Tap chi Tin hpc \ii DiSu khtSn b(?c, T.27, S.4 (2011), 317-328
NGHIEN CUTU VA THLT NGHIEM LAP TRINH GEN TRONG BAI TOAN TIM CAC XAP x i HAM Q-FUNCTION*
DAO NGOC PHONG', NGUYEN X U A N H O A I ^ NGUYEN THANH THUY^, NGUYfiN QUANG UY'
' 5 d Thong tin Truyen thong thdnh pho Hd Ndi
^ Vien nghien cicu vd phdt tric'n CNTT - Dg.i hgc Hd Ngi
^Dai hoc Cdng ngh$ - D(n hoc Qudc gia Hd Noi Hoc vien Ky thudt qudn sU
Tom tat. Bdi bdo dl xuat nghien cUu viec siS dyng hai hdm thich nghi khdc nhau, MAE va RAE, ap dyng he lap trinh GEN djnh huPng bdi vSn phgm noi cay (TAG3P) trong vigc giai quyet bai todn xap xi ham Gaussian Q-function. I\6t qua da chi ra r ^ g vi^c sil dung cac hdm thich nghi khac nhau se cho cac ket qua khac nhau \ l chat lupng ciia cdc ham xap xi. IVSn CO sd d6, cd thi su" dung hdm thich nghi phii hpp vdi d^c thii bdi todn xkp xi hdm Gaussian Q-function dl tien hanh tim cac xip xi theo hai dang: dang ham tu do va dang hdm mu. Ldi giai tim ra bdi TAG3P tdt hon tat ca cdc ham xip xi da dUpc de xudt trudc day bdi cac chuyen gia (cac nhd todn hpc) vk hon thi nUa nd lai CO ciu true don gian. Kit qua nky se dUOc Ung dyng vao thuc tiln, Ik cd sd cho vifc tilp tuc nghien ciiu va ling dung TAG3P (va GP) trong giai quyet bai toan xap xi hkm Q-function.
Abstract. In this paper, we investigate the use of two different fitness functions, MAE and RAE, for tree adjoining grammar guided genetic programming {TAG3P) with local search in solving the problem of Gaussittn Q-function approximation. The results show that these different fitness functions have different effects on the quality of approximations. Based on the appropriate fitness function, we have discovered good Q-function approximations both in free-form and exponential form. The results found by TAG3P are better than all of the human expert designed approximations in the literature.
This encourages further studies into the application of TAG3P (and GP) in solving the problem of Q-function approximation and applying it in the field of communications.
1. M 6 DAU
TVong linh vuc viln thong, ham Gaussian Q-function (sau day se gpi t i t la ham Q-function) CO y nghTa r i t quan trpng trong nhilu bai todn. Tuy nhign, ham Q-function chi duy nhit dUdc dinh nghia dudi dang tich phan suy rdng, dilu nay se gdy nhieu khd khkn cho nhiing phan tich sau nay dSi vdi vdi cdc hg thong viln thong [1-6]. Do do, r i t cin thilt phai tim ra cdc bilu diln tUdng minh (dang giai tich) ciia ham Q-function.
*Bki bSo dirge thyc hien vdi sy h5 trfl tif quy phdt trien KHCNVN
318 OAO NGQC PHONG, NGUVfiN XUAN HOAl, NGUvlllN THANH THUY, NGUV&N QUANG UY liiy nhien, drn I lu'ii dilm tiiiy, chua co giiii phdp chinli xdc tuypt dii cho dang tudng minh cvia hkm Q-function, vk do dd, cdc hkm x'lp xi Ik li/a chpn duy nhit [G|. M?lc dii c6 nhieu ham xfvp xi dgng tudiig iniiili ciiii hkm Q-hinction da dupc dfi xuit bdi cdc chuyen gia (mpt s6 ham xip xi pliP biln sc ditpc Iriiili liiiy trong phan tiqi theo), nhUng do tfnh chat quan trpng vd phd biln ciia hkiii CJ-runction trong iTnh vUc vifii Ihflng, viCc tim kiltn cdc hkm xap xi tot hdn, cd dgng gi;ii lich vii cd nhilu dgc dilm quan Lrpng (chdng hgn nhU sU ddn gian, tinh kha tich,...) van iic|> luc dUpc tlii,rc hien.
Lgp trinh gen (GP) cd iliO dUpc xem nhu phUdng phdp hpc mdy giai quyet cdc bki todn thong cpia vice lim ra ciic ldi giai dudi dgng chUdng trinh iiiiiy tfnh dUa tren qud trinh chpn lpc tu nhien |7-12l. Tit nhirng iigiiy dau, hoi quy ky hipu (symbolic regression), thdng qua vigc tim cdc xiip xi hkm dudi dgng kv hipu, dgng I ifdng minh vk dgng gi;ii tich Ik mpt trong cac Ung dyng ciuan trpng ciia GP Trong cuon sdch ciia inlnh, Knza da chi ra tai sao GP cd the dupc sd dung d l hpc cdc hkm dudi dang tUdng minh cho cdc bki todn hoi quy ky hieu va tich phkn [7|. liiv nhien, hign tiu chua cd b i t ky nghien (liii nko sii dpng G P trong viec tim dgng x i p xi ciia hkm Q-fimction.
He lap trinh Cil-^N dinh hudng bdi vdn phgm noi cay (TAG3P)- su md rpng cua GP, trong dd vdn phgm noi cay dUdc sii" dung d l djiili hUdng qud trinh ticn hoa [9-llj- TAG3P duoc trinh bay trong |10l tot hdn GP trong mpt so bki todn hoi (piy ky hieu vk tich phan. Cdc nghien edu gan diiy da chi ra ring viec bieu dien dua tren TAG Ik tot hdn cac bigu dien dua trgn vdn pham phi ngii canh trong tim kiem tiln hoa md rpng [8]. Dilu nay lam cho T.\G3P Ik cong cu tilm ndng de giai quylt cac bki todn hIi quy ky higu vk tich phan nhU viec xip xi ham Q-function.
Kit qua nghign ciiu trong bai bdo hudng tdi 2 myc dich. Thii nghigm nhiing anh hudng din kit qua khi sd dpng eac hkm thich nghi khdc nhau vk tit dd lUa chpn hkm thich nghi phii hop vdi bai toan xap xi ham Q-function. Thii 2 Ik Ung dung Igp trinh gen djnh hudng boi vdn pham noi cdy (TAG3P) sii dung tim kiem dia phUPng vdi hkm thicli nglii phii hdp de giai quylt bai todn x i p xi ham Q-function.
2. LY T H U Y E T CO B A N
Ttong phin nky, se dinh nghia bki todn x i p xi hkm Q-function. Sau dd mpt s6 ham xip xi dupc de xuat bdi chuyen gia se dupc rao ta clii tilt. Cuoi ciing Ik gidi thigu ve TAG3P.
2.1. B a i t o a n x a p xi ham Q-function
Hkm Q-function cd vai trd r i t quan trpng trong vi?c phdn tich hi0u ndng ciia mpt si b ^ todn thupc linh vUc vien thong [14]. Trong linh vUe viln thong, cdc nhk nghign ciiu gia thuylt ring hg thing nhilu e6 dang biln Gaussian ngau nhien. Vigc tinh todn xdc suit iSi trong hg thong thong tin so thudng gin lien vdi vi^c tinh todn dUa trgn hkm loi Q{x), dUdc dinh nghia nhu sau:
Q{x) = j^^e-y'lUy. (2.1)
Dang hkm Q-function nay cd hai vin de. Trudc tlgn, xgt v l gdc dp tinh toan, nd doi hoi su chkn tren ciia gidi han v6 cue khi sii dung tich phan so ho9c ky thukt tinh todn. Van dl thd hai gdp phai Ik tham so ciia hkm Ik gid tri dudi cua hdm tich phkn, do dd, nd se gdp khd
NGHIEN cCU VA THLT NGHIEM L A P T R I N H GEN TRONG BAI TOAN TIM xAp XI 3 1 9 khdn vl mat giai tich khi tham s6 nay phy thupc vko mpt biln s6 ngiu nhien ddi hdi trung bmh thing ke tren phan bd xdc suit. Trong nhimg trudng hpp nhu vdy, ta c;in cd dgng hkm Q{x) vdi tham so dau vko khong nhiing khpng phai Ik gidi hgn tren hay dudi cua hkm tich phkn, mk cdn xuit hign trong ham tich phan nhu Ik tham s6 diu vko ciia cdc hiim cd ban.
Dang thiic mong muon van Ik dang ma gidi hgn ciia tham so dpc lap Ik hiiu hgn, dong thdi, hkm tich phan van duy tri can true tu nhign ciia nd.
Cdc bieu diln tudng minh ciia ham Q-function r i t cin cho cdc bki todn lien cjuan din higu ndng ciia he thing vien thdng, ddc biet Ik loi trung binh, bit vk xdc xuit khIi 16i vk hdm mu so nguyen vdi biln ngau nhien thu dupc trong mdi trudng Fading. Muc dich cd ban ciia vigc x i p xi ham Q-function la nhdm dUa ra dgng dun gian vk tudng minh ciia hkm Q-function, tien ldi trong vice phdn tich todn hpc ciia higu ndng hp thong viln thdng [151.
2.2. M o t so d a n g x i p xi d o chuyen gia de x u a t
Din thdi digm nky, cd mpt so dang x i p xi dupc dc xuat bdi cdc chuyen gia (cdc nhk todn hpc) trong Hnh vuc vicn thdng vl cdc xap xi ham Q-function dgng tudng minh. Trong [1] dua ra mpt tap cdc x i p xi ciia ham ldi bii, lien quan din hkm Q-function. Xip xi ham Q-function trong [1, eqn(9)] dUpc dinh nghia nhu sau:
Q(x)
«%J
,.(2.2)
V27r\/1 -I- x^
Trong [2] sii dung x i p xi PBCS vdi ham x i p xi trong [2, eqn(13)]
vdi a = 0.339, b = 5.510 se dupc ky hieu la x i p xi OPBCS.
TVong [3], mdt dgng x i p xi don gian vk hilu ich khac dUdc dua ra. Tiiy nhien, nhUng xap xi nay cd sai so ldn vdi tham so diu vko nhd, chinh vi vay nd it phii hdp vdi kenh fading trgn trung binli. Hkm x i p xi cua Q-function trong [3] Ik:
Q{x)^^e-'^ + \e-'^, (2.4) trong bai bdo nay dUpc ky hieu Ik xap xi CDS.
TVong [6], cdc tdc gia da dl xuit dang thiic ddn gian khdc cua x i p xi hkm Q-function vdi sai s6 nho tren toan bp dai gia tri diu vao. Dg thugn tifn, ham x i p xi Q-function dupc dua ra trong [6] nhu sau:
trong do, gi4 tri tSi Uu cua biin so A vs. B Ik A = l,9S vl B = 1,135.
Tuy nhien, nhOng lian che cua xap xi dang t\} do niy IS, no c6 t h i rat chinh xSc nhirng lth6 ap dung trong thuc tS bai su phiic tap ciia dang ham xap xi. Do do, nhiing dang^xip xi, mSc dii CO do chinh x4c thap hdn nhung Co dang don gian thi van hiiu ich trong thuc te. Benitez vk Casadevall trong [7] da de xuat dang xap xi (duoc ggi la x i p xi dang ham mii) vdi dang tuong
320 D A O NGOC PHONG, NGUYEN X U A N HOAI, N G U Y A N THANH THUY, NGUYEN QUANG UY tu vdi hkm Gaussian, gpi Ik e^'^'''', trong dd P{x) Ik da thdc b ^ 2, vdi P{x) = ax^ -i-bx-\-c.
Vdi dgng x i p xi nky, trong khoang [0, 81, hp da Uiii dupc cite gid trj toi Uu ciia tham s6 a, 6 vd c tudng ling Ik -0,4098, -0,502G, vh -(),«111. Dgng xii]) xi nky it chinh xdc hon so vdi OPBCS, nhung kha nkng de tinh toiiii linn nd trd nen Inni ich trong bki todn phan tich hg thong vien thong.
2.3. L a p t r i n h g e n
Lgp trinh gen (GP) Ikiiint cong iij^lic tinh todn tiln lida tudpnggiiii quylt 1™ todn, khong ddi hdi phai bilt dgng thiic li.iy cau tnic ciia ldi giai [8, 131. GP cd t h i khai quat Ik phUdng thiic cd hg thong vk dpc lap vdi bki toj'm dg tu dpng gidi qiiycl bki todn tif nhflng djnh nghia c;\p cao. GP dua I rc'-ii y tudiig chhih Ik cdc chUdng trinh m;iy thih cd kha ndng tu tiln hda dl thuc hign mpt cong vi('c nko dd, dupc dUa ra bdi Koza vko iiain 1992. Ky thuat nky cung cip mpt iicii tang cho vice tgo ra nhrmg chUdng trinh mdy thih mpt cdch tu dfiiig.
Qua trinh tiln hda cua GP bit dau vdi vipc tgo ra ngau nhien quan t h i gom cdc cd thi (chudng trinh). C) nini tlic hg, dp lot ciia moi cd Ihc chudiig trinh (hkm s6) dupc ddnh gid tren cd sd miic dp tot eiia chUdng trinh (hkm Sd) trong vigc giai quyet bki toan ddt ra. Do dd, cae ca thg so dupc lua clipn thong cjua cdc loan tU di truyin (lai ghep, dot biln) dg tao ra cdc cd the con trong the hg til]) theo. Qud trinh nky Idp di Idp lgi cho din khi ldi giai duoc tim ra hodc khi gdp mpt so dilu kign dtfng nko dd (13]. Hg Igp trinh GK.\ cliuan se gom 5 thanh phan: bilu dien chUdng trinh, khdi tgo quan the, hkm thich nghi, todn tu di truyin va cac tham sd.
Nhu da trinh bky d trgn, GP dupe sii dung d l hpc cdch xdc dinh mpt hkm dua tren dii lieu man. Hg thing GP thudng thg hign dUdi dgng chuoi gen phiic tgp vdi kich thudc vk hinh dgng biln ddi. GP thknh cdng trong giai quylt nhilu bki todn trong thuc te bao gom cdc bai todn x i p xi tich phan ky higu ([8, 131). ^ V rihiOii. din thdi diem nky chua cd cdng bo nao sii dung GP trong vigc x i p xi hkm Q-hniction. Do dd, si' cd gid trj neu GP cd the giai quylt baj toan nik khong cin nhiing kiln thiic hilu bilt chuyen sdu.
Cd nhieu phUdng phdp hpc mdy dupc sir dyng nhu: c;iy quylt djnh, mgng nd ron nhan tgo, lap triiih GEN, iii;iy vccid ho trp - SVM Cdc phUdng phdp dupc tiep can theo hai mp hinh Ik hpp den (Blackbox) vk hpp trdng (W'hilcbox) liokc Iai gh6p giiia hai mo liinh tren.
Mang no-ron hay in;iv vecid ho Irp Ik mpt vi dp vl md hhih hpp den, do ldi giai thich cho kit qua qud phu:c tgp dc c6 t h i hilu dupc. IVoiig khi dd. l^p trinh GE.N tilp cgn theo md hinh hpp trdng vdi ldi giai d dgng giai [Inch vk dgng ddng. Chinh vi t h i , nhim myc dich tim cac xap xi 6 dgng giai tich vk tudng minh ciiii hkm CJ-fimctiuu, vigc sii dpng GP Ik phii hdp.
2.4. H e lap t r i n h G E N d i n h hUdng bdi vfin p h a m noi cay
Lgp trinh gen djnh hudng bdi vdn phgm (GGGP) (Whigham, 1995) vk lap trinh di truyen djnh hudng bdi vdn phgm noi cSy (TAG3P) (Nguyen et al., 2003; Nguyen, 2004; Nguyen et al., 2006) Ik cdc hg md rOng t lia Lgi^ trinh gen truyin thong dua trgn vigc sii dung v5n phgm phi nga canh vk vkn phgm noi cay d l djnh nghia ngon ngii, khdng gian bilu dien cdc doi tUPng dupc hpc (hkm, chUdng trinli mdy tinh, hay cdc thilt kl).
V&n pham not cdy (Tree-Adjoining Grammars - TAGs)
van pham noi cay da vk dang trd thdnh logi vkn phgm quan trpng trong xii ly ngon ngfl tu nhien (Natural Language Processing - NLP).^Muc dich cua TAG la chi ra mot cdch true
NGHIEN COU VA T H O NGHIEM U P T R I N H GEN TRONG BAI TOAN TIM xAP XI 321 tiSp cSch tao cau true ciia c4c ngon ngO tu nhien so vdi cic h? viet lai tren xau thugc cfc Idp cua Chomsky CSu true viin pham cua TAG dupc liinh thinh bdi hai tap hop cay con, cay khoi tao hay con goi la cay a. tuong ijng vdi cSc tliiinh phan co ban cua ngon ngii vk cay bo tro hay c6n goi l i cay /3 tuong ung vdi ciic nlifui t6 cd t h i eh6n them cua ngon ngS. Cac ciy con nay c5n dugc goi Ik cftc cay co ban. Cung gi6ng nhu doi vdi c4c vSn pham Cliomsky, cac nut ciia cay duoc gan bkng cac ky hi?u ket thiic (terminal symbols) v i khong k& thuc (non-terminal symbols), trong d6 cSc nut ben trong phai duflc giin bing cSc ky hieu kheng kit thiic, cac niit 1ft co the dupc gan bKng ca hai loai ky hieu ket thUc hoic khong ket thiic.
van pham TAG la mot bO nim thiuih phan {"£.. N. I. A, S), trong do:
• J2'. t i p hiiu han cac ki liieu ket thuc.
• N: tap him lian cftc ki hieu khong ket thuc.
• S tap phin biet cac ki liieu khdng kit thuc.
• /: tap cac ray khdi tao. TVong ray klidi tao, cftc niit ben trong phai duoc gin b4ng cic ky hieu khong ket thuc, cftc nut 1ft cd the dugc gin bing ea hai loai ky hi?u kit thuc hoac khong kit thuc. i\ut la cd ki hieu khong kit thiic cd dinh dSu i thi hien kha ning thuc hien phep the tai cic nut dd.
• .4: t i p cftc cay bd trg, TVong ciy bS trg, cac ndt trong dugc gftn bkng cic ky hieu khong kit thuc. Moi cay diu co chiia indt nut l i trimg ten vdi niit gOc (mang ki higu khong kit thiic). 0 niit l i nay dugc dinli dau bang ki liifu * v i dugc gpi l i nut chan ciia cay phu trg. .MOI cay phu tro chi co mpt nut chan.
V P * 1 a2 a3
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loved
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1
Hinh 2.1. Mdt vi du vl TAG
Cac phep viet lai tren cay chinh dUdc sd dung vdi cdc vdn phgm noi cay Ik phep nil cky (adjunction) vk phep t h i cay (substitution). Phep ndi cdy tgo ra cky d i n xuit mdi 7 tCr cay bd trp P va cdy r (cd t h i Ik mdt cay khdi tao hodc Ik cay dan xuit dudc da dupc tgo ra). Neu cay T CO mpt niit trong dupc gan nhan '^4' vk cky P \k cay dgng A, viee kit ndi P vk cdy r dupc thuc hign nhu sau: dau tien cay con a b i t nguIn tai A tam bi ngdt khdi r , sau dd P gdn vdi T d l thay t h i cho cay con dd, cudi cimg a dupc gin trd lgi mit chkn ciia p. Cay 7 Ik cay ddn dupc cudi cung dUdc tgo ra trong qud trinh nky.
322 D A O NGOC PHONG, NGUVfiN -XUAN HOAl, NGUVfiN THANH THUY, N G U Y £ N QUANG UY TVong phep t h i cay, nipt mil khong kit thiic -V ciia cky khdi tgo hodc cay d i n xuit dupc dupc thay thi Ixii mpt lAy khdi tgo dgng A' Cdc cay dan xuat dUpc hokn chinh trong T.AG (cay dan xuit cd cac mil Id diu dmc dan nhan boi cac ky tu kit thiic) tUdng Ung true tiep vdi nhiing cay dan xuit dUpc tgo ra trong Idp ngdn ngii Chomsky.
Lap trinh gen djnh hxidng bdi van pham noi cay
T.\Gs c6 gid tn itiig dpng cao dni vdi xu" ly ngon ngii tu nhign, d ^ bigt, mpt dilm mgnh trong cay dan xual ciia TAG Ik khi xda b i t ky mnl cay con nko ra khdi cay dan xuit, phan cdn Igi van la mpt cay dan xual TAG vk Ik <dy hpp Ip. Ndi cdch khdc, cay dan xuat trong TAG Ik cdy cd thii phan khdng c6 dinh. Hg (pia ciia dilu nky Ik kha ndng tgo ra cdc todn tut mdi giong nhu trong cdc \\v tinh todn phdng tiln hda sinh hpc dimg bilu dien tuyln tinh cho nhiem sic t h i mk van duy tri tinh chit Uu vi^t ciia bicu dien dgng hinh cdy.
Hp Igp trinh GEN dinh hudng bdi vdn phgm noi cay (TAG3P) Ik hg th6ng sii dung vdn phgm noi cAy de djnh hudng tien hda [10-121- TAG3P sii dpng vdn phgm noi cay ciing vdi vkn phgm phi ngU canh d^ ddt ra nliUng rkng bupc vl cu phdp cung nhu nhiing sai Igch khi rim kicm ciia cliUdng trinh tiln hda. Hg thong TAG3P cd t i t ca nhiing thudc tinh ciia cdc hg thing GP thdng thudng dUa trgn cdc bilu dien dgng hinh cky khdc.
Cfni tnic vkn phgm dupc xdc dinh bing tap hpp cdc cky a vk cky p. Ciu true quan thi Id cdc cay dan xuit tii vkn pham nay. Vigc lUpng gid dp tot ciia moi cd t h i dUpc thUc hign b^ing cdch tgo ra cdc cay dan xuit dudc tUdng iing tii cay dan xuit TAG, sau dd danh gid bilu thiie trgn cay dan xuit dUdc (tUdng tu nhu trong GP). Khdng gian tim kilm do dd dupe xdc dinll bdng vdn pham, tkp hpp t i t ea cdc cky bilu thiie GP dia do vdn phgm cho trudc tao ra vdi gidi han vl dd phiic tap ciia cay nay. Tliy nhign, ddc tinh thii nguygn khdng xdc dinh ciia cay giup kiem soat mdt each d l dang theo kich thudc ciia cky, do dd, kich thudc ciia cay dupc su dung d l kiem soat dd phiic tap cua cay trong TAG3P thay vi theo chilu cao cua cay nhu trong cdc hg GP khdc.
Ciing giong nhU trong hg lap trinh GEN chuin, TAG3P g6m 5 thknh phin Ik bieu dien chudng trinh, khdi tao qudn the, ham thich nghi, todn tii di tru>-lii \-k cdc tham so. Cu tlie nhu sau:
Bieu dien chuang trinh
Trong TAG3P, cau true vdn pham dUdc xdc dinh bdng tgp hpp cdc ca>' a vk cd>' p, moi ca thg trong mpt quan t h i Ik cay dan xuit eua LTAG, dudc d i n xuat tCr cdc vkn phgm nay.
Khdi tao quan thi
Thanh phin thii hai trong TAG3P Ik thuat giai dc khdi tgo ngiu nhien quan t h i (cdc cay dan xuit Glex). Day Ik qud trinh Idp di Idp l?u vigc tgo ra ngiu nghien mpt ca thg (rapt cay dan xuit Glex). Vigc nky dUpc thuc hign bing cdcli chpn mpt so ngiu nhien trong khoang cho trudc, sau dd lay ngau nhien cau o tU tap cay cd sd trong Glex d l tao ra cdy d i n xuit Glex.
Cay dan xuat nky se dUpc md rpng bdng phep noi vdi mpt ca>' P dupc clipn ngau nhien tU tap cay CP sd. Qud trinh nay kit thiic khi kicli thudc cay dgt tdi gid tri dupc clipn nhiu nhign 6 tren.
Hdm thich nghi
Dg danh gia su phii hpp eiia cd thi, diu tlgn ta se chuyin ddi thdnli cay dan xuit dupc trong Glex (cay d i n xuit cua G trong trudng hpp sil dung G). Sau dd, qua trinh tinh todn sii phii hpp cua cd t h i dupc thuc hien tren cay dan xuit dupc.
Todn tii di truySn TAG3P ciing cd cac toan tii di truyen chinh nhu GP cliuan la li;a clipn, tdi tgo, lai ghep vk dot biln:
NGHIEN CCU VA THtt NGHIEM LAP TRINH GEN TRONG BAI T O A N TJM XAP XI 3 2 3 Do tinh thii phkn khdng co dinh ciia bilu diln TAG, nhilu todn tii mdi dupc thilt kl. Viec sii dung TAGs giup cho TAG3P cd dupe nhiing todn til hiiu ich vk phdng tiln hod sinh hpc hdn so vdi cac toan tii cua hg lap trinh GEN vk hg lap trinh GEN dinh hudng bdi vdn pham khdc 111]. Viec thilt kg cdc todn tu: dd khd cd thg thirc hign trgn bilu dien chuin ciia GP vdi ciu true cay cUng nhu dii vdi Hg lap trinh GEN dinh hudng bdi vkn phgm su: dung ciu triic cay idn- xuat trong CFG. Mpt so todn tii dUpc thilt kl nhU: di chuyin (relocation), sao luu (dupHcation), vk dkc bigt Ik hai todn tiS chgn (insertion) vk xda (deletion). Nhiing ddc tinh nkng nky mang lai cho TAG3P nhiing Uu dilm trong vigc tim kiem m6 hinh d i giai quylt cdc bki todn hoi quy tupng tnmg vk ticli phan. Ben cgnh dd, trong TAG3P cbn thilt kl cdc toan tii dot biln tii vung.
Trong bai bdo nay, ta sii dpng 2 todn tii chen vk xda hogt dpng nhu Ik todn tii tim kilm dia phUdng tren cd sd kit hpp ciia vigc tien hda vk thukt todn tim kilm leo doi.
Chen: Neu kicli thudc cua cd t h i cdn nim trong gidi han tren cho phdp thi thao tdc chen dupe thuc hign bkiig cdcli thgm nut Id vko cd thi dd.
Xda: Nlu kich thudc ciia cd the cbn nim tren gidi hgn dudi thi thao tdc xoa dUdc thuc hien bdng cacli xda mit la ra khoi ca t h i dd.
Cdc tham so
Cdc tham so trong TAG3P bao gom: kich cd quan thi, sd the hg toi da, kich thudc t6i da va toi thiiu cua cky, s6 lin thuc hien toi da (thuc hien cdc thao tdc di truyin), xdc xukt thuc hign eke toan til di truyen, su sai khdc toi da vl kich thudc cua cdy con cu vk mdi khi thuc Iiign thao tac di truyen (lai ghep, ddt biln, ...).
3. X A C D I N H H A M THICH N G H I
Nhu da trinh bay d trgn, mpt he thing GP bao gdm 5 thknh phan Ik biiu dien chUdng trinh, khdi tgo quan thi, hkm thich nghi, todn til di truyen va cac tham s6. Trong dd, ham thich nghi Ik thknh phin rat quan trpng dl ddnh gia dp tot cua ca thi, lam cd sd cho vigc thuc hign qua trinh tiln hda. Cd nhilu dgng hkm thich nghi khdc nhau dUdc sil dung dl phu hpp vdi tiing bai todn khdc nhau.
Dang ham thich nghi hay dUdc sd dung vdi GP vk TAG3P la hkm dUa tren ldi tuyet d6i trung binh (Mean Absolute Error - MAE):
^*AE=^-^2l/i-yi|. (3.6)
Trong do, N Ik s6 lUdng dU lieu mau, fi Ik gid tri cua hkm Q-function vk yi Ik gid t n cua ham hpc dUpc tai m l u thii i trong tkp dii ligu miu. Tirong bdi bdo nky, tap dfl hen man gdm 400 gia tri, chia deu trong khoang [0-8] vk dUpc tao ra tUdng dng diia vao cong thiic (2.1). Ly do cua su gidi han nay Ik bdi ham Q-function se tiem can gid tri 0 khi a; ldn hdn 8, do dd it, cd gid tri hiiu dung trong linh vuc vien thPng.
Tiiy nhign, mkc dii hkm thich nghi nky chi ra dUdc tinh x i p xi vdi ham Q-function dudi gdo dp sai s6 tuyet doi, nhiCig nd khdng t h i hign dUpc tinh tot cua hkm x i p xi. Dkc bigt, vdi dgc dilm Ik ham Q-function se tiem cdn cdng gin gid tri 0 khi x cdng ldn. Chinh vi vky, nlu sii dung ham thich nghi dua trgn ldi tuyet dOi tnmg binh thi se khdng the hign dUdc ban chit
324 D A O NGQC PHONG, NGUYEN X U A N HOAI, NGUYEN THANH THUY, N G U Y £ N QUANG UY ciia hdm xap xi tim dUpc (do loi tuygt dni cd t h i Ik nhd nhung Igi Idn hdn nhieu so vdi gid trj ciia hkm). Do dd, d day ta thir nghiOm thfim TAG3P vdi hkm thich nghi khdc di;a trgn hdm loi tUPrig doi (Relative Absolute Error - RAI-'O uhu sau:
" • A E = i f : l ^ . (3.7)
trong dd, N, /,, y, dupc dinh ngliia tUdng tu nhU tren.
Thilt Ikp 2 tkp thf nghipni vdi TAG3P sii dgng 2 hkm thfch nghi d tren, moi tkp thi nghiem g6m 50 lin cligy. Khi (hilt Ikp thf nghipm, ta sii dung tdp hkm gIm cdc hdm (-i-,- ,*,/,exp,log,sqrt). Vigc lua chpn thgm cdc hkm exp, log, sqrt Ik dUa tren cd sd tham khao cdc dgng xip xi do chuyen gia dl xuat nhU da trinh bky 6 trdn. T^p kit dupc sur dung gom cd tham so X, n vk cd( hdng so ngau nhign dUpc sinh ra trong khokng (0,1).
Vdi cdc kit qua thu dupc sau khi chgy 2 tkp thi nghigm, ta lua chpn mpt s6 hkm xip xi t i t nhit tim dUpc vk ve do thi loi tUdng doi cua cdc hkm nky. Qua dd, cd nhan xet nhu sau:
klii sii dung MAE Ik hkni thfch nghi thi trong khoang [0-e], gid trj ham x i p xi dupc tim Id tot. Tuy nliien, khi x Idn hdn e, chit lUpng ciia cdc xap xi xau di ddng k l vk kha nhanh. Dilu nay ed thg ly giai bdi ddc digm eiia hkm Q-function Ik gid tri ciia hkm Q-function tien ve gia tri 0 r i t nhanh khi x ldn hdn e (nhd hdn 10"^). Do dd, vigc sii dung MAE Ik hkm thich nghi trong doan [e-8l cd thg dan den nhflng sai Igch. Ndi cdch khdc, do gid tri ciia hkm Q-function d nhiing digm nky la rat nho, viec cd gid tri tuygt doi nho khong dam bao cho vigc se cd loi tuong doi nho.
Ttong khi dd, vdi cdc kit qua thu dupc khi sii dung RAE Ik hdm thich nghi thi cluing toi nhgn thiy cdc xap xi nky cd loi tUdng doi nho dong deu hdn so vdi c4c xap xi dUpc tim ra khi sii dung hkm thich nghi MAE. Nhu vay ^i c6 dupc xap xi tot deu trgn doan [0-8l, RAE Ik Ii/a chpn phii hpp hdn.
Qua vigc so sdnh kit qua x i p xi nhgn dupc khi sil dung hai hkm thich nghi Ik MAE vd RAE, ta thiy ring vdi ddc diim cua hdm Q-function thi hkm thich nghi phii hpp dupc sii dung se Ik RAE.
4. T H I E T K E T H I N G H I E M
TVgn cd sd kit qua Ii/a chpn hkm thich nghi d trgn, ta tiln hknh thiet ke thi nghigm de tim cdc xap xi theo hai dgng: dgng hkm ty do vk dgng hkm mu.
Vdi thi nghiem tim x i p xi dgjig hkm t\r do, sii dung tkp hdm gom cdc ham (+, -, ", / , exp, log, sqrt). Cdn trong trudng hpp tim xap xi dgng hdm mii ta sil dung cdc tg,p hkm gIm -t-, -
*.
T^p kit dupc su: dyng gom cd tham s6 J;, II vk cdc hdng so ngiu nhien dupc sinh ra trong khoang (0,l).Phuong phdp chpn dupc suf dung d dky ik Itfa chpn cgnh tranh vdi kich c5 la 9.
D i ddnh gid hifu qua ciia cdc todn tu: tim kiem dja phuong trong giai quyet bdi todn xip xi hkm Q-function, nhu ciu hinh thi nghipm ngu trgn, ta tiln hknh chgy vdi 2 trudng hpp:
khong suf dung tim kiem dia phuong (TAG3P) vk cd sur dung tim kilm dia phuong (TAG3PL).
NGHlfiN c d u VA THLf NGHIEM LAP TRINH GEN TRONG B A I T O A N TIM XAP XI
Bang 1. Bang thong so dupc cki dkt cho mdi thi nghigm T h a m sd
So t h i hg Kich thudc quin t h i
Phuong phkp chpn Xdc xuat cdc todn tut Xdc xuit cdc todn til Hkm thich nghi Todn tu: tim kilm dia phuong Chien lupc tim kilm dia phUdng
Budc tim kilm dia phuOng S& lin chay thi nghigm
G i a tri 100
3000 vdi TAG3P vk 100 vdi TAG3PL Lua chpn cgnh tranh vdi kich cd Ik 9 Lai ghgp ^ 0.9, Dpt bien ^ 0.1 Lai ghep = 0.9, Dot biln = 0.1 Ldi tUdng doi trung binh (RAE) Ngdu nhien lua chon tokn t i chgn vk todn t^ xda
Leo doi 0 vdi TAG3P vk 30 vdi TAG3PL
50
5. K E T Q U A VA N H A N X E T
Di xac dinll cic xap xi tot ciia h i m Q-function, vdi moi lan chay, deu luu trii lai nhiing c i t h i cd gia tri RAE tugng dudng hoic tot hon gii t n RAE ciia cic xap xi dua ra bdi chuyen gia. Trong dd, gia tri RAE cda cic xap xi dUa ra bdi chuyen gia trong t i p dii heu huan luyfn la: CDS (0,2437409), GKAL (0,0614184), PBCS (0,0340417), OPBCS (0,0017471) vi EXP(0,0348177). Dua tren gia tri nay, c6 th'S ghi lai nhung xap xi tim dugc cd gia tri RAE nho hon 10"^ (tile la tot hon cic x i p xi dua ra bdi chuyen gia, ngoai trtt xap xl OPBCS) va nhiing x i p xi tim dudc cd gia tri RAE nho hon 10"' (tde la tot hon tat ca cac x i p xi dua ra bdi chuyen gia).
TVong thi nglliem tim xap xi dang ham tU do, khi chay 50 lin vdi trudng hgp TAG3P (khang sil dung tim kiim dia phUdng) thi ket qua thu dugc khdng cd x i p xi nio cd RAE nho hon 1 0 " ^ Con trong so 50 lin chay vdi trudng hgp TAG3PL (cd sil dung tim kiem dia phuong) thi cd 20 lin chay cho kit qua RAE nho hdn lO"^ (trong dd cd 1 trudng hgp cd RAE la 0,001 tuong duong vdi RAE ciia OPBCS la 0,0017), nhung khong thu dugc x i p xi nio cd RAE nho hon 1 0 " ' . Xap xi t6t nhit tim dugc la:
TAG FREE= /(x)
91W + 92W + h{x). (5.8)
trong dd
91W =
f(x) = 0,5122881 - 3.152892, 1,351156
0,7883671 + a:3e(i+3,292595)(i,a4825ii^D,04573ii_,.5_141592a;a_xH2,I41592a:+0,455729 0,088717 92(1) = 0,003616a:' + 0,087906x2 - 0,538290x + 0.096699,
h(x) = -0,4777591^ - 0,512288x + 0.450333.
(5.9) (5.10) (5.11) (5.12)
326 D A O NGOC PHONG, N G U Y £ N XUAN HOAl, NGUVliN THANH THUY, NGUYEN QUANG UY Cdn trong thi nghiCin tim x i p xi dgng hkm mn, klii chgy 50 lin vdi trudng hpp TAG3P thi kit qua thu dupc duy nhit mpt x i p xl c6 RAE iilin hdn 10"^ vk khdng cd x i p xi ndo c6 RAE nhd hdn 10"^. D9c bigt, trong s6 50 lin chgy vdi TAG3PL thi cd 30 lin chgy cho kit qua RAE nhd hon 10""^ vk cd 4 lin cligy cho x i p xi vdi RAE nhd hdn 10"^.
Nhu vdy cd the thiy, nlu dp dpng tim kilm dja phUdng thi diu tim dUpc (40-60% so I ^ cligy) cdc kit qua tot bing Iiokc hdn cdc x i p xl da d l xuit bdi chuygn gia. Dkc bigt, trong trudng hop x i p xi dgjig hkm mu vdi tim kilm dja phuOng thi thu dUpc 4 lin chay cho xip xi vdi RAE nhd hdn lO""*, trong dd, x i p xi t6t nhit tim dupc cd dgng nhu sau:
TAG EXP = e''' (5.13)
trong dd, cdc gid tr- hdng so nhU sau: a^0,0000018043; b=-0,00010'J; c=0,002238; d = • 0,023735: e --- -0,344644; f = -0,774128 vk g = -0,098740.
Hinh 5.2. Loi tUdng d6i ciia cdc x i p xi Q-function
Hinh ve tren Ik do thi hdm loi tUdng doi ciia x i p xi viia tim dupc bdi TAG3PL vdi dgng tu do (gpi tat Ik xap xi TAG-FREE) vk dgng hkm mi: (goi t i t Id x i p xi TAG-EXP) so vdi xip xi tot nhit dua ra bdi chuygn gia (xip xi OPBCS) vk x i p xi dgng hkm mu dua ra bdi chuyen gia (xip xi EXP). Cd thg thiy dp chinh xac cua x i p xi TAG-FREE la tuong dUdng vdi xap xi OPBCS (cd khoang TAG-FREE tot hdn nhung cd khoang thi ngijpc lgi), tuy nhign, dgng thiic ciia x i p xi nky thi lai r i t phiic tg.p.
Tirong klii dd, cd t h i thiy dp chinh xdc ciia x i p xi TAG-EXP Ik tUOng dUOng vdi OPBCS trong khoang [0-1,5] vk khi a: > 1,5 thi nd t i t hdn OPBCS. Hon t h i nQa, xap xi nky lgi c6
N G H I E N C C U VA THIS NGHIEM LAP TRINH GEN TRONG BAI T O A N T I M X A P XI 3 2 7
dang hkm mu nen r i t dOn gian trong qud trinh tinh todn, phti hpp vdi ddc dilm sii dung trong phkn tich hg thing vien thdng. Kit qua nay r i t quan trpng vi kl td khi xap xi OPBCS dUdc dua ra nam 1979 din nay chua cd mpt x i p xi nko tot hpn dupc tim ra. Ngoai ra, cung cd dang ham mu nhung so vdi xap xi EXP thi x i p xi TAG-EXP cd dp chinh xdc cao hdn rat nhilu.
6. K E T L U A N
Trong bai bdo nky, chiing tdi thuc hign thii: nghigm l^an d i u bing viec sii: dung cdc hkm thich nghi khdc nhau (MAE vk RAE) qua dd xdc dnih dUdc hkm thich nghi RAE Ik plm hdp nhit vdi bai todn x i p xi ham Gaussian Q-function. Trgn cd sd dd, ta thilt kl vk chay 2 tap thi nghigm sd dpng TAG3P vk TAG3PL tim cac xap xi dgng hkm tu do vk dang ham nm.
Kit qua thu dupc nhiing x i p xi tot hdn so vdi xap xi dupc dua ra bdi cliuyen gia, hdn t h i niia no lgi cd ciu true ddn gian, phii hdp vdi Ung dung trong hg thong vien thong.
Vigc tim them cdc x i p xi tot hdn nUa khi sii dyng TAG3PL vdi dang ham tu do vk dang ham mu, cCing vdi nghien cUu su dung phuPng phdp kilm sodt trkn ma nhu phupng phkp Tarpeian Bloat Control vk ihig dung cdc xap xi tim dupc vao bki todn phkn tich he thong vieu thong dl tao ra cdc mo phong cliiidi xdc cho cac hg thing ddc bigt se Ik npi dung nghien cUu cdc cong trinh tilp theo.
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328 D A O NGQC PHONG, NGUYEN XUAN HOAl, NGUYr;N THANH THUY, NGUYEN QUANG UY jlO| Nguyen Xuan Hoai, ".\ flexible representation for genetic programming: lessons from natural
language pioccssing", PhD Thesis, University of New South Wales, 2004.
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Nhdn bdi ngdy 10 - 11 - 2011 Nhdn l(ii sau siia ngdy 21 - 3 - 2012
