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

International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems
Vol. 7, No. 4 (1999) 347-361
© World Scientific Publishing Company

Int. J. Unc. Fuzz. Knowl. Based Syst. 1999.07:347-361. Downloaded from www.worldscientific.com
by UNIVERSITY OF QUEENSLAND on 02/03/15. For personal use only.

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

Institute of Information
Technology,
Center for Natural Sciences and Technology of
P.O.Box 626, Bo Ho, 10000 Hanoi, Vietnam
E-mail: { ncho, tdkhang} @ioit. nest. ac. vn

Vietnam,

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 recognized. 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 that our method is simpler and more effective.
Keywords: Linguistic-valued fuzzy logic, linguistic variable, fuzzy reasoning, hedge algebra.

63
347


348

H. C. Nguyen et al.

Int. J. Unc. Fuzz. Knowl. Based Syst. 1999.07:347-361. Downloaded from www.worldscientific.com

by UNIVERSITY OF QUEENSLAND on 02/03/15. For personal use only.

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 meaning 4 ' 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 to 4 ' 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 mathematical 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, More-or-less True, More-orless 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


Int. J. Unc. Fuzz. Knowl. Based Syst. 1999.07:347-361. Downloaded from www.worldscientific.com
by UNIVERSITY OF QUEENSLAND on 02/03/15. For personal use only.

Hedge Algebras, Linguistic- Valued Logic and Their Application

to . . .

349

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


350 H. C. Nguyen et al.

v, still the w-generated element x = hnhn-i...
h\u will still be smaller than the
^-generated element y = kmkm-i...
k±v. We will denote this relation by u When u is fixed, then we can describe such "^-generated" elements by their
canonical representations hnhn-i...
h\u is a canonical, i.e., representations in which
hnhn-i...
hiu ^ hn-i...
hiu.

Int. J. Unc. Fuzz. Knowl. Based Syst. 1999.07:347-361. Downloaded from www.worldscientific.com
by UNIVERSITY OF QUEENSLAND on 02/03/15. For personal use only.

Theorem l. 1 0 Let X_ = (X,C,H,<)
hold:

be an HA. Then, the following assertions

(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 = hnhn-i...
h\u, then there exists an

indexi j > i, we have hjX = x.
(iii) For every hedge operations h, k, if x < hx (resp., x > hx), then Ix (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 = km.. .k\u are canonical representations 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, in 11 , 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 property:
Theorem 3. 1 1 Each extended hedge algebra X_ = (X,C,He,<),
where He = 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, <).

^


Hedge Algebras, Linguistic- Valued Logic and Their Application

to . . .

351

Int. J. Unc. Fuzz. Knowl. Based Syst. 1999.07:347-361. Downloaded from www.worldscientific.com
by UNIVERSITY OF QUEENSLAND on 02/03/15. For personal use only.

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 generator 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 = hn...h\c
for some c € C, c ^ W, for which y can be represented as
y = hn... 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 contradictory element to the other. One can prove that W a contradictory 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 (v) x > y iff - x < -y;
(vi) x =» y = -y => -x;
(vii) x =^ (y =$> z) = y ^ (x => z);
(viii) x => y > x' => yf if x < x' and/or y >yf;
(ix) 1=^# = # ? # = ^ 1 = 1 , 0 = ^ # = 1 and x =^ 0 = — x;
(x)x=>y>Wiffx<Wory>W;
(x') x=>yiffy <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


352 H. C. Nguyen et al.

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)).

Int. J. Unc. Fuzz. Knowl. Based Syst. 1999.07:347-361. Downloaded from www.worldscientific.com
by UNIVERSITY OF QUEENSLAND on 02/03/15. For personal use only.

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).


Hedge Algebras, Linguistic- Valued Logic and Their Application

to . . .

353

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.

Int. J. Unc. Fuzz. Knowl. Based Syst. 1999.07:347-361. Downloaded from www.worldscientific.com
by UNIVERSITY OF QUEENSLAND on 02/03/15. For personal use only.

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
, . . . , ( r n , tn)
(Qi,si),...,(Qm,sm)'
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
l
}
((P,u),ahTrue)
((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(-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


354 H. C. Nguyen et al.

Int. J. Unc. Fuzz. Knowl. Based Syst. 1999.07:347-361. Downloaded from www.worldscientific.com
by UNIVERSITY OF QUEENSLAND on 02/03/15. For personal use only.

(RTI1) ^-P ~* h^aTrue^
(hP,crTrue)
(P -¥ Q,ahTrue)
(P ->
Q,ahTrue),(P,aTrue)
(RTI2)
(hP-*hQ,aTrue)
where Rule of modus ponens and modus tollens

(P -» Q,aTrue), (P, True)
(RMP)
(Q, a True)
( P -^ Q,aTrue),(^Q,
True)
(RMT)
(-iP.alrue)
An important intuitive property of implication, which has been accepted and examined 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 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.


Hedge Algebras, Linguistic- Valued Logic and Their Application

to . . .

355

(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:

Int. J. Unc. Fuzz. Knowl. Based Syst. 1999.07:347-361. Downloaded from www.worldscientific.com
by UNIVERSITY OF QUEENSLAND on 02/03/15. For personal use only.

(1) is(Univ(Robert),VeryHigh-ra),

PossTrue) (by assumption),

(2) (is(Unw(Robert),PossVeryHigh-ra),
(3) (study(Robert^RatherHard),
(4) (study(x,Hard)
tion),

True) (by (1) and Rule (RT)),

True) (by assumption),

A is(Univ(x),High-ra)

(5) is(Univ(x),High-ra)
Rule (RE)),

-> empl(x,Good),

-» (study(x,Hard) -» empl(x,Good)),

True) (by assumpTrue) (by (4) and

(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)
(6) and Rule (RMP)),

—> empl(Robert>Good)), True) (by (2),

(8) (study(Robert,RatherHard)
Rule (RT1)),

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

(9) (study(Robert,RatherHard)
(8) and Rule (RPI)),

—> empl(Robert,Rather Good), PossVeryTrue) (by

(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 method based on topology of hedge
algebras
Let us consider the following fuzzy model:
(1) IF X = Ax THEN Y = Bx
(2) IF X = A2 THEN Y = B2
(1)
(n) IF X = An THEN Y = Bn
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


Int. J. Unc. Fuzz. Knowl. Based Syst. 1999.07:347-361. Downloaded from www.worldscientific.com
by UNIVERSITY OF QUEENSLAND on 02/03/15. For personal use only.

356 H. C. Nguyen et al.

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 y 8 ' 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>zoixp(hx,x) _ p(hy,y)

• For all hjk £ H and for all &,y € X,
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,

Int. J. Unc. Fuzz. Knowl. Based Syst. 1999.07:347-361. Downloaded from www.worldscientific.com
by UNIVERSITY OF QUEENSLAND on 02/03/15. For personal use only.

Hedge Algebras, Linguistic-Valued

Logic and Their Application

to ...

357

we need to "quantize" the linguistic terms. So, we arrive at the following definition:
Let / be a function from PC into the unit interval [0,1] which preserves the ordering
relation on X. By a fuzziness measure fm{x) of a term x, we understand the
diameter of the set f(H(x)) = {f(u) : u € H{x)}.
For example, let X_ = (X,C,H,<)
be a hedge algebra. Assume that C = { True,
False} and H = {Less, Possible, More, Very}, where Less, Possible are negative
hedges with Less j Possible, while More and Very are positive hedges with More /
Very. In this example, Very and More are positive w.r.t. Very, More, and Less, but
negative w.r.t. Possible. The fuzziness measure of different terms x's, i.e., diameters
of the sets f(H(x)), which are shown in Figure 1.
.5
1

i

\

I

1

True
I I
*

!
V- , ,

1

,

-J

,

J

,

Diameter of

!

^

J

f(H^^yTrue))

J

'Diameter of f{H{Mo reT rue))!

1

Diameter of flHlFossibteTrue}}

D iam eterof f(H(L es sT ru e)}

«

!
\
\

D iam e ter off(H(Frue))
Fig. 1. Fuzziness measure of hedges

In reality, we only have linguistic terms, there is no fixed mapping / ; therefore,
we must define fm axiomatically:
Definition 4. A function fm : X -» [0,1] is called a fuzziness measure on X if it
satisfies the following conditions:
1) fm(c~) = fm(c+) = 0.5, where c~ and c + are negative and positive primary
terms, respectively.
2) Let hi < fi2 < • •. < hp < I < hp+i < ... < /i2p be a linear ordering of
all hedges, so that H~ = {h%, h%,..., hp} and H+ = {hp+i,...,

h,2V}. Then,
p

2p

y^ fm(hic)

= fm(c) and V j fm(hic)

i=l

i=l

2p

= 2_] fm(hic)

where c is either c~

i=p+l

or c+.
fmjhx) _ fmjhy)
fm(x) " fm(y) '
. fm(hx)
,
_
,
. . ,
i.e., the ratio ——-—~- does not depend on element x; this ratio is denoted by

3) For any two elements x and y in X, and for any hedge h,
jm\Xj

fm(h)

and called fuzziness measure of the hedge h.
73


358 H. C. Nguyen et al.

Proposition 2. A fuzziness measure fm has the following properties:
P

1) ^frn{hix)=:

2P

] P fm{hix)

i=l

i=p-j-l

P

2p

Int. J. Unc. Fuzz. Knowl. Based Syst. 1999.07:347-361. Downloaded from www.worldscientific.com

by UNIVERSITY OF QUEENSLAND on 02/03/15. For personal use only.

-fm(x)
1

2) / J / m ( A i c ) = Y j fm(hic)
«=i

=

1

= -rfm(c) = - 7 wftere c as either c~ or c+.

2=P+1

2p

3) £/m(/n) = l.
Definition 5. Let fm be a fuzziness measure on PC. Then, the function v : PC -> H
defined below is called a measure function:
1) «(c") = i / m ( 0 , «(c + ) = | +
2) v(hjhx) = v(hx) +

\fm{c+).

sign(hjhx) ^2ffm{hihx)

-

--fm{hjhx) , tfJ
i=3

3

and v(hjhx) = v(hx) + signQijhx)
where sign(h'hx)

y ^ fm(hihx)

j

-

-fm(hjhx)

ifj > P,

is recursively defined by the following equations:

a) sign(c~~) = —1? sign{c+) = -f-1,
b) sign(h'hx)

= —sign{hx)

c) sign{h'hx) = sign(hx)

if h! is negative w.r.t. h, and
if h! is positive w.r.t. h.

Proposition 3. Every measure function v has the following properties:
1) 0 < v(%) < 1 for any x £ PC
2) for any x, y G X, x < y implies that v(x) As we have mentioned before, every set (1) of IF-THEN statements can be regarded
as a fuzzy curve in the space PCx y. We now have two different approaches to solve
the multiple conditional fuzzy reasoning problems:
1) Fuzzy sets approach: Fuzzy multiple conditional reasoning methods which
based on fuzzy set theory.
2) Hedge algebra approach: Interpolation methods based on measure functions
fmx on PC and fmy on y.
To compare these approaches, we re-analyzed examples EX1, EX3, and EX6 given
in 2 ; all three examples describe a relationship between the input current intensity
(I) and the output rotation speed (N) of an electric motor. In EX1, the largest
error of a fuzzy-set based Cao-Kandel method is 300, but for our method, it is 183 9 .


Hedge Algebras, Linguistic- Valued Logic and Their Application

to . . .

359

Int. J. Unc. Fuzz. Knowl. Based Syst. 1999.07:347-361. Downloaded from www.worldscientific.com
by UNIVERSITY OF QUEENSLAND on 02/03/15. For personal use only.

In EX2 and EX3, the use of our method decreases the error, respectively, from 200
to 98, and from 100 to 55.
In Fig. 2-4, we show how the computation curves, one of which is computed
by our interpolation reasoning method using measure functions and the other is

obtained by Cao-Kandel's method using implication operator indexed by 25 (see 2 ),
approximate the given real curve in each example.

EX1

- Real curve
- Fuzzy curve
- Interpoiative curve

I (Current intensity)
Fig. 2.

EX3

- Real curve
- Fuzzy curve
- interpoiative curve

0 12

3 4 5 6 7

I (Current intensity)
Fig. 3.

75


360 H. C. Nguyen et al.

Int. J. Unc. Fuzz. Knowl. Based Syst. 1999.07:347-361. Downloaded from www.worldscientific.com
by UNIVERSITY OF QUEENSLAND on 02/03/15. For personal use only.

EX6

- Real curve
- Fuzzy curve
- Interpolative curve

0 1 2 3 4 5 6 7 8
! (Current intensity)

Fig. 4.

References
1. G. Birkhoff, Lattice Theory (Providence, Rhode Island, 1973).
2. Z. Cao and A. Kandel, "Applicability of some fuzzy implication operators", Fuzzy sets
and systems 31 (1989) 151-186.
3. N. C. Ho, "Fuzziness in the structure of linguistic truth values: a foundation for development of fuzzy reasoning", Proc. of Int. Symp. on Multiple- Valued logic, Boston
University, Boston, Massachusetts, May 26-28 (IEEE Computer Society Press, 1987)
325-335,
4. N. C. Ho, "A method in linguistic reasoning on a knowledge base representing by
sentences with linguistic belief degree", Fundamenta Informaticae 28, No. 3, 4 (1996)
247-259.
5. N. C. Ho and H. V. Nam, "A refined structure of hedge algebras", Proc. of the NCST
of Vietnam 9, No. 1 (1997) 15-28.
6. N. C. Ho and H. V. Nam, "A theory of refinement structure of hedge algebras and
its application to linguistic-valued fuzzy logic", in: D. Niwinski and M. Zawadowski
(Eds.), Logic, Algebra, and Computer Science (Banach Center Publications, PWN Polish Scientific Publishers, Warszawa, 1998).
7. N. C. Ho, H. V. Nam, and N. H. Chau, "Hedge algebras and their applications to

fuzzy logic and fuzzy reasoning", Proc. of VJFUZZY'98 (Halong Bay, Vietnam, 1998)
315-323.
8. N. C. Ho and T. T. Son, "On the distance between values of linguistic variable in hedge
algebra", J. of Comp. Sci. and Cyber. 1 1 , No. 1 (1995) 10-20 (in Vietnamese).


Int. J. Unc. Fuzz. Knowl. Based Syst. 1999.07:347-361. Downloaded from www.worldscientific.com
by UNIVERSITY OF QUEENSLAND on 02/03/15. For personal use only.

Hedge Algebras, Linguistic-Valued Logic and Their Application to ...

361

9. N. C. Ho and T. T. Son, "On the accuracy of a fuzzy model", J. of Comp. Sci. and
Cyber. 13, No. 1 (1997) 66-72 (in Vietnamese).
10. N. C. Ho and W. Wechler, "Hedge algebras: an algebraic approach to structures of sets
of linguistic domains of linguistic truth variables", Fuzzy Sets and Systems 35 (1990)
281-293.
11. N. C. Ho and W. Wechler, Extended hedge algebras and their application to fuzzy
logic", Fuzzy Sets and Systems 52 (1992) 259-282.
12. T. D. Khang, "Constructing measure function on hedge algebra and its application to
linguistic reasoning", J. of Comp. Sci. and Cyber. 13, No. 1 (1997) 16-30 (in Vietnamese).
13. H. Rasiowa and R. Sikorski, The Mathematics of Metamathematics (Polish Scientific
Publishers, Warszawa, 1968).
14. D. B. Rinks, "A heuristic approach to aggregate production scheduling using linguistic
variables", Proc. Internat. Congr. on Appl. Syts. Research and Cybernetics V I (1981)
2877-2883.

77