Inverses of fuzzy relations. Application to possibility distributions and medical diagnosis

Fuzzy Sets and Systems 2 (1979) 75-86 © North-Holland Publishing Company

I N V E R S E S OF F U Z Z Y R E L A T I O N S . A P P L I C A T I O N TO POSSIBILITY DISTRIBUTIONS AND MEDICAL DIAGNOSIS* E. SANCHEZ Laboratoire de Biomath~matiques, Statistiques et lnformatique M~dicale, Facult~ de M~decine, 27, Bd Jean Moulin, 13385 MarseiUe Cedex 4, France Received July 1978 Revised August 1978 In this paper we define and explore the properties of lower and upper inverses of fuzzy relations which extend multi-valued mappings. With the notion of degree of inclusion of non-fuzzy sets, we then relate the preceding notions to possibility distributions in natural languages and to problems of medical diagnosis.

Key Words: Multi-valued fuzzy relations, Lower inverses, Upper inverses. Ponderated inverses, Inverse possibility distributions, Medical diagno:ds assistance,

1. ln~oduction In approximate reasoning, fuzzy relations have shown to be of a primordial importance. Human judgments are often based on comparisons between couples of faced data. We study here a special class of fuzzy relations which extend multi-valued mappings, so that their membership (or compatibility) functions deal with subsets of given sets, rather than with single elements. For any fuzzy relation, a lower and an upper inverses are defined. By means of a SUP-MIN composition they are associated with relations of inclusion and intersection for non fuzzy sets. For practical purposes, the definition of a fuzzy relation of inclusion allows us to ponderate the upper inverse of a fuzzy relation. Such ponderated inverse contains the lower inverse and it is included in the upper inverse. Reasoning now in natural languages, from inverses of fuzzy relations we derive inverse possibility distributions, which are induced by propositions in the form " X is F", where F is a fuzzy relation and X a variable that takes values in the cartesian product of power sets of universes of discourse. Finally it is shown how inverses of fuzzy relations may apply to a "medical knowledge" with degrees of associations between diagnosis and syndromes.

2. Fuzzy relations. Lower and upper inverses Let two sets X and Y be given. Any non-fuzzy relation R between the elements of X and the elements of Y may acquire a functional nature, in the * Paper presented at the I.E.E.E. Special Symposium on Fuzzy Set Theory and Applications, New-Orleans, December 7-9, 1977. 75

E. Sanchez

76

sense of a multi-valued mapping from X to Y, by specifying the elements y in Y which are related to a given element x in X, i.e. such that (x, y) belongs to R. This simply means that the relation is being regarded as a correspondence from X to Y and our purpose is now to study fuzzy correspondences between non-fuzzy sets. Definition 1. A multi-valued fuzzy relation r from X to Y is characterized by its membership (or compatibility) function /xr " X x 2 v ---~[0, 1]

where 2 v stands for the power set of Y. /~r is not a multi-valued mapping, but the fuzzy relation R is said "multi-

valued" for its membership function associates a point in X and a subset of Y with a degree of compatibility in the interval [0, 1]. ~ r is an extension of fuzzy or non-fuzzy relations from X to Y, but the main point of view here is '.hat P-r generalizes multi-valued mappings from X to Y. Definition 2. For ea:h subset A of X, we define

VB ~_Y,

Izr(A, B) =SUP I~r(x, B). x~A

Note that, by convention, SUPxcAI~r(x,B)=O if A =~. Delinition 3. Let r be a "multi-valued" fuzzy relation from X to Y and let A be a fuzzy subset of X, we define R oA, a fuzzy subset of 2 v, by

VBc_ Y,

I~RoA(B)=SUP[pA(X)AtXR(X,B)]. x~X

Definition 3 still holds when A is a non-fuzzy subset of X and in such a case, from Definition 2 we derive

Izr(A, B)= lzroA(B), and

V x e X , VBC_ Y,

I~r({x},B)=l~r(x,B),

so that we shall now only consider fuzzy relations from 2 x to 2 v. When writing that R is a fuzzy relation from X to Y it will be understood that its compatibility function tLr is defined on 2 x ×2 v and that it is [0, 1] valued. The notation R c 2Xx 2 v will be used for it. Definition 4. Let R be a fuzzy relation from X to Y. We define its lower inverse as the fuzzy relation R . from Y to X ( i . e . R . c_2 v x 2 x) characterized by

VBC_Y,

VAt_X,

I~a.(B,A)=

SUP p.r(A,C). C~_Y, C ~B,C~_B

(1)

Inverses of fuzzy relations

77

Defmilion 5. Let R be a fuzzy relation from X to Y. We define its upper inverse as the fuzzy relation R * from Y to X (i.e. R*~_2 v × 2 x) characterized by

VB ~_ Y, V A ~ X,

/ ~ r . ( B , A ) = SUP I~r(A, C).

(2)

C~_Y, CNB

#fl

For R and S fuzzy relations from X to Y, defining

RcS

iff V A G X ,

VB~_Y,

IzR(A,B)~Vcs(A,B),

one then has

(3) (4)

R c_ S::~ R . ~_S., R c_S:~R*c_S*. With B~ and

12 being subsets of Y, for any fuzzy relation R from X to Y, one has

B1 g BE=:>YA ~ X,

/~R.(Bt, A)~< VLR.(B2, A),

(5)

B~ c 82=)' VA c X,

tza*(B1, A) ~
(6)

The following property holds for any fuzzy relation R from X to Y.

VA'c_X,

VBc_ Y,

I~R(A,B)<-WR.(B,A)<~t~R.(B,A),

(7)

where the inequality in the right is equivalent to R . ~ R*.

(8)

Remarks. Definitions 4 an6 5 are an extension of the definitions of lower and upper inverses of multi-valued mappings. The reason why we have introduced two kinds of inverses, a lower and an upper one, is because the fuzzy relations we study in this paper are considered as extensions of multi-valued mappings. In Definition 4, the supremum is taken over the non void subsets C of Y which are included in B but, depending on applications, one may choose an order relation different from the one of inclusion in 2 v. Example. ~ Let X = Y be the: set of possible positions in the game of chess. A position consists of the coordinates of the different pieces on the chess-board and the player whose move is next. The set X is then the union of three disjoint sets X1, X2, Xo; X1 is the set of positions in which White can move, X2 is the set of positions in which Black can move and Xo is the set of positions of checkmate or stalemate, when it is not possible for either to move. ~ThJs example is derived from an example in [2], where the author considers multi-valued mappings.

E. Sanchez

78

If x e XI (resp. X2) we shall denote by/~R(x, B) the degree of combination of x with the set of positions B, B ~ X , such that points in B are immediately reachable by White (resp. Black) after position x; this determines a fuzzy relation from X - X o to X. Starting with the position x,p,a.(B,{x})=SUPc~v.c~o,c~al~R(x,C) denotes the higher degree of association of x with sets of positions which can only be in B in the following move, while p,R*(B,{x})=SUPc~v, cnB~-,I.~R(x, C) denotes th,~ higher degree of association of x with sets C of positions that will have a common part with B in the following move. To take into account the role that such common parts may play in relation with the extent of C and B, it already seems natural to introduce weights to ponderate the expression of the upper inverse (see Section 5 for it).

3. F'wst and second monotonodty. Composition o| fuzzy relations Definition 6. Let R be a fuzzy relation from X to Y. We say that R has the property of first monotonocity or R is 1-monotonous if, and only if, for A~ and A2 subsets of X,

AIc_A2::~VBc_ Y,

I.~R(A1,B)<~IxR(A:,,B).

(9)

Definition 7. Let R be a fuzzy relation from X to Y. We say that R has the property of second monotonocity or R is 2-monotonous if, and only if, for B~ and B2 subsets of Y,

BI ~_B2=>VA c_X,

I.~R(A,B1)<~g.R(A, B2).

Let R be a fuzzy relation from X to Y, the following properties hold: R , and R* are 1-monotonous.

(11)

If R is 2-monotonous, then

VB c_ Y,VA ~_X,

/~R.(B, A) = ~R (A, B)

(12)

/~R.(B, A) = g-R(A, Y).

(13)

If R is 2-monotonous, then

VB c_ Y,VA c_X,

Definition 8. Let Q be a fuzzy relation from X to Y and let R be a fuzzy relation from Y to Z we define R o Q, a fuzzy relation from X ~to Z, by VA c_ )~,VC c_ Z,

I~Roo(A,C) = SUP [/~o(A, B ) A / ~ ( B , C)]. B~Y

(14)

Inverses of fuzzy relations Proposition. Let O be a fuzzy relation from X to Y and let R be a from Y to Z. If" O and R are 2-monotonous, then

79

fuzzy relation

(R o 0 ) , = Q , o R , . Let us now define two non fuzzy relations S# and S #, from Y to Y by VC_~ Y, VB~_ Y, {~

tZs,(C, B) = t~s*(C, B~ =

for C~_B, C#O, elsewhere,

{~ for

CNB¢O,

elsewhere.

Let R be a fuzzy relation from X to Y, then V B c Y , VAc_X, I.tR.(B,A)=IXS#oR(A,B).

(15)

(16)

(17)

VB c_ Y, V A ~_X,

/xR*(B, A) =/XS-oR(A, B).

(18)

S#, S #, S,~o R

S#oR

(19)

and

ace 2-monotonous.

R , = (S#o R ) .

(20)

R*=(S#oR),

(21 ~,

If R is 2-monotonous, then

S#o R = R.

(22!

4. Fuzzy relation of inclusion Let us now define a 'uzzy relation S, from Y to Y, which characterizes the degree of inclusion of couples of subsets of Y. It is introduced for applications, mainly for the ponderation of the uppcr inverse of a l uzzy relation, 2 for all (non-fuzzy) subsets C and B of Y, /~s (C, B) .a__ degree to which C _ B.

(23)

:;uch a relation S will be called a fuzzy relation of inclusion in Y. We require S to verify the following four conditions (i) / ~ s ( C , B ) = 0

for

CNB=O,

(ii) p s ( C , B ) = I for C ~ B , C~O, (24) (iii) 0 < / ~ s ( C , B ) < I (with the understanding that ps(C, B) expresses the degree to which C _ B) for C N B ~ O and C not included in B, (iv) If B,c_B2, then lzs(C, B1)<~ps(C, Bu), which is a natural condition, according to (23). This condition means that S is 2-monotonous. 2The symbol ~ stands for "denotes" or is "equal by definition".

80

E. Sanchez

For example, if Y is a finite set and if n(D) denote,,; the cardinality of a (non fuzzy) subset D of Y, the fuzzy relation S from Y to Y defined by for all subsets C and B of Y,

[n(CnB) for

C~0,

for

C=O,

~s(C,B)=l~--~(~i

(25)

verifies the four conditions in (24). Remarks. The relation S#, defined in (15), is a non frizzy case of S. With S# we only deal with inclusion or not, but with S we allow a possible continuity of variation for the degrees of inclusion. Let y be an element o~ Y and let B be a subset of Y. ~s({Y}, B) -n degree to which {y}cB, i.e. degree to which y e B. {~ ~s({Y}' B) = Xs(Y) =

for for

y~B y~B,

where Xa is the characteristic function of B, Xa: Y - ~ {0, 1}. In our example, see (25), {y} ~ 0, so that

~s({y},B)=

n({y}NB)

n({y}) =n({y}nB)=

{ 1 for 0 for

y~B, y~B.

For some applications, h may be useful to introduce the following type of relation of inclusion: T, fuzzy relation to type "x",, with compatibility function /Jr : 2v x [0, 1]v ._.>[0, 1], where [0, 1]v A the fuzzy subsets of Y, and such that for all (non-fuzzy) subset C of Y and for all fuzzy subset B of Y,/~r(C, B)A- degree to which C c B. For y ~ Y and for B ~[(), 1]v, ~r({Y}, B) can be assimilated with the gxade of membership of y in B, say tLs(y) in the usual notation. Let us remark that in some problems, it may be more convenient to descr~ibe fuzziness in terms of a relation of inclusion, rather than in terms of a relation of membership. Let us now note some properties related to S.

S# c S c_S#. VB c_ y, v c c_ y,

(26)

as.(B, c ) = ~s(C, B).

(27)

!For all fuzzy relation O from X to Y, S o Q is a 2-monotonous fuzzy relation from X to Y and t~(Soa),(C, A ) = ~SoQ(A, C) for all subsets A of X and C of Y.

81

Inverses of fuzzy relations

5. Ponderation of the upper inverse of a fum~ relation

For each subset B of Y, let us define

YI(B)={Cc_ YI Cc_~n, C~O}, Y2(B)={Cc_ YI Cr, B~O

(29)

c n},

and

(30)

Y3(B)={C c_ Y JCf~ B =O}.

(31)

From (24) we derive, for/3: c y, /xs(C, B) = 1

for

C ~ Y1(B),

(32)

/zs(C, B) ~ ]0,1[

for

C~Y2(BL

(33)

/~s(C, B) = 0

for

C ~ Ya(B).

(34)

Moreover, 2 v = YI(B)U YE(B)U Ya(B),

for all

B c_ y.

(35)

From Definitions 4 and 5 we now derive, tor all B c y and for all A _c X, ~R.(B, A) = SUP p,~ (A, C),

(36)

CE YI(B)

p~a.(B,A)=I~R.(B, A ) ' ¢ [ SUP

p,R(A,C)].

(37)

C ~ Y2(B)

In the right-hand member of (37), the supremum is taken over the subsets C of Y which have a common part with B and which are not included b~ B. The need to take into account the way such C's intersect with B lead us to introduce the following ponderation. Definition 9. Let R be a fuzzy relation from X to Y. We define its ponderated upper inverse as the fuzzy relation ~R*' from Y to X (i.e. R * ' c _ 2 V x 2 x) characterized by, for all B _c y and for all A c_ X,

SUP

C),,

B))],

(38)

C ~ Y2(B )

where Y2(B)is defined in (30) and S is a fuzzy relation of inclusion verifying the four conditions in (24). It follows easily that

R,c_R*'~R*.

(39)

Proposition. If R be a fuzzy relation from X to Y, then

R*'=(SoR),.

(40)

E. Sanchez

82

One may note that (40) is equivalent to

VB c_ Y, V A ~_X,

p,R,,(B,A)= IXSoR(A,B),

(41)

which has to be considered in parallel with the similar expressions given in (17) and (18). 6. Fuzzy relations and possibility distributions

The theory of possibility related to fuzzy sets was introduced by Zadeh in [13]. Let X be a variable which takes values in a universe of discourse U with " X = u " signifying that X is assigned the value u, u ~ U. Let F be a fuzzy subset of U, characterized by a membership function/x~, then the proposition " X is F " induces a possibility distribution, llx, which is postulated to be equal to F. Correspondingly, the possibility distribution [unction associated with X (or the possibility distribution function of //x) is denoted by 1rx and is defined to be equal to IxF. Poss ( X = u I X

is

F) = IxF(u)= 7rx(u)

(42)

with the understanding t h a t / / x is induced by " X is F". For example, Poss (Height (John)= 1751 John is m/l)= ix,au(175), where the universe of discourse U represents heights in cm, "Height" is an implied attribute ot X (i.e. John), and tall is a fuzzy subset of U, characterized by a given membernhip function /~,au. Let us now consider propositions involving n variables X ~ , . . ~ , X,, with Xi taking values in Ui, i = 1 , . . . , n, which are n possibly different universes of discourse. Let F be a usual fuzzy relation in the cartesian product U = U1 × " " x U,, and let X = ( X ~ , . . . , X,). Then the proposition " X is F " , translates i n t o / / x = F X

is

F ~ H ( x , ..... x.~=F,

(43)

where H(x: .....xn) is an n-ary possibility distribution which is induced by the proposition " X is F " . Correspondingly, the possibility distribution function associated with X is given by V ( u l , . . . , u , ) ~ U,

It(x, .....x . ) ( u l , . . . , u , ) = / x F ( u l , . . . , u,),

(44)

where /x~ is the membership function of F.

Poss(Xlx

is

F) =/~nx = ~rx =/~r~.

(45)

Let us now relate possibility distributions to inverses of 2-ary fuzzy relations. Let X = (X1, X2) and let R be a usual fuzzy relation from U~ to /-/2, X

is

R ~ H(x,.x_,)= R.

(46)

Inverses of fuzzy relations

83

Our purpose is to define (Hx)-t, an inverse possibility distribution of Hx. Let X - " &(X2, X~) and let us consider a proposition of the form " X -~ is R -~'', where R -~, the inverse 3 of R, is a usual fuzzy relation from U2 to U~ which is characterized by

V(u~_, ul)~ O2× Ul,

IXR-,(U2,U~)= p.R(Ul, U2).

(47)

The inverse possibility distribution of Hx, associated with X-~, is postulated to be equal to H x , . In other words, from H~x,.x9 = R one may deduce //(x2,x0 =

R-1

or

_t/x-, = ( H x ) -I

(48)

where R -~ is characterized in (47). Let us now suppose that Ut and U2 are two ~aower sets of universes of discourses, i.e. U~ = 2 v' and U2 = 2v', and that R is a fuzzy relation from V~ to V2, in the sense given in the preceding sections (R_c-2 v, ×2v0. Starting from Hoc,.x~, = R, H~x~.x,) is no more defined uniquely, like in (48). From //tx,x.~ = R we now postulate the existence of two inverse possibility distributions of Hx, which are associated with X-~, ,t lower and an upper one (H.)tx~.x0 = R , ,

(49)

* _ R=,, (11)~x~.x,)-

(50)

where R , c_ U2 x U~ and R* ~ U2 × U:, (recall that Ul = 2v, and U2 = 2v-) are characterized by /xR,(u2, ul) =

SUP

/zR(ul, w)

(51)

w~U2

/zR*(u2, ut) = SUP ttR(u~, w).

(52)

w E U:,, wNu2#fl

7. Fuzzy relations and medical diagnosis

The following application is based on fuzzy relations in t sense given in the preceding sections. We assume here that the source of imprecision is not in the nature of symptoms and diagnosis, but rather in ttae correspondences which associate syndromes with groups of diagnosis. In a given pathology, let ~ be a set of symptoms and let f)~ ,e a set of diagnosis, all being well-defined. We shall denote by s (resp. d or d ' generic subsets of S¢ (resp. ~). a We recall that such inverse relations are of a great use in the resolution of SUP-MIN composite fuzzy relation equations, see [4-7].

84

E. Sanchez

Let R be fuTzy relation from ,V to ~ (i.e. R c s ~ex 2 ~ ) , with VLR(S,d) representing the degree of association of the syndrom s with the clinical charactedsti~ of the diseases in d. The fuzzy re;ations R . , R * and R*' (from ~ to ,Y) enable a complete study of inverse probleras which consist in the determination of degrees of combination of diseases with ~roups of symptoms. Such fuzzy relations are characterized by

s) = SUP m (s, a')= s u P d ' c_d, d' #~

a')^

a)],

(53)

d'

WR*(d, S)= SUP IXR(:~,d') =SUP[/xR(s, d')A/~s#(d', d)], d' d'Nd#~

(54)

d'

/~R*,(d, S)= bl,R.(d, S)V SUP [/£R(S, d')A ~s(d', d)]

(55)

d, d ' N d ~t~ d'aed

or R*' = (S o R).. Let us note that when d consists of a single diagnosis, e.g. d = {8}, the following property holds

tLR.(& S) =

(% 8).

(56)

Moreover, we always have for all d G ~,

V~d,

Iza(s,6)<~t~a.(d,s).

(57)

We suggest to only use R . and R*', but not R* in which the supremum involved in its definition takes into account some intersections which may not be of paramount significance. In terms of medical diagnosis assistance we finally indicate a formulation which shows some analogies with already proposed models, see [5, 7, 9]. Let ~ be a set of diagnosis, 8e a set of symptoms, [9 a group of l;.atien*s. The relation of inclusion between subsets is here replaced by "implications". For s c_ ~ and d c_ ~, s--> d & all patients, with observed symptoms in s, must present the diagnosis in d. Moreover, for s c_ S/' and d c_ ~, s ~ d & [s --->d or there exists s' G s such that

s'---> d]. Defining S# c_ 2se × 29 and S # c 2 ~e× 29 by ~s,(S, d ) = 1 for =0

(58)

elsewhere,

V,s-(S, d ) = 1 for =0

s --->d,

s ~> d,

elsewhere

(59)

ln'~erses of [uzzy relations

85

and for R being a fuzzy relation from ~ to ~¢, we derive

/.~a.(d, p) = SUP pR(P, s) = SLIP [pa (P, s)A Ps.(S, d)] s---*d

~R*(d, p) = SUP ~R(P, s)= SUP [~R(p, s)^ ~s-(s, d)] s~,d

(60)

s

(61)

s

which allows one to infer associations of patients with diagnosis from observed symptoms.

8. Concluding remarks For further developments of the notions of inverses of fuzzy relations we suggest an exploration of "belief relations", see [10] for belief functions, for problems encountered in combinations of evidences, where "belief relations" may be associated with possibility distributions. Finally, in the theory of possibility, the compatibility of possibility measures (in different spaces) with fuzzy relations deserves to be investigated together with particularized possibility distributions which ihvolve SUP-MIN compositions see [131.

Acknowledgment This work was initiated while the author was a Visiting Research Associate at the Department E.E.C.S. (University of California, Berkeley) and it was stimulated by seminars given by Pr. Zadeh. Moreover, the author would like to thank Dr. J. Gouvernet from his Department in MarseiUe as well as the referees for helpful comments.

References [1] R.E. Bellman and L.A. Zadeh, l,ocal and fuzzy logics, Memo. No. ERL M584, University of California, Berkeley, CA (1976). [2] C. Berge, Topological Spaces, Including a Treatment of Multi-va~.ued Mappings, (The MacMillan Company, New York, 1963) [3] A. Dempster, Upper and lower 9robabilities induced by multi-valued mappings, Ann. Math. Statist. 38 (1967) 325-339. [4] A. Kaufmann, Introduction h la theorie des sous--ensembles flou~i, Tome IV (Ed. Masson, Paris,

1977). [5] E. Sanchez, Equations de relations floues, ThEse d'Etat de Biologic Humaine, Marseille (1974). [6] E. Sanchez, Resolution of composite fuzzy relation equations, Information and Control 30 (1976) 38--48.

86

E. Sanchez

[7] E. Sanchez, Solutions in composite fuzzy relation equations. Application to medical diagnosis in Brouwerian logic, in: Fuzzy Automata and Decision Processes (North-Holland, New York 1977). [8] E. Sanchez, On possibility--Qualification in natural languages, Memo. No. UCB/ERL M77/28, University of California, Berkeley, CA (1977). [9] E. Sanchez and R. Sambuc, Relations floues. Fonction O-floues. Application h l'aide au diagnostic en pathologie thyroidienne, Medical Data Processing Symposium, IRIA, Toulouse (1976). [lO] G. Shafer. A Mathematical Theory of Evidence (Princeton Univerist~ Press, Princeton, N J, 1976). [11] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, lnfommtion Sciences Part 1, 8 (1975) 199-249; Part II, 8 (1975) 301-357; Part III, 9 (1975) 43-80. [12] L.A. Zadeh, Theory of fuzzy sets, Memo. No. UCB/ERL M77/1, University of California, Berkeley, CA (1977). [13] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Memo. No. UCB/ERL M77/12, University of California, ~erkeley, CA (1977).