[ieee 2007 ieee symposium on foundations of computational intelligence - honolulu, hi, usa...

Proceedings of the 2007 IEEE Symposium onFoundations of Computational Intelligence (FOCI 2007)

Fuzzy Agg ation Techniques in SituationsWit out xperts: Towa s A eN usificaton

'Hung IT NguyenDepartment Of Mt'hetmatical SCienwcs

New WMxco State UivrsityL ':CeaLJLL New MeXicoL 80()3X !SA

Filiai 1 tiugLven nmsu6edi!

Akbtra- Fu'zq tineiqrd Iiavc been originally invuntodas a methodology that trmisfirms the knfwkidget cxpertstormulatcd in terms f natural hangagC itnto a precse iumputer-Winipin tablc formu ThLr>m ar>e many succ I app on ufthis methodilogy to situations in which expert kniowedf cist,the most wdi known is an appliutimn to fnIy contr.

In some case, futiz mnthodblogy 'i aptied cen hno xpt knoIwldglg d! iiAtheIId 6f trxing to appr Mautethtiunknown Control funetiln by spIir, p03nulmiaks or byany other traditional aprimation tehniqu. rscareher tryto appruixniatc it hy guessing and tuning the cp ruls.SupripnglSy this approximathnOflten works fine.

In thisLpep we gw a rathematial cxplanaithn tarftlitkphpnonwnon, anid Show that approximation by using fu;timdethod I ks itndci (il Some minaohab nsc) the

4, y h th bw~~~~~~~ ~S SCJ . _

NTIRODIUCTIO1N

AI F07f W Midque&s a bi f i; d

Fuiziy tehiIn1iqules C benor

lTly inWYLid aWaeL thadology that timnsron s the knowledge 6f expers trmu-At in ltms 14 natua languagc int a sre conputcirpi lenbiah k fmi Thet adr manysuctd Ssfil applicationsof this Mnethodoogy 'to situAt'ons in which exTrt knowled eexist the most well known is d VppIcation to fuuty contrnise , g iIeWI.2 ..1 1. 1.

B. ata rmitharro .trts

A guarantee (if Suteess conKS hoin th& iCt that ftiiyysle-1119 Sre mlLrvuI aWp i natwr in tlh Sense thiat br ev8ryContinou0inlsftnn ( ) and fbr >1 thereexists a set bf r6lo 1w which the corrtepontdhg inpot0otptuuincution doF-ClOge to f eg. 1[31]I [141! [61 [7] 191[IO] [ 12 1r 4and eiei aces thereinl

C hPuz ilewlthagx t flUe ste tu Withtout anye iet Lt U Ig

1ti Snoe casSc Iuzzy methodlogy i applied aven whenno cxt*rt hnowledoe cxIstsl rstond 6f tryini to approximatethe u-nknown ConiMM inctiKOP by splinhs, pQ1ynojmaiK or by4ny oth traditional approximaiUon tchnique. rWesrteher tlvr)w4POIatc it by guessing and tun ig the ex rt rules

Surpri§in , thiis appximation often wirks fi e,

Vlwadhk K einovich)epartuent of Comip0ior ScionceUniversity of Texas t i PASoRI Ptig6 Thkxiis 79968. USA

DX What we filin to don tihis pappr we give a mathetnatIcal explanation tor this

phenomnon adnd we sow tha dpprokximatin by using fizzynmethodogy is inded (in mwne reasandblescnen the hest.

C6hhactM In this papet WAe huild upon oLur pOerliinmrv 0e§1tsrhblighJd an [RJ4

IL IN MANY PRAft(CAL APl1AClIONS, DATAPROCFSS ING SVE) ISIMPORA N

XW have uentionrld thIt one of tho mini applications offiizzv hiithod1o6gy iq to inFtollcient contrbLl

In ApPlicafionsg to du)torntic coitrOL thie coiipitelr niistconstantly Loniopta the c rrLn: UlAhjS otf &titioL ThL value ofthe onttol dcendg on the stAte 6f the 666nth led ohect (calledpMat In contlh&6r)mS6o to got A high uahfycintl w&must measure as maly LhaIa teistics XI X, 6f theik LW ntgtatc as we can T'he more chatnst s WL measurc themor tumhbr Wo have to processo. ithe mow comtputIionsteps we must p0ertbran The re§UN 6f those conpU tatlnsmost be iady n111 tinie, horel we start the feXltround of'measurements So. automatic eontrl| espcially high.qualityautomatic M tmrL ig a realtime comptitltion pihblern with asenous thrie pressa.=

111. PARAI L.I1 C0MPPt7rN(, IS AN ANSW E

A natuaI way to inceae the speed 6f the compuptatIongis to N im computatioing hi pUn1k on swvye1a pir essomTo md e th conputtitons really fst, we ntmugt divide thealgonrthm into parAlIlIehihible steps oa h of whioh icquwts asmall atrmount of tirenWhat are thoseteps.

MV DI sc P'inoN o' 11 FAjSTST POSsi xICNToROLi()I Ni" 1ARAlDPA1R . CO(MPUTi R

A. 1hheft 1raiabl , rheJi4stAs We htave diady nienttiotd. the iain reason why conUtoI

Algofititubs Are computationdfly' totpiicatd is that Wc mnustprocess Many inpus. For ex;atpIe0 6ntrofling a is easierthan contrIing a phrano bhcause the pIlne (as a 3-1) ohoct)'has nioi claratenistis t take care ol, l re charterist',s to

1-4244-0703-6/07/$20.00 ©2007 IEEE 440


mneasutt and hKJICL more~Arctae isics to process. ControlIlino a spc shutt'Le especuaI1y durinig t'he fit-off and laindingisevn a mrer cornplicated, task, usually perfoiJlmuLd by severaII

groups ot peopte w'ho controll the tr4jectory, ternperatureroot]MOtn, IIn shorL the miore numnbers we need to process.the moi 1mhcted the ti1gotthin Thcrdoefi mooiwe wanttin d&omupos~ r MkofIfith into fh test pos fibi modulc~Wet"uA fnAikah iidu1e to pWCess as fi W nuinhiets it pi~§ibl&eB. F41n rwrio qfo6 e wanwble aie not! sitL(t

tddlx]lyj We sould Only istche tfkaluke that conipiteuftil6Ufin of one v4ridbIl& HowevCI idWw nly~have fLmtwnrii6fone v\dlI&ahL (ioc piecd6to with one input and one

66tput). tb&n~no rnatterh61 W e 6obninne tihb W& will diwdsLend Up w uh funtai6ts of k6 VWarIch Since Out uIirtniatL'oal is to compute the 'crntrotl funt fn (XI xhthuit JOepnds oiitinv vaialexii1dh1r we 616 thb ichreeniibl o.uwr proc~essos to cotniiput at leagt one luinotion of twoor WmoIC variables

\Vha functinLons 6f t\wo vatiablos should we &htoose

C C/Wo6in ftumr0ol~f two( r inowrtwdqbIe~inside the com.putori dLh ftunc-tin srprnd as a

se,QfLL f hardwa i emp n~ted -opedtions. The ffi-stestfunctions amc those that am~comnputed by-- a single hardwareoperation. The basijo hardware supNoutd ope!auons are- anhmetCit op [ s~O a 1), a - b. a ob, a b.6 andp'imm a b)arnd jnax(a. b), The tanic required fbr each operatiOm cirudel,sponkinfa corresponds to the hnumiber Of bits (pLratioml that

*Division is done~by successivc mhiplticauin cormpari gnfarnd. suihtractio11 bah si Ay~in th same w y as we do itmaInulilly~) so. it IS a~muc~h klower ope~ration than

*Muhaplication is implemented its a sNueuncc 6l tdditions(again~bd§ienily ifl thLe ~ 1fln inanner as W& d it nianffu-AUly), o I is nuclh §1wer thAn* nd + ae usually impkmchnit6d, in ih& sante way. ToAdd twVo U h'it bitiary nutob&r~we nced u hbit Additin6§and Ako poi rly~n bit iidditidhs ftoic c TW(A1Ywo need about 2ni hit 0p04tions.

*mi 6t two u hbt bhAr htifub&hcr bah done in nbnroporati6nis we cornpare thebhits ftrom th6bighlest to tho

OpNsed to 1 isthe dossimd mi i iiUMI, the uinuiMMrMof 0.10101 and 0.1lOW is ()1W1() . bcLause itn ihe thid,bict his numhLr has 0 as opposed to I.

*Similarly, nix is an rNht operation1So~the lahste piassihle fbnLtmons of two variables arc min andtIUa\ Slm Idi lv list, ~i com~putng the. tinipum and maxtUMuof Seveim (InWre than two) wtAl humbhc Ther~dbre we willchoose 'these functions lrourcUontrol oriehted conlputer

SumImarizing the ahbovc ~given analysis we can conc1uide,tlut our c.ornip ter Will COr~nti niodiuiks I two type;

,* mnodules that computec Iuntions of one variable* imodi les that compute nuini and i iax of two or Several

D) Huhe to Coimbinw /est irt dul9Wc want to cormbine thcbse modules in §uch ai waty th t the

resultng~comnpuatfions are as fat as possiblL TIw firnb tihd isrequred for Anar g1onuhih crudd6y iOmportionaI t6 the uh6mer(4 k06u6nitd1 steps that it tdk& Wec ani descrihc this numluieOf tLps, in cliear g~*OlttfltC tem'

* -at the beginnun& th& input nirmbur aiit -pm&kd 'bysoune pmce. iso thesci pfice~ssiorthm th fint 464IivOf

* ih~~~~~~~~~~~~~~~~ic~~~~~~~~~~~~~uIts~~~~~~~~~~~~d is processing mnaytn o dlitprocessom~that formthik ewid lIler(;

*the resblts ol tih s.ond nyr (prc in' O1lQIIthird1se,ar fprcss 9g jno

u

SO. weC WoLUl like to c'ombine tb& OmCCessm intW t wsmdlaIIst POWbIhk nutober oflahyersNow, we are -re~id hr the, fonmAl ddfinitions.

\V DMIiNIl uN AN imIH MA04 RmilutU~iet give an inductive &khnition of vvhatt it neans ftr

a tnetion to be comptiuhble by a k-layer ConihputenDcfibition.

* VW4 sa that a map t-6 rm i 3" h bomnputiAh1O bya I litytr copuc dt£ k1/,i w the fiunk tu fcoun iws wih uud or wi1uimiax.

* ht kA I he 4W hit(get 14 si that afiun noniXi,. is compoutahke by a (A1 1) Iayer cuampUlter

tf01f ~((ikt1o n c~te;h nuits c trluc. f iz

bkhI dadl h( J junp-aft r.a

irlet( all fiui nsi am( (114 ptoipud h Alaxe

* onaxq~i 'x( eqoher all tun tons q, aa nprI b vccI~~~~~pauurte

Coniui A mputer ~ I fini.pruivson machibe- so, mrostilts 6t the cormputadtions arL fl vLI dhsolLitCI nr is , Also,a computer is hun 4c rn~ the ~siie of iU niumhbr So weLcan onlyIConpute ~A ftU6tion appiroxjmutely. dnid ornhy an ab;wkitd r~ing& ThctJor~ wbhWen say that wv can cotpuite aniAtbhtritry tnim m,n we kmpiol mheAn that for*i arhitraty rangeTI for irn 41hiway continuous funcuQn j di .In R an dfbr an arhitry a cuc 0~We can cornput t functionthut iN tFo19 to If Mr ihk given range. In ihi snsehe we WillshOW tht not, evciy tintwLol can be comlputed a~n a 2 laycrcomuputer hut that 3 layers ame ahIeady sfiin

PrOpWIti4rn1. IIOu xist w~aI ; nbers T and >0, anid d4 Oithitknisf4heturnEW Jf R sTIWth ino furn )U

ClIose to f 6i [--ITI can be comtpatwd ( a 24IAwr

441


CmnwIn ' To make the text morv r adahble WX- present bhproo& in the Iag scti I.Fwevert we WilI I nake one coonmenthere. The funltion thtt will ibe pmved to be not computahie oniL 2 Layer comnputer is ThtAext at all it i f ( =rion the domain i and the Prpniitin is tre mr - I

Iheorem. Mrt eviert1 iI nnml TI ad Dl, andfir eveO*onniuow finvton f [1 RH, dinw euisur a tfa" onf that is £4av!O 1foJOh 1 7] and thtat is 0Fwittaltdkona 35*Imer morn7ter.

C aoit In the-r word,0tuc ons conipaed 1h a 3-Earrmp,ip ar i qMt p ;aron.

ReOAtla o fitm/ cntvi- As We Will ftcon tie pOof, theapproximatin h:netiaa / k 6f the type wlI4&(A 41, ,

whcrL A1 -uiii(Jit ( .)j -j ( Thk& functions cors>pond the W ctl1cd f4 coftailt [2ji 131] [11) nd0dJ lkt

aL d1tineij I T ( ),11 lxSttr

and- G.w ,

Jq,t, #

oXi -,rei ) LI w _

Let us now assu;me tiat the rules base that desefies the expertrecommonddation fotr cOntr6l co6Sets of exactly two ruI6S-

" iIf one of the conditions ( istru' , tIhenc -* USSl _

"

wieW ekh conditlon rmeanM that thc fbllowing it condtions

* x t sAtkfiea the prperty (b (dskribed by a mniemberhip

* kAtidfi6s§ ihe property q; (dk§cnibed by a muIeihirhiphi. t n i

x. sati .Cst the rt CS4g(des ibed a OJOliNtshIplunclitton x

In lo- ca terins, the condition C oe iz = U hiis the Frml

(b &.i tV.f )

If weWuse MIDrfor and jiiax fr V (these Ar the §imll'estLhoice§ in ftizzy control etithod6logy). ttin the dcgre pCwitih Which we beflcO in a condition (C = C V v a canbe &xpr§sed as

C6i §0p6dibi ly thi deoreoL f bdidi in a conditionfitI(. s I e Accordr ftoizzy cnntml methutIo1gy, wec

Most uk. dtuifi47 awn to d6intle tlLe kcttat colltrilwhich in this cae kLd to t1he choic of

[ p1) 1) 1 p

BCa se of1 ou1 choice of oneca1nW eLsily seethal this expressionl coincLides exaxtty w'th the IInction

ltna-xA t ,t t1A ) where A1 = ll X ,.SO, we get exactiy the expreS iots that tem fio)m the fuzzycotnrol rnethodologvyCa4i tdSiftl. our 3 1A' expression Lkehib6s th& fastestposibl computation tooL We- Can conclude that tC nrrolj tbkifsrthe ffistert poiWbklazmal &v Utdm on OneuoriVrepont t uAsngfit*" hlIthbioIThis stuhl explains Why htzzy rnethoology is someturn

used (And used sucesfulIly) wittiout any expert knoWledgebeing presen as an extrapolation tool for knwn)tim nton.

mmfitnent W have cobnsderd digt i ra1lel bo mputerm Ifwe USO ah4w4 procsors -instedd then mmni And tlka stopbeint ithesim&pIest bnctiui lnsdtd the sum is the sithp31estIi we Just join the two wires -togObht then thc msutifingcurtrnt is equaI to tiE sutn o6f the two input turents. In thiscasc if we ute a sutn (nd tiorv general linear icotination)instead of uuAn and maX, 3 1zaycr cotnputers aic a1ko universalapproximators; the, cor1sCpfldi11g cot.nputers cor)spond toneurteM ltoik~ D!]

VI. DISC SSION

'15fti e6tonrw( , f we areedv 6king tVr t gentet controlm)ethdlolyv ixe! a methiodol1ogy thma enaibles us to itilcaewnt(within a given accuracy) an ar1itrary control strteUty then tiheuinctins corresponding to this methodology must b univrsalapproximatom,

tFront this viewpoinat the known thu that fuzzy controllersaru univer&al appmximatorS i, ofle of the reasons why ftiZiyL )ntrrklIet a inded tised ir many prActieAl situatiorns

Sevendrlid( pes f con rLe& iw ihav a uniitVasOiprwinqat n pWeqt The universcr approximation characterof fUzzy conte61lers does not impt 1eu that fuzzyLcntr6liers dr the only possible cladss of cotrtolersg indeed thereare iany other universail approimatrer polynomiial%neural networks. etHen c if our only requirement on the cont )Il imtholo](oY,

ig that this ieuhodigy bh geteral (un veral) we can alsotISe, e.g. ("1ior tiadifinuAl) polyhomiil conrollers or neura1

b/Niru l J ev fia, etarlMd vr Oftui bertt Rom thevIeWpoint of the univer al app :irn,ation propenlytiadtiona:lor Cizzy contmliers are good asfizzy contir s. Howtevlrin -many priatiAl situations., Izy controllers peeflrm better

hit' te6hdh not eaiIz lead to I dt oflthtI,tey dutalUbIahk t to ]we; omwptU thte IWerd miottrot tn manyptdLtical sittiAtifll Ifaiiy cbiOblre1r pnfbrn betthe Betterin Whait sen§9 In ditIetOnt pldfcid sjituations, we may hAVediflbfOt reqiwrcmets to a contr(1ir irnd thus diIIrLnt ctvUerakfr gauging how gbOd a introUer is. wr Ox4ple, we inayw1n1 o l0ok iF a LontrOl which ;5 sftioothororwIchwl mnfObu§t or which is niore stabl

442

( O',

WIl]~& , * * &tVl.t


In some practcal situatiofns. IZ-y cnmtrolIen do havethese dvantalges :e.g.,. in 'ay cases, a uizizy conltrolIer istono r bust than the traditional one. Howerm in2ffost CaseFfizzy contr6lier is 1iso computtiionadly simpLer and thus. itscoulput:ltions am much fNcr Fra exnnplek for non linearsystem. computing a hizzy Contrhl te!UiS ffl explicit US dsinple inctlons while, .g. to apply a nore tadifionbii ionliheaf controller we fmay nhed tIo s6iw& systems lf ojuaUions.

pip,d a trhuit zal 0lnau1 ftrhfierhis e tpric lPewutomrnJ In this apei we expldinthitbt zy contiollcg are

indeed in sote easonable seneX tgter Thiq resLIt exlainstheAbove eImpi.l.C.01 tct thtt fiiy contmoli rs ofn enhIesus t1 compute control ister

7IheLr may be other iiscses )ffafst cotitomIkr. The fat thatffizy cofntrtllers are aio ig the fitest does not necosanivtnean that the class of iizy Controllers ig the odxr i§t9stOlaks there MAY be other nontiu,y ubntrollrs which re Alkocomputable by A 341ler com-puiteC

In ou-r prof of the.theoreT, we use 3Iayer C)mpuIters co-sponding to tiii, contoml, but tbere could be difdtint prrOO

of the universal apprxmultionrprpey 34l ty coiimputers6ok whhich would usc didtcrnt type of co6ntro1mPTn other woMsd While we proVe thdl ixu,y controllers are

a reasona:ble OWas, tSere 014 be ot1her clMses 6f c:ontrollerswhich am a roasornbL (dtnd as Ist)H cxiAMpIC In our proofsitfnilArIy to Most prO1 that

iWu,y systenim are un versaI approxinators - th coistruc-tioi fuiixy syqetb is based on the valucb of the tuflction

L(t.1* *E) diW ItptL t = (k,) . k

Futty cmt)filers provid a continuoustuaifitionbltWciw thecorresponditng valuesI howecr itemid we can simply use the16ok-up t4dh and aim on. to cah tpk x. the value f(at the parest selctIe tuplle x(). INS lok-up tahble &a 6lsoleas to a co uputationall1y lowx lost although the dkeontifnuty f the iCsuliing approxiluAing pPcW wise litnion 'is, ifnmniy prtical appciitionsl, a dhfinctO d1i;adhntage

VII PROOFSA. Pr(qf qth PY WtOon00. 1et ug prof fb reduction to a contradition) that iF afitm;fon f s0.4);U1 tof1otuJ xf( on

1J], then f cannot be computed on a 2Aayer corputcrIndeLd. suppou thAt i' Then aecording to the DT)ifitinth-e intion fI t.X is ot ine of the fo1lowino three l'rrs:

* g(hh - ) where h is cornputAble on a I 'Idtyer comn

ID i I X) qj (x, w) vhere Al I the ftctonsa e co utitahie on a Ilaer computer

* 6k( i ) t0 i ) whee all the fnutibonsgj ar coniodtahle6in ld LIer d.morruter

L t us shm case hyLcase that all these thre cases are impos-

inthIn e i rst case - (h xX( ) whee hi Scompu1ldb1e otn a 1-1Ayer computer Be,I fid ition this mea.nsihat h isether a ftmetionm of onef van,able or in n or naxLet us consider dl these three sub-cases.

,thn tho ftincti6on f dphdso. tf It panicdIi).

OM1Y On bltn A=,ulai

14iIiic fis197~Olse3 Ii i) 3(01')~ 2

1 ......) f(

aod

J({ 1)..... ..U

I ; )=)<...tN'.l'F-l ....... 0 4 > 06 > (Aatld~~~(OI

(I)i{- iI

.

So f(0 1) 6 e 0 I()1hhnce (0o J(I)which tofnttAdws to (I) So this gu'h-case is impssibI.similaly, it is irpossible to have hi dPendin' only on xo

L2? L t as nomWIrtIe sithccas when

......__ . .t_ _ \

In ibXii Suh £tef

] f- ) ' i I I )- It .1-itl1 ..........1.

Add

B Ut

r I1) (f f1 + .40|I1 =I(1

And

j( I~t) Itp I. I). (- 014 -014>H_16

sm -the equaLitlv (2) is dis Upossible1 Le1 6Ii nim cond ilkthe sheoas

J(r1 'Si) - ax iZ Z3)

1n this sohca

~ ia

.........

ButJf It 1 J<f II 0(1f 014 I4,

And

f0II > ft1YA 2. 01l IG 01.o4

,the equA ity (3) Also impossible

443


20O Tn the woond easItC'

f(kr X')= (I'tjiL.t f ,

wmhcr all the inctions gq are computable on a Il-ayerc(tmputen hFr this case. the impossihilby Hows mm thefo lowing sequence oft steps

) I if one of tihe inct-ions q of the type mu(; )-thten we can vWn!1e

asui~~(j q- (J(I 9 , ( I~

wbe g (Xi )x l` rfi" tbat is clArIy cow3)putdble on a 1 layer compWu&e Alter wci Mke Wucl trlnshwluatioims we get an exprsion br f tht only contaltis in xAnd tffInditbb of obe varjAale

2 2 .1 ct us AbW thAt this expressn canflot cOUL n XiaUIndeed if it d6s then

Ufix Xi )zO.In pdrtieu . f1(I ) < iij 1 ) But we bulkt have

f((I) f(1, i 2-04U 1 >

a contradiction with (4)The contradiction AmhsW that the seond case is also imtpos

i hbl-

I[n th6tL lid1

whei d11 thi tn ctians i arm compitAihk on a I'I4yer com-puw rw this c , the impo:sibility (sillitdvly to fht s§e&Odc&M&a 116w§ ftom lhe u16wing squence of stps

*l.Ii I one oi the onislgi k ft the tvpe ax.,x

then weLcaf rLwrfit

(1.9) (,2.)

wheit q X1 6') - is a lnicblian that iesclarly cornpltdhe on aI lJaye coompuie i AIwr wni akee such trdnA ob-

/,dMl C Pl f Xl~~fit ( fil .~ I L .fitdIl 111buatibtsf we, get an expvssion th1at Only erixtains I'lillAnd ffncuons of O ea able

8 20 Lt us Amu that this exprssion canInt eontain toiiimIndoed if i Jdo then

The, cbtrAdition hOws that Iu& tbnntb one 4f theibnctions 9.

23f So each unetion 9 d¢e nds only o oneOf oidhIlj It MIof th1em dcpend on one rd thL SBne vRAuiah y thenthe entirc tunction f dpnds 6nly n one< Ivahilcl Ad Whae riIeady prvedd (in th6 prwA of the itt &6§e) thait A sImposNiNO So SOflk Iudmlo q dc1ond on xi and wOM olthe I nitiong dIdlpndi on LetId dendtO byhy(b1 ) thbminmutnim of al ftinctions q Lhtd d*pOnd on d'nd by h (the m0nliflmum of All tbe inctions g that depend on x2$ Thenhwe can re fentf ds f )r....3.-.u.. h (: )hh (

24I TogcI a c ntrdidmon Jet UfItaIke xi 1and1. Then.

f(I tI 2 0 4-1.6

Sinice the lbi-nimum of the two nunbciB > b1-6, We Lancbnliu thit otich of thetia is > L6 iet, iiha hj(I) > 6Ga,dfhl) I(i> hSFr1 I Hmd r W-t nwhae

r -13 mlll(hj1(I h2(=' ...,J(1)) < + 0-4......unIi (I).

Since hj(1.) 1(1.6, wo conedothat f 1 1) hI( )Roim

L 1) >f:l 1)-

,1 1t S I3 tXf).

In pari uiar

I I...

2 A01 16 1-L

Th cOfltrddwtion .AOwMftthdidns (/-

that 1111 cannot be Ofel o' the

8 3& SO; e 'h inctoion q depnbd only an one vitrAhl If allof them depend on one and the samie vaiable, say x thenthe entimr inction If dpends orly on one v\afihle and wehbve aleady pmved (in the pr0 of ffte Sust ciase) thait it iS

'3iblbe So, sotnt finctions g. detpnd on xi., adsoni1of thl fUfLtUnJls y dej*nd o'i I...1ea usi denokt hhy1 (i'the mfaXtmuL m of all lundaid ons q, mat d&0pend on XMjiind by

XO the maxitmuiii 6 all thc ffni nsMq- thit dpend on

Thient we can ieprsent f as

1,X 1ii(h.x 1 2)

we can now conclWdei t h (" I )..,(,j Ipmvu tlmt hIi ( 1) 0. 14 Henc,e

f(--I =I tutri (h (I)-,h-h(Bbf

I 10 ' j( - 1 +

Simliad*y o e can

0 14

34}o1 To get a conitrdi tion ltLisu flrs tdkoe a1io - - L Then;

9? -4 014 - lI6 </'014 j(-i 1. 1)+

444

I .and

f(i 1 ....= mnax(hi -1 h(i 1))

9 ... 0 4 1.6

Ix - lnltjk.. . 0llaXGrt x,P i.XO 7

> ruiti(xi X,T .

11 2 ( 1."


Since the mna.iammu of the two nunbers Is < (1i6 we cancyn&LOIId thdit each of them is < 1. G tiett hb1) <1

-.i6 and 3 -.6. For x I and, w ve

f -1) 0.4.

A 1. 111( 11 .li' H1l J It I in(*)3!ig kiobL§ to flb pro thAta iI Pr v the fll.Winrg to in ai

4 I6ot i11xZ I o ft S

6, we cod6 ltudc that i (1 1-) = hi(1F all x-

IP,17 ) -< Rtg 1)\I

we can 'ow conclude that h,.IM <prove tht b(1) < O.A Hence,

E 0 41

04. Sibmilarly one

1-(1 1 ) =: tri xx (h1 Ii) - )( ;

ILet u firt prove tho tit hlnuality. AMOMO that we havepo~ni r i Fr evry 1I I,T _h q "wtl|

n)dehot the int1gcri thAt is thO docsNt to f0e Then,

, ]tf LI- 2 0 14 L (L41

whieh contradits to(5h.The umtad'i tion Shows thiA the third case iis a s) impisi

b>. lef

4i In j11 them cawse we have SheWn that the assump onthat f can hbe co"Ultd on a 2 Lver computer leadS to acontradltion. SO, f cannot bc thus com1puted Q FI)

B. Pmo f qf the Iheomr n

StncL the Inction ]! is continuou§ there cxists a 6 > 0 Wuchthat if 1a- ,'J then

f (xi., xOtt fI(91: Y0<tE.|

Let US mark the grid point on thj gnrd of sw'iz e a1I 6IpoinitS Ir wIich 0hb RiM Ait tht fn

tr integer q- (e_ we nmrk th'e pints with co0rana tes 0 ±i:g

O(i each cooodinate. thus in-a-rk P 2T16 pointS. SO,totally we mark 270Y)' gLd ponts Lt ug denote ihtkitot nunhmer of arid points by k and the points themselves

BR mnt let us deitIe the TUinimum ofMJI~ ~~~~~~~~~~~~~ f (7

T; Tl~~~~~~~~~~~~~~~~~~,,g,... k.'"i~T7..Ha ul.......t

fi iF E * f*

For eacl grid p01int !'.e. wil'l foimpiwleWWSe l1near finctions hi(, ) as f]iows:

* I x 07 6 then

*! nf

i1f O#6 6 < 1|. < 0 7 k th&_( I

h xi)? ="l mfj + j(f n;- Mfo7 0-liSt,,-at

0(7 6 0.6

It uct show tb tt tbheW; fnctioni hIL th function

(x. x [ix(A 1, A )

|zf=.f4l; .................These vues qddetemine a gnid point i, -Witl coordiniltes :r, 6. For this j rnd -or every

li v

thh.fbt yb dfinifinn 6f . we have l

T cret(uc,

H-noiz ~ ~ .......in Aj>

B Itsi& Ixrll hI.fW& h4ve (If i t

hJf(X , tI

S.t1 <... 0.5 o

S ) - fIIP.And hmcnx-W

;qn)

III,

6ibyb hcthohe(4oe, ( !

r ThLrd(,r f(A)E.,~~~ ~ I. _

J;f>x: f(1131{), >~f(t' Si_.

It as -now proVe thit s4ecqnd irequality Acording to uFr

dfin ition o1f the vtlu of Jx iAwa in betweenand IA and thi val iI d i fenti lftno i 0o Iy for the gnidpoints 1, br wih ch 1x.7z < 0 E 6. The dlue

A1 u-----ti(hJ (

ig thus diYeneit frmr m only if alll the viles (X)arediMrent from n.. i.e.. when Izr. x < 0 7 or l I i. R)r

gnrd point thercorm

Ifi( ) f(i = *=

And he ffPf'

.xv& ,t M-# f . I_.g ior f# vJjk. t )..J- By definition offI

/f ( IV) iS this ik Oa=f6t i] hAve

f(. ¼) < f(XI, hT n', (x,S

X. (,i

f6or' I Oth grld ptoits I' Wc

d gl 6iVLf (1,~ Since mf hA§ dcfirmd A bnl,III fJ have

445

si6Le h

JR60i

wbo

rn"

BuIlt

h,h GiJ )

J,J

Proceedings of the 2007 IEEE Symposium on

Foundations of Computational Intelligence (FOCI 2007)

So. tr all grid points we have

and theretore

tf~X iai XX (A A >j < tf(eRx-j *j 11 lEi a)WrJ.eRwl

The xeond inequablity is also pro enSo, both anequalities aie true and hence f is close to f.

VITIT CoNCLJ.SIONFuny techniques hae been orIginally invented a admethmod

o1ogy tha transorums the knowlge of experts (Ibrm-Iatdl in etenus of nlatuiral inag ) into a precise computerimplieptehie formt I erb ar many5succeessfl aplicationsof this methoology to situation in Which exprt knOWledexist. the most well known (and most successful) arc applications to Ifuzy contro

In some casc fuizzy methodology is applied even when noexpert knowledge exiss. In such cases, instead of trvint tappoxitinat6 the unknown ct.ntrol tiction by splines poly-notnialk or by anyv other traditional apprximation techniqueltewthers try to approxiitc t by guessing and tuning theexpert rtiesP Surpisigly thisalppr xi natI siten xv ks hne

In thi s paper we give a mathematical Pexpjnatio for thllsellmpirical phenomenon. S6ifithcally wxe how that apPrOxIa-tiOn by tsing fuizy methodo1ogy is indeed the bst (in sonreasonahbl sense)

At[NOWLItEDIMENTS

This Wbik was supported in part by NASA under co

oipertive agFCeIeEmnt NCC:52N NSF aitAnts EXRt0225670and DMS O53265.. Star Axard from the University of TxasSystem and Txas Iparttnt of Transpoationgrt t N. (I5453,

The authors are th to he anonytmious t s tvaluable suggesiona

[j i ii l$kOY7. isupnp iypt. co"tROlt Are uriVinsal conhu lbs" Hit5ek tilid 5v kI- 1993 o 2

[2] A Kandel gdGd( tawholtz (edst t)itF tunaI S its. LRC Plsts1Bee Ratoin Fii1, 1994.

f3 (. Kltrrr5KYnjt hizc"vewaifegyfta -tnhe ndismti taris(pteii c HtI, tdpor Svlddl Riv& NJ, 1995)

f4 B K ko tts--F tts a iun tr app _tt1njus f' ifth st mttEE Imi m tmiaj awu 6i tl i stoe Stat) t)go

(CA 199` jpp. I I165I[5] V IKreintvlsih ard A Hertnat t1aiALel i tdrilm tottrl roth lid

[iUtt ai inatloui titi"1hia (76"a a t i 1994 No. 3) 6-62[6] V K(iisei0tc ( C. Muecird. 1NH tugu1 rule

haeiamodelitng as a univesltappoi nuttini6h10" ItI- T N uyenand MSi Sogno (edsj11, DOMi dieliig aii Id (Ci iItt ISuWmtbnT MA, 1JM998hPiL35--il95

IL V Krejinoith I NguyennttY YarFi4Iu yS tii /Alo a A tu tatu ra5iit lietnAn. l stY alabl j1ama Ao. ijip anSIlo 6r~fi) A@X50Snbo Fundkin A.hd lts D&WtWO%VSfor iinnematiiia.ta m tflirti iii ii no 9 Vol, 15. No. 6.ppr 6P f4

I81 RK.N La ant V Krienoviti rt'rtifciigent Cortn Makts SrwFetcWIthiiit Expert Krnwedigentm xriimnnt 5 }tI/iaW(5t outin'. 1995.Supplemtent (Ixtendid Abst s 6f AliCt95; I,n i[i (inlortst poppnA00lations of lointeal Coiputatiros EI Paso 1 X. [et-h151995 P 40-144

r91 H N tyi and V Kreiclt "'On i i[r ftal idoficto byfsz systenmst, jtr FiebWnl-ariF ysiiA antOnit4i 6wti .mav Seoul Koaea July 1993 pi l44-14

11 H11 E Nge y niaTt E. A Wa kerAfie i ijurwa La(R( helBiia Raiton. FIidarda5 2005L '.pffilievaandv Ke itAn,-A sw itLniarsal A,proxiaink-mRstilt

For Fii Sys itits, Whi0h f'flet (CNF4I)NF lt1tvi 0ii iifiXlkit ot it i f (h V002T N epo, I 2l t 30

[12] 1 -XN Wait 'Iuusystleias are unit usa appioximatotrst 1m9 tnafit (IE theminimal 0itmni,oiia Am 5rem SanlDi ('A199 p II631. I 169

31 N .ngAindal MtdL ran f,Jam as nhaji ai/enMc dtvVitXh e#01"IkaK ULn.hwrity rf Se ni* C61itbi0tga., Stilfail And 111X5aWPun. ssinh Instiuruot WhItaural Rexu FLSC-SIt I # 169 199I

[14 Ptr R. Vi at V Kreinvaichit" aLiversali Appoxixati n T1 tAtbrtrinifrtllaBar FItzv SystelosMolion hm tt Si aod SiumsNtB Va. l)Np,p. 31 1-9

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