Several Results on de Morgan Algebra and Kleene Algebra of Fuzzy Logic

8/3/2019 Several Results on de Morgan Algebra and Kleene Algebra of Fuzzy Logic

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Several Results on De Morgan Algebra and Kleene Algebra ofFuzzy Logic

Jianming DENGDepartment of Computer Science andEngineering, Southeast University,Sipailou 2, Nanjing, 210096, ChinaEmai l address: jmdeng@seu. du. cn

Tetsuhisa ODADepartment of Marketing andInformation System,Faculty of Management and Information Science,Aichi Institute of Technology, JapanAbstract

In this paper, we present several results on de Morganalgebras and Kleene algebras of fuzzy logic. The main re-sults include (1 ) a necessary and sufficient condition for ade Morgan algebra to b e a Kleene algebra, (2 ) equivalenceof some expressions, (3 ) some properties about de Morganalgebras with sup W + and inf W - , (4 ) some properties of thede Morgan algebras which satisfy certain special conditions,(5 ) a method to obtain all fixed points of a de Morgan al-gebra and (6 ) extensions of some theorems on complete deMorgan algebras that have fixed points to the general case.Keywords: fuzzy logic, de Morgan algebra, Kleene algebra, lattice, fixed point

I. INTRODUCTIONVarious operations on fumy logic have been proposed

after L A .Zadeh presented his logical operations in fuzzyset theory [9]. A discussion of the properties of these fuzzylogical operations needs its associated algebras, such as deMorgan algebras and Kleene algebras [4]. In this paper,we present some results on de Morgan algebras and Kleenealgebras of fuzzy logic.

The algebra systems discussed in the paper arc at least deMorgan algebras. A de Morgan algebra L is a distributivelattice with the greatest element 1, he least element 0 anda strong negation operator N th at satisfies the de Morganslaw, tha t is, Vu ,b E L ==+N ( aV b)=N(a )AN(b ) , ( aAb)= N ( a )V N ( b ) , where V and A are the binary operatorsinduced by a partially-ordered relation 5 of L. A Kleenealgebra is a de Morgan algebra that satisfies the Kleeneslaw, that is, Vu,b E L ==+ ( a A N ( a ) ) A ( b V N ( b ) ) a A N ( a ) ,(U A N ( u ) )V ( bV N ( b ) )= b V N ( b ) .

The notations in this paper are listed as follows:N ( A ) { N ( z ) l z A } .W+= { X I . E L , x 5 N z } .w- E L , x 2 N z } .WO = {zlz E L ,zllNx}

(z11y means that both z 1 and z 5 9 do not hold).

Motohide UMANODepartment of Mathematics andInformation Sciences,College of Integrated Arts and Sciences,Osaka Prefecture University, Japan

Yushi UNODepartment of Mathematics andInformation Sciences,College of Integrated Arts and Sciences,Osaka Prefecture University, JapanU,+ = {XI. E L , x 5 c} .wc- = {zlz E L , z 2 c}.

c - zlz E L ,zllc).m ( A ) { a [a E A and (z 5 a and z E A ) ===-+ z = a }(a set of minimal elements of A that belong to A ) .M ( A )= {a1 a E A and (z 2 a and z E A ) ==+z = a }

(a set of maximal elements of A that belong to A ) .Fixed point, c of L: c E L and c = Nc .F ( L) :a set of all fixed points of L.

11. SOME RESULTSON NECESSARYND SUFFICIENTCONDITIONSF KLEENEALGEBRA

In this section, we discuss some necessary and sufficientconditions for a de Morgan algebra to become a Kleenealgebra and some concerning problems.

The problem is usually encountered whether a de Morganalgebra satisfies the Kleenes law or not. Several theoremsfor judging the problem are already proposed in [2][7][10].We add a simpler theorem that can bring us some con-veniences in dealing with de Morgan algebras and Kleenealgebras. The theorem is as follows.

Proposition 1: If L is a de Morgan algebra, then W - ={z V N z z E L } and W += {z A Nzlz E L}.

Theorem 1: A necessary and sufficient condition for a deMorgan algebra L to become a Kleene algebra is bx E W +& vy E w-* 5 y.

If a de Morgan algebra L satisfies the conditionIm(W-)l = 1, hen L is a Kleene algebra, as pointed out in[lo].This condition can be substituted by other equivalentconditions in the following theorem.

Theorem 2: In a de Morgan algebra L , the followings arcequivalent.(El) There exists an element c E L s.t.w; = W - .(E2) There exists an element c E L s.t. w$ = W+.

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(E 3 ) There exists an element c E L s.t. c 5 Nc and(E4) There exists an element c E L set. c 2 N c and(E5) here exist two elements a,2 E L s.t. c1 5 cp and( E 6 ) There exists an element c E L s.t. Vz E L *

Vx E L+ A N x 5 c.Vx E L+ v N x 2 c.VX E L ===+ X A NX5 C ~ , X NX2 ~ 2 .xA Nx 5 c 5 xV Nx.

( E 7 ) Im(W-)l= 1.(E 8 ) IM(W+)l= 1.The two elements c1 and c2 in (E5)m be the same. In

addition, this theorem leads to the following corollary.Corollary1: If a de Morgan algebra L satisfies one of t he

conditions (El)- E8) , hen L is a KIeene algebra.Note that the converse of Corollary 1 is not valid, for

example, the system (L1 = {l/n,-l /nln = l , Z , - . .},V, A , N, 1,-1) (Vx E L1, N x = -x) is a Kleene algebrabut does not satisfy (El)- (E8) .

Some Kleene algebras that satisfy the conditions (El) -(E8)have fixed points, but some do not. We discuss nec-essary and sufficient conditions for a Kleene algebra withthe conditions (El)- (E8) o have a fixed point.

Theorem 3: For a Kleene algebra L tha t satisfies the con-ditions (El) - (E8), he following conditions (a) - (e) areequivalent.

(a) L has a fixed point c.(b) There exists an element c that satisfies both condi-

tions (El) and ( E 2 ) .(c) There exists an element c that satisfies both condi-

tions (E3)and (E4).(d) If c1 and c2 satisfy (E5),hen c1 = cp.(e) There exist only one element c that satisfies (E6).Therefore, the conditions ( b ) , (c ) , ( d ) and (e) are all

necessary and sufficient conditions for a Kleene algebra Lwith the conditions ( E l )- (E8) o have a fixed point.Next, we discuss the relation between the Kleene alge-bras that satisfy the conditions (El)- E8)and the Kleenealgebras tha t have sup W+ and inf W - . First, we have thefollowing proposition.

Proposataon 2: In a de Morgan algebra L , if sup W +(or inf W - ) exists, then inf W - (or supW + ) xists andN(sup W +)= inf W - and N(inf W - )= sup W+ .

As pointed out in [7],he condition sup W + 5 inf W -holds if L is a Kleene algebra with sup W + and inf W - .Using this result and Proposition 2, we obtain the followingproposition.

Proposation 3: If sup W + (or inf W - ) exists in a Kleenealgebra L, then L satisfies conditions ( E l )- (E8).

Therefore, a complete Kleene algebra satisfies conditions(El)- (E8).The converse of Proposition 3 is also true asshown in the following.

Proposition 4: If a Kleene algebra L satisfies conditions(El)- (E8), hen sup W + and inf W - exist in L.Consequently we have the following theorem.Theorem 4: L is a Kleene algebra that satisfies condi-

L is a Kleene algebra in whichions (El) - (E8)sup W+ and inf W - exist.Obviously, the existence of sup W+ and inf W - does notguarantee the Kleenes law. In the following, we discussnecessary and sufficient conditions for a de Morgan algebrawith supW+ and inf W - to satisfy the Kleenes law. F irst ,we have the following three propositions.

Proposition 5: In a de Morgan algebra L , sup W+ E W+inf W - E W -.

Proposition 6: If sup Wf (or inf W - ) exists in a Kleenealgebra L , then supW+E W+ and inf W - E W -.

Proposition 7: If supW+ E W+ (or in f W - E W - )holds in a de Morgan algebra L, then L satisfies theKleenes law.

Combining Proposition 6 with Proposition 7 , we obtainTheorem 5: For a de Morgan algebra L in which sup W+

and inf W - exist, t he following two conditions are equiva-lent:

the following theorem.

(1)L satisfies the Kleenes law,(2 ) sup W +E W + (or inf W - E W - ) holds in L.Using the Propositions 3 - 7, the following corollary isCorollary2: In a de Morgan algebra L , the following(1)L satisfies the conditions (El) (E8).(2 ) sup W+ exists, and sup W+E W+.(3 ) inf W - exists, and inf W - E W -.

also obtained.three conditions are equivalent:

111. SOME RESULTS N DE MORGAN LGEBRASHATSATISFY ERTAIN ONDITION SIn this section, we discuss de Morgan algebras that sat-

isfy some special conditions. F irst , let us see the followingcondition that is proposed in (31.(E9)Vx, y E L, either x and y, or x and N y are comparable

It is proved that a subset of the set of Interval TruthValue with some logical operations, which are closed on thesubset, forms a de Morgan algebra and that the condition( E 9 ) s a necessary and sufficient condition for it t o becomea Kleene algebra [3].

We discuss the condition (E9) in a more general casewhere L is an ordinary de Morgan algebra. We find out( E 9 ) is also a sufficient condition for a de Morgan algebrato become a Kleene algebra.

by the partially-ordered relation 5 of L.

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Fig. 1. A Kleene Algebra That does not satisfies condition (E9)

Proposition 8: If a de Morgan algebra L satisfies condi-tion (E9), hen L satisfies the Kleene's law.

Note tha t th e converse of Proposition 8 does not hold.The system shown in Fig. 1 is a finite Kleene algebra,but neither c and d , nor c and Nd are comparable by itspartially-ordered relation.Although (E9) s also a sufficient condition fora de Mor-gan algebra to become a Kleene algebra, it is not equ ivalentto the conditions (El) - (E8). The finite Kleene algebrain Fig. 1satisfies (El)- E 8 ) from Proposition 3 but doesnot satisfy condition (E9) from the above discussion. Onthe other h and, the systemL1 in the previous section satis-fies the condition (E9)but does not satisfy conditions(El)

However, for a de Morgan Algebra that satisfiesWO = 0,Proposition 9: If a Kleene algebra L satisfies the condi-

- (E8).(E9) is a necessary condition to satisfy the Kleene's law.tion WO = 0, then the condition (E9) holds.

Therefore, th e following theorem is obtained.Theorem 6: If L is a de Morgan algebra that satisfies thecondition WO = 0, then (E9) is a necessary and sufficient

condition for L to become a Kleene Algebra.Note that WO is not necessarily an empty set when the

condition (E9) s satisfied. The algebra L2 = ( 0 ,a ,N a ,1)is such an example, tha t is, Lz satisfies the condition (E9) ,but its WO (= {a ,N c Y } )s not empty.

Next, we discuss the relation between the conditions(El) and (E2) and the following condition.(E10) There exists an element c E L such that 4 = WO.

The condition(E10) s similar to the conditions(El)and(E2) in appearance. And it is proved tha t the cond itions(El), ( E 2 ) and (E10) are equivalent in a Kleene algebrathat has a fixed point [2]. The problem is what is theirrelation when L is a de Morgan algebra.First, let us see two algebra systems & = ( 0 ,a ,N a , 1 )and L3 = {O,a(=Na),,B(= N,B),l} . The concerning setsin these two systems are as follows.

L2: w- (1) = w;,w + { O } , WO = { a , N a } ,L3 : w- {a ,@,}, w+ ( a , P , O } , WO =w: = 0,= { N a } ,w g a = { a } wy =WO" = 0.

w, = { a , 1,u p = { P ,I},w; = (11,W O = {a,P, , O } .

From these sets, the system & satisfies the conditions(El) and (E2) but does not satisfy the condition (E10).On the other hand, L3 satisfies condition (E10) but doesnot satisfy the conditions (El) and (E2). Therefore, th econditions (El) (or (E2))and (E10) do not imply eachother. However, if we add some restrictions, t he implica-tion (E10) ==+ (El) is obtained. We need some prepara-tions to show the relation between (El) and (E10).

NxllcJ x I J N cJ llNc.Proposition 10 : If w: = WO, then Vx E L , x ) ( cxJINxBy using these equivalent formulas, we can obta in a the-

Theorem 7: In a de Morgan algebra L, wz = WO @

For an element c with w," WO,we have c # WO sincec w: always holds. Therefore, either c 2 N c or c 5 N cmust hold for c with w," WO. In othcr words, c and Ncare comparable when th e condition (E10) is satisfied. Inaddition, we know that w k c = WO also holds in this casefrom the Theorem 7. Therefore, if c satisfies w," WO andmax{c, N c } s denoted as a, we have a 2 N a and w," WO.Hence, we can prove the following proposition and theorem.

Proposition 11 : In a de Morgan algebraL , if there existsan element c that satisfies the condition wz = WO, thenx V N x 2 a and x A N x < N a hold for every x E WO,where a = max{c, Nc}.

element c that satisfies the conditionw," WO # 0, hen

orem as follows.

w g c = WO.

Theorem 8: In a de Morgan algebra L , if there exists an(1) V x E L* V N x 2 a , x AN x 5 Na,( 2 ) N a 5 y 5 a(y E L )* = a or y =Na,

where a = max{c,N c} .In other words, if we have w," = WO # 0, hen thereexists no element between the elements c and Nc . With

this result, we obtain a relation between (El) and (E10)as follows.

Theorem 9: In a de Morgan algebraL, if there exists anelement c tha t satisfies the condition w: = WO # 0, henthe condition (El) is satisfied.

Note tha t th e converse of this theorem is not correc t, forexample, the algebra Lz satisfies the consequcnt but notthe condition of the theorem. In addition, the followingresult can be derived from this theorem and Corollary1.

Corollary 3: In a de Morgan algebra L , if there exis ts anelement c tha t satisfies the conditionw: = WO # 0, then Lsatisfies the Kleene's law.

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Iv. SOME RESULTS N DE MORGANALGEBRAS HA THAVEFIXED OINTS

Fixed points have great influences to constructions ofalgebra systems. Therefore it plays an important roleto check whether a system has any fixed points or not.Some results on this topic have already been presented in[2][7][10].However, we have some other methods to checkit.

Theorem 10: If L is a de Morgan algebra, then W- nW+ = F (L ) ,where F ( L ) s a set of all fixed points of L.If W - nW +# P), then m(W-) = M(W+) .

A necessary and sufficient condition for a de Morgan al-gebra L to have fixed points is easily obtained from thistheorem.

Corollary 4: A necessary and sufficient condition for a deMorgan algebra L to have fixed points is W- nW+ # 0.

Another necessary and Sufficient condition for a de Mor-gan algebra L to have fixed points is given in the followingtheorem.

Theorem 11: In a de Morgan algebra L , L has a fixed3c E L s.t. N(w;) = w t * c E L s.t.In [a],we have some properties on complete algebra sys-

tems t hat have fixed points. However, some of them holdeven without the completeness.

Theorem 12: If a de Morgan algebra L has fixed pointscz (i E I ) , then WC, U w ; is a Kleene algebra with fixedpoints cpfor every i E 1.

Theorem 13: In a de Morgan algebra L that has a fixedpoint c, w; = W-

Therefore, we obtain the following corollary from theabove theorem.

Corollary 5: If c is a fixed point of a de Morgan algebraL and c satisfies the condition w; = W- (or w z = W+),then L has no other fixed point.

Theorem 14: If a de Morgan algebra L has fixed pointsc, ( i E I ) , hen WO c {z (zE L,zllc, (V i E I)}.

Theorem 15: If a Kleene algebra L has a fixed pointc, then there exist inf W-and sup W+ such that c =inf W - = sup W+.

From this theorem and Proposition 3 , we have a relationbetween the Kleene algebras that have fixed points and theKleene algebras that satisfy the conditions ( El ) - (E8) .

Corollary 6: If a Kleene algebra L has a fkcd point, thenL satisfies the conditions (E l ) - (E8).

Note that t he converse of this corollary is not true. Whena Kleene algebra L satisfies the conditions (El) - (E8) andhas a fixed point C Y , L satisfies the conditions in Theorem 3and a is the only element in L tha t can satisfy the condition

point cN ( w ,+)= w;.

U,+= W+.

- -

(El ) - (E8).

Finally, we have two simple necessary and sufficient con-ditions for a de Morgan algebra that has a fixed point tobecome a Kleene algebra.

Theorem 16: In a de Morgan algebra L tha t has a fixedpoint c, the following conditions are equivalent:(1) sup W+= inf W-(E L ) .

(2) sup W + s a fixed point of L.(3) L is a Kleene algebra tha t has a fixed point.V. CONCLUSIONS

In this paper, we present several results on de Morganalgebras and Kleene algebras of fuzzy logic. The main results include (1) a necessary and sufficient condition for ade Morgan algebra to be a Kleene algebra, which is sim-ple and hence brings us some conveniences in dealing withde Morgan algebras and Kleene algebras, (2 ) equivalenceof some expressions, (3 ) some properties about de Morganalgebras with sup W+ and inf W - , (4) ome properties ofthe de Morgan algebras which satisfy certain special con-ditions, (5 ) a direct and convenient method to obtain allfixed points of a de Morgan algebra and (6 ) extensions ofsome theorems on complete de Morgan algebras tha t havefixed points to the general case.

REFERENCES[l] T.S.Blyth and J.C.Varlet: Ockham Algebra, Oxford Science

Publications, 1994[2] M.Morioka: On the Propertiesof Kleene Algebraand De MorganAlgebra with C enters, Journal of Japan Soc ie ty for fizzy T h e -ory and Systems, Vo1.7, No.3, pp.572-584, 1995 (in Jap ane se)

M.Mukaidono and H.Yasui: Algebraic Structure of Fuzzy Inter-val Truth Values, T he l l ans ac t io w o f t he I n s t it u t e o f E l ec t ron -i c s , I n f om a t i on and C om m un ic a t ion E ng ine e r s, Vol.J81-D-I,No.8, pp.939-946, 1998 (in Japan ese)

[4] M.Mu!&dono: fizzy Logic, The Nikkan Kogyo Shimbun Ltd.,Tokyo. Japa n, 1993 (in Japanese)[5] M.Mukaidono and H.Kikuchi: A Proposal on fi zz y IntervalLogic, J ourna l o f J apan Soc i et y f o r f i z z y T he ory and Sy s t e m s ,V01.2, No.2, pp.209-222, 1990 (in Japane se)N.Nakajima and M.Morioka: Fixed Cores of De Morgan Alge-bras and Kleene Algebras, Proceedings of 14th f i zz y Syste mSy m pos ium by J apan Soc ie t y f o r f i z z y T he ory and Sy s t e m s ,pp.711-712, 1998 (in Japanes e)

[7] N.Nakajima and M.Morioka: On the Characterization of DeMorgan Algebra and Kleene Algebra by t he Fixed Core, Jour-n a l o f J ap a n S o ci e ty f o r f i z z y T h eo r y an d S y s t e m , Vo1.9, No.6,pp.988-994, 1997 (in Japanes c)(81 K.Ots uka: M.Em oto and M.Mukaidono, Proper ties of 'Rape-zoid Truth Values in Fuzzy Logic, Journal of Japan Soc ie ty fo rf i z z y T he ory and Sy s t e m s , Vol.11, No.2, 88.338-346, 1999 (inJapanese)L.A.Zadeh: T he Concept of a Linguistic Variable and Its Appli-cation t o Approximate Reasoning-I,II,III, I n fo rm a t ion Sc i e nc e s ,

[lo] K.Zhang, K.Hirota, and Y.Nakagawa: On t he Kleene Algebraand De Morgan Algebra with Centers, Journal of Japan Soc ie tyfor f i z z y T he ory and Sy s t e m s , Vol.9, No.1, pp.131-139, 1997(in Japanese)

[3]

[SI

[9]V01.8, pp.199-249, pp.301-357, V01.9, pp.43-80, 1975

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Several Results on de Morgan Algebra and Kleene Algebra of Fuzzy Logic