AbstractA new algebraic approach to the problem of equivalence, reduction and minimization of some kinds of fuzzy-automata is given. A system of necessary and sufficient conditions for equivalence of weakly initial fuzzy-automata is formulated. Some algorithms considering the equivalence for two fuzzy-automata are constructed

Peeva, Ketty G

Topencharov, Vladimir V

Elsevier - Publisher Connector 

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 84. 270-28 1 (198 1) 
Equivalence, Reduction and Minimization 
of Finite Fuzzy-Automata 
VLADIMIR V. TOPENCHAROV AND KETTY G. PEEVA 
Cenrer of Applied Mathematics. 
P. 0. Box 384. 1000 Sofia, Bulgaria 
Sub&ted by A. 0. Esogbue 
A new algebraic approach to the problem of equivalence. reduction and 
minimization of some kinds of fuzzy-automata is given. A system of necessary and 
sufficient conditions for equivalence of weakly initial fuzzy-automata is formulated. 
Some algorithms considering the equivalence for two fuzzy-automata are 
constructed. 
One classical problem in the theory of automata is equivalence, reduction 
and minimization. The problem is completely solved [ 121 for the cases of 
deterministic automata: for stochastic automata it is studied in detail in 
13, 5, 7); an attempt for the case of fuzzy-automata is given in 19, lo]. 
The theoretical foundation [ 81 of the well known algorithm of Even [ 5 ] 
and the analogies between some aspects of the theory of rings (resp. 
modules) and the theory of semi-rings (resp. semi-modules) indicate the way 
for asking a general solution in the case of fuzzy-automata. Constructing the 
notion of noetherian semi-module, an algorithm for equivalence of some 
kinds of fuzzy-automata is exhibited. 
In the following, all sets are supposed to be tinite; if C is a set. we denote 
by lC] its cardinality and by C* the free semigroup of words on C with the 
empty word A E C* as unity. The length of the word H’ E C* is denoted by 
/(rc) E PI and we express two words U, 11 E C* having the same length k E N 
writing /(u/c) = k. The terminology and the notations not especially 
indicated in the paper are according to 171. 
1. BASIC NOTIONS 
We recall the definitions of semi-ring and semi-module [ 1,4, 61 and some 
notions of the theory of fuzzy-automata (9, 11, 141 in form appropriated for 
the following. 
Let C be a set with two inner binary laws of composition 
270 
0022~247X~Xl; I10?7~12%0?.00;0 
CopyrIght C 1981 by Academic Press, Inc. 
All rights of reproduction in any form reserved. 
EQUIV.ALENCE OF FUZZY-AUTOMATA 771 
ki : C x C + C, i = 1, 2. We call the algebra (C. k,, 17~) a (commurafirre)semi- 
ring if (C, k,) and (C. k,) are (commutative) semi-groups with unity and k, 
and k2 are distributive one after the other. 
Let E be a set. C be a semi-ring and let k’, : E x E + E and k’, : C x E -+ E 
be two laws of composition, the second being external. The algebra 
(E. k’, , k;) is a left semi-module over C if for each a. b E C and X. J E E the 
following conditions hold: 
(SM.1) (E, k’,) is a commutative semi-group with unity: 
(SM.2) k;(a, k;(x, 1’)) = k’,(k;(a. x). k;(a. y)). 
k;(k,(a. b), x) = k’,(k;(a. x), k;(b, x)): 
(SM.3) k; (a, k;(b. vu)) = k;(k& b). x). 
The mapping k: (E, k’, , ki) + (E”, k;. k;) is called morphism cf semi- 
modules if the following holds 
h(k’,(x, 4’)) = k;(h(x), h(y)) and h(k;(u, x)) = k;‘(a, h(x)). 
By the same way the notion of right semi-module is defined. If C is a 
commutative semi-ring we talk about semi-module. The category of all C- 
semi-modules is denoted by C-SMod. 
Let M be a C-semi-module. The set X E M is a system of generators for M 
if X generates M. A quasi-base is the minimal system of generators for M. If 
the quasi-base is finite, the dimension of M (denote dim M) is the number of 
its vectors (elements of E). 
Let X be a set, not necessarily finite and let C be a semi-ring. Putting 
vx= x a, . x. u, E c, .Y E x, 
XEX 
where a, # 0 only for a finite number of elements x E X. it is easy to verify 
that VX is a semi-module according to the laws of composition of the semi- 
ring, called free semi-module. The set X is a minimal system of generators 
for VX. 
DEFINITION 1. The C-semi-module M should be called noetheriun if M 
is a noetherian object [6] in the category C-SMod. 
The following two results are important for the theory and its 
applications; the proofs are omitted. 
PROPOSITION 1. For a semi-module M E C-SMod the following 
conditions are equivalent: 
(a) M is a noetheriun C-semi-module; 
4W84 1 I8 
‘71 - h TOPENCHAROV AND PEE\;/4 
(b) Each increasing sequence of sub-C-semi-modules of M, i.e.. M, c 
M,c...cM,c... , such that Mi # Mi _, , is jinite; 
(c) For each sub-C-semi-module of M ‘there exists a finite minimal 
system of generators; 
(d) Each non-empty set G of sub-C-semi-modules of ME C-Mod 
contains a maximal element. 
PROPOSITION 2. If X # 0 is a Jnite set, the free semi-module VX is 
noetherian. 
EXAMPLES. (1) Let be given the closed interval I = 10. I ] c F; let us 
consider the binary operations k, = max and kz = min in I. according to the 
natural order in I; the algebra (Z, k,, kz) = ([0, 11, max, min) is a 
commutative semi-ring. 
(2) Let L be a distributive lattice; the algebra (L. max min). 
constructed and studied in [lo] is a commutative semi-ring. 
(3) Let X be a finite set and VX be the free semi-module generated by 
X over the semi-ring from the Example 1; the operations in the free semi- 
module are defined as follows: 
k’, = + : VX x VX-+ VX, x a, . x + x 6, . x = x max(a,, b,) . x. 
XEX XEX XES 
k’, = . : 10. 11 x VX+ VX, y x a, . x = x 
( 1 
min(y, a,) . x. 
IE.Y .‘( E x 
When X is finite, VX is a noetherian semi-module (see Proposition 2). 
DEFINITION 2. The quadruple A = (X, Q. Y, h), X, Q. Y being finite sets 
and h: X x Q x Y x Q -+ (0. 1 ] being a map, should be called a fuzzy 
(shortly F-) automaton. 
As usual, X is the input alphabet, Y is the output alphabet. Q is the set oj 
states for the F-automaton A; the map h is called membership function and 
we write h(?si, qj, yr, qk) = a;! E [O. 11. 
It is easy to show that the classical definition of fuzzy-automaton 
[9. 13, 141 gives an automaton according to Definition 2 (see [ 1, 101). For 
our purpose, however, this definition is preferable. 
If the interval [0, l] is replaced by the distributive lattice L (see 
Example 2) we obtain the more general notion of L-automaton, closely 
related to F-automaton. 
Every F-automaton A defines the free semi-modules V(X x Q) and 
V( Y x Q) over the semi-ring [0, 11. The membership function 
h: V(Xx Q)- V(Yx Q) 
EQUIV.-ZLENCE OF FUZZY-AUTOMATA 273 
is defined such that h(.ui, qj) = xrqk a:( ~1,) qk); its corresponding matrix 
M, = ]]a;?]] characterize the work of the F-automaton. 
Let us consider the words u=x, .x,. ... *xpfzx*, 
L’ = ,‘, . yz . . . . . JD (g y* and the matrices M(xi/Yrj = Il~mjk(Xi/JJr)ll~ 
m,,(~,/~~,) = ~7:. With the maxi-min product of matrices denoted by 0, we 
obtain the expression 
M(u/u) = M(x,/y,) 0 M(xJyz) 0 ... 0 M(x,/y,). 
If P = P(A/A) is a matrix-column of the type IQ] x 1 which elements are 
equal to 1. the following composition is defined: 
P(u/c) = M(u/v) 0 P. 
Let A be an F-automaton with (Q, E) as the F-set of initial states, sub-F-set 
of Q; c(q) E (0. I] defines the membership of q E Q as an initial state of A. 
We denote 
,+(q’)= 1 if q=q’ &;(q’) = 0 if q = q’ 
= 0 if q # q’ =a E [O. l] if qfq’ 
with the supplementary condition xVEQ e:(q) # 0 for E:. The F-automaton A, 
denoted in this case (A, E:) (resp. (A, E)) is called initial (resp. weakly initial) 
if (Q. E:) (resp. (Q, E)) is a sub-F-set of Q. 
For the F-automaton A we define S,(U/U) = E 0 P(u/c), an entry indicating 
the maximal degree of membership for the input word u and the output word 
L’. (Q. e) being fixed. 
Let A = (X, Q, Y, h) and A’ = (X, Q’, Y. h’) be F-automata; (Q, E) and 
(Q’. E’) are sub-F-sets of Q and Q’, respectively. 
DEFINITION 3. Two initial automata (A, E) and (A ‘, E’) are equivalent 
(notation (A, E) - (A’. E’)) if S,(U/C),~ = S,.(U/U),~ for all u E X-* and 
L’ E Y*. In particular: 
-let A =A’=(X,Q, Y,h); if (A, E) - (A’, E’), then E and E’ are 
equivalent on Q (notation E - E’); 
- if (A. EE/) - (A ‘, E;!). then the states q E Q and q’ E Q’ are equivalent 
(notation q - q’); 
-A = (X, Q, Y, h) is equivalently embedded into A’ = (X, Q’, Y, h’) if for 
each q E Q there exists an equivalent state q’ E Q’ of A’ (notation A ZA’); 
-A is weakly equivalently embedded into A’ (notation A 5 A’) if for each 
6: Q + (0, I ] there exists E’: Q’ + [0, l] such that (A, E) - (A’. E’); 
-A and A’ are equivalent (notation A -A’) if A ,C A’ and A Z! A’; 
-A and A’ are wekly equitjalent if A &A’ and A’ &A (notation A z A’). 
‘74 ToPENCHAKo\'AND PEEC'A 
DEFINITION 4. Let A be an F-automaton; 
---A is in reduced form if for each 9. q’ E’Q the relation q - q’ implies 
q = q’: 
--A’ is called a reduct of A if A’ is in reduced form and equivalent to A; 
-A is in minimal form if for each &ii (i < (Q 1) there does not exist F:, 
(i < / Ql) such that (A. &ii) - (A, .cii); 
-A’ is called a minimal of A if it is in minimal form and if A =: A’. 
The above defined notions are in concordance with the classical theory of 
automata [7. 121 and coincide with the usual terminology in the cases of 
deterministic. nondeterministic and stochastic automata. For the F-automata 
[91 this is an attempt to unify the terminology. 
2. EQUIVALENCE OF F-AUTOMATA--AN ALGEBRAIC APPROACH 
For each F-automaton A =(X. Q, Y,h) we define a map 
f: V(X* x Y*)+ VQ as follows: 
W,A)= y q. l(u. L’)= \’ qEa PjC”it) qj if I(u) = I(c) 
se0 
=; if I(U) f l(c). 
It is easy to verify that t is a morphism of semi-modules. Let us denote its 
corresponding matrix by M,. 
We construct the sequence E, c E, c ... c E of subsets of E = X* X Y* 
obtained as follows: 
E, = ((A, A)};..., 
Ei = Eim ,U {(u, ~1); u EX*, LI E Y*, l(u) = I(c) = it. 
PROPOSITION 3. The following statements hold: 
(a) VE, is a sub-semi-module oJ‘ I/E,+, . for each i = 0. l,...: 
(b) IftVEi=tVEi, ,, then tVE, = tVEilp for each p = 0, l,...: 
(c) The quasi-base of tVE contains at most n elements; 
(d) WE,-, = rVE, = ... = tVE. 
ProoJ: (a) According to the construction of E,, E,,..., which are sets of 
generators (quasi-bases) for the semi-modules VE,, i = 0, l,.... we have 
VE,cVE,c... c VE. (b) The morphism of semi-modules I being a linear 
operator, the image of the sequence VE, c VE, c .+. c VE is the following 
EQUIVALENCE OF FUZZY-AUTOMAT.4 275 
sequence tVE, c tVE, c ... c tVE; hence (see Proposition 2) for i E Ed and 
p = 0, l..... we have tVE, = tVEitg. (c) The semimodule VQ being 
noetherian and since tVE G VQ. holds dim tVE < n. If tVE = VQ. the 
equality holds. (d) Let us consider the sub-semimodule tVEi; it contains a 
certain number of vectors of the quasi-base. If tVE, c tVEi+, , the sub-semi- 
module contains at least one supplementary vector of the quasi-base. Hence 
if tVE = VQ and since each quasi-base tEi contains exactly a new 
(supplementary) vector of the quasi-base of tVE, we obtain: 
tVE,ctVE,c...ctVE,-,=tVE,=...=tVE. 
having obviously dim t VE,- , = n. 
This result reinforces some statements and algorithms of [lo]. 
THEOREM 1. Let (A, E) and (A’. E’) be two bveakly initial F-automata. 
(A. E) - (A’, E’) lfl E o t = E’ o t’. 
Proof. If 1(u) = I(v) for (u, u) E X* x Y*, according to Definition 6: 
s,w~7),, = S,W~L ’ 
and since S,(u/tl) = E 0 (M(u/tl) 0 P). we obtain 
& 0 (M(u/u) 0 P) = E’ 0 (M’(u/v) 0 P); 
this expression is equivalent to E 0 t(u, u) = E’ o t’(u, u) for each couple 
(u, tl) E X* x Y* such that I(U) = f(v); if 1(u) # I(u), according to the 
definition of t, holds t(u, c) = t’(u, ~1) = 0, i.e., E o t = E’ o t’. 
Conversely. let e 0 t = E’ 0 t’; obviously E o t(u, U) = E’ o t’(u, tl) for each 
couple (u. ZJ) E X* x Y* and hence E 0 t(u/u) = E’ o t’(u/u); but 
M,(u/c) = M(U/L~) 0P if j(u) = 1(u) 
=o if l(u) # I(v); 
it follows E 0 (M(u/o) 0 P) = c’ 0 (M’(u/o) 0 P), i.e., S,(u/u), = S,,(u,lc),, for 
each (u, ~1) E X* x r* such that I(U) = I(U). 
A similar result is given in [9, lo]. 
COROLLARY 1. Let A be an F-automaton. E - E’ gS,(u/tl), = S,,(U/V),~ 
for each (u, c) E X* x Y* such that /(u/u) < n - 1. 
ProoJ If (A, E) - (A, E’), then S,(U/U),~ = S,(u/u), for /(u/u) =: 0, l,...; 
hence /(u/v) < n. If S,(u/u), = S,(U/L~)~ for each (u, u) E x* x 1% such 
that /(U/Ll) < n - 1, according to Proposition 3(d) it follows 
E 0 (M(u/c) 0 P) = E’ 0 (M(u/v) 0 P); hence (A, E) - (A, E’). 
276 TOPENCHAKOV.aND PEEV.A 
This is the fuzzy-interpretation of the well-known Carlyle theorem [3] for 
equivalence of stochastic automata. 
COROLLARY 2. For a given F-automaton A the following statements are 
equivalent: 
(a) & - E’: 
ib) E~M~=E’~M~. 
COROLLARY 3 [lo]. For a given F-automaton A the following 
statements are equivalent: 
ia) &ii - q: 
(b) The ith’bnd jth rows in the matrix M, are identical. 
ProoJ Let E:, - &ii; according to the Corollary 2 of Theorem I, 
4, ~M,=E i. 0 M,: but by the construction of E: this means that the ith and 
jth rows in’ M, are identical, hence (a) * (b). The inverse implication 
(b) 3 (a) is directly verified. 
The following auxiliary result is an important criterion to ascertain the 
equivalence of two F-automata. 
LEMMA. If (A, E) - (A’. E’) then dim(Im t) = dim(Im t’). 
Proof. According to Theorem 1, E o f = E’ 0 I’ o E 0 M,, . This matrix 
equality leads to dim(Im t) = dim(Im t’). 
Let A and A’ be two F-automata. 
THEOREM 2. If E is given, the problem of finding E’, if it exists. such that 
(A, E) - (A’. E’) is algorithmically decidable. 
As a proof we give the algorithm (see Fig. 1). 
The computing program is not easy to realize, useful standard programs 
are missing. 
3. REDUCTION AND MINIMIZATION OF FUZZY-AUTOMATA 
The problem of reduction and minimization of F-automata is a conse- 
quence of the theory of equivalence of F-automata, but they have a high 
importance in applications. This part is a completion of very rich ideas of 
1101. 
Closely connected with the problem of reduction of F-automata is the 
following statement: 
EQUIV.ALENCE OF FUZZY-AUTOMATA 277 
TRUE 
I 
PRINT E’ 
I 
PRINT \A,61 IS PRlNT NO EXISTS E 
EQUIVALENT TO I A”, E ” 1 %CH THAT IA.6 1 IS 
EQUIVALENT TO I A”, E” ) 
I 
FIGURE I 
THEOREM 3. Let M, be the matrix associated to the F-automaton A. If 
M, contains two identical rows, there e,uist wo F-automata A’ and A”, with 
1 Q 1 - 1 states each, such that A - A’ and A -A”. 
Proof: Let in M, the rows corresponding to the states qi and qj be iden- 
tical and let Q’ = Q - (qi}, Q” = Q - (qj}; the corresponding matrix M,, 
(resp. Ml,,) for the F-automaton A’ (resp. A”) is obtained by M, eliminating 
the ith (resp. the jth) row. We shall prove that A -A’ (resp. A -A”). The 
equivalent state to q E Q, qi # q f qi is q E Q’ (resp. q E Q”) and vice versa, 
because E;’ o M, = E;’ o M,, (resp. E: o M, = E:’ o AI,,,). The equivalent state 
to q = qi, qj E Q respectively is the state qi E Q’ (resp. qj E Q”). The state 
equivalent to qi E Q’ (resp. qj E Q”) is qi E Q (resp. qj E Q). The proof in 
these conditions is a consequence of the definition of E;, of the construction 
of M, and a direct verification holds. 
‘78 'TOPENCHAKOV ‘AND PEE\'.4 
COROLLARY. For every F-automaton there exists a reduced F-automaton. 
All reduced F-automata associated to a given F-automaton have sets of 
states with the same cardinality. 
THEOREM 4. For finite F-automata the relation of equivalence is 
decidable. 
The block-scheme (Fig. 2) of the algorithm proving the equivalence of two 
F-automata A and A’ is in fact the proof of the Theorem 4. 
The following result is connected with the existence and the explicit 
construction of a minimal F-automaton to a given F-automaton. 
THEOREM 5. Let A = (X. Q. Y. h) be an F-automaton. If E:* ‘v E:,, and 
PRINT A IS NOT 
EQUIVALENT TO A’ 
TRUE 
PRINT A IS NOT 
EOUIVALENTLY 
EMBEDDED INTO A’ 
END ) 
FALSE 
FIGURE 2 
EQUIV;\LENCE OF FUZZY-AUT0MAT.A 279 
I FALSE 
FIG. 2-Continued. 
’ contains 1 E [O. 1 ] as a component, there e.xists an F-automaton 
?L (‘7, (3, Y. /fi), with 1 Q 1 - 1 states, such that A z 2. 
Proof Let (Q, E~J and (Q, E~J be a sub-F-set of Q: let 
&i, = (E, , El . . . . . Eb ~, . 0. Eb + , ,..., E,) 
verify the condition of the theorem and 
E ;, = (0 ,..., 0, 1, 0 ,...) 0) 
be equivalent to $. We construct the F-automaton A= (X, 0. Y, i) as 
follows: 
Q=Q- (qt,t, h: qxx Q)+ V(Yx (3). h(-Yi 3 qj) = 1 E$( ?‘,3 qk). 
r.k 
280 TOPENCHAROV AND PEEVA 
where c?;! = max(a$, min(et, a;!)). We shall show the states qj. j f 6, with 
the same indices in A and x are equivalent. For the words with length I= 1 
we have 
Writing the last equality we have in mind that sib contains 1 E [O, 11 as a 
component. I.e., max,,,(min($, a;;)) = ai;, because 
Suppose the states qj, j # b, Ml-equivalent, i.e., for arbitrary words u E X*, 
L’ E Y* such that l(u) = I(U) = ~3, the following holds: pj(~/~!),7= JI~(u/L~).~. 
According to the hypothesis max kth(min(sz, P~(u/z~)) = P~(u/L~). Then we 
obtain 
min(uz, Tz;(min(s:‘. P~(u/L~)))) 
= m;x(min(uT!. pk(U/U))) = pj(.UiU/~,O), 
i.e.. the states with the same indices for automata A and x are (LC + 1). 
equivalent and thus equivalent. For each E= (E,, E? ,..., tzhm,, cb+ , . . . . . E,) for 
A. there exist an equivalent E = (E, . sz ,..., E*-, , 0. Ed+, . . . . . E,) for the 
automaton A. For a given e = (E,. s?,..., E,) for the automaton A. the 
corresponding equivalent E = (E, , Ez ,..., C,,) for x is defined by the correspon- 
dence fi = max(si, min(e,, ey,)). i # b. where E:. is the ith component in the 
vector sib. Indeed, having in mind the definition, we obtain 
S,(u/tl), = 6 0 P(u/tl) = p~x(mitl(&i, p(U/V)) 
= max(m3$min(ei. p,(u/u)j, min(s,, max($, pi(U/U))))) 
= m:$min(max(c,, min(e,. EP))), pi(U/C)) 
= m+a;(min(Ci, pi(U/U))) = S,-(U/U),. 
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Equivalence, reduction and minimization of finite fuzzy-automata 

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Equivalence, reduction and minimization of finite fuzzy-automata

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