by M. DAVIO and A. THAYSE

Transcription

1 VOL. 28 No. 2 APRIL 1973 Philips Rsarch Rports EDITED BY THE RESEARCH LABORATORY OF N.V. PHILlPS' GLOEILAMPENFABRIEKEN. EINDHOVEN. NETHERLANDS R822 Philips Rs. Rpts 28, , 1973 REPRESENTATION OF FUZZY FUNCTIONS by M. DAVIO and A. THAYSE Abstract This papr is dvotd to th study of fuzzy functions of n variabls. Th uniqu rprsntation of a fuzzy function as th join-irrdundant sum of join-irrducibl lmnts is prsntd: two algorithms ar givn for obtaining that rprsntation ithr from any fuzzy xprssion or from a truth tabl of th function. A paramtrical rprsntation of th lattic of fuzzy functions compatibl with a givn Boolan function is studid through som disjunctiv and conjunctiv diffrntials introducd arlir for Boolan functions. Finally two algorithms for computing th prim implicants and th prim implicats of a fuzzy function ar drivd. 1. Introduction Th concpts of fuzzy sts, fuzzy languags, fuzzy automata, tc. hav bn introducd in a rathr rcnt past 1-3) in an attmpt to provid mathmatical modls for classs of ill-dfind vnts. Th algbraic structur of soft algbra, axiomatically introducd by Prparata and Yh 4.5) rprsnts a first tool for approaching this goal: it dviats vry slightly from th structur of Boolan algbra sinc it satisfis all th axioms of th lattr but th laws of complmntarity. It sms rathr clar that th structur of soft algbra is too rigidly dfind to ncompass all of its suitd goals: howvr, its study throws som light upon nw problms that aris whn on lavs th dtrministic ralm of Boolan algbra. This papr is dvotd to a spcific soft algbra, namly th family of functions of n variabls ranging in th closd intrval [0, 1]. Ths functions, calld fuzzy functions, hav bn studid in grat dtail by Prparata and Yh whos main rsults ar brifly rcalld in sc. 2. Sction 3 studis th uniqu rprsntation of a fuzzy function as th joinirrdundant sum of join-irrducibl lmnts: two algorithms ar prsntd for obtaining that rprsntation ithr from any fuzzy xprssion or from a truth tabl of th function. Sction 4 shows th usfulnss of th disjunctiv and con-

2 94 M. DAVIO and A. THAYSE junctiv diffrntials, introducd in rf. 6 for Boolan functions, in th study of fuzzy functions: th main rsult of this sction is a paramtrical rprsntation of th lattic of fuzzy function compatibl with a givn Boolan function; this rsult, asily drivd thanks to th aformntiond diffrntial oprators, is a usful tool for driving fuzzy proprtis from th corrsponding Boolan proprtis. Finally, sction 5 prsnts two algorithms for computing th prim implicants and th prim implicats of a fuzzy function from any xprssion of that function: th improvmnt with rspct to rf. 5 is that it is no mor ndd to driv th canonical-pair rprsntation of th function. Th first of ths two algorithms is th application of a gnral lattic-thortical algorithm 9) to th cas of fuzzy functions. It provids th prim implicants, th prim implicats or both, at will. Th scond on obtains sparatly th V- prim implicants and th P-prim implicants. Th formr ons ar asily sn to b ssntial and ar thus immdiatly discovrd in any disjunctiv normal form ofth function. Th computation ofth P-prim implicants is thn rducd to th rsarch of th prim implicants of a st of Boolan functions. 2. Prliminaris On invstigats wll-formd xprssions using th symbols XI> +,. and ', Th variabls Xi tak thir valus in th closd intrval I = [0, 1] and th oprators hav th following maning: Xi + Xj = max (XI> Xj), Xi Xj = min (XI> Xj), (1) To thos xprssions functions ar associatd, gnrally calld fuzzy functions, which will b dnotd f (xq, Xl>, Xn-l); a fuzzy function f is a mapping f:r-+l. In viw of th proprtis of th oprators max and min, th st of fuzzy functions constituts a boundd and distributiv lattic. Th fuzzy functions ar gnralizations of Boolan functions in that thy satisfy all th axioms of th lattr ons xcpt th laws of complmntarity, i.., X + x' ='1 and X X' = 0. Th following thorm, which appars to b of basic importanc for th charactrization of fuzzy functions, has bn statd by Prparata and Yh 4.5). Thorm 1 (1) Th valu assumd by an xprssion j'(xj, Xl>, x n - 1 ) at th vrtics of th n-cub can only b or 1; at th cntr of a k-cub (k :::;;n), it can only b 0; 1/2 or 1. (2) If th valu assumd by an xprssion at th cntr of a k-cub Cis 1 or 0, th sam valu occurs at th cntr of ach (k- Ij-cub containd in C.

3 REPRESENTATION OF FUZZY FUNCTIONS 95 (3) Th valu of an xprssion at th cntr of a k-cub whos vrtics hav mixd valus is 1/2. (4) Th valu of an xprssion j'(xs, Xl', Xn-l) is compltly dtrmind by th valus of f at th vrtics of th n-cub and at th cntrs of ach of its subcubs. Th following qualitis, also du to Prparata- Yh, will b usd furthr on in this papr:. Xl X/ Xj + Xl X/ x/ = Xl X/, Xl Xj + x/ Xj + Xj x/ = Xl Xj + x/ Xj, Xl + X/ Xj X/ = Xl + Xj x/. (2) According to Prparata, it will b convnint in th following to subdivid th product trms into two catgoris, P-trms and V-trms, dpnding upon whthr thy contain or do not contain, rspctivly, symmtric pairs of th typ Xl x/. A V-trm {3 dos not contain any symmtric pair, a P-trm {3 y is th product of a V-trm (3 and a product y of symmtric pairs. 3. Th uniqu rprsntation of a fuzzy function Th two following thorms, du to Prparata- Yh 4.5) and to Birkhoff 8) rspctivly, will b usd in th following. Thorm 2 (rfs 4, 5) Th st of fuzzy functions is a boundd and distributiv lattic. Lmma 1 (rf. 8, p. 58) If p is a join-irrducibl lmnt in a distributiv lattic, thn p ~ ~ / 1 implis p ~ / 1 for som i. Thorm 3 (rf. 8, p. 58) In a distributiv lattic of finit lngth, ach lmnt has a strictly uniqu rprsntation as th join of a join-irrdundant st of join-irrducibl lmnts. Lmma 2 If pand q dnot products of litrals, thn Proof (1) p <=» = qr, (2) p < q ~ p = q r whr r is a non-void product of litrals that do not blong to q. (l.a) If p ~ q, thn p q = p and th lttrs of q which do not blong to p play th rol of r. (l.b) If p = q r, thn by dfinition p < q. (2.a) If p < q, thn p = q r; morovr, if r is mpty, thn p = q.

4 96 M. DAVIO and A. THAYSE (2.b) If p = q r and if r is non-mpty, thn p :::;;; q. Morovr, if th valu of th lttrs of q and r ar such that q = 1/2 and r = 0, thn p = 0 at this point. ' Thorm 4 In th lattic of fuzzy functions of n variabls th join-irrducibl lmnts ar (1) th products of litrals containing no symmtric pair X x'; (2) th products of litrals containing ach of th n variabls at last with on polarity. Prooi (1) By dfinition, ach join-irrducibl lmnt may b rprsntd by a monomial. (2) Lt p b a monomial which dos not contain any symmtric pair of th typ XI x/. Lt us giv to th lttrs of p valus such that p = 1 and to all th rmaining lttrs th valu 1/2. In viw of th lmma, ach monomial strictly smallr than p contains at last on lttr having th valu 1/2 or on lttr having th valu O. Each monomial has thus th valu 0 or 1/2 at th considrd point and th maximum of such monomials cannot rach th valu 1. (3) Lt p b a monomial containing all th lttrs. Lt us giv to th lttrs blonging to symmtric pairs of th form XI x/ th valu 1/2 and to th rmaining lttrs a valu 0 or 1 such that thir product is qual to 1. Each monomial strictly smallr than p is obtaind by substituting a lttr Xk in p (Xk dos not blong to a symmtric pair of p) by th symmtric pair Xk Xk' Th monomial so obtaind taks th valu 0 at th considrd point and th union of such monornials cannot giv th valu 1/2. (4) In viw of rlations (2) a monomial which contains at last a symmtric pair and dos not contain all th lttrs is not join-irrducibl. In what follows, th join-irrducibl lmnts containing no symmtric pair ar calld V-join-irrducibl lmnts. Th othr join-irrducibl lmnts ar th P-join-irrducibl lmnts. For illustrativ purposs, th Hass diagram of th join-irrducibl lmnts for n = 2 is shown in fig. 1. Two algorithms will now b givn in ordr to obtain a canonical form for th fuzzy functions as th union of thir join-irrducibl lmnts. Algorithm 1: Obtaining th uniqu rprsntation of a fuzzy function givn by an arbitrary xprssioni (1) Rduc I to a normal disjunctiv form using D Morgan's laws and th distributivity. (2) Dtct th non-join-irrducibl lmnts and multiply thm by a factor XI + x/ for ach of th missing litrals. Rduc th nw xprssion to th normal disjunctiv form..

5 REPRESENTATION OF FUZZY FUNCTIONS 97 x x'y! x'yy' xxlyy' Fig. 1. Hass diagram of th join-irrducibl lmnts for n = 2. (3) Discard th products containd in othr products by applying th absorption law. Exampl 1 (rfs 4, 5) f = X3 {[X2' (X2 + x/)]' + X3' X2 Xl x/} + X3' X2' (X2 Xl X/ + X3)' (1) f = X2 X3 + Xl X2' X3 + X2 Xl Xl' X3 X3' + X3' Xl X/ X2 X2' + X2' X3 X3" (2) All th trms ar join-irrducibl xcpt th last on which is substitutd by X3 X3, X2'( Xl + Xl ') = X3 X3 "+ X2 Xl X3 X3 '" X2 Xl' (3) Sinc X3 X3' X2 Xl X/ < X3 X2 and X3 x/ X2' Xl < X3 X2' Xl' th uniqu rprsntation is Algorithm 2: Obtaining th uniqu rprsntation of a fuzzy function f givn by its truth tabl or by a graphical display (1) For k = n, n - 1,..., 0, xamin th valu assumd by f at th cntr of ach k-cub. (a) If th valu at th cntr of a givn k-cub is or 1/2, go to th nxt k-cub. (b) If th valu at th cntr of th k-cub x(c) is 1 and has not bn discardd during a prior stp, thn (i) introduc th V-trm TI Xt( t ) in th rprsntation, (ii) discard th 1- and 1/2- valus of f implid by th slctd V-trm in viw of thorm 1, (iii) go to th nxt k-cub. (2) For k = 1, 2,..., n- 1, xamin th valu assumd byf at th cntr of ach k-cub.. (a) If th valu at th cntr of a givn k-cub is go to th nxt k-cub.,

6 98 M. DAVIO and A. THAYSE (b) If th valu at th cntr of a k-cub x(c) is 1j2 and has not bn discardd during a prior stp, thn (i) introduc th P-trm TI Xt(l) Xj x/, (ii) discard th 1j2-valus off implid by th slctd P-trm in viw of thorm 1, (iii) go to nxt k-cub. 4. Th lattic of fuzzy functions compatibl with a givn Boolan function Dfinition 1 A fuzzy function f will b said to b compatibl with a Boolan function fb if f and fb tak th sam valus on th vrtics of th n-cub. Thorm 5 Th st of fuzzy functions compatibl with a givn Boolan function fb(x) is a sublattic of th lattic of fuzzy functions. Proof In viw of th lattic charactr of th oprations min and max th sum and th product of two fuzzy functions compatibl withfb(x) ar thmslvs fuzzy functions taking th sam valus as fb on th vrtics of th n-cub. It must howvr b pointd out that th st of fuzzy functions compatibl with a givn Boolan function is not a Boolan algbra. Lt b th canonical disjunctiv and conjunctiv xpansions of th Boolan function fb' Lt furthr b th rprsntations of fb as th disjunction of all its prim implicants and as th conjunction of all its prim implicats. Th following thorm thn holds. Thorm 7 (1) Th soft functions t; = ~ f) TI xt(l) and fm = IT [f () + ~ x/i)] c ar th minimum and th maximum lmnts rspctivly of th lattic of fuzzy functions compatibl with fb' (2) Th soft functionfmm = ~ Pt = IT qj is maximum on th 1-subcubs of fb and minimum on th O-subcubs of fb' (3) (4)

7 REPRESENTATION OF FUZZY FUNCTIONS 99 Proof (1) Any fuzzy function compatibl withfb contains th fuzzy function rprsntd by a mintrm x<c) of fb. Indd, 1/2 is th minimal valu at th cntr of ach subcub containing th vrtx whr x<c) = 1. (2) Th proof of (2) dirctly follows from th covring proprtis of th prim implicants and prim implicats of a Boolan function. Lt us rcall th following dfinitions 6). Dfinition 2 (1) Th function qfb/qx C is th conjunction of th prim implicats of fb indpndnt of x. (2) Th function p fb/px c is th disjunction of th prim implicants of fb indpndnt of x. Dfinition 3 Th q-diffrntial qfb and th p-diffrntial p fb of a Boolan function fb hav bn dfind in rf 6. as O<~2n-I, (5) C (2) PfB = I PfB - (dx)<c), pxs On knows that th subcubs with O-valuation of fb ar charactrizd by th quation qfb = O. (7) Lt us considr th Boolan function obtaind by substituting in th disjunctiv xpansion of qfb th xprssions qfb/qx C by thir complmnt, that is PfB' = ~ ( qfb )'(dxyc). L,; (qxs)' Th fuzzy function obtaind by rplacing in this last xprssion dxi by XI xt' is a function idntically 0 at th vrtics of th n-cub and at th subcubs with l-valuation offb. It is 1/2 at th cntrs of th subcubs with O-valuation of fb. Lt us now rqust that th cntrs of th subcubs with O-valuation of fb may arbitrarily tak 0- or 1/2-valus according to th valus of som paramtrs. A paramtrical rprsntation of th lattic of fuzzy functions compatibl withfb is asily drivd from (8): xpand th (qfb/qx C )' in thir mintrms with (6) (8)

8 100 M. DAVIO and A. THAYSE rspct to th variabls xc' and multiply ach of ths mintrms by a binary paramtr IXI' Th function so obtaind will b dnotd by q*f Clarly th l-valu of a paramtr IXI corrsponds to a I/2-valu at th cntr of th corrsponding subcub, whil th a-valu of th paramtr corrsponds to a a-valu at th cntr of th sam subcub. Th function f=fmm + q*f (9) is a fuzzy function, maximal in th subcubs of 1 of fb and arbitrary in th subcubs of O. This immdiatly follows from th way in which q*fwas built. On knows similarly that th constant subcubs with l-valuation of fb ar charactrizd by th quation PfB = 1. (10) Th fuzzy function obtaind by substituting in th disjunctiv xpansion of PfB, i.. dxi by XI x/, is a function which is idntically 0 at th vrtics of th n-cub and at th subcubs with a-valuation of fb' It is 1/2 at th cntrs of th subcubs with l-valuation of fb' A paramtrical form of this fuzzy function will thn b obtaind in th sam way as for q*f; it will b dnotd by p*f Th following thorm dirctly follows from th way in which th functions q*f and p*f wr built. Thorm 8 Th lattic of fuzzy functions compatibl with a Boolan functionjj, is givn paramtrically by (11) f(x, x, (3) = fm",(p*f)' + q*j, (12) whr xand (3 ar vctors of binary paramtrs. Exampl 2 fb = Xo Xl X2 + Xo Xl X2' + Xo' X/ X2 + Xo' Xl X2 + Xo X/ X2' fmm = XOXI + X2' PfB' = X/ X2' dxo (dxi)' (dx2)' + Xo' X2' (dxo)' dxi (dx2)" q*f = IXOX/ X2' Xo Xo' + IXI Xo' X2' Xl X/, PfB = X2 dxo (dx.)' (dx2)' + X2 (dxo)' dxi (dx2)' + Xo Xl (dxo)' (dx.)' dx2 + + X2 dxo dx, (dx2)" p*f = ({Jo xi' X2 + (Jl Xl X2) Xo Xo' + ({J2 Xo' X2 + (J3 Xo X2) Xl X/ + + {J4 Xo Xl X2 X2' +, X2 Xo Xo' Xl Xl"

9 REPRESENTATION OF FUZZY FUNCTIONS 101 Two possibl applications of th paramtrical rprsntation of th st of fuzzy functions compatibl with a Boolan function will now brifly b suggstd.. First of all lt us obsrv that this rprsntation can lad to th dsign of a particular logic modul capabl of ralizing any fuzzy function compatibl with a givn Boolan function. An illustration of such a ralization for xampl 2 hrabov is givn by fig.2. Mor gnrally, lt us xtnd to fuzzy functions th classical problm of dsigning th st of Boolan functions of n variabls by a singl logic modul. Clarly th paramtrical rprsntation of th st of fuzzy functions compatibl with a univrsal Boolan function dirctly lads to th synthsis of th complt st of fuzzy functions of n variabls. Anothr possibl application ariss from th fact that th fuzzy functions X o xi x 2 Fig. 2. Logic modul for fuzzy functions compatibl withfb.

10 102 M. DAVIO and A. THAYSE compatibl with a Boolan function bhaviours of this Boolan function. ar modls for th possibl transint 5. Minimal normal forms for fuzzy functions In what follows, w ar daling with two dual substs of th lattic of fuzzy functions. Th first on is th st of products of lttrs, th lmnts of which ar calld classs. It is closd undr th product and vry fuzzy function may b xprssd, at last in on way, as th sum of som classs. Th scond on is th st of sums of lttrs, th lmnts of which ar calld anticlasss. It is closd undr th sum and vry fuzzy function may b xprssd, at last in on way, as th product of som anticlasss. A class smallr than a fuzzy function f is an implicant off A maximal implicant of f is also calld a prim implicant of f An anticlass largr than a fuzzy function f is an implicat of f A minimal implicat of f is also calld a prim implicat off Th following thorm has bn statd by Davio and BiouI 9 ). Thorm 9 (1) Ltf and g b two lattic functions rprsntd by th sts of thir prim implicants {PiU)} and {Pi(g)} rspctivly. Th prim implicants of th conjunction (or product) fg ar found by forming all th conjunctions PiU) PJCg) and by dlting in th obtaind list th classs smallr than othr classs. (2) Ltf and g b two lattic functions rprsntd by th sts of thir prim implicats {qiu)} and {qjcg)} rspctivly. Th prim implicats of th disjunction (or sum) of f + g ar found by forming all th disjunctions qiu) + qjcg) and by dlting in th obtaind list th anticlasss largr than othr anticlasss. In what follows, th rsarch of th prim implicants of an anticlass and that of th prim implicats of a class is calld auxiliary problm. Whnvr th solution of th auxiliary problm is known, th following gnral algorithm allows us to obtain th st of prim implicants and of prim implicats of a lattic function. Algorithm 3 (rf. 9): Obtaining all th prim implicants and all th prim implicats of a lattic function f (1) Rplac ach class by its prim implicats, using th solution ofth auxiliary problm. (2) Comput th prim implicats of f by prforming disjunctions of implicats, using thorm 9.

11 REPRESENTATION OF FUZZY FUNCTIONS 103 (3) Rplac ach prim implicat (anticlass) by its prim implicants, using again th solution of th auxiliary problm. (4) Comput th prim implicants of I by prforming conjunctions of implicants, using thorm 9. Th solution of th auxiliary problm for fuzzy functions is givn by th following thorm. Thorm 10 (1) (a) Th prim implicats of a join-irrducibl class ar all its litrals. (b) Th prim implicats of a non-join-irrducibl class ar (i) all its litrals, (ii) th disjunctions Xj + x/ corrsponding to all of th missing litrals in th givn class. (2) (a) Th prim implicants of a mt-irrducibl anticlass ar all its litrals. (b) Th prim implicants of a non-mt-irrducibl anticlass ar (i) all its litrals, (ii) th conjunctions Xj x/ corrsponding to all th missing litrals in th givn anticlass. Prooi (1) (a) Th litrals of a join-irrducibl class ar implicats of that class, by dfinition of th conjunction, and thy ar prim in viw of lmma 2. Considr now an implicat of th givn function. As an anticlass it is a disjunction of litrals, and, again by lmma 2, if it contains a litral of th class, it can only b prim if it is rducd to that litral. If it contains no litral of th class, two cass ar possibl: ithr th class contains a symmtric pair and th candidat implicat thn only contains with th opposit polarity lttrs apparing in th class with a singl polarity, but, in this cas, thr xists a point whr th class and its candidat implicat hav valus 1/2 and 0 rspctivly, or th class contains no symmtric pair, and in this cas a similar rasoning shows th xistnc of a point whr th class and its candidat implicat hav valus 1 and 1/2 rspctivly. (b) A similar proof holds for th non-join-irrducibl class. (2) Dual statmnt of (1). Exampl 1 (rfs 4, 5) Prim implicats of I obtaind aftr stps (1) and (2) of algorithm 3: (X3 + X3') (X2 + X3) (X2' + X3) (Xl + X3) (Xl' + X3) (X2 + X2') X X (Xl + X2 + X3') (Xl + Xl' + X2)'

12 104 M. DAVIO and A. THAYSE Prim implicants of I obtaind aftr stps (3) and (4) of algorithm 3: Prparata and Yh hav subdivisd th implicants into two catgoris, P-implicants and V-implicants dpnding upon whthr thy contain or do not contain, rspctivly, complmntd pairs of th typ Xi x/. A V-implicant fj dos not contain any complmntd pair; a P-implicant fj y is th product of a factor fj which is an unat product of litrals and a factor y built up with complmntd pairs only. Lmma 3 Th maximal V-join-irrducibl lmnts of a fuzzy function I appar in vry disjunctiv normal form of f Proal Lt I = I:mi and lt p b a maximal V-irrducibl lmnt of f. Assum that p is not containd in th st of mi. This is impossibl sinc on would hav p :::;;I:mi and p :::;;mj for at last on j. If p =1= mb P is not maximal. Thorm 11 thn immdiatly follows from lmma 3. Thorm 11 In ordr to obtain th V-prim implicants of a fuzzy function I it is ncssary and sufficint to obtain a disjunctiv xprssion of I by applying D Morgan's laws and distributivity. Th V-prim implicants ar thn th fj-trms that rmain in th xprssion aftr application of th absorption law. It must b notd that a thorm vry similar to thorm 11 was alrady statd in a papr by Siy and Chn 7). Th goal of this papr was to show that vry prim implicant of a fuzzy function is a trm of any of its normal forms and that its minimum canonical sum-of-products form is th union of all its prim implicants. Clarly, howvr, this assrtion is only tru for th V-prim implicants so that th main rsults by Siy and Chn ar wrong. In th following, it will b shown that th P-prim implicants ar, gnrally, not obtaind by simply applying an absorption rul and that, morovr, th union of all th P-prim implicants dos not usually constitut a minimum sum-ofproduct form for th fuzzy function. Th following notation, charactrizing a P-trm, will b usd furthr on: Thorm 12 y(xo) = TI XI XIEXO Each of th trms of th fuzzy function x/.

13 REPRESENTATION OF FUZZY FUNCTIONS 105 I q/b -Y(X C ), qx C 0<'::;;2 n -I, is an implicant of any fuzzy function I compatibl with IB' Prooi Th subcubs of I having mixd valus on thir vrtics ar charactrizd by th quation /j./b//j.x c = 1. Th subcubs ofihaving l-valus on thir vrtics ar charactrizd by th quation p/b/px c = 1. Both typs of subcubs hav a valu ~ 1/2 at thir cntr, so that th trms of /j./b + P/B) y ( XC ) -_ I[( P/B)' -- q/b + -- P/BJ y ( XC) /j.xc pxs pxs qxs pxs I( C ar implicants of f = I q/b --y(x c ) qxs C Algorithm 4: Obtaining all th prim implicants of a fuzzy function (1) Obtain a disjunctiv xprssion of I by applying dualization and distributivity. (2) Th V-prim implicants ar th p-trms that rmain in th xprssion aftr application of th absorption law (thorm 11). (3) If q/b/qx = 1, slct th P-trm y(x C ) and discard th substs XE2 x. If q/b/qx C =1= 1, thn for th st of trms containing y(x C ), i.. Pi Y(X C ), j = 0, 1,..., m- 1, obtain th minimal form for th xprssion m-l (:~: +IPi) Y(X c ) i=o and writ it in disjunctiv form, i.. (13) y(x c ) ~ Pk*(X c ). k (14) It must b notd that, in viw of rlations (2), th cofficint of y(x ) may b minimizd as if it wr a Boolan xprssion. (4) Th P-prim implicants ar th py-trms that rmain in th xprssion ~ (V-prim implicants) + ~ y(x c ) [~ Pk*(X c )] k aftr application of th absorption law (thorms 11, 12).

14 106 M. DAVID and A. THAYSE Exampl 2 f = (XO Xl + X2) (XO' + Xt' + X2 + XZ') (XO + XO' + Xl + Xl' + X3') + + Xt' XZ' Xo XO' + XO' XZ' Xl Xt'. V-prim implicants off obtaind aftr stps (1) and (2) of algorithm 4: Stp (3): obtaining th xprssion (14): Xo xo' + Xl xt' + Xz xz' + trms of th form XI x/ XJ x/ which can alrady b absorbd. Stp (4): obtaining th st. of implicants: f= Xo Xl X2' + Xo X2 + Xo' X2 + Xl Xz + xt' X2 + Xo Xo' + Xl xt' + Xz Xz'. It could asily b vrifid that th P-prim implicant Xz X2' is not an ssntial on and might thus b droppd in a minimal form off. 6. Conclusion This papr brings som improvmnts to our knowldg of fuzzy functions; it lavs howvr many opn problms: among ths, th simplst ons ar probably of computational natur; lt us.g. mntion th rsarch of irrdundant normal forms of fuzzy functions which has up to now not bn copd with and which appars to b of a rathr ovrhlming complxity. But th main and most difficult opn problms ar crtainly of concptual natur: to tak an xtrm xampl, th procssing of that kind of "fuzzynss" arising in natural languags still sms a non-ralistic goal. MBLE Rsarch Laboratory Brussls, April 1973 REFERENCES 1) L. Zadh, Information and Control 8, , ) L. Zadh, Fuzzy sts and systms, Proc. of th Symposium on Systm Thory, Polytchnic Inst. of Brooklyn, ) P. N. Marinos, IEEE Trans. Computrs C-l8, , ) F. Prparata and R. Yh, On a thory ofcontinuously valud logic, Confrnc Rcord of th 1971 Symposium on th Thory and Applications of Multipl-Valud Logic Dsign, pp ) F. Prparata and R. Yh, J. Computrs and Systm Scincs 6, , ) A. Thays, Philips Rs. Rpts 28, 1-16, ') P. Siy and C. Chn, IEEE Trans. Computrs C-21, , ) G. Birkhoff, Lattic Thory, AMS Colloquium Publications, Providnc, R.I., 1967, vol ) M. Davio and G. Bioul, Philips Rs. Rpts 25, ,1970.