interval control system is:

ELSE

Bo = P r o j v { A o ' A 1 "B1 + A o ' ~ A 1

If A 2 T H E N B 2

ELSE

"Az'B2

+ Ao'-qAI"~A2"A3"B3

IF A 3 T H E N B 3

+ Ao" ~ A 1 "-qA2" ~ A 3 " A4" B4 + A o ' ~ A 1 " - q A z ' ~ A 3 '-qA4}.

ELSE (6.1) where Ai, B; are predetermined propositions that will be called as defining propositions. Note that although Ai's are different propositions, some of the Bi's can be the same in some propositions. The aim of the fuzzy control system is to find the output of the controller Bo, for the measurement input Ao. The implication "IF A T H E N B ELSE { ... }" was expressed as " ' A . B + - q A ' { ... }" in Eq. (5.1) above. Using this expression successively in Eq. (6.1) we obtain, R = A I ' B 1 +--lAa" {

6.2. More on properties o f propositions 6.2.1. Mutually semi-exclusive propositions Definition. The propositions A, B in the same universe U are said to be mutually semi-exclusive if

for a l l u i n U

(6.6)

(plus sign means arithmetic addition).

A 3 . B 3 + ~A 3. {

+ A k ' B k + ~ A k } } ... }.

As stated earlier, the design of a fuzzy control system starts by determining the propositions Ai, Bi. How close and how far apart these "defining propositions" should be selected will be discussed in Section 6.3 after the development of some additional properties of propositions.

pA(U)+pB(U)~< 1

A 2-B 2 +--1A 2'{

(6.5)

Lemma 6.1. T w o propositions A, B are mutually semi-exclusive iff (6.2)

(a) A ~<-qB

(6.7a)

Turban ~iftfiba~l / Fuzzy Sets and Systems 85 (1997) 325-354

341

g,u

or equivalently

(b) B ~<-nA.

(6.7b)

.

T-x

A1

/~5

,, _

.

.

.

.

.

Proof. (a) pA(U)+#~(U)~ 1

for a l l u

<, pA(U) ~< 1 --/~B(U) for all u

A ~<--nB hence the proof. Proof of (b) is obvious as the counterpart of this proof. In Fig. 2, the propositions Aa and A 2 a r e mutually semi-exclusive since the truth function curve of A 2 is below that of-hA1. Lemma 6.2. Two f u z z y propositions A , B are mutually semi-exclusive iff A.--3B = A

( B ' - n A = B)

(6.8)

which is a direct consequence of Lemma 6.1 and of sub-proposition property.

Lemma 6.3. I f two fuzzy propositions A, B are mutually semi-exclusive then A. B ~< (0.5).

(6.9)

Proof. F r o m Lemma 6.1 we have A <~--qB

by " A N D " operation via B on both sides (see Lemma 2.3(a))

Fig. 2. Examples of exclusive and complementary properties of fuzzy propositions (A1 ,A3) are exclusive, (At, A2) are semi-exclusive (A 2 is below ~ A l ), (A 3, A4) are complementary, (A 1, A 5) are semi-complementary (As is above -7 A 1)-

6.2.2. Mutually exclusive propositions Definition. The propositions A, B in the same universe U are said to be mutually exclusive if

A-B = (0).

(6.10)

In Fig. 2 the propositions (A1,A3), ( A 1 , A 6 ) , (A2, A3) and (A2, A 6 ) a r e mutually exclusive pairs of propositions. Lemma 6.4. Exclusive property implies semi-exclusive property. ProoL

A .B = (0) ~ Min{#a(U), #B(u)} = 0 for all u ~ pA(u) + pB(u) ~< 1 for all u. Hence the proof. 6.2.3. Semi-complementary propositions

A" B <~-1B" B <~(0.5).

Hence the proof. Note: Eq. (6.9) cannot be used as the definition of semi-exclusivity, because for specially selected propositions (6.9) can be satisfied while more restrictive and meaningful (6.7) is not satisfied. In Fig. 2 the propositions (A3, A6) and (A2, Av) are not mutually semi-exclusive pairs although (6.9) is satisfied.

Definition. Two fuzzy propositions A , B in the same universe U are said to be semi-complementary propositions if

pa(u) + #s(u)/> 1

for all u in U.

(6.11)

Lemma 6.5. Two f u z z y proposition A, B are semicomplementary iff (a) A /> ~ B ;

(6.12)

342

Turban ~iftfiba~t /Fuzzy Sets and Systems 85 (1997) 325-354

Proof. A,B complementary ~ A + B = (1)

or, equivalently, (b) B ~> ~ A .

Max{pa(U), #B(u)} = 1 for all u

(6.13)

#A(U) + #B(U) ~> 1 for all u. Hence proved.

Can be proved easily as in Lemma 6.1. In Fig. 2, the propositions A1 and A5 are semi-complementary since the truth function curve of A5 is above that of ---nA~,

Lemma 6.9. Two fuzzy propositions A, B are both

Lemma 6.6. Two fuzzy propositions A, B are semi-

A =--nB

mutually semi-exclusive and semi-complementary iff they are negations of each other, i.e. (or equivalently B = ~ A ) .

(6.17)

complementary iff A +~B=A

(B+~A

=B)

(6.14)

which is a direct consequence of Lemma 6.5 and of the sub-proposition property. Lemma 6.7. I f two fuzzy propositions A, B are semi-

complementary then A + B >~ (0.5).

Proof. [~A(U)

-[- [AB(U) ~ 1 and

]AA(U) -I- /JB(H) ~ 1

~=~ pa(u) + ttB(u) = 1 ¢> /tA(u) = 1 - p~(u). Hence the proof. Lemma 6.10. Two propositions A,B are mutually

(6.15)

Proof. A,B semi-complementary ~ A >~ ~ B by "OR" operation on both sides via B (see Lemma 2.3(b)):

semi-exclusive (exclusive) iff ~ A , ~ B are semi-complementary (complementary). Proof. Obvious by definition.

6.2.5. Set of semi-exclusive propositions A+B~>~B+B/>(0.5). Definition. A set of propositions {A1, A 2 . . . . . A,},

Hence the proof.

Note: Eq. (6.15) is a necessary but not a sufficient condition for semi-complementary property. The negations of A3 and A6 in Fig. 2 (~A3 and -hA6) satisfy Eq. (6.15), although they are not semicomplementary.

represented shortly as {A~}, is said to be a semi-exclusive set if Ai and ~2j #iAj are mutually semi-exclusive for all i. 52 sign is used for OR ( = maximum) operation. Lemma 6.11. A set of propositions {Ai} is a semi-

exclusive set iff

6.2.4. Complementary propositions

~ A i >~ ~, Aj for all i

Definition. The propositions A1, A2 in the same universe U are said to be complementary propositions if

or, equivalently,

A + B = (1).

(6.18)

j#i

(6.16)

In Fig. 2 the propositions (A3, A4), (A4, A5) are complementary pairs.

~ Aj >~ Ai for all i.

(6.19)

jT'i

Proof. Obvious from the definition and mutual semi-exclusive property of two propositions. L e m m a 6.12. A set of propositions { Ai } is a semi-

L e m m a 6.8. Complementary property implies semi-

complementary property.

exclusive set iff each pair (Ai, A j) is mutually semiexclusive.

Turhan ~iftqiba~l / Fuzzy Sets and Systems 85 (1997) 325-354

Proof. A~ and ~ Aj

semi exclusive for all i.

j~i

Lemma 6.16. A set of propositions {Ai} is semicomplementary iff 7 A i <~ ~ Aj for all i

~-TAi~> ~A~

343

(6.20)

j¢i

for alli.

or, equivalently, ¢~-7Ai > / M a x A j

for all i.

-n ~ Aj ~ Ai for all i.

(6.21)

j¢i

<~>-7A~/> A~ for all i ~ j . Hence the proof.

Proof. Obvious from the definition of the semicomplementary property of two propositions.

Lemma 6.13. In a set of semi-exclusive propositions {Ai}, at most one proposition can have truth value > 0.5 for any statement u.

Lemma 6.17. I f a set of fuzzy propositions {Ai } is semi-complementary, then Ai ~> (0.5).

Proof. Obvious from Lemma 6.12. Lemma 6.14. Consider a set of semi-exclusive propositions { Ai}. I f for a statement u, one of the propositions has truth value ~ > 0.5, then all other propositions should have truth value less than or equal to (1 - ~).

(6.22)

i

Proof. From Lemma 6.16 Aj >7--7Ai for all i

(6.23)

j¢i

by "OR" operation via Ai on both sides (Lemma 2.3(b))

Proof. Since each pair of propositions should be mutually semi-exclusive, then I~Ai(U)+ I~Aj(U) <<.1 and the result follows.

Ai + ~ Aj >~ Ai + ~ A i >~(0.5).

6.2.6. Set of exclusive propositions

Lemma 6.18. Let {Ai} be a set of semi-complementary propositions. For each statement u, at least one proposition should have truth value greater than or equal to 0.5.

Definition. A set of propositions {A~} is said to be an exclusive set if each pair of propositions are mutually exclusive. Lemma 6.15. A set of exclusive propositions is also a semi-exclusive set. Proof. Obvious since each pair should be exclusive, therefore semi-exclusive due to Lemma 6.4. 6.2.7. Set of semi-complementary propositions Definition. A set of propositions {Ai} is said to be a semi-complementary set if A~ and

~ Aj j~i

are semi-complementary for all i.

j¢i

Hence the proof.

Proof. Otherwise for this u, Lemma 6.17 will not be satisfied. Lemma 6.19. For a semi-complementary set of propositions in a universe U, if for a statement u, a proposition has truth value e < 0.5, then at least one proposition should have truth value greater than or equal to (1 - e) for the same statement u. Proof. Let Ai has truth value e < 0.5 for statement u. Then ---~Ai has truth value (1 - e) for the same u. From Lemma 6 . 1 6 - - 1 A i < ~ j e i A j for all i ~ ~ j ~ i A j has at least one statement having truth value greater than or equal to (1 - ~ ) . Hence the proof.

Turhan (ififiba~l / Fuzzy Sets and Systems 85 (1997) 325-354

344

6.2.8. Set of complementary propositions

6.2.11. Non-fuzzy basis

Definition. A set of propositions {Ai} is said to be complementary if

Definition. A special case of fuzzy basis is a nonfuzzy basis, which has either "1" or "0" truth values for all statements of basis propositions.

A~ = (1)

(6.24)

i

Lemma 6.20. Complementary property implies semicomplementary property for any set of propositions.

6.2.12. Crisp basis Definition. A special case of nonfuzzy basis is the crisp basis, in which the basis propositions are all crisp propositions. Obviously, in the continuous case the crisp basis should have infinite number of elements (propositions).

Proof.

{Ai} complementary ~

Obviously, a set of propositions constituting a nonfuzzy basis is exclusive and complementary.

~, Aj = (1) J

~ A i + ~ Aj = (1) for all i j~i

Ai, ~ Aj are complementary pair for all i

Example 6.1 (Finite discrete universe). Consider the universe set U = {hi, u2, u3, u4, us}. (a) The set of propositions

Ai and ~ Aj semi-complementary for all i j¢i

{Ai } semi-complementary. 6.2.9. Set of fuzzy basis for the universe Definition. A set of propositions {Ai} is said to be a fuzzy basis for the universe if (a) The set is a semi-exclusive set. (b) The set is a semi-complementary set. (c) For each Ai, truth value is > 0.5 for at least one statement u.

Lemma 6.21. {Ai} ~ j ~ i A j f o r all i.

is a fuzzy

basis iff Ai=

U 1 = {1 0 0 0 0},

n2={01000},

n3={0

0 1 0 0},

U 4 = {0 0 0

us={0

0 0 0 1}

1 0},

constitutes a crisp basis for the universe. (b) The propositions

.1={10000},

.2={01000},

U3 = {0 0 1 0 0},

u4 = {0 0 0 1 1}

constitutes a nonfuzzy basis for the universe. (c) The set of propositions

ul = { 1

1 0.8 0.2 0},

u2 = {0 0 0.2 0.8 1} = ~ u 1 Proof. Obvious by definition, L e m m a 6.11 and L e m m a 6.16.

constitutes a fuzzy basis for the universe U. (d) The set of propositions

6.2.10. Summary of some important points

ul = {0 0.3 0.8 0.3 0},

Let {Ai} be a fuzzy basis. Then (1) If for any statement u, truth value is ~ > 0.5 for one proposition, then truth value for all other propositions are (1 - c~) for the same u. (2) If for any statement u, truth value is 0.5 for one proposition, then the truth value for all other propositions are also 0.5.

U 3

=

n2 = {0 0.7 0.2 0.7 1},

{1 0.3 0.2 0.3 0}

constitutes another fuzzy basis for U. (e) The set of propositions ul = { 1 0.5 0 0.5 0},

u3 = {0 0.5 0 0.5 1}

u2={0

0.5 1 0.5 0},

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Turhan ~iftqiba~t / Fuz~ Sets and Systems 85 (1997) 325-354

constitutes a n o t h e r fuzzy basis for U. (f) The set of propositions

Ux = { 0 0.1 0.4 0.6 1}, u3={1

u2={0

0.1 0.6 0.4 0},

0.9 0.4 0.4 O}

I

]

~Ai

A

o

A3

m

is also a set of fuzzy basis for the universe. Example 6.2 (Continuous case). A set of propositions constituting a fuzzy basis is given in Fig. 3(a). Triangular propositions have been used in this example. Bell shaped or different types can be used, as long as the set is semi-exclusive and semicomplementary. A set of propositions constituting a nonfuzzy basis is shown in Fig. 3(b). 6.2.13. Expressing a proposition in terms o f a crisp basis set Let {fig}g_ 1,, be a set of crisp basis for the discrete universe U. Then a set A can be expressed in terms of this basis set as follows:

(a) A3

[J) (b) Fig. 3. Fuzzy basis sets in continuous universe, a) A set of propositions constituting a fuzzy basis, b) A set of propositions constituting a non-fuzzy basis.

A = (0(1, ~ 2 , ~ 3 , --- ,0~n) = ( ~ , 0, 0 . . . . . +

...

0) + (0, ~2, 0 . . . . .

where fi is the crisp proposition having truth value ' T ' at u, and "0" otherwise.

0)

+(0,0,0,...,~,)

6.3. Selection o f defining propositions

= ( ~ l ) v ' ( 1 , 0 , 0, ... , 0 ) + ( ~ 2 ) v ' ( 0 , 1,0, ... ,0) +

'"

The decision rule was obtained in Eq. (6.2) as follows:

+ ( ~ , ) v ' ( 0 , 0 . . . . ,0,1)

R = A I "B1 + -hA1 "A2 "B2 + ~A1 "~Az'A3"B3

=~(~g)ufg i

A = E (~A(Ui))U "Ui

+ (6.25)

i

This representation is equivalent to the "representation of a fuzzy set as a collection of singletons" in fuzzy set theory by Z a d e h as A = ~il~A(Ui)/ug. Besides, Eq. (6.25) is a mathematical equality rather than a sole representation. F o r the c o n t i n u o u s case, similarly, one can obtain:

A = fu (~A(U))u" ~l,

(6.26)

"'" + ~ A I " ~ A 2

---~A,-I"A,'B,

+ -"IA1 " ~ A 2 " ~ A 3 "'" ~ A n

(6.27)

L e m m a 6.22. I f the defining propositions {Ag} constitute a basis in the universe, then Eq. (6.27) can be simplified up to a small term as R = A 1 . B 1 + A z . B 2 + A3"B3 + "" + A . ' B . .

(6.28) Proof. F r o m L e m m a 6.2 (mutually exclusive set property), all terms - q A , , A , are simplified as A,. The last term is a small term since its inverse is

346

Turhan ~!ft¢iba~t / Fuzzy Sets and Systems 85 (1997) 325-354

a large term by L e m m a 6.17 (complementary set property). Hence the proof. It is also useful to investigate what happens if the defining propositions are selected "closer to each other = not semi-exclusive" or "far from each other = not semi-complementary". Choosing the defining propositions closer to each other makes the last term (error term) smaller which is an advantage, but the terms ~ A m ' A, will not be simplified as in Eq. (6.28). For example if A 1 and A2 are very close to each other, ~ A ~ "A2 is very close to (0), hence it is ineffective in the decision process• Choosing the defining propositions very far apart (exclusive) will not give any additional advantage for the simplification of ~ A , , . A , terms to A,. However, the last term becomes a proposition close to (1) which will not make any selection between A~, B~ pairs. 6.4. Decision rule for 2-input, 1-output system

The implication statements for 2-input, 1-output control system are given as follows in the literature. IF A1 A N D B1 T H E N C1

A 3-B 3.C 3 + ~ ( A 3-B3).{

+ Ak'Bk "Ck + ~(Ak" Bk)}} ... } (6.30) R = AI"BI"C1 +-7(AI"B1)'A2"B2"Cz

+-7(A 1 "B1)~(Az" B2)" A3 "B3"C3 + "-'7(A1 "B1)7(A2" B2)"--7(A3 "B3)" A4' B," C a + ... + - 7 ( A 1-B1)-~(A 2.B2) •.. ~ ( A k - 1 "Bk-

1)' Ak" Bk" Ck

+~(AA "B1)-7(A2 "B2) "" ~ ( a k - 1 "Bk- 1) • -7(Ak" Be).

(6.31)

By composing the relation via the measurement imputs Ao and Bo, and by taking the projection into the output universe, the output will be obtained: Ro = A o ' B o ' R Ro= Ao•Bo'AI"BI'C1 + Ao'Bo'~(AI"B1)'{ A2" B2"C2 +-7(A~" B2)" {

ELSE

A 3"B 3 ' C 3 +--q(A 3"B3) •{

IF A2 A N D B2 T H E N C2 ELSE + Ak" Bk" Ck

IF Ak A N D Bk T H E N Ck,

(6.29)

where the defining propositions A~ in U, B~ in V are input, and Ci in W are output propositions• The indices of Ai, B~, C~ have been used to denote 1st, 2nd, etc. rule number of the implication rules, not necessarily representing different propositions. A~, B~, Ci can be the same or different, and all combinations of Ai and B~ should be used to define the behaviour of the system completely. By a similar discussion as in the 1-input case, the following decision rule is obtained from Eq. (6•29): R = AI" BI" C1 -~- --I(AI" B1)" { A 2"B 2"C 2 + ~ ( A 2"B2)'{

+ ~(Ak" Bk)}} "'" }

Co = Projw{Ro}

(6.32) (6.33)

The above steps are a summary of the technique that has been used in the literature. Eq. (6.31) has been simplified in the literature as: R = A 1 • B 1 • C 1 + A 2 • B 2 • C2 + A3" B3" C3 + "'" + Ak'Bk'Ck.

(6.34)

Again Ai, Bi, C~ represent different propositions, containing all combinations of Ai, B~ once. Mathematical validity of this simplification can be shown by a sufficient condition given below:

Turhan (iftgiba~t / Fuzzy Sets and Systems 85 (1997) 325 354

347

Lemma 6.23. A sufficient condition that the relation in Eq. (6.31) be simplified to (6.34) is that the defining propositions be selected as f u z z y basis sets in their universes. u

Proof. The terms except the last one of Eq. (6.31) can be simplified as in Eq. (6.34), due to mutual exclusivity of all pairs of basis sets in each universe. The fact that the last term becomes small can be shown by A1 +

A2 +

"" +

l Fig. 4. Simpler defining propositions which are not basis elements, but will work equivalently for decision process.

Ak >~ (0•5)

({Ai} is a semi-complementary set)

which can be shown in table form as:

B1 + B2 + ... + Bk >~ (0.5) ({Bi} is a semi-complementary set) and by Lemma 2.3(d) AIB1 + AIB2 + AxB3 + ... + A1Bk + A2B1 + A2B2 + A z B 3 + "'" + A2Bk + AkB1 + AkB2 + .." + AkBk >~(0.5).

By taking the negation of this term it is found that the last term of Eq. (6.31) is a small term. Hence the proof. 6. 4.1. A simpler set o f defining propositions Note that the only properties that are used in the simplification of the "defining propositions" are: (a) Each pair is mutually semi-exclusive, (b))~i Ai = ~> (0.5). Therefore simpler defining propositions can be used as given in Fig. 4, which are the ones being used in the literature. This selection of defining propositions will work equivalently• As an example of a fuzzy control system, consider the relation matrix for a typical two-input, one-output system defined in the form: R = A 1 "BI"C 1 + Ax'B2"C 2 + AI"B3"C 2

R

B1

B2

B3

A1 A2 A3

C1 C: C2

C2 C3 C3

C2 C3 C4

where universe of {Ai } has three intervals, universe of {Bi} has three intervals, and universe of {Ci} has four intervals (defining propositions). Note that (6.35) is a logical expression which is equivalent (with error less than (0.5)) to a collection of if-then rules using propositions given in the above table. If Ao and Bo are given input (measurement) propositions in the universes U and V, then by composition of R, Ao, and Bo propositions; the logical proposition in U x V x W is obtained as Ro = R - A 0 - B 0 .

Projection of Ro into universe W will give the possibility function Co in this universe• Co = Projw{R0} = Projw{R'Ao'Bo} = P r o j w { A I " B I " C I " A o ' B o + A1 "B2"C2 "Ao'Bo + AI'B3"C2"Ao•Bo + A2"B1 "C2"Ao'Bo + A2"B2"C3"Ao'Bo + A:

+ A:" B 1 " C 2 -~- A2" Be" C3 + A2" B3" C3 + A3" B1 " C2 + A 3 " B2" C3 + a 3 " B3" Ca

(6.35)

•B3"C3"Ao'Bo + A3"BI"C2"Ao'Bo + A3"B2"C3"Ao'Bo + A3"B3

• C4" Ao" B0 }.

(6.36)

Turhan (2iftfiba~t / Fuzzy Sets and Systems 85 (1997) 325-354

348

The resultant operations, although straightforward, need long matrix calculations. A simpler method has been practically developed by Mamdani [5] and Larsen [8] in the literature, that gives the output in a simple way based on a graphical technique. This method will be proved in Section 8 by using the mathematics developed in this paper. For the development of this technique, some properties of projection and shadow operations will first be developed in Section 7 below.

7. On projection and shadow 7.1. Properties o f projection and shadow operations

As discussed in the previous sections, the decision rule in Boolean algebra gives the result in the extended universe. Since the result is always requested in the output universe, projection or shadow of the result should be taken. Noting that both "Projection" and "Shadow" operations are inverses of the "Extension" operation, the following properties are important in the sequel: Lemma 7.1, Consider the fuzzy propositions A and B in the universes U and V, respectively. Let {A}u ×v and {B}u ×v denote the extended universe representations of A and B. Projections of these propositions into the universes U and V are given as: Projv{A}v×v = A,

Projv{B}u×v = B, (7.1a, b)

Projv {A} v× v = (C0v,

Projv{B}v×v = (fi)v,

(7.2a, b) where ~, fi are the maximum values of the truth functions of A and B; and (e)v, (fl)v are f u z z y propositions in V , U with all truth values equal to ct, fl, respectively.

Proof. Eqs. (7.1a, b) are obvious, since extending a proposition and then projecting into the original universe gives the same proposition. For the proof of (7.2a), assume that the maximum truth value of A is at the kth element with value e. "Representing A in U x V" (copying the

column vector A to all columns) leads to the fact that the kth row of this extended universe matrix is all e, while the other rows have smaller values. Projecting into the universe V gives (C0v. Eq. (7.2b) is proved similarly. The above properties are also valid for the multiuniverse case, i.e. if fuzzy proposition A is extended into multi-universe and then projected into the same or different one-universe, the above results are still applicable. Lemma 7.2. Consider the f u z z y propositions A and B in the universes U and V, respectively. Let {A}v× v and {B}u× v represent the extended universe representations of A and B. Shadows of these propositions into the universes U and V are given as: Shadv{A}c,×v = A,

Shadv{B}vxV = B,

(7.3a, b) S h a d v { A } v x v = (e)v,

Shadv{B}v×v = (~)v,

(7.4a, b) where e, 6 are the minimum values of the truth functions of A and B; and (e)v, (6)v are fuzz),' propositions in V, U with all truth values equal to e, 3, respectively.

Proof. Eqs. (7.3a, b) are obvious, since extending a proposition and then taking shadow into the original universe gives the same proposition. For proof of (7.4a), assume that the minimum truth value of A is at the kth element with value ~. "Representing A in U x V" (copying the column vector A to all columns) makes the kth row of this extended universe matrix all e, while the other rows have larger values. Taking shadow into the universe V gives (e,)v. Eq. (7.4b) is proved similarly. The above properties are also valid for the multiuniverse case. The projection (shadow) expressions (7.1) and (7.3) can be explained as "returning back into the original one-universe after multi-universe extension". Expressions (7.2) and (7.4) are "projection (shadow) into another one-universe, after multiuniverse extension", and will be called as "indirect

Turhan ~ifrfiba~t / Fuz~ Sets and Systems 85 (1997) 325 354 projection (indirect shadow)" from a "one-universe"

into another "one-universe". Lemma 7.3. The following property which is similar to the De-Morgan law connects the definitions of Projection and Shadow operations: Proje{-nA} = ~Shadv{A}

(7.5)

Shade{~A} = ~Proje{A}

(7.6)

349

B without any change by taking projection or shadow of these propositions into the universes U and V? The answer is given by the following lemmas: Lemma 7.5. I f the maximum truth value of the proposition B is greater than 0, say fl, the projection of A + B into U is different from the proposition A such that the truth values smaller than fl are increased to ft. The counterpart is also applicable, i.e. in Boolean expression:

Proof. Obvious by definition. Proje{A + B}u×v Lemma 7.4. For any two f u z z y propositions X, Y in the universe U1 x U2 x ... x Uk,

= Projv{A}v×v + Projv{B}vxv = A + (fl)v

P r o j v , { X + Y } e , × e . . . . . . e~

Projv{A + B}e×v

= P r o j v , { X } e , ×e . . . . . . v~ + P r o j e , { Y } e , × v : x .... e~

= Projv{A}e×v + Projv{B}exV

(7.7a)

S h a d e , { X . Y } u , × e ..... ×u~

= (X)v + B.

(7.8b)

Proof is obvious by Lemma 7.4.

= Shadv,{X}e~ xv~x .... u~ "Shadu,{Y}e~xe~× .... u~

(7.8a)

(7.7b)

i.e. in words, projection operation is transitive for " + " (OR) operation, and shadow operation is transitive for " - " (AND) operation. Proof. Obvious since both "taking projection" and "OR operation" are performed by taking the maximum value. Similarly, both "taking shadow" and "AND operation" are performed by taking the minimum value. 7.2. Projection and shadow o f the relations "A" B " and "A 4- B " into one-universes U and V

Consider the fuzzy proposition A defined in the universe U, and fuzzy proposition B defined in the universe V. The proposition A, B is obtained by AND operation and the proposition A + B is obtained by OR operation in the universe U x V as given above. What is the projection/shadow ofA. B and A + B in the original universes U and V? When is it possible to get propositions A and

Note that if fl = 0 (the proposition B is a null proposition) then the result is trivial. Lemma 7.6. / f the minimum truth value of proposition B is greater than 0, say 6, then Shade {A" B} e x v is different from proposition A such that the truth values above 6 are decreased to the value 6. The counterpart is also applicable, i.e. in Boolean expressions: Shade{A'B}e xv = S h a d v { A } e × v " S h a d e { B } v × v = A "(6)v (7.9a)

Shady {A" B }v ~v = Shadv{A}e×v'Shadv{B}e×v

= (e)v" B (7.9b)

Note that if6 = 1 (B is the identity f u z z y proposition) then the shadow is the original proposition A in U.

Proof is obvious by Lemma 7.4. Eqs. (7.8) and (7.9) above states that "projection of OR operated propositions" and "shadow of A N D operated propositions" have distributive property.

Turhan ~iftfiba~l / Fuzzy Sets and Systems 85 (1997) 325 354

350

What can be said about the "projection of A N D operated propositions" and "shadow of OR operated propositions"? In general, distributive law is not applicable. However, if the propositions in U x V are in fact the extensions of one-universe propositions, distributive law is applicable as will be seen in the following lemmas:

we obtain, Projw{A'B'C}v×v×w

= Projw{Projv×w{A'(B'C)}v×v× w} = Projw {(~)v×w}' Proj w {Proj v× w {B" C} v× v× w } = (COw'Projw{{B' C}v×w}.

L e m m a 7.7. Consider the fuzzy propositions A, B in the universes U, V respectively. I f the maximum truth value of the proposition B is less than "1" say [1, then the projection P r o j v { A ' B } v × v is different from proposition A such that the truth values above [1 are decreased to the value [1. The counterpart is also applicable. That is,

Pr~v{A'B}v×v = Pr~v{A}v×v'Pr~v{B}v×v = A -([1).

(7.10a)

Projv{A" B}v×v = Projv {A}v× v "Projv{B}v× v = (CO)v"B

(7.10b)

Note that if[1 = 1 (maximum truth value of B is 1) then projection o f " A " B'" into U gives the original proposition A in U.

Hence

Projw{A'B'C}v×v×w=(~)w'([1)w'C

is obtained. Therefore, both ~ and [1 are upper limits on the values on the fuzzy proposition C. L e m m a 7.8. Consider the fuzzy propositions A, B in the universes U, V, respectively. I f the minimum truth value of the proposition B is greater than O, say 6, the shadow of A + B on U is different from the proposition A such that the truth values below 3 are increased to 3. The counterpart is also applicable i.e. in the Boolean expressions: Shadv{A + B}v×v

= Shadv{A}v×v + Shadv{B}v×v = A + (3)v

Proof. Let the m a x i m u m element (say kth element) of B is [1. In the extended universe representation of B into the universe U x V (copying the row vector B into all rows) the kth column of the extended matrix is all [1 (this column is a constant fuzzy proposition ([1)v), while the other columns are smaller. Extension of fuzzy proposition A into U x V makes all columns copies of the column vector A. A N D operation of these two extension matrices makes the kth column A'([1)v, while the other columns are smaller. Projection into V gives (7.10a). Eq. (7.10b) can be proved similarly. The above lemma has been given for the twouniverse case, but is also applicable for the multiuniverse case. This can be shown inductively by considering the three-universe U x V x W as U x (V x W) and A N D operation as A "(B' C) and by using the fact

Projw(R)v×v×w = Projw{Projv×w{R}u×v×w}

(7.11)

(7.12a)

Shadv{A + B}v×v

= Shadv{A}v×v + Shadv{B}v×v = (e)v + B

(7.12b)

Note that if3 = 0 (minimum with value of B is zero), then shadow of A + B into U gives the proposition AinU. Proof. Can be made simply as in L e m m a 7.7. The above lemma is also applicable for the continuous case as can be intuitively seen, and is also applicable for the multi-universe case which can be shown by a similar argument as in the previous lemma. Hence for three fuzzy propositions A, B, C in different universes U, V, W: Shadw{A + B + C}v×v×w = (e)w + (3)w + C. (7.13)

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Turhan ~iftfiba~l / Fuzzy Sets and Systems 85 (1997) 325-354

Therefore both e and 3 are lower limits on the truth values of the fuzzy proposition C. Note that the Lemmas 7.7 and 7.8 are only applicable if U, V, W are distinct one-universes. If there is "dependence between universes" this is not applicable. What is meant by "dependence" can be presented intuitively, by the example that the universes U x V and U x W are dependent, and the above lemma is not applicable if the original propositions are in these universes. This concept needs further development.

8. Transforming the equations 8.1. One-input one-output case

Consider the resultant equation (6.34) (singleinput single-output case), with four intervals in which the propositions Ai are in the universe U (input) and, Bi are in the universe V (output):

B 0 = (~Xl)v. B 1 +

((Z2) V"

B 2 + "-" + (~Xn)v'Bn, (8.2)

where (Cq)v are the constant propositions with ~1 = Max(Ao" A1)v, ~2 =

~, = Max(Ao" A,)v.

Max(Ao' A2)u, ... ,

(8.3) Hence the decision process can be summarized as follows which is already used in the literature depending on practical developments: (I) ai terms will be calculated after A N D operation in the universe U. (II) The propositions B~ will be limited from above by ~i. (III) Bo will be calculated by "OR operation" of the limited B/s in the universe V. 8.2. Two-inputs, one-output case

R = AI"B1 + A z ' B 2 + A3"B3 + "" + An'Bn

Ro = A0' R Ro = A o ' A 1 "Bx + A o ' A 2 " B 2 + ... + A o ' A , ' B n

Bo = ProjvRo = P r o j v { A o ' A l " B 1 + A o ' A 2 " B 2 + ... + A o ' A , ' B } .

+ ... + P r o j v { A o ' A , } ' P r o j v { B , } ,

Consider Eq. (6.35) for the two-input, one-output case, where Ai, Bi are in the input universes U and V, and Ci are in the output universe W: R = A1 "B1 "C1 q- AI "B2"C2 + A1 "B3"C2

(8.1) + A2" B1 • C2 + A2" B2" C3 + A2" B3" C3

This equation can be transformed directly into the output universe by using the results of Section 7 as follows: (i) Since projection operation is transitive with respect t o " + " (OR) operation (see Lemma 7.4 and 7.5):

Ro = Ao" R

Bo = Projv{Ao" A1 "B1} + Projv{Ao" A2" B2}

Ro = A o ' B o ' A I ' B I " C 1 + A o ' B o ' A 1 "B2"C2

+ ... + P r o j v { A o ' A 4 " B 4 } .

(ii) Since projection operation is transitive with respect to " . " (AND) operation if the operands are originally one-universe propositions (see Lemma 7.7): Bo = Projv{Ao'A1}'Projv{B1} + Projv{Ao "A2}'Projv{B2}

+ A3" B1 " C2 + A3" B2" C3 + A3" B3" C4.

(8.4)

+ Ao" Bo" A1 "B3"

C2 ~- Ao" Bo" A2" BI" C2

+ Ao" Bo" A2" B2" C3 + Ao" Bo" A2" B3" C3 + Ao'Bo'A3"BI'C2 + Ao'Bo'A3"B2"C3

+ Ao" Bo" A3" B3" C4 Co = Projw Ro.

(8.5) (8.6)

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Thisequation can b e t r a n s ~ r m e d directlyinto the outputuniverse by usingtheresultsofSection 7 as: Co = Pr~wRo

= Pr~w{Ao'Bo'AI'BI"C1} + Pr~w{Ao'Bo'AI'B2"C2} + Pr~w{Ao'Bo'AI"B3"C2} + Pr~w{Ao Bo'A2.B1.C2}

Hence the decision process can be summarized as follows as also known in the literature: (I) cq terms will be calculated by AND operation in the universe U. (II) /~ terms will be calculated by AND operation in the universe V. (III) The propositions C~ will be limited from above by c~i and/?i. (IV) Co will be calculated by "OR operation" of the limited C~'s in the universe W.

+ Pr~w{Ao Bo'A2.B2"C3} Example 8.1. Consider one-input, one-output decision relation given by

+ Pr~w{Ao Bo.A2.B3.C3} + Pr~w{Ao Bo'A3.B1.C2}

R = A I ' B 1 + A a ' B 2 + A 3 " B 3 + A,,'B4,

+ Pr~w{Ao Bo'A3"B2"C3} + Pr~w{Ao Bo'A3"B3"C4}

(8.7)

Co = Projw{Ao "A1}" Projw{Bo "B1}" Projw{C, } + Projw{Ao "A, }" Projw {Bo "B2} PrOw{C2} + Proj w{Ao "A1 }" Projw {Bo "B3} Pr~w{C2} + Projw{Ao'A2}'Projw{Bo'B1} Pr@w{C2} + Proj w{Ao "A2}'Projw{Bo "B2} PrOw{C3} + Projw{Ao'A2}'Projw{Bo' B3} Prow{C3} + Projw{Ao'A3}'Projw{Bo'B1} PrOw{C2} + Proj w{Ao "A3}'Projw{Bo "B2} PrOw{C3} + Projw{Ao "A3}'Projw{Bo 'B3} PrOw{C4}

(8.8) Co = (~l)W'(fll)w'C1 -1-(~l)W'(/~2)w'C2 "~ (@I)W'(/~3)W" C 2 -t- (~Z2)w'(fll) W" C 2 -~ (o~2)w'(/~2)w" C3 -~ (o~2)w'(/~3)w" C3 -1- (o~3)w" ( f l l ) w ' C 2 -]- (~3)w'(fl2)w'C3 -1- (~3)w" (/J3)w" C4,

(8.9)

where :q = Max{Ao'A1},

fll = M a x { B o ' B 1 } ,

~2 = Max{Ao'A2},

f12 = M a x { B o ' B 2 } ,

~3 = Max{Ao-A3},

f13 = M a x { B o ' B 3 } ,

~4 = Max{Ao'A4},

[t4 = Max{B0'B4}. (8.10)

where Ai and Bi are given in Figs. 5 and 6, For the given crisp input proposition Ao = fi0 (having truth value 1 at Uo, 0 otherwise), ~ constants can be found by Eq. (8.3), which are shown in Fig. 5. By using these ei values, the proposition in the output universe can be found as expressed in Eq. (8.2). (Each c~ limits B; and "OR" combination of "limited Bis" gives Bo). The resultant proposition is shown in bold lines in Fig. 6. If the measurement input Uo were at the value u2 (having maximum truth value for truth function A2, zero truth value for other truth functions), then the technique shows that ~2 = 1, and the proposition B2 would not be limited (hence completely selected), while the other c~'s are zero, other B~'s would be suppressed to zero value, leading Bo to be exactly B2. |n case the measurement input Uo is between two steps as in the above example, then the output truth function contains more components from the closer proposition. If the defining propositions are triangular as in the above example, "Defuzzification" of the output truth function will give the result which will make a kind of interpolation in the output universe. Therefore, fuzzy decision process is a "Rule based system with interpolation for intermediate inputs" for such cases. However, it is stronger and more general since it can be applied to the multi-input case, and also applicable to the complicated truth functions.

Turhan ~iftfiba~t / Fuzz), Sets and Systems 85 (1997) 325 354

with the relation R given as:

~Ai 01

353

A1

ul

A2 AO

A3

u2 u0

A4

u3

u4

Fig. 5. Truth (membership) functions of the basis propositions and the crisp measurement data in the input universe U for a typical four-interval decision process.

R

B1

B2

B3

AI

C1

C2

C2

A2 A3

C2 C2

C2 C3

C3 C3

If the measurement inputs are fuzzy given as: A o = ( 0 0 0.4 1 1 0.4 0) B o = ( 0 0.4 1 1 0.4 0)

in U

in V

]~Bi B1

B3

B2

B4

then

Ro = A o ' B o ' A I " B I " C 1 + Ao'Bo'A1 "B2"C2 + Ao'Bo'A1 "B3"C2 + Ao'Bo'A2"BI"C2 + Ao" Bo" A2" B2" C2 -~- Ao" Bo" A2" B3" C3 vl

v2

v3

v4

Fig. 6. Truth (membership) functions on the output universe V for a four-interval decision process and the calculated output Bo plotted in bold lines.

+ Ao" Bo" As" BI" C2 ~- Ao" B 0 • A3" B2" C3 + Ao" Bo" As" B 3 "C3 From Eq. (8.7), we have: Ao A1 = ( 0 0 0 0 0.4 0.4 0)

Example 8.2. Consider the two-input, one-output decision system given below:

Ao A2 = (0 0 0.4 1 0.6 0.4 0)

Input universes:

U = {ul u2 u3 u4 us u6 uT}

Ao A3 = (0 0 0.4 0 0 0 0)

V = {V1 U2 V3 U4 U5 U6}

Bo B1 = (0 0 0 0 O.4 0) Bo B 2 = (0 0 0.5 1 0.4 0)

Output universe: W = {/4)1 W2 W3 W4 W5 W6 W7}

AI=(0

0 0 0 0.4 0.6 1)

in U

A2 = (0 0.4 0.6 1 0.6 0.4 0) A 3 = ( 1 0.6 0.4 0 0 0 0) BI=(0

0 0 0 0.5 1)

B 2 = ( 0 0 0.5 1 0.5 0) B 3 = ( 1 1 0.5 0 0 0)

in U

in U

in V

C 3 = ( 1 0.8 0.2 0 0 0 0)

Co = (0.4)w'(0.4)w" C1 + (0.4)w" (1)w" C2

+ (0.4)w'(O.5)w'C2 + (1)w'(0.4)w" C2

+ (0.4)w" (0.4)w" Cz + (0.4)w'(1)w' C3

in V

0.2 0.8 1 0.8 0.2 0)

Therefore

-t- ( l ) w ' ( 1 ) w ' C 2 + (1)w'(0.5)w" C 3

in V

+ (0.4)w" (0.5)w" C3

C 1 = ( 0 0 0 0 0.2 0.8 1) in W C 2 = (0

B 0 B 3 = (0 O.4 0.5 0 0 0)

in W

in W

C o = ( 0 . 5 0.5 0.8 1 0.8 0.4 0.4) is the output truth function in universe W. Defuzzification will give w4 as the most possible output.

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Turhan ~iftfiba~l / Fuzzy Sets and Systems 85 (1997) 325 354

9. Conclusions

Constructing the fuzzy decision theory upon fuzzy propositional logic instead of fuzzy set theory, and by looking at the problem of fuzzy decision structure from a higher universe, the confusions that were present in the developing phases of fuzzy control system mathematics are removed. The facts that are already known have simpler interpretations with this structure. All known techniques of Boolean algebra are now ready for use in fuzzy system theory. Some of the further contributions that are expected on this line of development can be listed as follows: • The terminology offered in this paper should be generalized for different definitions of OR and AND operations. • Properties of basis should be generalized for the multi-universe case, and counterparts of the vector space notions should be developed. • Selection of defining vectors in the input universe has been investigated here; but the selection in the output universe has not been developed. The effect of choosing the defining propositions in the output universe as nonexclusive or noncomplementary should be investigated. • Previous applications can be investigated using this approach, hence the reasons for unexpected results if any, can be observed and be removed.

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