fuzzy operators for possibility interval sets

Fuzzy Sets and Systems 22 (1987) 215-227 215 North-Holland

F U Z Z Y O P E R A T O R S FOR POSSIBILITY I N T E R V A L SETS

James BUCKLEY Mathematics Department, University of Alabama at Birmingham, Birmingham, AL 35294, USA

William SILER Carraway Medical Center, Birmingham, AL 35234, USA

Received May 1985 Revised January 1986

The problem of estimating a certainty level for the AND and OR of two assertions when certainty levels are known for each assertion individually is discussed with reference to the construction of a fuzzy expert system shell. The importance of prior associations between the assertions is pointed out. To deal with this problem a 'possibility interval set' is proposed in which the grade of membership in a conventional fuzzy set is replaced by a two-element vector, the minimum and maximum grades of membership possible given existing knowledge. Operators are proposed for possibility interval sets corresponding to the conventional AND, OR and NOT which have desirable mathematical properties, and which can be used in a fuzzy expert system shell to avoid dit~culties posed by the Zadeh and Lukasiewicz operators. An association measure is proposed which lends itself to informed guesses regarding prior associations and which can be used to reduce the gap between lower and upper certainty levels.

Keywords: Expert systems, Operators, Fuzzy sets (kind of).

1. Introduction

The implementation of an expert system shell based on fuzzy logic [6] virtually requires that certainty levels be assigned to complex logical statments, each element of which is a (fuzzy) logical assertion. This leads to interpreting a certainty level in the Sugeno sense [12], i.e. as a measure of belief that an assertion is true in the context of the current contents of the system's working memory. Interpreting certainty levels in this way, one can bring to bear the armamentarium of fuzzy set theory on the problem of constructing a fuzzy expert system shell.

Basing themselves on the fundamental work of Zadeh on possibility theory, Martin-Clouaire and Prade [5] give an excellent discussion of fuzzy logic in which the certainty level is two-valued: the lower limit is the necessity of the datum, or the extent to which the data support the truth of the item; the upper limit is the possibility of a datum, or the extent to which the data do not refute the datum. Ruspini [7, 1] uses slightly different but isomorphic terms, plausibility instead of possibility, and support instead of necessity. Ruspini also is concerned with the relationship between fuzzy logical operators (AND/OR~NOT) and prior associations between the logical operands, and the use of interval-valued logic in an expert

0165-0114/87/$3.50 (~) 1987, Elsevier Science Publishers B.V. (North-Holland)

216 J. Buckley, W. Siler

system shell. In this paper we will concern ourselves with the problem of economical logical operators in a system for automated reasoning, and the inclusion of prior knowledge of associations among variables in these operators, as an aid in the development of fuzzy reasoning methods analogous to Bayesian methods.

Let us set out the terminology which we will follow in this paper. A fuzzy production rule consits of a number of assertions logically connected

by ANDS or ORS, possibly with NOTS as unary operators. When the rule is evaluated, a truth value or certainty level is assigned to each assertion using the contents of working memory at the time of evaluation. The certainty levels of the individual assertions are combined to yield a posterior certainty level for the rule indicating its degree of applicability given the current state of working memory. Each assertion may be viewed as a fuzzy set whose members have grade-of- membership (truth value) determined by the contents of working memory, or system state.

More specifically, suppose we have two fuzzy sets A and B with members u. Suppose further that a member u corresponds to an indentifiable real-world state, given (for a fuzzy expert system) by the contents of the expert system's working memory. Fuzzy set A corresponds to an assertion, one component of a production rule; our certainty that this assertion is true in a given real-world context u is given by gm(u, A), the grade of membership of u in A; and fuzzy set B corresponds to a different assertion, another component of the same production rule, with certainty in the truth of assertion B in a real-world context u given by gm(u, B). Then the grade of membership of u in (A A B) is the certainty that the assertions which define A and B are simultaneously true, and the grade of membership of u in (A v B) is similarly the certainty that at least one of the assertions which define A and B is true.

However, a certainty level can also be interpreted, at least qualitatively, as the probability that a given assertion is true. While the circumstances under which a probabilistic interpretation of a certainty level is strictly true are extremely restricted, we would like to achieve something approaching an order-preserving mapping between certainty levels and probabilities.

In evaluating all but the simplest expert system production rules, we must evaluate the certainty that assertion A and assertion B are simultaneously true (A A B), or that at least one of them is true (A v B). In dealing with the problem of certainty in (A A B) and in (A v B), we must consider the effect of any (in general, unknown) dependencies among the truths of A and B. This is a problem encountered in a fuzzy expert system. In evaluating the production rule for the current state of working memory, we must define how we 'AND' the two assertions if they are logically connected by an ANt), or how we 'OR' the two assertions if they are logically connected by an OR. Of course, the unconditional probabilities of (A A B) and (A v B) depend on any associations which may exist between the truths of A and B in the real world.

As an example of a rule antecedent, suppose we have

if A A (B v C) then D.

Fuzzy operators for possibility interval sets 217

The truth values of A, B and C individually must be combined (possibly using prior knowledge of any associations between their truths that might exist in the real world) to yield the certainty with which we can assert D.

As Ruspini [7] has pointed out, if all prior associations are known, Bayes' theorem can give unequivocally correct answers for the grade of membership of u in both (A A B) and in (B v C); unfortunately, the prior associations are seldom accurately known. The controversies which have attended the use of Bayes' theorem in the absence of complete prior knowledge for the past century and a half are legendry: the controversies continue to the present day in the field of expert system shells which use certainty measures [4]. Applebaum and Ruspini [1] in their fuzzy inference engine ARIES permit the user to specify the way in which he prefers to deal with prior knowledge; the problem itself will not go away.

Ruspini [7] also notes that when the (unknown) prior associations range over all possible values, the true value of the grade of membership of u in (A A B) has the Lukasiewicz [3] fuzzy logical AND, L(A A B) = max(gm(u, A) + gm(u, B) - 1, 0), as a lower limit and the Zadeh [13] fuzzy logical AND, Z(A A B ) = min(gm(u, A), gm(u, B)), as an upper limit; and the true value of the grade of membership of u in (A v B) has the Zadeh fuzzy logical OR, Z(A v B)= max(gm(u, A), gm(u, B)), as a lower limit and Lukasiewicz fuzzy logical OR, L(A v B)--min(gm(u, A ) + g m ( u , B), 1) as an upper limit. This is most un- fortunate. In an expert system, we would like to be consistent about the possibility of a complex statement being true; it is quite desirable to retain consistently the minimum or maximum degree of certainty, or possibly something in between these limits. If we select the Zadeh AND/OR/NOT operators, we choose the maximum grade-of-membership value for the AND of tWO assertions, but choose the minimum for the OR; if we select the Lukasiewicz operators, we choose the maximum value for the OR, but choose the minimum for the AND. The selection of Zadeh's AND and Lukasiewicz's OR gives a pair of operators which do not satisfy De Morgan's theorems. Since our fuzzy expert system shell FLOPS [10, 11] uses primarily the AND fuzzy logical connective, we have chosen the Zadeh operators as the best suited to our purpose, but we are not totally happy about this choice.

To illustrate the problem, consider that we have two fuzzy sets A and B, corresponding to two assertions about the real world, in which the truth of both assertions is to be evaluated using the current state of working memory, and that we wish to define a method of evaluation of A A B and of A v B. Table 1 gives some possible numerical values of the probabilities involved for some hypothesized associations between the truths of A and of B.

Table 1 shows that for all combinations of unconditional probabilities of A and B the Zadeh combination rule gives a value for the truth of Z(A A B) which is the maximum possible value, and the Lukasiewicz rule yields for L(A A B) the minimum possible value. For cases 1, 2 and 3, p(A n B) has a maximum value of 0.5, yielded by the Zadeh operator, and a minimum value of 0, yielded by the Lukasiewicz operator. The maximum value for p(A ^ B) is found when the association between A and B as measured by the Pearson correlation coefficient r(A, B) has its strongest possible positive value, and the minimum value of


Table 1. In this truth table, 0 = FALSE, 1 = TRUE. T h e first f o u r r o w s give the hypothesized t r u e

probabilities that the truth table values f o r A a n d B will be observed. The second four rows give the ' t r u e ' probabilities for A t r u e , B t r u e , A A B t r u e a n d A v B t r u e . (Note that p(A = 1, B = O) + p(A = 1, B = 1) must equal p ( A ) ; p(A = 0, B = 1) + p ( A = 1, B = 1) must equal p(B).) The t h i r d f o u r rows give predicted values f o r A A B a n d A v B under the truth-functional logics of Zadeh and L u k a s i e w i c z . T h e last two rows give two measures of association between A and B : r is the Pearson correlation coefficient; R is a measure of association discussed in the text based on the maximal

positive and negative associations possible given the unconditional probabilities p(A) and p(B)

p(a) = p ( 8 ) = 0.50 p(A) = p ( B ) = 0.05 p(A) = p ( B ) = 0.95 p(A) = 0.1; p(B) = 0.9

Case Case Case Case Case Case Case Case Case Case Case Case A B 1 2 3 4 5 6 7 8 9 10 11 12

Hypothesized true probabilities for truth table values of A and B p for 0 0 0.5 0.25 0 0.95 0.925 0.90 0.05 0.25 0 0.10 0.05 0 p for 0 1 0 0.25 0.50 0 0.0475 0.05 0 0.0475 0.05 0.80 0.81 0.90 p for 1 0 0 0.25 0.50 0 0.0475 0.05 0 0.0475 0.05 0 0.01 0.10 p for 1 1 0.5 0.25 0 0.05 0.0025 0 0.95 0.9025 0.90 0.10 0.09 0

Hypothesized true values for unconditinal probabilites of A, B, A A B, A v B p(A) 0.5 0.5 0.5 0.05 0.05 0.05 0.95 0.95 0.95 0.10 0.10 0.10 p(B) 0.5 0.5 0.5 0.05 0.05 0.05 0.95 0.95 0.95 0.90 0,90 0.90 p(A A B) 0.5 0.25 0 0.05 0.0025 0.0 0.95 0.9025 0.90 0.10 0.09 0 p(A v B) 0.5 0.75 1.0 0.05 0.0975 0.10 0.95 0.9975 1.0 0.90 0,91 1.0

Truth-functional values for A A B and A v B according to Zadeh and Lukasiewicz Z(A A B) 0.5 0.5 0.5 0.05 L(A A B) 0 0 0 0 Z(A v B) 0.5 0.5 0.5 0.05 L(A v B) 1.0 1.0 1.0 0.10

Measures of association between A and B r(A, B) 1.0 0 - 1 . 0 1.0 R(A,B) 1.0 0 - 1 . 0 1.0

0.05 0.05 0.95 0.95 0.95 0.10 0,10 0.10 0 0 0.90 0.90 0.90 0 0 0 0.05 0.50 0.95 0.95 0.95 0.90 0.90 0.90 0.10 0.10 1.0 1.0 1.0 1.0 1.0 1.0

0 -0 .05 1.0 0 -0 .05 0.11 0 - 1 . 0 0 - 1 . 0 1.0 0 - 1 . 0 1.0 0 - 1 . 0

p(A A B) is found when r(A, B) is a negative as possible. Anything between the minimum and the maximum value is possible in the real world.

For the OR operation, the situation is reversed, and the Zadeh OR yields the minimum possible value, and the Lukasiewicz OR yields the maximum possible value. In cases 1, 2 and 3, the maximum value for p(A v B) is 1.0, yielded by L(A v B), and the minimum value is 0.5, yielded by Z(A v B). The maximum value is found when the association between A and B is the most negative possible, - 1 , and the minimum value is found when the association is the most positive possible, +1. Again, anything between these limits is possible in reality.

Note that in Table 1, although the degree of association between A and B covers the maximum range possible for the unconditional probabilites p(A) and p(B) given, the Pearson correlation coefficient r(A, B) only covers its complete range from - 1 to +1 for one of the four groups of cases: the correlation between A and B which can be observed is limited in range by their unconditional probabilities. (The association measure R(A, B) will be discussed later.)

We have then the problem of predicting the value of A n B and A v B when information on true prior associations between A and B may be wholly or partially lacking, and to do this in a fashion which is suited to an expert system


shell. Clearly, neither the Zadeh nor the Lukasiewicz operators are totally satisfactory; we are confronted with Hobson's choice. To overcome this problem and to facilitate the use of simple crisp interval possibility theory [7] in expert systems applications such as ours, we suggest in this paper the use of a crisp possibility interval set with a certainty measure which is contained within a lower and an upper limit. A possibility interval set is a generalization of a fuzzy set in which the grade of membership of an object is redefined to be a two-vector in which the first element is the minimum and the second the maximum of all possible values. We propose possibility interval set operators for AND, OR and NOT which have desirable mathematical properties and which retain through a sequence of operations the minimum and maximum possible values in the probabilistic sense. We further propose a measure of association which is adapted to prior 'guessing'.

2. Possibility interval set

A possibility interval set is defined to be a triple

A := {gmmin, gmmax, X} for all x in X

in which A is a possibility interval set, gmmin is the minimum possible grade of membership of object x in A, and gmmax is the maximum possible grade of membership of x in A.

We further define a universe and a zero set

X : = { 1 , 1 , x} for a l l x i n X ,

0 :={0 ,0 , x} f o r a l l x i n X .

where in our application to an expert system X is the set of all possible states of pertinent working memory.

3. Logical operators

Consider two fuzzy sets F~ and F2 in which an object u has grades of membership gml and gm2 respectively. Let ZAND(gml, gm2) and ZOR(gml, gm2) be operators which yield the grade of membership of u in the intersection and union of F1 and F2 defined according to Zadeh's rules. Then

ZAND(gml, gm2) := min(gmm, gm2),

zoa(gml, gm2) :---- max(gml, gm2).

Let LAND(gml, gm2) and LOR(gml, gm2) be similarly defined operators for the intersection and union of F~ and F2 defined according to Lukasiewicz' rules. Then

LAND(gml, gm2): = max(gml + gm2 - 1, 0),

LOR(gml, gm2) := min(gml + gm2, 1).


Now, given two possibility interval sets

A = {minA, maxA, x} for all x in X,

B = {mins, maxB, x} for all x in X,

with minA and max3, minB and maxB the minimum and maximum grades of membership of x in the corresponding set, we define the intersection of A and B by the possibility interval operator PAND:

A PAND B : = {LAND(mina, minB), ZAND(maxA, maxn), x} for all x in X;

the union of A and B by the possibility interval operator POR:

A POR B := {ZOR(minA, mine), LOR(maxA, maXe), x} for all x in X;

and the complement of A by the unary possibility interval operator PNOT:

PNOTA := {1 -- maxA, 1 -- minA, x} for all x in X.

(In the above, paralleling fuzzy sets, objects not in the set and objects with both grades of membership zero are equivalent.)

The possibility interval operators PAND, POR and PNOT then operate on the vector grade of membership possessed by all elements of a possibility interval set(s) to yield the vector grade of membership in the resulting possibility interval set. They have the following properties:

(a) Commutativity holds:

A POR B -- B POR A, A PAND B -- B PANDA.

(b) Associativity holds:

A PAND (B PAND C) = (A PAND B) PAND C,

A POR (B POR C) = ( a aoa B) POR C.

(c) The binary operators are not mutually distributive: in general,

A PAND ( n POR C) :#: (A PAND B) POR (A PAND C),

A POR (A PAND C) 5/: (A POR B) PAND (A POR C).

(d) They are not idempotent: in general,

A PANDA #: A, A POR A 4= A.

(e) Identity elements exist: for PAND, {1, ], X), and POR {0, 0, X}. (f) For all x in X,

A PAND {0, 0, X} = {0, 0, X}, A PAND {1, i, x} = A,

A POR {0, 0, X) = A, A POR {1, 1, x} = {1, 1, X}.

(g) Absorption does not hold: in general,

A PAND (A POR B) ::# A, A POR (A PAND B) :# A.

(h) De Morgan's laws hold. (i) The PNOT satisfies involution: PNOT(PNoTA)= A.


(j) The excluded middle law does not hold: in general, for all x in X,

A PAND (PNoTA) ~ {0, 0, X},

A POR (PNOTA) :/: {1, 1, x}.

(k) Addivity does not hold: in general,

(A PAND B) q- (A POR B) ¢ (A d- B).

where + indicated vector addition of the minimum and maximum grades-of- membership.

(l) If the grades of membership of an object in a possibility interval set be considered as minimum and maximum probabilities that the object belongs to the set, then the minimum and maximum grades of membership in the union or intersection of two sets are the minimum and maximum probabilities possible over the entire ranges of any prior conditional probabilities that might exist consistent with the given grades of membership in the two sets individually.

The proofs of the above are straightforward but tedious, and are omitted because of their length.

As a simple example, apply these operators to the simple production rule given above,

i fA A (B v C) then D

where A, B, C and D are possibility interval sets with grades of membership evaluated for u, the current state of working memory. Suppose also that A, B and C are possibility interval set versions of fuzzy sets, with equal minimum and maximum grades of membership. Then

A = (a, a, u), B = (b, b, u), C = (c, c, u),

resulting in

D = (minD, maxD, u) = A PANO (B POR C)

which produces

mind = LAND(a, ZoR(b, C)), maxD = ZAND(a, LoR(b, C))

where a, b and c are both minimum and maximum grades of membership of u in A, B and C; and mind and maxo are the minimum and maximum grades of membership of u in the consequent possibility interval set D.

The proposed operators yield only a crisp certainty interval, and that is the widest possible in the absence of prior knowledge. If prior knowledge of the associations between assertions exists, the possibility exists of using that knowledge to narrow the certainty range. This problem has been known for well over a century in connection with Bayes' theorem; of particular interest is the work of Dempster [2] and Shafer [9]. One possible implementation of prior knowledge for that purpose involves the use of simple measures of association. We propose an approach based on an association measure which facilitates 'guessing' the unknown actual values, and which may serve to bracket the values for A ^ B and A v B to more reasonable limits than the minima and maxima possible.


4. Proposed measure of association

The association measure R proposed is based upon the lower and upper limits possible provided by the Zadeh and Lukasiewicz combination rules when the unconditional truth values for A and B are given. This measure R has as value of +1 when A and B are positively associated as strongly as possible; a value of - 1 when A and B are negatively associated as strongly as possible; and a value of zero if A and B are independent. This corresponds to the actual situation in a fuzzy expert system; we will be able to evaluate the unconditional certainty levels in assertion A and assertion B with no trouble; these unconditional certainty levels place restrictions on the degree of association possible between A and B.

Consider the association measure R(A, B) defined to be 1 when A and B are positively associated as strongly as possible; - 1 when negatively associated as strongly as possible; and 0 when A and B are independent of each other. We implicitly define R(A, B) by mapping R and the truth values of A = a and of B = b onto the truth values RAND for A A B and ROR for A v B by piecewise linear functions. Let

Z A = ZAND(a, b) = min(a, b),

P A - - a * b

LA = LAND(a, b) = max(a + b - 1, 0)

ZO = ZOR(a, b) = max(a, b)

PO = a + b - PA

LO = LOR(a, b) = min(a + b, 1)

Then define:

(Zadeh AND), (Probabilistic AND),

(Lukasiewicz AND),

(Zadeh OR),

(Probabilistic OR),

(Lukasiewicz OR).

If R t> 0, then

RAND(A, B, R) = PA + R * (ZA - PA)

RoR(A, B, R) = PO + g * (ZO - PO)

If R < 0, then

RAND(A, B, R) = PA + R * (PA - LA)

RoR(A, B, R) = PO ÷ R * (PO - LO)

(Restricted AND, R I> 0),

(Restricted OR, R I> 0).

(Restricted AND, R < 0),

(Restricted OR, R < 0).

Here a and b are the unconditional truth values of A and B; R is the value of the proposed association measure, with maximum and minimum values of +1 and - 1 , and a value of 0 if A and B are independent; and P.AND(A, B, R) and RoR(A, B, R) are the values for the truth of A A B and A v B as restricted by the association measure. ZAND and LAND are the truth-functional predictions for the truth of (A A B) given by the Zadeh and Lukasiewicz operators; ZOR and LOR are the truth-functional predictions for the truth of (A v B). The functions above give a piece-wise linear map between R and the resulting AND and OR values. (Alternatively, a curvilinear relationship could be used.) By specifying R to be +1, the Zadeh operators will result; by choosing R to be zero, the probabilistic


operators will result; and by choosing R to be -1 , the Lukasiewicz operators will result.

The association measure R has properties similar to that of a correlation coefficient, having a lower limit of -1 and an upper limit of +1. It is different from the correlation coefficient in that it is a measure not of the absolute degree of association between A and B, but of the degree of association which exists within the limits possible, given the unconditional probabilities of A and B. R will always have lower and upper limits of -1 and + 1 no matter what restrictions on the conditional probabilities of A and B are placed by their unconditional probabilities. Note that an R value of zero for A and B means that A and B are statistically independent.

Let us now define a two-valued association measure RX = (R~, R2) where R1 and R2 are the lower and upper limits possible for R. These limits can easily be used to restrict the ranges of the minimum and maximum grades of membership required by a possibility interval set. Even the simple restriction "any possible association between A and B is positive", with R1 = 0 and R1 = 1, will serve to reduce the gap between the minimum and maximum possibilities which can be realized. If both upper and lower limits for R are chosen to be equal to + 1, the Zadeh operators will result; if the both limits are chosen to be zero, the probabilistic operators will result; and if both limits are chosen to be -1 , the Lukasiewicz operators result.

5. Restricted possibility interval set operators

If prior knowledge exists of an association between the truth values of the members of possibility interval sets A and B, and if this knowledge can be specified as a lower limit R1 and an upper limit R2 which comprise the vector association measure RX defined above, then restricted possibility interval set operators can be easily defined to make use of this knowledge.

The definitions of RAND and aoa above lead to operators RXAND(A, B, RX) and RXoR(A, B, RX) similar to the PAND and eOR operators which will produce possibility interval set grades-of-membership restricted by the specified lower and upper limits on RX(A, B). Let the prior limits on R be R1 and R2, where R1 ~<R2.

Now defined the operators RXAND and RXOR by

A RXAND B : = {RAND(A, B, R1), RAND(A, B, RE)} = RXAND(A, B, RX),

A axon B : = {RoR(A, B, R2), RoR(A, B, RI} = RXoR(A, B, RX).

These operators have the same properties as PAND and eoa, but can yield certainty intervals which reflect prior knowledge in a painless way and which may be much reduced from the intervals yielded by the PAND and POR operators. In cases where the association is known to be perfectly positive or negative, with RX values of (+1, +1) or ( -1 , -1) , we obtain either the Zadeh AND and OR or the Lukasiewicz AND and OR. Associations which are only known to be positive (R1 = 0, R2 = 1) or negative (R 1 = - 1 , R 2 = 0) reduce the certainty interval.


Table 2. Certainty intervals using possibility interval set operators. Unconditional probabilities p(A)= p(B)= 0.5; various values of prior lower and upper limits of the association measure RX(A, B). The lower limit for RX(A, B) is R1; the upper limit is Rz. Z(A A B) and Z(A v B) are the Zadeh AND and OR operators; L(A A B) and L(A v B) are the Lukasiewicz operators. PAND, POR, RXAND and RXOR are the possibility interval operators discussed in the text which yield lower and upper bounds for (A A B) and (A v B). RXAND and RXOR are constrained by the specified lower and upper limits for the association measure RX(A, B); FAND

and POR are unconstrained

Lower and upper limits for RX(A, B) as (R1, R2) (unconditional probabilites p (A) = p (B) = 0.5)

(1, 1) (0, O) (-1, -1) (0, 1) (-1, O) (-1, 1)

L(A A B) 0 0 0 0 0 0 Z(A A B) 0.5 0.5 1.5 0.5 0.5 0.5 L(A v B) 1.0 1.0 1.0 1.0 1.0 1.0 Z(A v B) 0.5 0.5 0.5 0.5 0.5 0.5

PAND lower 0 0 0 0 0 0 PAND upper 0.5 0.5 0.5 0.5 0.5 0.5 RXAND lower 0.5 0.25 0 0.25 0 0 RXAND upper 0.5 0.25 0 0.5 0.25 0.5

POR lower 0.5 0.5 0.5 0.5 0.5 0.5 POR upper 1.0 1.0 1.0 1.0 1.0 1.0 RXOR lower 0.5 0.75 1.0 0.5 0.75 0.5 RXOR upper 0.5 0.75 1.0 0.75 1.0 1.0

Table 2 gives examples of the values yielded by the various opera tors for A A B and A v B for uncondi t ional probabil i t ies p ( A ) and p ( B ) both equal to 0.5. Of course , the Z a d e h and Lukasiewicz opera tors ZAND, and ZOR, LAND and LOR, make different assumpt ions about pr ior associations be tween A and B; as expected, the values yielded by these two sets of opera tors are quite different f rom each other . Also as expected , the PAND and POR opera tors yield a range of possible values for A A B and A v B which are b racke ted by the Zadeh and Lukasiewicz opera tors . The RXAND and RXOR opera tors yield ranges for A A B and A v B cons t ra ined by the specified range of the association R X ( A , B) ; these ranges vary f rom zero difference be tween min imum and max imum values when the lower and limits in R X ( A , B) are the same, to the same spread be tween min imum and m a x i m u m values as given by the PAND and POR opera tors when the limits in R X ( A , B) are ( - 1 , +1) . W h e n the lower and upper limits in R X ( A , B) are (0, 0), RXAND and RXOR yield values for A A B and A v B which lie be tween the Z a d e h and Lukasiewicz values.

As an example of the foregoing, consider a simple stylized rule:

rule (husband wan t s_baby ' Y E S ' ) AND (wife wan t s_baby ' Y E S ' )

m a k e decision h a v e _ b a b y ' Y E S ' ;

Fuzzy operators for possibility interoal sets 225

Suppose that the certainty we have that the husband wants a baby is the single value 0.6, and that the certainty for the wife is 0.8. Now let us tabulate the certainty values for a truth table under three assumptions: (1) husband and wife tend strongly to agree with each other, with R = 1; (2) husband and wife tend strongly to disagree, with R = - 1 ; and (3) husband and wife hold independent opinions, with R = 0. We have four logical possibilities: neither husband nor wife really want a baby; wife does, but husband doesn't; husband does, but wife doesn't; and finally, both really want a baby. (We implicitly assume additivity of certainties.)

The lower and upper values in Table 3 for husband AND wife want_baby are precisely those produced by the PAND operator. Now consider the simple decision rule given above, with the RXAND operator employed. Suppose the grade_of membership of member YES in possibility interval set husband wants_ baby is (0.6, 0.6), and of YES in wife wants_baby is (0.8, 0.8) as in Table 3. Then with no restriction on prior association between husband's and wife's feelings, i.e. RX = ( - 1 , +1), the rule yields grade of membership of member YES in decision have_baby of (0.4, 0.6). If we know a priori that the husband's and wife's feelings are independent, with RX = (0, 0), the rule results in grade_of_membership of YES in decision have_baby of (0.48, 0.48). If husband and wife tend to agree, with RX = (0, 1), limits for YES in have_baby are (0.48, 0.6); if husband and wife tend to disagree, with RX = ( - 1 , 0), limits for YES in have baby are (0.4, 0.48).

If our simple rule is used for decision making, the extent to which husband and wife tend habitually to agree or disagree is clearly of importance. Suppose, for example, that we normally fire a rule if the net certainty in the antecedent is at least 0.5; then the rule which ANDS the feelings of husband and wife is not fireable if husband and wife tend to disagree, or if their feelings are independent. In the absence of prior information on assoication between the feelings of husband and wife, a system decision must be made on whether to fire on the lower limit, the upper limit or somewhere in between. This decision determines what we call the

Table 3. Truth table for husband's and wife's desire for a child, with grades of membership of possible combinations of truth values in fuzzy sets of actual feelings. Unconditional certainties: husband wants_baby

0.6, wife wants_baby 0.8

Possible values for wants_baby

Husband Wife

Truth values for possible values of wants_baby, given prior associations below between husband's and wife's feelings

Strongly Strongly Opinions agree: disagree: independent: R = I R = - I R = 0

No No 0.2 No Yes 0.2 Yes No 0 Yes Yes 0.6

Husband AND wife 0.6

0 0.08 0.4 0.32 0.2 0.12 0.4 0.48

0.4 0.48


personality of the system. Firing on lower limits is very conservative; firing on upper limits is danderously venturesome. (Note that a certainty interval of (0, 1) represents complete ignorance.)

6. Discussion

Negoita [6] has considered the problems which can occur with the use of new operators, and stresses the need for caution; he has used the theory of categories as a structured approach to the need for rigor.

While the proposed operators do not have all the excellent mathematical properties of the Zadeh operators [13], they retain the most important properties; associativity, commutativity, involution and satisfaction of De Morgan's laws. Importantly to us, they permit implementation in a fuzzy expert system shell of the OR operator without running the danger of assigning a certainty level lower than the data permit, and of easy rational reduction of the certainty interval when rough (or accurate) prior knowledge of associations exists.

Schwartz [8] has recently proposed use of crisp interval possibilities in a manner similar to what is here proposed, except that he employs the Zadeh operators on prior certainty intervals to generate new certainty intervals. Schwartz developed his method as an alternative to the use of possibility distributions which are fuzzy numbers; we developed our method as an alternative to the use of single-valued certainties. Aside from differences in motivation, the methods are quite similar. Our feeling is that the method here proposed has the advantage of focussing attention on a root cause of uncertainty in certainties, namely the lack of knowledge of prior associations, and deals with minimum and maximum certainties which can be expected when knowledge of such prior associations is wholly or partially lacking.

Our fuzzy expert system shell FLOPS does not yet have the above operators implemented, although in the case of fuzzy comparisons of numbers we made use of the Zadeh AND, Lukasiewicz OR pair. FLOPS has been nevertheless been successful in preliminary tests on image interpretation [10], but the inability confidently to implement the OR in production rules places an undesirable restriction on the rules we can write. The proposed operators remove that restriction.

In their implementation of a fuzzy inference engine Appelbaum and Ruspini [1] have utilized the equivalent of the PAND and POR operators to establish lower and upper certainty intervals for rule antecedents. The above treatment then amounts to a formalization of their method, with the addition of the RX operators to incorporate prior knowledge to reduce the certainty gap.

Note that a conventional fuzzy set can be considered as a crisp possibility interval set in which the minimum and maximum grades of membership are identical. This permits fuzzy sets to be combined with each other or with possibility interval sets to yield possibility interval sets, but possibility interval sets cannot be combined to yield fuzzy sets. Initially, in an expert system, it is far more likely that fuzzy sets will be defined rather than possibility interval sets;


possibility interval sets will in most cases first arise as the result of combining fuzzy sets.

The proposed association measure is not necessary to the use of possibility interval sets, nor is it necessary to reduce the gap between lower and upper possibility limits when using prior knowledge; any other convenient way of using prior knowledge to reduce this gap may be used, or the lower and upper limits may be used without modification if prior knowledge is completely lacking. The proposed measures does seem to offer a convenient way to estimate prior associations.

The reasoning in this paper may perhaps be criticized as being a composite of probabilistic and fuzzy logical reasoning, neither flesh nor fowl. Certainly we have used probabilistic analogies rather freely. It is our conviction, however, that the resulting chain of thought will stand as fuzzy reasoning, motivated and directed by probabilistic considerations, but purely fuzzy in the end.

Acknowledgment

We thank a reviewer of this paper for suggesting the 'husband-wife-baby' example to clarify the ideas here presented.

References

[1] L. Appelbaum and E.H. Ruspini, ARIES: a tool for inference under conditions of imprecision and uncertainty, in: Proceedings SPIE Technical Symposium East '85 (Society of Photo-Optical Instrumentation Engineers, BeUingham, WA, 1985).

[2] A.P. Dempster, Upper and lower probabilities induced by a multi-valued mapping, Ann. Math. Stat. 38 (1967) 325-339.

[3] B.R. Gaines, Fuzzy and probability uncertainty logics, Inform. and Control 38 (1978) 154-169. [4] C. Glymour, Independence assumptions and Bayesian updating, Artificial Intelligence 25 (1985)

95-99. [5] R. Martin-Clouaire and H. Prade, Managing uncertainty and imprecision in petroleum geology,

in: J.J. Royer, Ed., Computers in Earth Sciences for Natural Resource Characterization, Coll. Int., Nancy (9-14 April 1984).

[6] C. Negoita, Expert Systems and Fuzzy Systems (Benjamin Cummings, Menlo Park, CA, 1985). [7] E.H. Ruspini, Possibility theory approaches for advanced information systems, Computer 15

(1982) 83-91. [8] D.G. Schwartz, The case for an interval-valued representation of linguistic truth, Fuzzy Sets and

Systems 17 (1985) 153-165. [9] G. Shafer, A Mathematical Theory of Evidence (Princeton University Press, Princeton, N J,

1976). [10] W. Siler, D. Tucker, J. Buckley and V.G. PoweU, Artificial intelligence in processing a sequence

of time-varying images, in: Proceedings SPIE Technical Symposium East '85 (Society of Photo-Optical Instrumentation Engineers, Bellingham, WA, 1985).

[11] D. Tucker, W. Siler and J. Buckley, FLOPS: a fuzzy expert system, Unpublished manuscript (1985).

[12] M. Sugeno, Fuzzy measures and fuzzy integrals: a survey, in: M.M. Gupta, G.N. Saridis and B.R. Gaines, Eds., Fuzzy Automata and Decision Processes(North-Holland, Amsterdam, 1977) 89-102.

[13] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353.

fuzzy operators for possibility interval sets

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