DSpace at VNU: Hedge algebras, linguistic-valued logic and their application to fuzzy reasoning

DSpace at VNU: Hedge algebras, linguistic-valued logic and their application to fuzzy reasoning tài liệu, giáo án, bài g...

Vol 7, No 4 (1999) 347-361

© World Scientific Publishing Company

H E D G E A L G E B R A S , LINGUISTIC-VALUED LOGIC A N D

T H E I R A P P L I C A T I O N TO FUZZY R E A S O N I N G

NGUYEN CAT HO TRAN DINH KHANG

Institute of Information Technology, The National Center for Natural Sciences and Technology of Vietnam,

P.O.Box 626, Bo Ho, 10000 Hanoi, Vietnam E-mail: { ncho, tdkhang} @ioit nest ac vn

HUYNH VAN NAM

Department of Mathematics, Quinhon University of Pedagogy, 170-Nguyen Hue, Quinhon, Binhdinh, Vietnam

NGUYEN HAI CHAU

Mathematics-Mechanics-Informatics Faculty, College of Natural Sciences, Hanoi National University

Received 15 January 1999 Revised 21 June 1999

People use natural languages to think, to reason, to deduce conclusions, and to make decisions Fuzzy set theory introduced by L A Zadeh has been intensively developed and founded a computational foundation for modeling human reasoning processes The contribution of this theory both in the theoretical and the applied aspects is well recog-nized However, the traditional fuzzy set theory cannot handle linguistic terms directly

In our approach, we have constructed algebraic structures to model linguistic domains, and developed a method of linguistic reasoning, which directly manipulates linguistic terms In particular, our approach can be applied to fuzzy control problems

In many applications of expert systems or fuzzy control, there exist numerous fuzzy reasoning methods Intuitively, the effectiveness of each method depends on how well this method satisfies the following criterion: the similarity degree between the conclusion (the output) of the method and the consequence of an if-then sentence (in the given fuzzy model) should be the "same" as that between the input of the method and the antecedent

of this if-then sentence To formalize this idea, we introduce a "measure function" to measure the similarity between linguistic terms in a domain of any linguistic variable and to build approximate reasoning methods The resulting comparison between our method and some other methods shows t h a t our method is simpler and more effective

Keywords: Linguistic-valued fuzzy logic, linguistic variable, fuzzy reasoning, hedge

alge-bra

347

63

1 Computational approach to human reasoning

Fuzzy set theory was introduced in 1965 as a method of modeling human reasoning From the mathematical viewpoint, the main idea of this approach is to embed the

finite linguistic domain of linguistic variables into the set of all functions T(U, [0,1]) defined on a universe U Based on the rich computational structure of T(U, [0,1]),

ones can create several methods for reasoning This embedding has led to numerous successes, but it also has a problem: there are only finitely many linguistic terms and infinitely many functions, so, although we can represent each term by a function, most of the functions do not have direct linguistic meaning4'10

In our paper, we will try to recover an algebraic structure of linguistic domains

or, algebraically, to embed these domains into natural algebraic structures, which only contain linguistic terms (and no other elements) Then, we will introduce a linguistic reasoning method for directly handling linguistic terms By equipping the resulting algebra with a metric, we can analyze different methods of fuzzy multiple conditional reasoning

This paper is an overview of our research results; for more details, we refer the reader to4'5'6'1 0'1 1

2 Hedge algebras as algebraic models of linguistic domains

Mathematical structures on a given set of truth values play an important role in studying the corresponding logics Let us therefore find an appropriate mathemat-ical structure of a linguistic domain of a linguistic variable

As an example, let us consider the linguistic variable TRUTH with the domain dom(TRUTH) = {True, False, Very True, Very False, less True, More-or-less False, Possibly True, Possibly False, Approximately True, Approximately False, Very Possibly True, Very Possibly False, } This domain is a partially ordered set (poset), with a natural ordering a < b meaning that b describes a larger degree of

truth From the algebraic viewpoint, this set is generated from the basic elements

(generators) C = {True, False} by using hedges, i.e., unary operations from a set

H = {Very, Possibly, Approximately, More-or-less, } So, this domain can be described as an abstract algebra X_ = (X, C,H,<), The meaning of each term from X_ is described by its relation with other elements

Let us formulate the natural properties (axioms) that such an algebra should

satisfy In the above example, each element of the set H either increases or decreases the degree of truth Therefore, it is natural to assume that all elements of H are ordering operations, i.e., that for every h £ H, either hx < x for all x £ X, or

hx > x for all x We say that two operations h,k £ H are converse if Vx £ X (hx <

x iff kx > x), and compatible if V# £ X (hx < x iff kx < x) These relations divide the set H into two subsets H + and H~ so that every operation in H + is converse

w.r.t any operation from H~ and vice-versa, and operations belonging to the same

subsets are compatible with each other

64

If h modifies the linguistic terms stronger than x, i.e., if \/x G X (hx < kx <

x or hx > kx > x), then we denote it by h < k It is reasonable to assume that the sets (H + , <) and (H~,<) are modular lattices with the greatest elements denoted correspondingly by V (Very) and L (Little, or Less) In each of these lattices, the least element is the identity i" (for which Ix = x for all x)

When k and h both belong to B~~, we get kx < x and h(kx) < kx, so we get

a complete description of the order between the three elements x, kx, and hkx: hkx < kx < x When k € H~ and h E H + , then kx < x and hkx > kx, but it is not clear whether hkx < x or hkx > x Intuitively, it is reasonable to assume that the operation h changes the degree in the relation between kx and x, but should not change the relation itself, i.e., that we should have kx < hkx < x Generalizing this intuition, we require that for each pair of hedge operations h and k, either h is positive w.r.t k meaning that \fx € X {hkx < kx < x) or \/x € X (hkx > kx > x),

or h is negative w.r.t k, meaning that V# £ X (kx < hkx < x) or \/x € X (kx > hkx > x) For example, Very is positive w.r.t Very, More, Less, and negative w.r.t Possibly, Approximately, More-or-less

(Al) Each hedge operation is either positive or negative w.r.t the others, including itself

Another natural requirement is semantics heredity of linguistic hedges: since hedges

are modifiers or intensifiers, they inherit the meaning of terms they act on Hence,

the meaning of Less Possibly True inherits that of Possibly True and the meaning

of Possibly Less True inherits that of Less True As a result, from Possibly True

> Less True, we can conclude that Less Possibly True > Possibly Less True To formalize this idea, let is denote, by H(w), the set of all terms generated from term

w by different hedges

(A2) If terms u and v are independent, i.e., u £ H(v) and v £ H(u), then for all

x € H{u), we have x (£ H(v) In addition, if u and v are incomparable (i.e.,

u <£v and v jtu), then so are x and y, for every x € H(u) and y E H(v) (A3) If x 7^ hx then x £ H(hx) and iih^k and hx < kx then h'hx < k'kx, for all

h, k, hi and k' in H Moreover, if hx / kx then hx and kx are independent (A4) If u £ H(v) and u < v (resp., u > v) then u < hv (resp., u > hv), for any hedge h

As a result, we arrive at the following definition:

Definition 1 An abstract algebra X_ = (X,C,H,<), with H decomposed into

H + and H~ as above, is called a hedge algebra (HA, for short) if it satisfies the properties (A1)-(A4)

Some pairs of values (u,v), e.g., True and False, have the property that u is

"much weaker" than u in the sense that not only u < v, but also if we apply

an arbitrary string to hedges to u and an arbitrary (different) string of hedges to

v, still the w-generated element x = h n h n -i h\u will still be smaller than the

^-generated element y = k m k m -i k±v We will denote this relation by u <C v

When u is fixed, then we can describe such "^-generated" elements by their canonical representations h n h n -i h\u is a canonical, i.e., representations in which

h n h n -i hiu ^ h n -i hiu

Theorem l 1 0 Let X_ = (X,C,H,<) be an HA Then, the following assertions hold:

(i) If x is a fixed point of an operation h, i.e., if hx = x, it is also a fixed point

of all other hedge operations

(ii) If x has a representation of the form x = h n h n -i h\u, then there exists an indexi <n such that hi h\u is a canonical representation of x and, for all

j > i, we have hjX = x

(iii) For every hedge operations h, k, if x < hx (resp., x > hx), then Ix <C hx

(resp., Ix ^> hx), and if hx < hx and h ^ k then hx <^ikx

Note that (i) is simple but interesting and intuitively reasonable, since it shows that if the meaning of a vague concept can not be changed by applying a specific

hedge, then it cannot be changed by using other hedges either The following theorem give us some criteria for determining the relative position of elements in a

HA

Theorem 2 Suppose that x = hn .h\u and y = k m .k\u are canonical

repre-sentations w.r.t u Then, there exists an index j < min{ra,n} + 1 such that for

every i < j , we have hi = k{, and:

(i) x < y iff hjXj < kjXj, where Xj = hj-i h\u;

(ii) x = y iffn = m = j and hjXj = kjXj;

(iii) x and y are incomparable iff hjXj and kjXj are incomparable

Although HA models the natural structure of linguistic domains rather well, but,

in general, HA is not even a lattice It can be, however, extended to a lattice, in

a manner similar to the construction of real numbers by adding "limit elements"

to the set of rational numbers Namely, in11, the authors added limit elements to

HA algebraically via operations inf and sup whose intuitive meaning is that inf (x)

and sup(#) are, respectively, the infimum and the supremum of the set H(x) in the

poset X As a result, they obtained the algebraic structure of the extended hedge

algebra (EHA, for short)

For such extended hedge algebras, we can prove the following important prop-erty:

Theorem 3 1 1 Each extended hedge algebra X_ = (X,C,H e ,<), where H e = H U

{inf, sup}, is a complete lattice Therefore, we can add the lattice operations join U

and meet f) to the structure of JL and write X_ = (X, C, if, U, D, <) ^

3 Symmetric hedge algebras and linguistic-valued logic

As we have said above, the structure of the set of truth values plays an important role in determining characteristics of a logic It is therefore important to study linguistic domains of linguistic truth variable, which contain two primary terms

True and False

Let us consider a HA which has two generators / < £, and an additional

gen-erator W lying in between fadt and satisfying the condition hW = W for every

h € H We say that y is a contradictory element to x if there exists a representation

of x as x = h n h\c for some c € C, c ^ W, for which y can be represented as

y = h n h\d for some generator d € C which is different from c and W For example, each of two terms VeryPossiblyTrue and VeryPossiblyFalse is a contra-dictory element to the other One can prove that W a contracontra-dictory element to

itself

If y is a contradictory element to x, we write y = —x In general, every element may has several contradictory elements If X_ has the property saying that every element has a unique contradictory element then we call such HA symmetric In a

symmetric HA, — is thus a new unary operation The following is a characterization

of symmetric HA:

Theorem 4 A HA X is symmetric iff for every x in X_, x is a fixed point if and

only if —x is a fixed point

Now, we can introduce an implication operation in a standard way: x => y = — xUy, for any two elements x } y € X The following theorem shows that each symmetric

EH A can be taken as a logical foundation for reasoning methods:

Theorem 5 For every symmetrical extended HA X_ = (X, C, H, — ,U, f\ < ) , the

following properties hold:

(i) —hx = h{—x), for any h € H;

(ii) - - x = x; - 1 = 0, - 0 = 1? and -W = W;

(hi) —(xUy) = (—x fl — y) and —{xf\y) = (—x U — y);

(iv) xH-x <W <yU -y;

(v) x > y iff - x < -y;

(vi) x =» y = -y => -x;

(vii) x =^ (y =$> z) = y ^ (x => z);

(viii) x => y > x' => y f if x < x' and/or y >y f ;

(ix) 1 = ^ # = # ?# = ^ 1 = 1 , 0 = ^ # = 1 and x =^ 0 = — x;

(x)x=>y>Wiffx<Wory>W;

(x') x=>y<W iffy <W and x>W;

(xi) x => y = 1 iff x = 0 or y = 1

This theorem shows that logics based on symmetric EHA's are non-classical The property (iv) shows that these logics are Kleene algebras, and (ii), (x), and (xii)

show that the set of generators C is a three-element Lukasiewicz algebra which is a

symmetric subalgebra of the original EH A

67

Many multiple-valued logics use elements of the interval [0,1] as truth values For symmetric EHA's, the use of this interval is justified by the following result:

Theorem 6 11 Suppose thatH + andH~ have the same number of elements and are linearly ordered Then, there is an isomorphism ip from X_ = (X, C, H, —, U, f\ <) into [0,1] for which y> preserves the ordering of X , (p(—u) = 1 — (f(u), (f(u U v) = max((p(u),(p(v)), Lp{uC)v) = min(cp(u),cp(v)), and (p(u ^ v) = max(l — ip(u), (p(v))

4 A refinement of hedge algebras

Let us describe how the hedge algebra X_ = (X,C,H,<), where H = H + U H~, can be refined Let H' be either H~ or H + We have assumed that H' is a modular lattice Therefore, H' can be graded by its height function (see, e.g.,

Birkhoff1), and then, decomposed into grades H^, i = 1 , , /', each of which consists

of incomparable elements It is reasonable to assume that any two elements of

H' belonging to different grades are always comparable Let LH\ denote a free distributive lattice generated if we take the set H\ of mutually incomparable hedges

as the set of its free generators We will call LH\ i-th class of hedge operations Let

LH = LH + ULH~ be a union of all such classes Then, RX = (RX, C, LH, <) is a new algebra, with C as the set of generators and LH as the set of unary operations; the original algebra X_ is algebraic substructure of this new algebra

The following axioms describe natural properties of linguistic values in this new algebra:

(Rl) Every operation in LH + is converse w.r.t every operation from LH~, and the greatest V in LH + is either positive or negative w.r.t every operation in

LH

(R2) Property of unambiguous meaning and semantics heredity:

(i) For any h,k € LH, if hx ^ kx, then hx and kx are independent, i.e hx £ LH(kx) and kx £ LH(hx) In particular, all generators are independent (ii) If u, v are independent, then x £ LH(v), for all x € LH(u)

(hi) If x ^ hx then x $ LH(hx)

(R3) Comparability and incomparability: For any h, k € LH such that h ^ k,

(i) If hx, kx are incomparable, then so are u and v, for any u € LH(hx) and

v € LH(kx)

(ii) If hx < kx and either h, k do not belong to the same class LH{, for some

i, or hx = kx, then h'hx < k'kx, for all h!, k' in LH

(iii) If h and k belong to the same LH[ and hx < kx, then for every string

of operations a £ LH*, we have ohx < akx and My € LH(kx) (ahx <

y iff akx < y) and \/z £ LH(hx) (z < akx iff z < ahx)

Definition 2 An algebra RX = (RX, C, LH, <) which containing X_ =

(X, C,H,<) with the same ordering relation as a substructure is called a refined hedge algebra (RHA, for short) if it satisfies axioms (R1)-(R3)

For RHA's, several important results can be proven which are similar to the above results about HA's; see, e.g.,5'6 The main result about RHA's is as follows:

Theorem 7 6 Every refined HA RX = (RX,C,LH, <) is a distributive lattice

5 Reasoning by directly handling linguistic t e r m s

We shall give a natural representation of human knowledge and, then, inference rules

to handle linguistic terms Normally, a basic element of human knowledge consists of

two components: a vague sentence and a truth (belief) degree which is also expressed

in linguistic terms An elementary vague sentences can be expressed by p(x;u), where x is a variable, u is a vague concept, and p(.;.) is a linguistic analog of the

classical predicate For example, the phrase "Robert is studying hard" is formalized

as P = study (robert,hard), and the phrase "It looks like Robert is studying hard" is formalized as a pair (P,RLooksLike) In general, by an assertion, we mean a pair

A = (p(x;u),t), where p(x;u) is a vague sentence and t is a linguistic truth degree for which t > W By a knowledge base, we mean a finite set K of assertions From the given knowledge based, we can deduce new assertions by using inference rule of

the type

, , ( rn, tn ) (Qi,si), ,(Q m ,s m )' where (P{,ti),ti > W,i = 1 , ,n, are premises and (QJ,SJ),SJ > W,j = 1 , ,ra, are conclusions The following are the natural inference rules:

Rule of moving hedges: For any string of hedges s and any hedge h,

((P,hu),aTrue) ((P,u),ahTrue)

1 j ((P,u),ahTrue) l } ((P, hu), a True) According to these rules, from (Mary is VeryAttractive, Possibly True), we deduce that (Mary is Attractive, Poss Very True) and that (Mary is Poss Very Attractive, True) These rules are easy to apply One can show that they give the same results

as the rules following from the composition rule in fuzzy set theory

Rule of moving hedges for implications To describe this rules, we need to

introduce several relevant notions Algebraically, a valuation is a homomorphism val from the set of all formulas to a hedge algebra of linguistic truth values (a

homomorphism here is a mapping which preserves all logical connectives considered

as operations) For P = -*hQ, where h is a hedge, we shall write h-^Q if for every val such that val(-^hQ) = ac, we have val(-<Q) = ahc Similarly, for P = hQOhQ f , where 0 is an arbitrary binary logical connective, we shall write P = h(Q 0 Q')

if for every val for which val(P) = ac, we have val(Q 0 Q') = ahc, where c

is a generator True or False Formulas P which have these properties are called distributive w.r.t hedges For such formulas, the following deduction rules are

reasonable:

69

(RTI1) ^- P ~* h ^ aTrue ^ (hP,crTrue)

(RTI2)

(P -¥ Q,ahTrue) (P -> Q,ahTrue),(P,aTrue) (hP-*hQ,aTrue)

where <r, a are strings of hedges

Rule of m o d u s ponens and modus tollens

(P -» Q,aTrue), (P, True)

(RMP)

(RMT)

(Q, a True) ( P -^ Q,aTrue),(^Q, True)

(-iP.alrue)

An important intuitive property of implication, which has been accepted and ex-amined by many researchers, will be stated in our study in the following form:

(aP(x, u) -¥ aQ(x, v), a True)?

where a, a are strings and P, Q are distributive w.r.t hedges

Rule of substitution:

(RSUB) g ^ ,

P{a, u)

where # is a variable and a is a constant

Rule of equivalence formulas substitution:

P^Q,(F(P),aTrue) {1Xrj) (F(Q/P),aTrue) ' where F(P) is a formula containing P as a subexpression, and F(Q/P) denotes the formula obtained from F by replacing all occurrences of P in F with Q

The notion of a derivation which enables us to deduce, from a given set K (knowledge base) a conclusion (P, t) by means of above rules, can be defined in a

similar way as in traditional mathematical logic; see, e.g.,13 In this case, we write

K h (P,t) Let C(K) denote the set of all possible conclusions: C(K) = {(P,t) :

K \- (P, £)} A knowledge base K is called consistent if, from K, we cannot deduce two assertions (P, t) and (P, / ) with the same vague sentence P for which t > W and f <W The following theorem justifies these definitions:

Theorem 8 4 Let K be a knowledge base Then:

(i) ifKh(P,t) thent>W;

(ii) if K is consistent, then so is C{K);

(iii) if there exists a two-valued valuation for K then K is consistent

Let us illustrate deduction by an example In this example, the knowledge base consists of the following three statements:

(i) The sentence "If a student works hard and his university is high-ranking, then

he will be a good employee" is True

(ii) The sentence "The University where Robert studies is very high-ranking" is

Possibly True

(hi) The sentence "Robert is studying rather hard" is True

We build a derivation as follows:

(1) is(Univ(Robert),VeryHigh-ra), PossTrue) (by assumption),

(2) (is(Unw(Robert),PossVeryHigh-ra), True) (by (1) and Rule (RT)),

(3) (study(Robert^RatherHard), True) (by assumption),

(4) (study(x,Hard) A is(Univ(x),High-ra) -> empl(x,Good), True) (by

assump-tion),

(5) is(Univ(x),High-ra) -» (study(x,Hard) -» empl(x,Good)), True) (by (4) and

Rule (RE)),

(6) (PossVery is(Univ(Robert),High-ra) —> (PossVery (study(Robert,RatherHard) -> empi(Robert,Good)), True)) (by (5) and Rules (RPI), (RSUB)),

(7) (PossVery (study(Robert,RatherHard) —> empl(Robert>Good)), True) (by (2),

(6) and Rule (RMP)),

(8) (study(Robert,RatherHard) -» empl(Robert,Good), PossVery True) (by (7) and

Rule (RT1)),

(9) (study(Robert,RatherHard) —> empl(Robert,Rather Good), PossVeryTrue) (by

(8) and Rule (RPI)),

(10) (study(Robert,RatherGood), PossVeryTrue) (by (3), (9) and Rule (RPM)), (11) empl(Robert,PossVeryRatherGooa), True) (by (10) and Rule (RT)),

(12) empl(Robert,Good), PossVery Rather True) (by (11) and Rule (RT))

6 Multi-conditional fuzzy reasoning m e t h o d based on topology of hedge algebras

Let us consider the following fuzzy model:

(1) IF X = Ax THEN Y = B x

(2) IF X = A 2 THEN Y = B 2

(1)

(n) IF X = A n THEN Y = B n

Informally, each IF-THEN sentence defines a fuzzy "point" and, therefore, this

model describes a fuzzy curve C in the Cartesian product X x y , where X and

y are linguistic domains regarded as hedge algebras of the linguistic variables X

and y , respectively Then, the fuzzy reasoning problem "For a given fuzzy model (1) and an input A, find an output B corresponding to A" can be considered as

an interpolation problem for a given fuzzy curve C This analogy with traditional numerical extrapolation suggests that we introduce metrics in X and y8'1 2, and apply metric-motivated interpolation methods to find a solution to the above fuzzy

reasoning problem Crudely speaking, to get the output corresponding to A, we must "combine" the outputs B{ corresponding to those inputs Ai which are close to

A It is reasonable to require that the closer A{ to A, the more "weight" we should give to the corresponding output B{ in the resulting combination Therefore, we

need to be able to describe to what extent different terms are close to each other

In mathematical terms, we need a metric on the set of all possible terms

Another idea is that if we have two rules for which the inputs Ai and Aj are close

to A, and Ai is "less fuzzy" (more crisp) than Aj, then we should give more weight

to the rule for which the input is less fuzzy and therefore, for which, hopefully, the output is less fuzzy either To describe this idea, we need to define a numerical measure of fuzziness of a term

We will show how these numerical measures (metric and fuzzy measure) can be defined, and we will show that the resulting methods lead to better results that the existing non-metric fuzzy interpolation techniques

By a metric on a hedge algebra X, we understand a function p from I x l t o

[0, oo) such that:

• p(x, x) = 0,

• p(x,y) = p(y,x),

• p(x,z) = p{x^y) 4- p(y,z) for all #, y, and z for which y is in between x and z (i.e., x>y>zoix<y<z)

• For all hjk £ H and for all &,y € X , p(hx,x) _ p(hy,y)

p(kx,x) p(ky,y)'

The last requirement formalizes the intuitively reasonable idea that the relative

meaning of h in comparison with k does not depend on linguistic terms to which

they applied

Definition 3 We say that a HA X with a metric p is similar to the HA X' with

a metric p' if there exists a one-to-one mapping f from X_ onto ]C_ such that y is in between x and z iff f(y) is in between f(x) and f(z), and y ' , = l( , ' ( ,

for all x^y,z

Recall that H(x) is the set of all elements in X generated from x by using hedges

Proposition 1 For every x,y, the sets H(x) and H(y) are similar

Note that if hu < x = h'u < h"u, then H{hu) < H(x) < H{h"u) Therefore, Proposition 1 is a basis for constructing various metrics of X

The next task is to define a "fuzziness measure" of a linguistic term The more the hedges change the term, the fuzzier it is Therefore, intuitively, as a measure of

fuzziness of a term x, we can take the "diameter" of the set H(x) of all the terms obtained from x by using different hedges To describe this diameter numerically,