Approximate reasoning based on generalized disjunctive syllogism

Fuzzy Sets and Systems 61 (1994) 143-151 North-Holland

143

Approximate reasoning based on generalized disjunctive syllogism Swapan Raha Department o f Mathematics, Visva-Bharati Universitv Santiniketan, Birbhum ZI1 235, India

K u m a r S. R a y Electronics and Communication Science Unit, Indian Statistical Institute, Calcutta, India

Received June 1991 Revised January 1993 Abstract." In this paper we present a purely semantic approach to approximate reasoning based on the extended concept of the law of disjunctive syllogism. The concept of generalized disjunctive syllogism is discussed. Deductive models of approximate reasoning are formulated based on that concept. Simple but concrete examples are cited to illustrate the models. Keywords: Approximate reasoning; fuzzy logic; fuzzy set; generalized disjunctive syllogism; possibility theory.

Introduction

In any logic, approximate reasoning is the process or processes by which an approximate/inexact conclusion is deduced from possibly approximate/inexact information (Premises). It has the remarkable ability to solve the most complex and/or otherwise insolvable problems to some desired degree of accuracy (feedback/iteration). In processing ill-defined classes of events that are rampant in everyday life such as the recognition of a pattern of information, where the symbols found are rarely precisely known, we use techniques of qualitative reasoning. Simply because we find binary-valued logic unsuitable for the description of objects which are not of black or white type and the concept of multivalued logic is insufficient as truth values of grammatical sentences are qualitative rather than quantitative. Only continuous-valued logic provides a way to represent such classes of problems adequately. Motivated by this we consider in this paper, approximate reasoning based on fuzzy logic [8, 9], a continuous-valued logic. Approximate reasoning based on fuzzy logic has been interestingly applied in different fields of soft-sciences such as biology, psychology etc. and in particular, in the design of expert systems, in medical diagnosis, in risk analysis, in controlling processes and in many other fields [2-4, 7]. By approximate reasoning, from now onwards, in this paper we shall mean approximate reasoning based on fuzzy logic. Since we are dealing with inexact/vague concepts, in this paper we consider semantic representation ]8, 9], i.e., a set exemplifying the concept and then use laws of the underlying set theory to manipulate them. We use fuzzy sets for representation and theory of fuzzy sets for manipulation. Related work goes back to 1973 when a first formal description of such a deductive process was published by Zadeh [7]. Mamdani and his associates [4] then successfully applied it in the design of fuzzy logic controllers. Since then different forms of approximate reasoning have been discussed by many researchers [2-7]. Correspondence to: Dr. Kumar S. Ray, Electronics and Communication Science Unit, Indian Statistical Institute, Calcutta, 203, B.T. Road, Calcutta 700 035, West-Bengal, India.

0165-0114/94/$07.00 © 199~-Elsevier Science B.V. All rights reserved SSDI: 0165-0114(93)E0187-W

144

S. Raha, K.S. Ray / Disjunctive syllogism

Zadeh's concept of approximate reasoning is an application of the theory of possibility [9], a model termed as generalized modus ponens. Here the inexact information is represented in terms of possibility distributions which are found to be equal to some fuzzy subsets. Therefore, manipulation of possibility distributions is performed as manipulation of fuzzy sets. As probability measure is found to be unsuitable because of normalization criteria, the approach is based on possibility measure. Rather than justifying definitions by typical proofs of expected results as is done in mathematics, in this paper, our attempt is to model a deductive process 'Generalized Disjunctive Syllogism' through mathematical formulation and illustration only. The approach is almost similar to that of Zadeh's except that here an inexact proposition is also attached with an inexact truth value like 'Arnab is tall is quite true'.

Mathematical preliminaries In this section we present some definitions and basic concepts related to terms that are frequently used to develop the literature.

Fuzzy set In representing human understanding of the real world activities, Zadeh in 1965 introduced for the first time the concept of a fuzzy set. Like classical set, a fuzzy set is also characterized by its members. But unlike classical set, members of the universe may or may not belong 'to some degree'. Let D denote the universe of discourse and /z :D--> [0, 1] (here instead of [0, 1] one may take any partially ordered set L) be any mapping. Then a fuzzy set over D is conveniently represented by the collection of pairs of the form {(d, ~(d)) I d ~ D,/x(d) > 0}. /z (d) are gradual membership vallaes to represent the degree with which generic elements d ~ D may belong to the set of elements identified by the fuzzy set. Since membership function takes values other than 0 and 1 it is possible to handle vagueness and since there may be more than one local maximum of the membership function it can cope with ambiguity as well. Thus through fuzzy sets it is possible to represent linguistically expressed inexact concepts adequately.

Fuzzy logic Fuzzy logic is the logic for inexact reasoning and is considered by many as the logic for future generation computer systems. The underlying set theory is the theory of fuzzy sets. Classical theory of sets and formal logic are dual representations, of the same information. Often we say that "something is a member of the set" in order to mean "it is true that this thing has the defining property of the set". But this is not the case with fuzzy sets and fuzzy logic. Here degree of possibility and degree of truth are not considered as the same. Fuzzy sets possibly has no precise boundary and the truth values are qualitative rather than quantitative and hence they are possibly inexact and sometimes conflicting in nature. A first formal description of fuzzy logic was given by J.A. Goguen in 1968 [1]. Fuzzy logic is formally defined as an algebraic system (L, ^, v) where L is some pseudo-complemented distributive lattice, A and v are the well-known min and max operators. We take, for convenience, L to be the unit interval [0, 1] and d = 1 - a, Va E L. The mostly used rules in fuzzy logic are the conjunction principle, the projection principle and the entailment principle [9].

145

s. Raha, K.S. Ray ] Disjunctive syllogism Conjunction principle

Let p and q be two propositions whose translations are expressed as and

p ~ H(x,,x2....,x,) = S

q --~ [ I ( x , . x 2 , . . . , x m , r,,r2,...,Vk ) = T

( m <~ n; m , n, k < ~ ) .

S and T are fuzzy relations. X~, X2, • • •, X,, are the variables which appear in both premises p and q. U, (i = 1, 2, . . . , n), b (j = 1, 2 . . . . , k) are the universes of discourse associated respectively with the variables Xz and Yj. Let X=(Xl,X2

.....

Xn) ,

Xt=(Xl,

X2,...,Xm)

and

Y=(Y~,

Y2,...,

Yk).

S, T respectively denote the cylindrical extension of S and T in D, the Cartesian product /31 × U2 × • • • × U, × 1/1 x V2 x • • • × Vg. Then the conjunction principle asserts that r may be inferred from p and q according to the following scheme: p---> I I x

= S

q--> I I ( x , . r ) = T r ~-- H ( x , r )

= S N 7" .

Projection principle

Let p be a fuzzy proposition whose translation is expressed as p ~ II(x,,x2,...,xo) -- F, a fuzzy relation and let X ' = ( X 1 , X 2 , • • •, X m ) denote a sub-variable (m ~
7L'x.(Um+I,Urn+ 2. . . .

, Un) =

sup

141,U2,...,Um

I&F(Ul . . . .

, Urn 1, Urn, U r n + l , ' ' ' ,

Un),

ui is a generic element of Ui. Let q be the retranslation of the above possibility assignment equation. Then the projection principle asserts that q may be inferred from p according to the following scheme: p ~ H(x,,x~,...,xo) = F q * - - H ( x ........ x,) = Proju, xu2×...×vm F,

where the right hand side of the latter equation is a fuzzy relation defined by the induced possibility distribution function. It should be mentioned here that a proper combination of the application of the conjunction principle followed by an application of projection principle results in the well-known compositional rule of inference due to Zadeh [9]. Entailment principle

The entailment principle asserts that from any fuzzy proposition another fuzzy proposition can be inferred if the possibility distribution induced by the inferred fuzzy proposition contains the possibility distribution function of the former (from which it is inferred) one. F o r m u l a s in f u z z y logic

In this paper we consider a simple fuzzy formula as follows:

X i s F ; TisC. X and T are two linguistic variables of which T explicitly denotes the truth value. F and C are fuzzy

S. Raha, K.S. Ray ] Disjunctive syllogism

146

subsets of respective universes. Thus if ( X I , X 2 , . . . , Xn) are X = (X~, X2 . . . . . An) then we consider a fuzzy formula in fuzzy logic as

n-linguistic

variables

and

F(X); C(T) where F is a fuzzy relation defined over U1 × U2 × • • • × U,,; Ui being universes of discourse of the linguistic variables Xs (i = 1, 2 , . . . , n). This formula actually determines a possibility assignment equation

H(X;T)= R

(say)

where R is a fuzzy relation defined over/-/1 × U2 × • • • × Un × V; V being the universe of discourse of T. Consider a fuzzy proposition X is F; T is C. Let G be a set (fuzzy) such that F c_ G. Then the truth value of the proposition (fuzzy) X is G on the assumption X is F; T is C will be given by C' where C' = P r o j v R ; U being the universe of discourse of X. R is a fuzzy relation defined by R = G A (F × C); is the cylindrical extension of G over U × V, V being the universe of discourse of T. Thus,

c(v) = sup{.~(u) ^ ~ × d u , v)} //

= sup{/xa(U) ^ tZF(U) ^/Xc(V)} u

= sup{/ZF(U) ^/Xc(V)}

[since F c_ G]

U

=

tXc(V)

if (3U)vIXF(U)

~>sup{/xc(V)}. v

Thus C' = C if we choose F to be a normal fuzzy set. Disjunctive syllogism

In two-valued logic the law of disjunctive syllogism can be stated as 'Given a disjunction and a negation of any of the disjuncts, the other can be inferred'. Symbolically, preml : p v q prem2 : - p Concl : q . In using this rule the user must be sure that the disjunct as appears in prem2 is exactly the negation of the one that appears in p r e m l i.e., they must be contradictory pairs. No doubt, such condition is a restriction so long as the real-life problems are concerned. Here we often find certain pairs of information which are not completely contradictory but certainly close to the same. In this case, the above framework is not admissible. For this let us remove this restriction on exactness and generalize this concept to the case where the disjuncts are inexact/imprecise in nature and hence the 'degree of contradiction' is not absolutely specified. But as the first premise is a restriction of the disjuncts, and the user may have some possibly inexact knowledge about any of the disjuncts, whatever the level of contradiction may be, it is always possible to infer the induced possibility distribution of the other disjunct. Thus we get preml : p vq prem2 : p ' Concl : q' and q' is close to q as p ' is close to - p .

S. Raha, K.S. Ray / Disjunctive syllogism

147

Mathematical formulation Let X, Y and T be three linguistic variables that take values from the domains U, V, W respectively of which T explicitly denotes the truth values. We consider the derivation of an inexact conclusion r from two typical knowledges (premises) p and q according to the following scheme:

p • XisAorYisB;TisC q • XisA';TisC' r ~---YisB';TisC" where the A's, B's and C's are approximations of possibly inexact concepts by fuzzy sets over U, V, W respectively. The deduction of r follows the following basic steps. Let

U =

ui,

V =

i=1

vi,

W =

i=1

wi. i=1

Then a possible translation of p and q can be given by the following possibility assignment equations:

P ~ II(x. Y:r) = R = ~ ~'. ~ p.R(U,, Vj; Wk)/(Ui, Vj: Wk) i

j

k

and

q --" H(x:r, = S = ~'~ ~'~/xs(u,; wk)/(ui; wk) i

k

where

tXR(U, V; W) = min{1 -- IXA(U), tZ,(V), /Xc(W)} and

tXs(U; w) = min{/xA,(U), /Xc,(W)}.

It should be noted that other meaningful interpretations of R and S are possible as well. The conjunction of R and S, to be denoted by R 0 S, is given by

Rn

:

: Sl

= • Z Z uRo. (u,, i

j

k

wkl/(u,,

wk)

where tZRng(U,, Vj; Wk) = inf[/XR(Ui, Vi; Wk), tZs(Ui, Wk)]. Projecting R A S on V × W we obtain the relational matrix for conclusion as (induced)

n¢v:,.) = Projv×w[R A S] such that re(v; w ) = sup,,[p.Rng(U, V; W)]. In order to obtain the above conclusion in the form of r project II¢y:r) separately on V and W. Thus, B' = Proj[H(v:r)] = sup sup[tZRns(U, V; W)] V

W

tl

and C " = Proj[H(v:r)] = sup sup[genS(u, v; w)]. W

v

u

Now, since

II, x,y:T)[II(x:T) = Q] = R n Q, set

O = ~?~ ~ tzo(ui; wk)/(ui; wk) i

k

where p.o(ui; wk) = inf{/zA,(U), /Xc,(W)}

S. Raha, K.S. Ray / Disjunctive syllogism

148

as the possible translation of an inexact premise q: X is A'; T is C' into a possibility assignment equation. Hence, if we choose A' = n o t A and C' = C then H~v;r) = Projv×w(lI~x,v;T)[H~x:T)= O])

= sup{/XR(U, V; W ) ^ IZo(U; W)} tt

= sup{inf(1 - tZA(U), IZB(V), I X c ( W ) ) ^ inf(1 - tZA(W), /Zc(W))} It

= sup{inf(l -/Za(U),/~B(v),/Xc(W))} u

= sup{(1 u

-- ~.~A(U)) A /..£B(V) A/.£c(W)}

= {p,n(V),/zc(w)}

if sup(1 - I~A(U)) >i {/zs(v), /-~c(W)}. ll

Then projecting H(y;T) on V and W respectively we obtain after retranslation (Y is B; T is C). Next let Xi (i = 1, 2 . . . . . n) be n such variables and V~ (i = 1, 2 . . . . . n) be the respective universes of discourse. Let Ji

Ui~-Eu~, j

i=l,2,...,n.

1

Consider a second model where from premises p and q we derive a conclusion r of the form p : XlisAlorX2isA20r...orX,

isAn;TisC

q : XlisA'~;TisC' r ~- X2 is A2 or X3 is A~ or . . . or X, is A',; T is C". Here Ai, A; (i = 1, 2 , . . . , n) are possibly approximate representations of inexact concepts by fuzzy sets over Ui ( i = 1, 2 , . . . , n ) . The translation of the logical relation between sentences appearing in the premise P into a fuzzy relation gives p ~ H~x,,x2,..,x,,.r) = R ~_ UI × U2 × " " " x Un × W

where tXe(Ul, u 2 , . . . ,

u,; w) = inf{1 -/ZA,(U,), /ZA,(U2). . . . , tZA,(U,), /Xc(W)},

and translation of q gives q ~ H~x;r) = S

where I~s(U; w) = inf{/Xai(U), /Zc,(W)}.

The conjunction of R and S, denoted by R fq S, is given as follows: R N S = IIcx,,x2,...,x,,;r)[IIcx;y) = S]

such that ~ R n s ( u l , u2 . . . . .

u.; w) = {~R(u,, u2 . . . . .

u.; w ) ^ ~s(u~; w)}.

Then projecting R N S on U2 x U3 × • • • × U, × W we obtain, for conclusion, a relational matrix H~x~,x ...... Xo:T) = Projv~×v~ .... ×vo×w[R fq S] = m

(say)

S. Raha, K.S. Ray / Disjunctive syllogism

149

where ~(u2,

u3 . . . . .

u . ; w) = s u p { ~ ( u ~ ,

u~ . . . . .

u~; w)}.

Ul

For a m o r e meaningful inference we project M over u~ (i = 2, 3 . . . . .

n) and then W separately to obtain

r ~---X2 is A; or X3 is A; o r . . . or X , is A'~; T is C" where A ~ = P r o j u , M;

i=2,3 .... ,n

and

C"=ProjwM.

Let us n o w consider a third model: p : Xi is A~ or X2 is A2 or . . . or X m is q :

Am;

T is C

(m <~n)

X1 is A'~ or X,, is A;, or . . . or X,~ is A~ or Xm+~ is A,,+~ or . . . or X , is A~; T is C '

r <-- X2 is A2" o r . .. or Xm is A " or X m + j is A;,,+I or . . .

or

X n 1s' A,,," T is C"

w h e r e the s e q u e n c e {Sl, s 2 , . . . , s k } is a subsequence of {2,3 . . . . ,m}. In this case, as before, the translation of p into a possibility assignment e q u a t i o n is given by p ~ ll
where g R ( U l , U2 . . . . .

Urn; W) = inf{a -- tZA,(UO, /ZA2(U2). . . . .

IXA,,,(Um), /.~,.(W)},

and the translation of q into a possibility assignment e q u a t i o n is given by q ---" H~x,,x,~,x~2....,x~,x,,,+,,x,,,+~,...,x,,:r) = S

where ~ s ( U l , u,.,u~2 . . . .

, u~k, Um+l . . . . .

U.; W)

= inf{txai(ul),/~A;~(Us,),. • •,/Xa;~(Usk), tZa,o_,(Um+l) . . . . .

IXAm(U,), /Xc,(W)}.

NOW l e t / ~ and S respectively d e n o t e the cylindrical extension of R and S o v e r the d o m a i n U~ x U2 × " " × UN × W.

T h e n the Conjunction of R and S, d e n o t e d b y / ~ fq S is given as II~x,.x2.....x,j)[llx,x, ......X~,,X,,+,.....X,,:r~ = S] = R Yl S = m

(say)

where # M ( U l , U2 . . . . .

U,; W) = inf{/xR(ul, u2 . . . . .

Urn; W), ~ s ( U l , Us, . . . . .

U~k, Um+l . . . . .

U,; W)}.

Projecting M over 1-12x ~ x • • • × U~ × W we obtain, for conclusion, H(x,.x~.....x,,:r) = Proju2×rz,×...×uo×w.

F o r a m e a n i n g f u l inference of the f o r m r project Proju2 ........ ×~ M over U~ and W (i = 2, 3 . . . . . by one to obtain r ~---X2 is A~ o r . . .

or X m is A " or Xm+l is A ' + I , o r . . . or X , is A',; T is C"

where A " = Proju~[Proj~× .... uo×w W],

i = 2, 3 . . . . .

A~ =Proju,[Proje~×...×u,×wM],

i=m

C~' = Projw[Proj~× .... v,×w M].

+ 1.....

m, n,

n) one

S. Raha, K.S. Ray / Disjunctive syllogism

150

Examples In this section we consider examples to illustrate the models presented in this paper. We consider variables that range over finite sets or can be approximated by variables ranging over such sets. Example 1. Consider the premises P : X is large or Y is small; T is true q : X is not very large; T is very true in which X, Y, T range over U, V, W and large, small etc. are defined (approximately) by large = 0.1//Xl + 0.45//z2 + 0.95/tt3 + 0.9/u4, not very large = 1/ul + 0.8/uz + 0.l/u3 + 0.2/u4, small = 0.9/vl + 0.95/v2 + 0.45/v3 + O.]/v4, true = 0.l/w1 + 0.45/w2 + 1/w3 + 1/w4, very true = 0.01/wl + 0.2/wz + 0.75/w3 + 1/w4. In terms of these definitions, the translations of p and q may be expressed as

p ~ FI(x,Y;T ) = R

(say)

and

q ~ H(x;T) = S

(say).

Then, for conclusion, we have II(v,;r) = Projv×w(R fq S). Hence, B = PrOjv[H(v:T)] such that /zB(v) = sup sup[/zRn~(u, v; w)] -- sup sup[inf{/zR(u, v; w), tZs(U; w)}] w

U

w

u

and C = Projw[I-l(v;r)] such that

tZc(W) = sup sup[inf{/zR(u, v; w), tZs(U; w)}]. V

U

After simple comparisons we find

B = 0.9/vl + 0.9/v2 + 0.45/v3 + 0.1/v4

and

Thus from premises p and q we infer r as given by

r*--Y is B; T is C. Example 2. Consider the premises p : XliSAaorX2isA2;TisC q : X~isA~orX3isA3;TisC' where

A, = O.1/u~ + 0.5/u~ + 0.75/u~ + 1/u~4, A'~ = 0.68/ui + 0.29/u~ + 0.13/u I + O/u~,

A2 = 0.7/u~ + 0.95/u 2 + a/u 2 + 0.82/u], A3 = 0.86/u 3 + 0.97/u 3 + 0.75/u~ + 1/u], C = 0.01/wl + 0.2/w2 + 0.75/w3 -t- 1/w4, and C ' = 0 . 1 / w I + 0.45/w2 +

1/w3 + l/w4.

C = 0.01/wl + 0.2~we + 0.75/w3 + 0.9/w4.

S. Raha, K.S. Ray / Disjunctive syllogism

151

With these definitions the translations of p and q may be expressed as p~II(x,,x~:T)= R

(say)

and

q~H(x,,x~:r) = S

(say).

Then, for conclusion, as given in the third model H~x~,x~:r~ = Proju,×u~×w(R N S) and conclude X2 is A; or X3 is A~; T is C" where A'2 = Proju2[Proju~×u~×w(R N S)], A~ = Proju~[Proju~× u~×w(/~ fq S)], and C"= Projw[Proju:×v~×w(R N S)].

Conclusion The present work shows a new direction in approximate reasoning based on fuzzy logic. As it is based on the extended concept of the law of disjunctive syllogism, therefore, it may help to derive a resolution principle in fuzzy logic. More work is in progress such as how to tackle fuzzy quantifiers in this framework.

Acknowledgments The authors wish to thank the unknown referees for their useful comments on the earlier version of this paper. Authors are also thankful to Maya Dey for preparing this manuscript.

References [1] J.A. Goguen, The logic of inexact concepts, Synthese 19 (1969) 325-373. [2] M.M. Gupta, A. Kandel, W. Bandler and J.B. Kiszka, Approximate Reasoning in Expert System (North-Holland, Amsterdam, 1985). [3] R.C.T. Lee, Fuzzy logic and the resolution principle, J. ACM 19(1) (1972) 109-119. [4] E.H. Mamdani and B.R. Gaines, Fuzzy Reasoning and its Application (Academic Press, New York, 1981). [5] M. Mukaidono, Fuzzy inference in resolution style, in: R.R. Yager (ed.), Fuzzy Set and Possibility Theory- Recent Developments (Pergamon Press, NY., 1982) 224-231. [6] S. Raha and K.S. Ray, Analogy between approximate reasoning and the method of interpolation, Fuzzy Sets" and Systems 51(3) (1992) 259-266. [7] L.A. Zadeh, Outline of a new approach to the analysis of complex systems and decision processes, IEEE Trans. Systems Man Cybernet. 3 (1973) 28-44. [8] L.A. Zadeh, Fuzzy logic and approximate reasoning, Synthese 30 (1975) 407-428. [9] L.A. Zadeh, A theory of approximate reasoning, in: J.E. Hayes, D. Michie and L.I. Mikulich (eds.), Machine Intelligence, Vol 9 (Elsevier, New York, 1979) 149-194.