421
INFORMATION SCIENCES
Note on Fuzzy Languages?
E. T. LEE
AND L. A. ZADEH
Department of Electrical Engineering and Computer Sciences
Universityof California, Berkeley, California
ABSTRACT
A fuzzy language is defined to be a fuzzy subset of the set of strings over a finite alphabet.
The notions of union, intersection, concatenation, Kleene closure, and grammar for such
languages are defined as extensions of the corresponding notions in the theory of formal
languages. An explicit expression for the membership function of the language Z,(G)
generated by a fuzzy grammar G is given, and it is shown that any context-sensitive fuzzy
grammar is recursive. For fuzzy context-free grammars, procedures for constructing the
Chomsky and Greibach normal forms are outlined and illustrated by examples.
1. INTRODUCTION
The precision
of formal languages contrasts rather sharply with the
imprecision of natural languages. To reduce the gap between them, it is
natural to introduce randomness into the structure of formal languages, thus
leading to the concept of stochastic languages [l-3]. Another possibility lies
in the introduction of fuzziness. This leads to what might be called j”uzzy
languages.
It appears that much of the existing theory of formal languages can be
extended quite readily to fuzzy languages. In this preliminary note, we shall
merely sketch how this can be done for a few basic concepts and results. More
detailed exposition of the theory of fuzzy languages will be presented in
subsequent papers.
Our notation, terminology, and reasoning parallel closely the presentation
of the theory of formal languages in Hopcroft and Ullman [4].
t This work was supported in part by the Army Research Office, Durham, DAHCO-469-0024.
fRformatio~Sciences 1 ( 1969), 42 I-434
Copyright 0 1969 by American Elsevier PubIi~ing
Company, Inc.
422
E. T. LEE AND L. A. ZADEH
2. BASIC DE~NITIONS
As usual, we denote by VTa set of terminals; by V, a set of non-terminals,
with Vr tl YN= $; by V,* the set of finite strings composed of elements of
V,; and by (VT U VN)* the set of finite strings composed of the elements of
Vr or V,. A generic string in VT* is denoted by x or, more generally, by a
lowercase letter near the end of the Latin alphabet.
Fuzzy language
A fuzzy language, L., is a fuzzy set? in VT*. Thus, L is a set of ordered pairs
L = {(x, P&)f9
XE v**,
(1)
where p&x) is the grade of membership of x in L. We assume that pL(x) is a
number in the interval 10, I].
A trivial example of a fuzzy language is the fuzzy set
L={(0,1.0),(1,1.0),(00,0.8),(01,0.7),(10,0.6),(11,0.5)}
in (0, I)*. It is understood that all strings in (0, I)* other than those listed have
the grade of membership 0 in L.
Union, intersection, concatenation, and Kieene closure
Let L, and L2 be two fuzzy languages in VT*. The union of L, and LZ is a
fuzzy language denoted by L, + LZ and defined by
EI.~,+&) = max oL&),
P&))~
XE vr*.
(2)
In effect, Ll +-LZ is the union of the fuzzy sets L, and L2, Employing v as
an infix operator instead of max and omitting the argument X, we can write (2)
more simply as
(3)
PLI+L2 = PLI v clL2.
The ~~te$sect~onof Lt and L2 is a fuzzy language denoted by L1 fl L2 and
defined by
pL, n&)
= min(k,(x),
cLL1(x)),
x E VT*,
(4)
t Intuitively, a fuzzy set is a class with unsharp boundaries, that is, a class in which the
transition from membership to non-mem~rship
may be gradual rather than abrupt. More
concretely, a fuzzy set A in a space XI= (x) is a set of ordered pairs {(x, pa(x))). where ~~(x)
is termed the gmde of membershipof x in A. (See [S] and (61 for more detailed discussion.)
We shall assume that pa(x) is a number in the interval [0, 11; more generally, it can be a
point in a lattice [7, 81. The union of two fuzzy sets A and B is defined by pLAv B(x) =
max(&~),p~(x)).
The intersection of A and B is defined by ~*,~(x) = min(&x),pB(x))_
Containment
is defined by ACB c-$-p,dx) Q p&x) for all x. Equality is defined by
A = B o &x)
= pa(x) for all x.
423
NOTE ON FUZZY LANGUAGES
or, employing A instead of min,
PL,
” L2 =
FL,
A PL2.
(5)
The concatenation of L, and L2 is a fuzzy language denoted by L, Lz and
defined as follows. Let a string x in V, * be expressed as a concatenation of a
prefix string u and a suffix string u, that is, x = UV.Then
I*.~,Ax) = s;p min(pL,(4, Pan),
(6)
where the supremum is taken over all prefixes u of X. Using v and A in place
of sup and min, we may express (6) in the somewhat simpler form
PL,
LAX)
=
yL,w
A IlL2(4).
(7)
It is of interest to note that (7) takes on the appearance of a convolution
when t’ is expressed as x - II, with v corresponding to the sum (integral) and
A to the product :
PL,
L&4
=
WL,W
u
A P&
-
II)).
Note also that (7) implies-by virtue of the distributivity of v and A-that
concatenation has the associative property.
Having defined the union and concatenation, we can readily extend the
notion of Kleene closure to fuzzy languages. Specifically, denoting the Kleene
closure of L by L”, we have as the definition of L*,
L*=E+LtLL+LLL+LLLL+.*-,
(8)
where Eis the null string. Note that the meaning of the multiple concatenations
LLL, LLLL, . . . is unambiguous because of the associativity of concatenation.
Fuzzy grammar
Informally, a fuzzy grammar may be viewed as a set of rules for generating
the elements of a fuzzy set.? More concretely, a fuzzy grammar, or simply a
grammar, is a quadruple G = ( VN, V,, P, S) in which V, is a set of terminals,
VNis a set of non-terminals (V, tl V, = $), P is a set of fuzzy productions,
and S E VN.Essentially, the elements of V, are labels for certain fuzzy subsets
of I’,* called fuzzy syntactic categories, with S being the label for the syntactic
category “sentence.” The elements of P define conditioned fuzzy setsf in
(VT U T/A)*.
More specifically, the elements of P are expressions of the form
where
a
cL(m+ B) = p,
p ’ 0,
(9)
and fi are strings in (V, U VN.)*and p is the grade of members!lip of
t Note that an element of a fuzzy set, L, is an ordered pair of the form (x, pa(x)).
: A fuzzy set conditioned on a is a fuzzy set whose membership function depends on C(
as a parameter.
Informarion Sciences 1 (1969), 421-434
424
E. T. LEE AND L. A. ZADEH
p given 01.Where convenient, we shall abbreviate &a -+ fl) = p to CI$ /3 or,
more simply, a -+ fl.
As in the case of non-fuzzy grammars, the expression a + p represents a
rewriting rule. Thus, if a L /3 and y and 6 are arbitrary strings in (VT U VB)*,
then
yaS -fl y/3S
(10)
and #S is said to be dire&y d~r~~~ble from yak
If al, . . . . CL,,,are strings in (V, U VN)* and a1 2 a2, . .., a,,,_] 2 am,
Pm > 0, then a1 is said to derive a, in grammar G, or, equivalently, a,,,
is derivable from ai in grammar G. This is expressed by aI 3 a,,, or simply
P29
*-*p
G
a, =S a,. The expression
Pa
al
Pm
-+a2
*-a
a,_,
+ a,
(11)
will be referred to as a derivation chain from al to a,.
A fuzzy Lamar
G generates a fuzzy language L(G) in the following
manner. A string of terminals x is said to be in L(G) if and only if x is derivable
from S. The grade of membership of x in L(G) is given by
pG(x)
=
supfi+@
+
OLd, da1
+
a2h..
., pb,,
+
x)h
(12)
where p&x) is an abbreviation for p LOGE and the supremum is taken over
all derivation chains from 5’ to x. Thus, (12) defines L(G) as a fuzzy set in
(I’, U V,)*. If L(G,) = L(G,) in the sense of equality of fuzzy sets, then the
grammars G, and G2 are said to be equivalent.
It is helpful to observe that if a production a 2 j3 is visualized as a chain
Pm
link of strength p, then the strength of a derivation chain CI~% a2 -1. -+ a, is
the strength of its weakest link, that is min@,, . . ., p,). Then, the defining
equation (12) may be expressed in words as:
&x)
= grade of membership of x in the language generated by grammar G
= strength of the strongest derivation chain from S to x.
Let &S + aI) = pr, p(ag -+ a2) = p2, . . ., p(cL, -+- x) = P~+~. Then, on
writing (12) in the form
pGtx)
=
vh
n P2
“*
A ,h+l)
(13)
it follows at once from the associativity of A that (12) is equivaIent to a more
general expression in which the successive cr’s are derivable from their
immediate predecessors rather than directly derivable from them, as in (12).
425
NOTEON~ZZY~ANGUAG~
Example. Suppose that V, = (0, I), VN = {A, B, S>, and P is given by
I*(S + AB) = 0.5
p(A + 0) = 0.5
I”@ + A) = 0.8
p(A -+ 1) = 0.6
/L(S + B) = 0,s
/L(B --f A) = 0.4
p(AB + BA) = 0.4
p(B + 0) = 0.2.
Consider the terminal string x = 0. The possible derivation chains for this
0.8
0.2
0.8
0.4
0.5
0.8 0.5
string are S + A + 0, S -+ B -+ 0, and S -+ B -+ A + 0. Hence
&O) = max(min(O.S, OS), min(O.S,O.2), min(O.S, 0.4,0.5)) = 0.5.
Similarly, the possible derivation chains for the terminal string _X= 01 are
0.2
S~A~~O~*~OA”~Ol,
0.5
S”:&$o:AAo:OAo:Ol,
0.4
0.4
0.5
S’:&#*:BA
-+
0.6
OA“2 01, and S -+ AB -+ BA -+ AA -+ OA -+ 01. Hence,
~~(01) = max(0.4,0.4,0.2,0.4)
= 0.4.
Given a grammar G, an important question which arises in connection
with the definition of L(G) is whether or not there exists an algorithm for
computing am by the use of the defining equation (12). If such an algorithm
exists. then G is said to be recursive.
3. TYP~O~G~MMA~S
Paralleling the standard classification of non-fuzzy grammars, we can
distinguish four principal types of fuzzy grammars.
Type 0 grammar
In this case, productions are of the general form CL> 8, p > 0, where E
and /3 are strings in (V, U V,)*.
Type 1 grammar (context-sensitive)
Here the productions are of the form c(iAclz z q /~cQ,p > 0, with tl,, a?,
and fl in (V, U VN)*, A in VN. and /3 # E. In addition, the production S -+ E
is allowed.
Type 2 grammar (context-free)
The allowable productions are of the form A 2 /3, p > 0, A E V,,
j3 E (V, U VN)*, /3 # E, and S -+ E.
Type 3 grammar (regular)
In this case the allowable productions are of the form A L aB or A 2 a,
p > 0, where a E V,, A, B E V,. In addition, S -+ c is allowed.
Znfor~ation Sciences 1(1969),
421-434
426
E.T.LEEAND L. A.ZADEH
In what follows, we shall focus our attention on context-free grammars.
However, there is one basic property of context-sensitive grammars that
needs stating at this point. Specifically, in defining L(G) we have mentioned
that G is recursive if there exists an algorithm for computing p&x). It is easy
to demonstrate that context-sensitive-and
hence also context-free and
regular-grammars
are recursive. This can be stated as an extension of
Theorem 2.2 in Hopcroft and Ullman [4].
THEOREM. Zf G = (V,,
G is recursive.
VT, P, S) is a fuzzy context-sensitive grammar, then
Proof. First we show that for any type of grammar the supremum in (12)
may be taken over a subset of the set of all derivation chains from S to x,
namely, the subset of all loop-free derivation chains. These are chains in
whichnocr,,i=I,...,
m, occurs more than once.
For suppose that in a derivation chain C,
i. Now consider the chain C’ resulting from
/I,+,
PI
PI+ I
PI+1
replacing the subchain CQ--+ ***+ crj ---+ CX,+~in C by CQ--f “j+i.
Clearly, if C is a derivation chain from S to x, so is C’. But
cci, say, is the same as aj, j>
min(pi , . . .,P~,P~+~~.. .,P~+~,. . .,pm+d < minh,.
. .,~PI+I~.
. .,fb+d
and hence C may be deleted without affecting the supremum in (12). Consequently, we can replace the definition (12) for pL(x) by
CL&) = sup minMS -+ ai), ~(a, -+ Q), . , ., PL(G + x)),
(14)
where the supremum is taken over all loop-free derivation chains from S to x.
Next we show that for context-sensitive grammars the set over which the
supremum is taken in (14) can be further restricted to derivation chains of
bounded length la, where lo depends on 1x1(length of x) and the number of
symbols in V, U V,.
Specifically, if G is context-sensitive, then because of the non-contracting
character of productions in P, we have
ifj> i.
(1%
ltrjl a Ia*1
Now let k be the number of symbols in VT U V,,,. Since there are at most k1
distinct strings in (VT U VN)* of length 1, and since the derivation chain is
loop-free, (15) implies that the total length of the chain is bounded by
I,,= 1 +k+.+.+k’“‘.
To complete the proof, we have to exhibit a way for generating all finite
derivation chains from S to x of length <Z,,.For this purpose, we start with S
427
NOTE ON FUZZY LANGUAGES
and using P generate the set Qi of all strings in (V, U I’,)* of length ~1x1
which are derivable from S in one step (that is, are directly derivable from S).
Then, we construct Q2--the set of all strings in (V, U V,)* of length <lx\
which are derivable from S in two steps-by noting that Q2 is identical with
the set of all strings in (V, U I’,)* of length ~1x1 which are directly derivable
from strings in $3,. Continuing this process, we construct consecutively Q3,
Qe . . .. Q, until I+= l,, or QF= 4, whichever happens first. Since the Qn,
x= f,..., F, are finite sets, their knowledge enabtes us to find in a finite number
of steps all loop-free derivation chains from S to x of length Z& and thus to
compute p&x) by the use of (14). This, then, constitutes an algorithm-though
not necessarily an efficient one-for the computation of&x). Consequently,
G is recursive.
4. FUZZY CONTEXT-FREE GRAMMARS
As was stated in the introduction, many of the basic results in the theory
of formal languages can readily be extended to fuzzy languages. As an illustration, we shall sketch-without
giving proofs-such extensions in the case of
the Chomsky and Greibach normal forms for context-free languages.
Chomsky normal form for fuzzy cantext-free languages
Let G be a fuzzy context-free
grammar. Then, any such grammar is
n
equivalent to a grammar G, in which all productions are of the form A & BC
or A L Q, where p z==
0; A, B, C are non-terminals, and a is a terminal.
It is convenient to effect the construction of G, in three stages, as follows.
First, we construct a grammar G, equivalent to G in which there are no
productions of the form A -+ B, A, B E V,.
Thus, suppose that in G we have productions of the form A -+ B, which
lead to derivation chains of the form
PI
P2
pII+,
pm+2
A+B,+B2~~~Bm--+B---+~,
where
a q! V,.
Then we replace all such productions:
A 2 B,, Bt 2 B,, . . .,
Pm+1
B,
--+
B in G by single productions of the form A z
p = min(p(A 3 B), p(B -+
a,
in which
a))
(16)
where
&4 5 B) = sup min@(A -+ B,), . . ., p(B,,, -+ B))
(17)
with the supremum taken over all loop-free derivation chains from A to B.
It can readily be shown that the resultant grammar, Gi, is equivalent to G.
information
Sciences
1 (1969), 421-434
E. T. LEE AND L. A. ZADEH
428
Second, we construct a grammar Gz equivalent to G, in which there are
no productions of the form A z B, Bz ***B,,,, p > 0, m > 2, in which one or
more of the B’s are terminals. Thus, suppose that B,, say, is a terminal a.
Then Bi in B1 B2 *s-B, is replaced by a new non-terminal C, which does not
appear on the right-hand side of any other production, and we set
~(A-tB,B,...B,...B,)=~(A-tB,BZ...C,...B,)
(18)
Furthermore, we add to the productions of G the production C, 1, a. On
doing this for all terminals in B1 - **B,,, in all productions of the form
A + B, - -*B,,,, we arrive at a grammar Gz in which all productions are of the
form A + a or A -+ B, -a-B,, m 2 2, where all the B’s are non-terminals. It
is evident that G2 is equivalent to G,.
Third, we construct a grammar G3 equivalent to Gz in which all productions
are of the form A + a or A + BC, A,B,C E V,, a E VT. To this end, consider
a typical production in Gz of the form A -f+ B, **-B,,,, p > 0, m > 2. Replace
this production by the productions
A:B,D,
1
0
(19)
-fBzDz
I
k-2
-+
&-,
&,
where the D’s are new non-terminals which do not appear on the right-hand
side of any production in G2. On performing such replacements for all productions m G2 of the form A 5 B, *e-B,,,, we obtain a grammar G3 which is
equivalent to G2. This establishes that G3-which is in Chomsky normal
form-is equivalent to G.
Example. Consider the following fuzzy grammar in which VT = {a, b} and
V, = {A, B, S}.
S’fbA
So:
B”:b
aB
A “2 bSA
0.2
A-+a
B “2 aSB.
To find the equivalent grammar in Chomsky normal form, we proceed as
follows.
First, S “2 bA is replaced by S y C, A, C, L b. Similarly, So2 aB is
429
NOTEONFUZZYLANGUAGES
0.3
replaced by So2 C, B and C, -!+ a. Also A ‘2 bSA is replaced by A + C3 SA,
0.5
C3 -!+ b; B “2 aSB is replaced by B + C, SB and C., i a.
0.3
Second, the production A 2 C3SA is replaced by A -+C3D,,
DI -&A;
0.5
and the production B “2 C,SB is replaced by B -+ C, Dz, DZ -k. SB. Thus,
the productions in the equivalent Chomsky normal form read:
S:C,A
1
C, -+b
A:C3D,
1
D, -+SA
S”2C2B
C, -+b
1
1
C2 -+ a
A
‘2,
Byb
B”: C,D,
I
D2-+SB
1
C, -3 a.
Greibach normalform
As in the case of the Chomsky normal form, let G be any fuzzy contextfree grammar. Then G is equivalent to a fuzzy grammar Gc in which all
productions are of the form A ---facr,where A is a non-terminal, a is a terminal
and LXis a string in V,*. The fuzzy grammar GG is in Greibach normal form.
Paralleling the approach used in Ullman and Hopcroft, it is convenient to
state two lemmas which are of use in constructing GG. We shall omit proofs
of these lemmas since their validity is reasonably evident and their detailed
proofs fairly long.
LEMMA 1.Let G be a fuzzy context-free grammar. Let A -+ CL,Ba, be a
production in P, A, B E V,, and al, a2 E (VT U V,)*. Furthermore, let
B-+131, . . .. B -+ & be the set of all B-productions (that is, all productions
with B on the left-hand side). Let Gr be the grammar resuit~ngfrom the replacement of each of the productions of the form A -+ aI BorZwith the productions
A +-=ljgla2, . . .. A -+ q &q, in which
p(A -+ a~BI ~2) = min(~(A + at BaJ, p(B + &)I,
i= I , -. ., r. (20)
Then G, is e~i~alent to G.
LEMMA 2.Let G be a fuzzy context-free grammar. Let A -+ Aa,, i = 1, . . ., r,
be the set of A-productions for which A is also the leftmost symbol on the righthand side. Furthermore, let A + pj, j = 1, . . ., s be the remaining A-productions,
Information Sciences 1(1969),421-434
430
E. T. LEE AND L. A. ZADEH
witha,,fijE(VT U V,)*,i= l,...,r,j=
l,...,s.LetGZbethegrammarresulting
from the replacement of the A -+ AC+in G with the productions:
A -+ /$Z,
j=l
, ***,s
(21)
i=l
,..., r,
(22)
Z-+~L,,Z+U~Z,
where
i=l ,‘.., r
P(A -+ PJ) = P(A -+ I$),
~(2 --f ai) = P(A + Aa,)
/_@ -+ uf 2) = &A --f Aq).
(23)
Making use of these lemmas, the Greibach normal form for G can be
derived as follows.
First, G is put into the Chomsky normal form. Let the non-te~nals
in
this form be denoted by A,, . . _, A,.
Second, the productions of the form A, -+ A,y, y E (V, U V,.J*, are
modified in such a way that for all such productions j> i. This is done in
stages. Thus, suppose that it has been done for i < k, that is, if
A, -fAjY
(24)
is a production with i g k, then j > i. To extend this to _&+,-productions
suppose that Ak+, -+ A, y is any production with j < k + 1. Using Lemma 1
and substituting for A, the right-hand side of each A,-production, we obtain
by repeated substitution productions of the form
A k+l
-+-A%
I>kSl.
(25)
In (23, those productions in which 1 is equal to k-t 1 are replaced by the
application of Lemma 2, resulting in a new non-terminal zk+i. Then, by
repetition of this process all productions are put into the form
Ati -+Al?‘,
l>k,yo(vNU
A, -+ ay,
a E V,
zk
-+
y
{Z,,..JJ)*
(26)
(27)
(28)
with membership grades given by Lemmas 1 and 2.
In view of (26) and (27), the leftmost symbol on the right-hand side of any
production for A,,, must be a terminal. Similarly, for A,_$, the leftmost symbol
on the right-hand side must be either A, or a terminal. Substituting for A, by
Lemma 1, we obtain productions whose right-hand sides start with terminals.
Repeating this process for Am_*, . . ., At, we eventually put all productions for
the Al, i= 1, . . .. m, into a form where their right-hand sides start with
terminals.
At this stage, only the productions in (28) may not be in the desired form.
431
NO~ON~ZZY~A~GUAGES
We observe that the Ieftmost symbol in y in (28) may be either a terminal or
m. If it is the latter, application of Lemma 1 to each
oneofthe&i=l,...,
Zr production results in productions of the desired form and thus completes
the construction.
Ex~m~ie.As a simple illustration, we shall convert into the Greibach
normal form a fuzzy grammar G in which VT = (a,b), V, = (A,, AZ, A,) and
the productions are in the Chomsky normal form :
0.2
0.8
4
-tA2A,
A2
“2
A3-+4Az
0.5
A3
Al
A, -+fIl.
0.6
A2-+b
Step 1. Since the right-hand sides of the productions for A, and A2 start
with terminals or higher-numbered variables, we begin with the production
A, -+ AL A2 and substitute A,A, for A,. Note that A, + A2 A, is the only
production with A, on the left.
The resulting set of productions is
0.8
A,
-+AzA,
AzozA3At
0.6
A2-fb
A3 “2 A2A3AZ
0.5
A3-ftl
Note that in A, “2 A2 A3 A2, 0.2 is 0.8 A 0.2,
Since the right-hand side of the production A3 + A,A,A2 begins with a
lower-numbered variable, we substitute for the first occurrence of A2 either
A,A, orb.
The new set is
A, “2 A2 A3
AzyA3A,
0.6
AZ-+b
A3 “2 A3 A, A, A,.
A3 “2 bA, A,
A,-+@
0.5
WenowapplyLemma2totheproductionsAi
-+ A3A, A,A,,A, -+ bA,A,,
and A3 -+ a. We introduce 2, and replace the production A3 + A3 Al A3 A2 by
A3 -+ bA3 A2Z3, A, -+ aZ,, 2, + A, A3 A2, and Z3 -+ Al A, A,&.
Infor~a%ion
Sciences10 969),421-434
432
E. T. LEE AND L. A. ZADEH
The resulting set is
0.6
AZ-+b
A3 ‘2 bA3A2
0.5
0.5
A,+a
A, -+ a&
Step 2, Now all productions with A3 on the left have right-hand sides that
start with terminals. These are used to replace A3 in the production A2 -+ A3 A,
and then the productions with A2 on the left are used to replace A2 in the
production Al -+ Az A3. The resulting set is
A3 “2 bA, A2
0.5
A, “2 bA, A,Z,
0.5
As-+-a
A3 -+ a&
Azo:bA,AzAl
A2:bA3A2Z3A1
0.5
A:! --f aA,
0.5
A2 -+aZ~A,
0.6
Az--+b
A, 2 bA3 Az A, A3
Al ‘:bA,AIZjAIAs
A, ‘~czA, A~
Ai ‘2 bAz
Z3 2 Al A3 A2Z3.
Z3
‘2A, A, A2
Note that in A2 “-;”bA3A,AI, 0.2 is 0.7 A 0.2. Likewise, in Ai “2 aA, A3, 0.5
is 0.5 A 0.8. The same holds for the way in which the membership grades of
other productions are determined (by the use of Lemmas 1 and 2).
Step 3. The two Z3 productions, Z3 [I; A, Ax A2 and Z3 2 A, A3 A2Z3, are
converted to desired form by substituting the right-hand side of each of the
five productions with Al on the left for the first occurrence of Al. Thus,
Z~‘~A~A~A~isrepla~dby
Z3 “2 bA3 A, AZ
ZsofbASA2A,A3AjA2
Zso~aA,A~A,A2
Zso~bAsA2Z3A,A3A3Al.
Z,“zaZ,lA,AIA3AZ
NOTE ON FUZZY
433
LANGUAGES
The other production
productions reads
for 2, is converted
A3 “1: bA, A2
similarly. The final set of
A3 “2 bA3 A2Z3
0.5
0.5
A, + aZ,
A3 +a
0.2
A2 +bA3A2A,
A2’<bA,A2Z3A,
0.5
0.5
A2 +aZ,A,
A2 +aA,
0.6
A2 +a
A, ‘<bA3A2A,A,
A, ‘:bA3A2Z,A,A,
A, +aA,
0.5
A3
0.6
A, + bA,
Z3 “2 bA3 A3 A2
Z3 “2 bA3 A3 A2Z3
0.2
Z3+bA3A2A,A3A3Az
ZJo:bA,A2A,AjA,A2Zj
Z3 ‘:aA,
Z, ‘:aA,
A3A,A2
A,A3A2Z3
Z3 O-1:bA3 A2Z3 A, A, A3 A2
Z,‘:bA
AZA
32313323
A A AZ
Z3’2aZ3A,A3A3A2
Z3 “2 aZ3 A, A3 A, A2Z,.
This concludes our brief description of the construction of the Chomsky and
Greibach normal forms for a fuzzy context-free grammar.
As was stated in the introduction, the purpose of the present note is merely
to illustrate by a few examples how some of the basic concepts and results in
the theory of formal languages can be extended to fuzzy languages. As the
reader can see, the extensions are, for the most part, quite straightforward
and require hardly any modifications in the statements of lemmas and
theorems. The proofs, however, are generally somewhat longer since they
involve not just the positivity of membership functions but their values in
the interval [0, 11.
The theory of fuzzy languages offers what appears to be a fertile field for
further study. It may prove to be of relevance in the construction of better
models for natural languages and may contribute to a better understanding
of the role of fuzzy algorithms and fuzzy automata in decision making,
pattern recognition, and other processes involving the manipulation of fuzzy
data.
Information Sciences 1 (1969), 421-434
434
E. T. LEE AND L. A. ZADEH
REFERENCES
1 P. Turakainen, On stochastic languages, Information and Control, 12 (April 1968), 304-313.
2 M. M. Kherts, Entropy of languages generated by automated or context-free grammars
with a single-valued deduction, Nuuchno-Tekhnicheskaya Znformatsyu, Ser. 2, No. l(1968).
3 K. S. Fu and T. J. Li, On stochastic automata and languages, 3rd Conference on
Information and System Sciences, Princeton University, 1969, pp. 338-377.
4 J. E. Hopcroft and J. D. Ulhnan, Formal Languages und Their Relation to Automata,
Addison-Wesley, Reading, Mass., 1969.
5 L. A. Zadeh, Fuzzy sets, Znformation and Control, 8 (June 1965), 338-353.
6 L. A. Zadeh, Toward a Theory of Fuzzy Systems, ERL Rept. No. 69-2, Electronics Research
Laboratories, University of California, Berkeley, June 1969.
7 J. Goguen, L-fuzzy sets, J. Math. Anal. Appl., 18 (April 1967), 145-174.
8 J. G. Brown, Fuzzy sets on Boolean lattices, Rept. No. 1957, Ballistic Research Laboratories, Aberdeen, Md., Jan. 1969.
Received August ZZ,I969