Gradual inference rules in approximate reasoning

INFORMATION

Gradual

Inference

SCIENCES

103

61, 103-122 (1992)

Rules in Approximate Reasoning

DIDIER DUBOIS and HENRI PRADE Institut de Recherche en Informatique de Toulouse, Uniuersite’ Paul Sabatier, 118 Route de Narbonne, 31062 Toulouse Cedex, France

ABSTRACT Gradual inference rules of the form “The more X is F, the more Y is G” for of similar forms with “the less” instead of “the more”), which express a progressive change of the degree to which the entity Y satisfies the gradual property G when the degree to which the entity X satisfies the gradual property F is modified, are often encountered in commonsense reasoning. A representation of such rules by means of fuzzy sets is proposed and discussed. This representation turns out to be based on a special implication function already considered in multiple-valued logic. Patterns of reasoning involving gradual inference rules are formalized. Their links with interpolation mechanisms are pointed out.

1.

INTRODUCTION

Gradual inference rules of the form “The more X is F, the more Y is G”, where X and Y are entities, F and G are gradual properties (i.e., liable to be satisfied to different degrees), or rules of similar forms (where “the less” is substituted to one or both occurrence(s) of “the more”), are often used in the expression of commonsense knowledge, and even in proverbs or mottoes. The so-called production rules in expert systems are most of the time stated under the form “if X is F, then Y is G” (with a possible degree of uncertainty) and then do not express how to modify our knowledge about Y when X changes. However, this progressive and continuous change of the value of an attribute of Y in relation with the value of one or several attributes of X is often present in expert knowledge. In other words, we may contrast a “dynamical” description of the relation between an attribute of Y and an attribute of X by OElsevier Science Publishing Co., Inc. 1992 655 Avenue of the Americas, New York, NY 10010

0020-0255/92/$5.00

104

D. DUBOIS AND H. PRADE

means of a gradual inference rule, with the “static” description of this relation by means of several rules of the form “if X is 5, then Y is G,” (i = 1, n). We first discuss the place of gradual inference rules (g.i.r., for short) among various kinds of rules. Then their representation by means of fuzzy sets is studied in detail; the role played in this representation by a particular multiple-valued logic implication function is emphasized. The mechanization of a deductive pattern of inference from a g.i.r. and a possibly imprecise or vague fact is then presented and the link with interpolation mechanisms is clarified. This paper is based on ideas recently proposed in a paper in French (Prade [20]) and stresses the ability of various multiple-valued implication functions to model different kinds of relational links in the framework of possibility theory.

2. VARIOUS RINDS OF RULES OF KNOWLEDGE

FOR THE EXPRESSION

Let x and y be two variables corresponding to the respective values of the attributes a and b for the entities X and Y. The “logical” expression of a relationship between x = a(X) and y = b(Y) leads to state one or several rules of the form if xEF

then LEG,

where F and G are subsets (possibly with a single element) of values of the attribute domains of a and b, respectively. A collection of rules of the above form, in general, gives a partial and rough description of a “functional” link of the form y = f(x) or more generally y E I’(x) in the case of a multivalued mapping F. For simplicity we have assumed that the value of b(Y) only depends on the value of a(X); more generally, we have rules with compound conditions involving several variables. Uncertainty and imprecision can be captured when F or G are fuzzy sets. First consider the case where F is an ordinary subset and G is a fuzzy subset whose membership function (from the domain of b into [O,l]) is denoted by po. Then, it expresses that, among the possible values of y when x E F, some are more plausible than others; in other words, the value u is all the more impossible for y as p,Ju) is small. This value is forbidden for y only if ho(u) = 0, whereas pa(u) = 1 means that u is completely possible as a value for y; this view extends the interpretation of the case where G is an ordinary subset. For more details see Dubois and Prade [9]. By contrast, if F is a fuzzy subset the rule “if x E F then y E G” may be understood in the following

GRADUAL

INFERENCE

RULES

105

way: “The more x E F, i.e., the larger the degree of membership of x to F, the more certain y E G,” i.e., the more certain is G a subset of more or less possible values for y (G may be fuzzy). Especially the certainty is total if the value u of x belongs to F with the degree 1 and decreases towards 0 when ~Ju) diminishes. See Buisson, Farreny, and Prade [2] and Lebailly, MartinClouaire, and Prade [1.5] for the preliminary treatment of such rules in the framework of fuzzy set and possibility theory, in practical cases. In this paper, we are interested in g.i.r.‘s of the form “The more x E F, the by means of more y E G,” in which the gradual properties are represented fuzzy sets F and G. The g.i.r. is then understood as “the larger the membership degree of x to F, the larger the membership degree of y to G must be.” The notion of a degree of membership is related to the gradual nature of the property under concern represented by means of a fuzzy set, and should not be confused with the degree of certainty considered in the previous paragraph for the estimation of the extent to which G could be regarded as a correct description of the more or less possible value of y. As already said, a g.i.r. roughly describes how the value of y depends on the value of x in a given area, whereas a conditional rule of the form “if x E F then y E G” only states that the possible values of y are restricted by G when x ranges in F, and, if F is fuzzy, that our certainty that the value of y is indeed in G decreases when x goes astray from the values which completely belong to F (i.e., with the degree 1). The reader is referred to Dubois and Prade [IO] for a formal study of the distinction between the two kinds of rules [i.e., gradual rules vs. rules yielding (more or less) uncertain conclusions]. Differences between “if.. . then” rules and g.i.r. have been demonstrated from the point of view of the theory of argumentation by Bruxelles and Raccah {I]. In the following the representation of g.i.r.‘s by means of fuzzy sets is presented and discussed, then this representation is used in order to formalize and mechanize deductive reasoning with g.i.r.‘s. 3.

REPRESENTATION

OF A G.I.R. BY MEANS OF FUZZY

Given two fuzzy sets F and G defined on the universes tively, the g.i.r. “the more x E F, the more y E G” translates constraint

SETS

U and V respecnaturally into the

(1) which defines

a (crisp) relation

between

the variables

x and y. It expresses

106

D. DUBOIS

AND H. PRADE

that when the degree of membership of x to F increases, then membership of y to G should also increase. In other words, stipulates that the membership degree of y to G cannot be membership degree of x to F. Constraint (1) comes close to so-called “fuzzy mappings” some authors, especially Goguen [141 and Negoita and Ralescu mapping f from U to V is such that VMEU,

the degree of constraint (1) less than the considered by 1181. A fuzzy

Ilc(f(u)) >CLF(U).

(2)

See also Dubois and Prade (141, p. 96) where an interpretation of the form “the more x is F, the more y = f(x) is G” is proposed for (2). Note that (21 can be viewed as a consequence of the extension principle (Zadeh 12411,which defines the image G of a fuzzy set F by a mapping f as

which clearly entails ~~(f(~)l z pF(ul, V u E Ii; see Dubois and Prade [5] for further discussions along this line. The proposed approach can be also applied to a g.i.r. with n-ary predicates (n z 21 (e.g., “the nearer x, and x2, the more y E G”) which translates into a constraint of the form pR(xl,. . . , XJ < p,J y). From (11, it is obvious that the g.i.r.‘s enjoy a transitivity property, namely, from “the more x E F, the more y E G” and “the more y E G, the more z E H,” we can deduce that “the more x i F, the more z E H,” since we have CLJX) < I+(Y) and CLJY) < pJz). Note also that fuzzy set inclusion defined by F L F’- ~~ < pLFris equivalent to the g.i.r. “the more x if F, the more x is F’,” which is satisfying. 3.2.

POSSIBILI7Y

DlSTRlBUTiON

ASSOCOl TED WITH A G. I. R.

Constraint (1) associates with each value u of x, a subset Gp+, = (u E V, &u) > ~~(~11 of values L’ which can be regarded as possible for y when x = u. This can be expressed by means of a conditional possibility distribution (Zadeh [26]) rrTTylX defined by

ry,x( u,u,=pGsF ,.,( u) = { Equality

1

if k(u)

0

otherwise.

a p,?(u)?

(3) means that if x = u, the only possible values of y, according

(3) to

GRADUAL

INFERENCE

RULES

107

Fig. 1.

(11, are the u’s which belong to GM,(uj. Figure 1 exhibits the proposed interpretation of a g.i.r. Although in many practical examples, the membership functions are monotonic (as in Figure 11, this is not at all required by the approach. 3.3.

DECOMPOSITION

OF A G.I.R. INTO

“IF..

THEN..

” RULES

A fuzzy subset F can be viewed as equivalent to the nested &eve1 cuts, which are ordinary subsets defined by

and which fulfill the “nestedness”

family of its

property

In terms of cr-level cuts, the g.i.r. “the more x E F, the more y E G,” represented by the constraint p,&n) < pa(y) appears then to be equivalent to the collection of rules VaE(O,l],if

xEF,

then BEG,.

The smallest a-level cut of F which includes the value u of X, and to which, due to (31, corresponds the most restrictive subset for the possible values of y, is obtained for (Y= pcLF(u).In other words, the subset GPFcu, = 1~ E V, pa(u) > p,Ju)) = ~6 l([~~(u), 1’)) includes all the possible values of the variable linked to x by the g.i.r. “the more x E F, the more y E G.” It can be checked that

D. DUBOIS

108

AND H. PRADE

where

(7) is the possibility distribution which represents the rule “if x E F, then y E G,“; (7) expresses the standard material implication in logic. (6) formally states that the g.i.r. is equivalent to the conjunctive combination of nested “if.. . then” rules. N.B. (3) can be rewritten ~Y,,(u,u) = ~tFF~uxll(~o(u)), where CL[~~(~)r1 is the characteristic function of the real interval [~Ju), 11. Under this form’it would be possible to generalize the approach to the case where IL&X) is a fuzzy number rather than a precise number (i.e., where F is a type 2 fuzzy set), see Dubois and Prade ([4], p. 41) for this question.

3.4.

GRADUAL

EQUIVALENCE

AND MODIFIERS

Observe that the g.i.r., “the more x E F, the more y E G” is not equivalent to the g.i.r. “the more y E G, the more x E F.” Indeed, the increase in the degree of fulfilment of the gradual property G by y [i.e., pG(y)I may be due to a cause other than the increase of pF(x). If it is known that we have both “the more x E F, the more y E G” and “the more y E G, the more x E F,” it means that x and y are changing conjointly. Using the interpretation (21, it leads to state the equality pF(x) = pa(y). It yields the possibility distribution

1 if pF(u) = k(u),

~~Ix(”‘) = (

0

othenvise

(8)

This may be considered as too rigid as long as we want only to express that pa(y) is completely determined by wF(x) and, conversely, at least in a certain area. As we shall see, this idea can be captured by generalizing (2) under the form

where m is a monotonically increasing mapping from [O, 11 to C-m, 11, which enables us to modulate the strength of “the more.. . the more” in the g.i.r. It

GRADUAL amounts

INFERENCE

to modifying

the fuzzy set F into a fuzzy set F’ defined by

and to state the constraint us consider two examples. EXAMPLE

109

RULES

p&x)

6 pa(y);

thus we formally recover (2). Let

1. If we take m(t) = t - E with 0 < E < 1, (9) gives

which expresses that po(yY) is constrained only if ~Jx) > E since non-negative. Note, in the example, that if X is not sufficiently nothing can be said about Y with respect to G. EXAMPLE

pG(y)

is F, then

2. Let us take m(t) = \/t; it leads to the constraint

which corresponds to a g.i.r. of the form “the more X if F, the more it is true that Y is very G, ” if we remember that the square of a membership function can be regarded as a sufficient model, in practice, of the effect of the modifier “very” on a fuzzy predicate (Zadeh [25]). The modulation introduced by the modifier m in (9) enables us to somewhat relax the equality constraint in (8). Indeed let us consider the g.i.r.‘s “the more x E F, the more y E G” and “the more y E G, the more x E F,” respectively associated with the constraints

m’(kAY)) G PAX). Let us suppose that m and m’ are one-to-one m ~ ’ and rn’- ’ exist and the above constraints

and

for the sake of simplicity; are equivalent to

(11) then

110

D. DUBOIS

which clearly expresses these intervals exist if

Vt,

an approximate

mym( t))

equality

< t, and

AND H. PRADE

(in a crisp way). Note that

m(m’(t))


(12)

This guarantees that the two constraints (11) will not lead to a contradiction by transitivity. When m = m’, (12) means Vt, m(m(t)> < t, which implies due to the strict monotonicity of m, m(t)< t. Indeed, if m(t)> t for some t, then t > m-‘(t) and by transitivity m(t) > m-‘(t), which contradicts (12). Hence m must express hedges of the “very” family; it ensures that (9) is a weakened form of (1) [i.e., pF(x) < pc(y) implies m(p,dx)) < p,(y) but not the converse, when m(t) < tl. 3.5.

NEGATIVE

G.I.R. AND

The complement

CONTRAPOSITION

-I F of a fuzzy set F is usually defined

by (Zadeh

[24])

The fuzzy subset -I F represen?s the vague predicate “not F.” Definition (13) can be understood as “the more x E F, the less x E 7 F” and, conversely, “the less x E 7 F, the more x E F,” or if we prefer “the less x E F, the more x E 7 F” and “the more x E 7 F, the less x E F.” It shows that “the more x E F” is equivalent to “the less x E 7 F.” Thus, the g.i.r. “the more x E F, the less y E G” understood as “the more x E F, the more y E -I G” is thus represented by the constraint

Similarly, inequality

the g.i.r. “the less x E F, the more y E G” is represented

by the

Since pF(x> < pG(y) can be written 1- pa(y) < l- pF(x), the g.i.r. “the more x E F, the more y E G” is equivalent to the g.i.r. “the less y E G, the less x E F,” which is intuitively satisfying. More generally, a g.i.r. of the form where (K, L) E {the less, the more}2 is equivalent to the “KxEF,L~EG,” g.i.r. “negL y E G, negK x E F” where “neg” changes “the less” into “the more” and conversely.

GRADUAL 3.6.

INFERENCE

111

RULES

CONJUNCTION AND DISJUNCTION IN G.I.R.‘s

A logical combination of elementary conditions may be present in a g.i.r. Thus, the g.i.r. “the more x E F and y E G, the more z E H” can be represented by

min(kAx),k(y))


(14)

where the logical conjunction is expressed by the operation “min” as usual with fuzzy sets (Zadeh [24]); see (Dubois and Prade [9]), for instance, for justifications and other possible choices. The logical disjunction of two g.i.r.‘s “the more x E F, the more z E H” or “the more y E G, the more z E H” entails (but is not equivalent to) “the more x E F and y E G, the more z E H”; this is due to

(VX, pF(x)

Q

CL&)) or (VY,k(Y)

-Vx,Vy,min(~AX),

LAY))

G cLff(z))


but the converse implication is false. This is satisfying, since for instance “the hotter and the drier it is, the yellower the grass” does not entail that “the hotter it is, the yellower the grass” or “the drier it is, the yellower the grass.” A more satisfying translation, in many cases, of the g.i.r. “the more x E F and y E G, the more z E H” is obtained by using the product in place of the minimum in (14), i.e., PF(X).k(Y)


(15)

The advantage of (15) is that p,(z) should increase as soon as pcLF(x) or increases; it is not the case with min. By contrast, observe that the g.i.r. “the more x E F or y E G, the more z E H” can be represented by pJy)

max(d4,

IGAY))

w&)

(16)

(where the operation “max” is used for the logical disjunction). It is equivalent to “the more x E F, the more 2 E H” and “the more y E G, the more z E H” since V x,V y, max(pF(x), pG(y)) d kH(z) *(V x, p&x) G ~.&N and (V Y, /.LJY)

d CLfJz)).

D. DUBOIS

112 4. MULTIPLE-VALUED MODELING OF RULES

IMPLICATION

In the conditional possibility multiple-valued logic implication,

FUNCTIONS

AND H. PRADE IN THE

distribution defined by (3), we recognize i.e., (3) can be written

a

where a+b=aAb aA,b=

and 1 0

V(a,b)E[0,112,

ifab.

This implication has been already considered in the fuzzy set literature; see Gaines [13] and Mizumoto [17, 161 especially. However, these authors do not particularly mention the capacity of this implication for modeling rules of the form “the more . . . the more.” It is worth noticing that other implication functions enable us to model other kinds of rules such as the ones discussed in Section 2. Particularly, let us consider the three implication functions defined by a&b=

1 b

ifab,

a$b=max(l-a,b), a&b=

1 l-a

if ab.

These three implication functions can be obtained from the conjunction min and/or the negation II defined by n(a) = 1 - a, through a generation process fully described in Dubois and Prade [8]. Namely, we have a~b=sup(sE[O,l],min(a,s)
Clearly using a similar generation

process, other implication

functions

can be

113

GRADUAL INFERENCE RULES

Fig. 2.

Cm)

T&.,u).

obtained starting with a conjunction operation different from min (e.g., the product). We limit the discussion to min-based implications for the sake of brevity and because they exhibits the main characteristic behaviors. In Figure 2 we have pictured the behavior of these implications, together with the implication corresponding to (3), when used in (17) for defining a conditional possibility distribution. Figure 2 exhibits the fuzzy sets of possible values for y when x = u in the case of the four implication functions considered above. It shows the differences in meaning of these implications. As already said in Section 3.3., (17) equipped with -$ expresses that if we take a value for x in the cu-level cut of F, the value of y is certainly in the cu-level cut of G (which is in agreement with the intended meaning of rules of the form “the more.. . the more,” expressing that the higher the degree of membership in F, the higher the

114

D. DUBOIS

AND H. PRADE

degree of membership in G). If we use the implication 3 in (17), we express that if the value of x belongs to the support of F [support(F) = {u E U, pF(u) > O]], then the value of y should belong to the support of G; moreover, if the value of x deviates from the core of F [core(F) = (u E U, pLF(u) = l]], the core of the possibility distribution vYIX(., u), which restricts the possible values of y when x = U, is enlarged (i.e., more values are regarded as completely possible for y). This can be viewed as a weakened version of A since when x belongs to an a-level cut of F, the values outside the a-level cut of G are no longer forbidden but have a degree of possibility which is smaller than (Y.And, indeed, 3 is obtained by solving a fuzzy relational equation that extends (2) changing function f into a fuzzy relation R, namely, min(p.,(u), p,&u, u)) f ~JLI) (see the remark below). It corresponds to the following interpretation of the gradual rule: “the more x is F and the more y is a possible image of x, the more y is G”; it presupposes that the rule expresses a fuzzy constraint between x and y rather than a crisp one [as in (3)]. This type of fuzzy rule is studied in Dubois and Prade [9], and implemented in the SPII system (Lebailly et al. [El). By contrast, (17) equipped with 3 expresses that we are no longer completely certain that the value of y is restricted by G (and in particular by its support) when the value of x is only in the support of F without belonging to the core; more precisely, we are all the more certain that the value of y is restricted by G as the value of x is nearer the core of F [I - p.,(u) estimates the possibility that the value of y when x = u is outside G and thus anywhere in V]; this corresponds to the interpretation of rules of the form “if x E F then y E G” when F is fuzzy, proposed in Section 2 (“the more x E F, the more certain y E G”). The implication 3 combines the effects of the implications 5 and -% (i.e., enlarging the core of G and introducing a level of uncertainty). REMARK. As already hinted above, (17) with “ 4” = “ 4” greatest solution in the sense of fuzzy set inclusion when functional equation

is known as the it exists of the

where rY,X is unknown; see Sanchez [211, Dubois and Prade [71, Pedrycz 1191, and Trillas and Valverde [231 for related discussions. This equation expresses that the image of F by vYIX is G, in the framework of possibility theory, i.e., that if x is restricted by F, then y is restricted by G. It is worth noticing that (17) with “ + ” = “ -$ ” IS . also a solution, provided that pF is continuous and onto. It is even the smallest solution of the above equation which is based on an “intuitionist” implication, i.e., such that n + b = 1 as soon as a < b.

GRADUAL

INFERENCE

115

RULES

Lastly, the implication “ 4 ” can be obtained by residuation from a conjunctionlike operation, just as “ 3 ” can be obtained from “min” (it is also true for 3 and 5 with other conjunction-like operations, see Dubois and Prade [81). Namely, we have a 4 b = sup{ s E [o, l]Q * s d b] ) with

a nonsvmmetrical (a,b);(0,1}2.

operation -

5. INFERENCE FACT

FROM

that reduces

A G.I.R.

In this section, we consider

AND

to an ordinary

conjunction

AN IMPRECISE

the following pattern

the more x E F, the more y E G x E F’

of deductive

when

OR FUZZY

inference

(18)

y EG’ where the fact x E F’ means that the possible values of x are known to be restricted by the subset (possibly fuzzy) F’ and where we are looking for the most restrictive subset G’, which gathers all the more or less possible values of y. Possibility theory applied to approximate reasoning (Zadeh [26]; Dubois and Prade [9]) enables us to obtain G’ by performing a conjunctive combination of the representations of the two premises, followed by a projection on the universe of y, i.e.,

Here the conditional possibility distribution n-y,X which represents the g.i.r. is given by (3). Figure 3 gives a graphical illustration of the construction of G’. The method for building G’, shown in Figure 3, is simple and can be easily justified since

116

D. DUBOIS AND H. PRADE

-

G'

with

r,

-_--G’

with

5

Fig. 3.

and then (19) reduces to @Ju) = sup u Ell,py(fj( ~~~~~~~~(~).It expresses that a value u belongs all the more to G’ as it exists, a value u having a high degree of compatibility with F’ and satisfying the constraint associated with the g.i.r. The construction method of pot (using implication -1,) appearing on Figure 3 can be summarized as follows, when pF and po are monotonically increasing on closed intervals of the real line: . For any value u of y compute pctc(u) = cc . If (Y= 1, then @Ju)= 1. l If (Y< 1, then compute t = IL;‘(~). Compute p =

-

sup(y&)lu

G t}. More

specifically:

If 1 a inf~u~~~(~) = l), i.e., f is not on the left of the core of F’, then @G’(U)= 1; Otherwise, ELo,(~)= p = w&t) (provided that pFf is unimodal and upper semicontinuous).

The following facts are noticeable: Ci) If F’cF but F’+ F, then G’cG but G’+:G. (ii) More generally, the conclusion G’ does not always contain G, as is patent on Figure 3. (iii) If 3 u E U, a! E (O,l], pF(u) = 0 and pLF’(u)= (Y,then p&u) B (2, V u. In other words, when F’ goes out of the support {ul/.~&) > 0) of F there is a level of uncertain~ on the conclusion. (iv) If F’ = F, using what is said in the remark at the end of Section 4, we obtain G’ = G (provided that ~1~ is continuous and onto). This can be considered as intuitively satisfying since if we know, for instance, that “X is large” without any further precision and that “the larger X, the larger IF,” it is natural to conclude that “Y is large.”

GRADUAL

INFERENCE

RULES

117

(i) and (ii) are typical of the implication involved in g.i.r.‘s, and distinguish it from implication 3 for which, by construction, F’ produces G’ that contains G, for any F’ (Dubois and Prade [7, 9]), while properties (iii) and (iv) are common to 4 and 3. Properties (i) and (ii) stress that the behavior of g.i.r.‘s is not in accordance with usual modus ponens for which a rule if X is F then Y is G cannot produce a more precise conclusion than “Y is G.” Pattern (18) enables us to obtain, for instance as in Figure 3, from “the larger X, the larger Y” and “X is somewhat large” a conclusion like “Y is at least somewhat large.” We do not consider, here, the linguistic approximation problem (see Eshragh and Mamdani [ll], for instance), which consists in generating the linguistic expression which is the best description of a fuzzy subset (here G’), from a vocabulary (interpreted in terms of fuzzy sets) and a formal grammar. Note that in the above example we conclude that “Y is at least somewhat large” and not “Y is somewhat large” (which allows for values greater than the ones compatible with “somewhat large”). For getting this latter conclusion we need to know also that “the more y E G, the more x E F.” Taking m = m’= identity in (ll), then we would have

%(u+) =

1 if k(u) (()

=PAu),

othemise

Figure 4 pictures this situation. As shown by Figure 4, the g.i.r. offers a “dynamical” description of the relation between y and X. For instance, assume that pF and pG express degrees of redness and ripeness, respectively, and that F’, as in Figure 3, expresses “rather red.” From “the redder the tomato, the riper the tomato” and “the tomato is rather red,” we conclude that “the tomato is (at least) rather ripe” and that “the tomato is rather ripe” if we have also the converse g.i.r. “the riper the tomato, the redder the tomato.” Such a conclusion is not obtainable if we have a rule

t

Fi

F;

lF

Fig. 4

118

D. DUBOIS

G’l

forXYW”)

1 if PF(u)s w;(v) = {

o

otherwise

AND H. PRADE

1 if PF(@= CIGIV) ; G’2 for ~XPJJJ)

= { o otherwise

Fig. 5

of the form “if x E F then y E G” (modeled by (17) with + equal to 3, 3, or 3 1 using the approximate reasoning scheme (191, as pointed out in Dubois and Prade [7] since either a level of uncertainty would appear or, in the case of 4, the conclusion is blurred with imprecision due to the property G L G’ (see the behavior of this implication on Figures 2 and 3). Indeed in our example we want to express that the degree of ripeness increases with the redness degree of the fruit; this is in agreement with the implication -$ which enables us to represent that “the higher the degree of redness, the higher the degree of a “proxripeness.” Deductive reasoning with g.i.r.‘s is a way of implementing imity” reasoning method (in the sense of Dubois and Prade [6]) in a rigorous manner, viewing a membership grade as a “distance” to the core of the fuzzy set, as suggested by Figure 5, where the conclusions obtained with the implication L and with the corresponding equivalence are compared. Note that in the case of the equivalence, we get the expected effect: the gap between the cores of F and F’ induces a gap between the cores of G and G’. However, this behavior occurs only if the support of F’ remains in the part of the support of F where membership is partial. functions, the Moreover, if pF and pG are not monotonic membership equivalence connective underlying the two g.i.r.‘s “the more X is F, the more Y is G” and “the more Y is G, the more X is F” may produce, an ambiguous response from X is F’ as Figure 5 clearly shows, since G’ has two peaks. This fact points out that g.i.r.‘s are most useful when pF and pG are monotonic. 6.

LINK WITH LINEAR

INTERPOLATION

Let us consider the case where the membership functions FF and pG are monotonically nondecreasing with linear transitions between 0 and 1, as in

GRADUAL

INFERENCE

119

RULES

Fig. 6.

Figure 6. Let a, (resp. b,) be the least upper bound of the set {u,p&u) = 0) (resp. {u,pJu) = 0)) and a, (resp. b,) be the greatest lower bound of the set {u, pcLF(u)= 11 Crew. (u, j+(u) = 11). S ee Figure 6. Let us suppose that the value of x is known and equal to uO, i.e., F’= {u& in (18); moreover, we assume a0 Q u0 Q al. It can be easily checked that if u0 = Ba, +(l8)u, (with 0 Q 0 < l), then the value u0 such that pu,(u,) = pF(uO) is given by u0 = Bb, + (1 - B)b,. This means that, when pLF and /*.G have linear transitions between 0 and 1, the deductive reasoning process from the two g.i.r.‘s “the more x E F, in the more y E G” and “the more y E G, the more x E F,” presented Section 5, is equivalent to a linear interpolation. This interpolation is generally no longer linear when the transitions between 0 and 1 of pF or pG are not linear. The interpolation process is naturally extended in our approach to the case where the value of x is not precisely known, i.e., when F’ is not a singleton and possibly a fuzzy set. Note that this idea of interpolation plays an important role in the success of fuzzy controllers as recently emphasized by Zadeh [27]. It can be implemented in fuzzy reasoning schemes in different ways. For example, an interpolation technique is provided by Sugeno and Nishida’s [22] approach to fuzzy control, as discussed by Farreny and Prade [12]. In their approach they use a family of rules of the form if xEFi

then y=g,,i=l,n

where the Fi’s are fuzzy sets, and the g(‘s precise values. The available information about x is also supposed to be precisely known; then the answer computed by the fuzzy controller for x = x0 is given by

120

D. DUBOIS

AND H. PRADE

A l-

F. I- 1

Fi

F.

I+1

l/2'

Fig. 7

Clearly in the particular case where the F;‘s are triangular fuzzy sets (with core {a#, such as the ones pictured in Figure 7, that form a fuzzy partition, that is, V X, Ci=l,n~~$x)= 1, since only two consecutive Fi’s overlap, we get y = B.g, with 0 = ~~{;(~a) when x = x,, E [a,,a,+,], i.e., we are perform+(l-ti)‘gi+r ing a linear interpolation. It suggests that the application of g.i.r.‘s to fuzzy control may be worth considering.

6.

CONCLUSION

This paper has proposed an approach to the representation and the treatment in deductive reasoning patterns, of gradual inference rules (g.i.r.‘s) of the form “the more x E F, the more y E G” in the framework of fuzzy set and possibility theory. These g.i.r.‘s have the distinguished ability to produce results that may be more specific than the conclusion G appearing in the rule when the input fact is more specific than the condition part F, and lies in the range of partial membership of F. G.i.r.‘s can be represented as such and dealt with in the same possibilistic framework as other kinds of rules, in the presence of imprecision or uncertainty. G.i.r.‘s enable us to have a rulelike description of how the value of a variable evolves as a function of the value of one or several other variable(s), in a rather qualitative way. Then they seem also appealing in order to model qualitative reasoning in artificial intelligence. The practical application of this approach presupposes that membership functions pF and po can be approximately identified as monotonic functions, and this in turn presupposes that variables x and y range in well-defined totally ordered (often continuous) sets. An example of application is presented in Buisson and Prade [3] where g.i.r.‘s express dependencies between parameters that characterize an individual (age, weight, physical activity, appetite) and the amount of needed calories.

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REFERENCES 1. S. Bruxelles and P. Y. Raccah, Info~at~on et argumentation: l’expression de la consequence. Actes Confkrence “Cognitiua”, CESTA, Paris, 18-22 May, 1987, Vol. 1, pp. 445-452. 2. J. C. Buisson, H. Farreny, and H. Prade, Dealing with imprecision and uncertainty in the expert system DIABETO-III fin French), in Proc. 2nd Int. Conf. on Artificial Intelligence, CIIAM-86, Marseille, Hermes, Paris, 1986, pp. 705-721. 3. J. C. Buisson and H. Prade, Un systeme d’inference pratiquant i’inte~olation au moyen de regles i predicats graduels-une application au calcul des rations caloriques, in Actes Convention Intelligence Artificielle 88-89, 23-27 Jan. 1989, Hermes, Paris, pp. 43-57. 4. D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic, New York, 1980. 5. D. Dubois and H. Prade, Towards the analysis and the synthesis of fuzzy mappings, in Fuzzy Set and PossibiIity Theory-Recent Developments (R. R. Yager, Ed.), Pergamon, New York, 1982, pp. 316-326. 6. D. Dubois and H. Prade, On distances between fuzzy points and their use for plausible reasoning, in Proc. IEEE Int. Conf. on Cybernetics and Society, Bombay-New Delhi, 30 Dec., 1983-7 Jan. 1984, pp. 300-303. 7. D. Dubois and II. Prade, Fuzzy logics and the generalized modus ponens revisited, Cybernet. Syst. 15:293-331 (1984). 8. D. Dubois and H. Prade, A theorem on implication functions defined from triangular norms, Sfochu.r&r VfII(3):267-279 (1984). 9. D. Dubois and H. Prade, (with the collaboration de H. Farreny, R. Martin-Clouaire, C. Testemalef, Possibiiity Theory-An Approach to Computerized Processing of Uncertuinty, Plenum, New York, 1988 (French editions, Masson, Paris 1985 and 1987). 10. D. Dubois and H. Prade, A typology of fuzzy “If... then...” rules, in Proc. 3rd Inter. Fuzzy Systems Association Congress, Seattle, Wash., 6-11. Aug. 1989, pp. 782-785. 11. F. Eshragh and E. H. Mamdani, A general approach to linguistic approximation, Int. J. fan-machine Stud., 11:501-519 (1979). 12. H. Farreny and H. Prade, Uncertainty handling and fuzzy logic control in navigation problems. Preprints, In?. Conf. on Intelligent Autonomous Systems, Amsterdam, The Netherlands, 1986, pp. 218-225. Proc. edited by L. 0. Hertzberger and F. C. A. Groen, North-Holland, 1987 13. B. R. Gaines, Foundations of fuzzy reasoning, in Furry Automnta and Decision Processes (M. M. Gupta, G. N. Saridis, and B. R. Gaines, Eds.), Noah-Holland, Amsterdam, 1977, pp. 19-75. 14. J. A. Goguen, Concept representation in natural and artificial languages: axioms, extensions and applications for fuzzy sets, Int. J. Man-Machine Stud 6~513-561 (1974). 15. J. Lebailly, R. Martin-Clouaire, and H. Prade, Use of fuzzy logic in a rule-based system in petroleum geology, in Approximate Reusoning in Intelligent Systems, Decision and Control (E. Sanchez and L. A. Zadeh, Eds.), Pergamon, New York, 1987, pp. 125-144. 16. M. Mizumoto, Fuzzy inference using max-A composition in the compositional rule of inference, in Approximate Rezoning in Decision Making (M. M. Gupta and E. Sanchez, Eds.), North-Holland, Amsterdam, 1982, pp. 67-76. 17. M. Mizumoto, S. Fukami, and K. Tanaka, Some methods of fuzzy reasoning, in Aduances in Fuzzy Set Theory and Applications (M. M. Gupta, R. K. Ragade, and R. R. Yager, Eds.), North-Holland, Amsterdam, 1979, pp. 117-136. 18. C. V. Negoita and D. A. Ralescu, Applications of Fuzzy Sets to Sysfems Analysis, Birkhluser, Base], 1975.

122

D. DUBOIS AND H. PFLADE

19. W. Pedrycz, On generalized fuzzy relation equations and their applications, J. Mar/r. Anal. Appl. 107:520-536 (1985). 20. H. Prade, Raisonner avec des regles d’inference graduelle-Une ensembles flous, Rev. Intell. Artif. 2(2):29-44 (1988).

approche baste sur les

21. E. Sanchez, Solutions in composite fuzzy relation equations: application to medical diagnosis in Brouwerian logic, in Fmzy Automata und Decision Processes (M. M. Gupta, G. N. Saridis, and B. R. Gaines, Eds.), North-Holland, Amsterdam, 1977, pp. 221-234. 22. M. Sugeno and M. Nishida, Fuzzy control of model car, Fuzzy Sets Syst. 16:103-113 (1985). 23. E. Trillas and L. Valverde, On mode and implication in approximate reasoning, in Approximate Reasoning in Expert Systems (M. M. Gupta, A. Kandel, W. Bandler, and J. B. Kiszka, Eds.), North-Holland, Amsterdam, 1985, pp. 157-166. 24. L. A. Zadeh, Fuzzy sets, Inform. Control 8338-353 (1965). 25. L. A. Zadeh, A fuzzy set-theoretic interpretation of linguistic hedges, J. Cybemer. 2(3):4-34 (1972). 26. L. A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets Syst. 1:3-28 (1978). 27. L. A. Zadeh, QSA/FL-qualitative systems analysis based on fuzzy logic, University of California, Berkeley, 1989. Received 25 July 1989; revised 30 August 1989