Form invariance of schema and exact schema theorem

Size: px

Start display at page:

Download "Form invariance of schema and exact schema theorem"

Cecily Allen
6 years ago
Views:

1 Vol. 46 No. 6 SCIENCE IN CHINA (Seres F) Deeber 2003 For nvarane of shea and exa shea heore YANG Haun ( ) & LI Mnqang ( ) Insue of Syses Engneerng, Tann Unversy, Tann , Chna (eal: navyy@vp.sna.o; Reeved Marh 0, 2003 Absra One of he os poran researh quesons n GAs s he explanaon of he evoluonary proess of GAs as a aheaal obe. In hs paper, we use arx lnear ransforaons o do, frs. Ths new ehod akes he sudy on ehans of GAs spler. We oban he ondons under whh he operaors of rossover and uaon are ouave operaors of GAs. We also gve an exa shea equaon on he bass of he onep of shea spae. The resul s slar o Sephens and Waelbroek s work, bu hey have novel eanngs and a larger degree of oarse granng. Keywords: shea heore, shea spae, lnear ransforaon, exa shea equaon. DOI: 0.360/02yf0384 Dfferen ehods have been used o analyze he evoluonary proess of gene algorhs (GAs). Holland s shea heore has been wdely eployed o explan he power of GAs and gene prograng (GP) [ 5]. However, has several laons. The frs s ha s no an exa equaon, bu an approxae one whh only provdes lower bounds for he expeed values of he nubers of a shea n he nex generaon. The seond s ha does no ake a shea as a subse of he searh spae, bu only onsder as an ndvdual. Ths paper wll ry o ake soe novel exensons o overoe hese laons. No only shea heores bu also oher odels an pred he generaon-based behavor of GAs, suh as Pre s Theore. Pol presened a aegory of dfferen odules of GAs n deal [3]. We propose o dsngush he dfferen approahes of GAs fro wo aspes: () wheher he approahes are approxae or exa; (2) wheher he onep spae s onsdered or no. I s known ha he Holland s shea heore s an approxaon and does no onsder he onepon of shea spae. Vose s odels and oher Markov han odels [6 8] were exa, bu hey only onsdered he ndvduals n he searh spae, whh dd no dep he haraerss of he subspae of he searh spae or shea spae. Alenberg forulaed an exa verson of he shea heore on he bass of Pre s heore usng sasal ehans [9]. Sephens e al. derved an exa expresson for he predon of he nex generaon for one-pon rossover [0]. Vose e al. presened soe new resuls on he onep of For Invarane. However, none of he ook sheaa ras no aoun fro he vew of shea spae. In hs paper, we wll gve a new expresson of shea heore based on sple gene al-

2 476 SCIENCE IN CHINA (Seres F) Vol. 46 gorhs (SGA) by onsderng s behavor on shea spae. By usng lnear ransforaon of searh spae and opology, we presen a new forulaon for exa shea heore. Ths paper s organzed as follows. Frs, we wll gve soe denoaons and oneps abou GAs. Then, we explan he operaons of SGA n usng lnear ransforaon n shea spae. Fnally, a new shea heore wll be gven by usng dfferenal equaon on he bass of shea spae. Shea and spae In hs paper, we wll ake he SGA as he general for of GAs. Whou loss of generaly, we use bnary bs o represen he hroosoes n he populaon. For bnary srng of lengh l, he se Ω represens searh spae. The se Ω s a fne se whh s oposed of n =2 l eleens. And ore, n = = Ω s sasfed, where Ω s he poeny of he se Ω. The searh spae Ω 2 l an also be defned as e 2 e e 2 2. We furher defne operaons and on he searh l spae Ω. : he wo eleens of Ω are bwse added od 2. : he wo eleens of Ω are bwse ulpled od 2. For an arbrary eleen u Ω, u s he opleen of u whh an be denoed as u. I s obvous ha u Ω. For operaon, Ω s an Abelan group. Wh respe o operaon and, Ω s a ouave rng, bu Ω s no a doan beause here does no exs any nverse eleen. The onep of a shea s defned as a eplae ade of, 0 and *. A shea s an Ω = 0,,* l, and s also a subse of searh spae Ω. eleen of he shea spae H { } So, we an now defne a subspae of he searh spae Ω : Ωk { Ω, k } Ω has relaon o a shea. For exaple, {*00 *} Ωk s a lnear spae. I s obvous ha k = =,where Ω =, where k = 00. If he unspefed posons of a shea are, suh as H = **, he subspae of Ω an be denoed as Ωkh, = { Ω, k = h, h Ω},whereh s a onsan and defned by H and k. When k 00, h 00 Ω = **. For a shea lke *0 *,we = =,hen k { } have Ω { Ω Ω} k h kh, =, k = h, h,where = 00, = 000.Forashealke *0*, { } Ωkh, = Ω, k = h, h Ω,where k = 00, h= 000. In oher words, for a shea, when he ndeernae posons are frsly replaed by 0, and deernae posons are replaed by, we wll oban k. Then aordng o he shea and k, we an ge h and Ω kh,. Ase represens a subspae of Ω for a shea H. I s a fne se whh s oposed of ΩH k

3 No. 6 FORM INVARIANCE OF SCMA & EXACT SCMA TOREM 477 n = 3 l eleens, and s also a lnear spae. There exss a one-o-any lnear appng fro shea spae ΩH no searh spae Ω, bu he appng fro Ω H no Ωkh, s a one-o-one appng. Proposon. For searh spae Ω,hereexs Ω,, Ω,, Ω,,, Ω, ha sasfy he dre su: Ω = Ωk, h Ω k 2, h 2 Ωk 3, h 3 Ω k, h. Proof. Sne Ω s a fne densonal lnear spae, any eleen an be denoed as veor T V = [ a, a2, a3,, a n ], where a = 0 or. For un veors e, e2, e3,, e n, where T [ 0,0,,,00, ] e =, and s he h poson, we have Ω = e e2 en.le Ω k, h = e. Then, he proposon s proved. I an also be proved ha he proposon s sll enable when Ωk, h s represened as a obnaon of e. 2 Sple gene algorhs (SGA) The SGA s a rando heurs searh (RHS) algorh wh hree an operaons as seleon, rossover and uaon. Populaon of sze r a e s denoed as P =[ V, V2,, Vr ],where V s a olun veor. By Proposon and prnples of algebra, here exss a se of bass veors of he searh spae Ω : e, e2,, en T,where [ 0,0,,,0,0 ] k h k h 2 2 e =, sh poson. 2. Seleon Proporonal seleon s deerned by he seleon funon, suh as roulee wheel. A seleon operaor s denoed as F(x), where x s he olun veor. Whou loss of generaly, sup- P = V, V,, V, V,, V, we wll ake he posng ha he nal populaon s represened as: [ ] F 0 2 seleon operaon as P P[ V, V,, V, V,, V ] r k h r,wherehe( + )h eleen of he populaon s replaed by he h eleen of he populaon under he fness funon f : T (where [ ] k h + Ω R and operaor of seleon. Sne e, e 2, e 3,, en e = 0,0,,,00, s he h poson) are he bass veors of he searh spae Ω, he seleve proess an be deped by usng ares as follows: P 0 = [ V, V2,, V, V +,, Vr ] P0 = V, V2,, V, V+,, Vr e, e2,, en P = V, V2,, V, V,, Vr e, e2,, en P [ ] V V2, V, V,, Vr. I s obvous ha he ransforaon P 0 P s a lnear ransforaon. Thus we ge he seond proposon as below.

4 478 SCIENCE IN CHINA (Seres F) Vol. 46 Proposon 2. The seleon operaon n SGA s a lnear ransforaon based on he exenson of ares. 2.2 Crossover In hs seon, we use one-pon rossover frs, on whh he resul of ul-pon rossover an hen be obaned easly. A frs, we onsder populaons before and afer he operaon of one-pon rossover: P = V, V,, V, V,, V, populaon before rossover: 0 [ 2 + r] populaon afer rossover: P = [ V, V,, V, V,, V ], 2 + r where V, V + are used o do one-pon rossover a poson k, and rossover probably s aken as. We wll oban V, V +, k n. = [,,,,,, ] P [ V, V,, V, V,, V e, e,, e ] P V V V V V r = r 2 n P = [ V, V2,, V, V +,, Vr e, e2,, en P = V, V 2,, V, V +,, Vr]. Wh e ( n) as bass veors of he searh spae, V, V + ( n) and V, V + an be represened as he lnear obnaon of e. Ths lnear ransforaon s a hooorphs of he spae, and s a lnear appng fro Ω o Ω. For a se of bass veors, a lnear ransforaon an be represened by a arx, whh eans ha P 0 s ulpled on he lef by a arx T : P = TP 0, where de( T ) 0. Wh respe o rossover operaons by generaon, P0 P an be denoed as P = T TP 0, where de( T ) 0,. Leng T = T T,hen P. = T P 0 Proposon 3. of ares. The rossover operaon n SGA s a lnear ransforaon on he exenson 2.3 Muaon In order o gve he algorhs he ably o searh for he opu soluon, uaon has been used for GAs runnng beyond he resron of he nal populaon. New ndvduals of he populaon are reaed by uaon operaons. Boh uaon and rossover an reae new ndvduals, bu hey are dfferen essenally. Muaon s appled o brngng varey o he populaon; however rossover ends o ge he populaon o be of hoogeny. Frs, le us onsder he populaon evoluon under one-pon uaon. We have p0 = [ V, V2,, V,, Vr], P = [ V, V2,, V,, Vr], where V s uaed o V hrough one-pon uaon a poson k ( k n), and uaon probably s µ. The proess an be represened by usng arx anpulaons as: P = V, V,, V,, V P = V, V,, V,, V e, e,, e [ ] [ ] 0 2 r 0 2 r 2 n [,,,,,,,, ] [,,,,, ] P = V V V V e e e P = V V V V. 2 r 2 n 2 r Wh he bass veors of he searh spae e ( n), V ( n) and V an be repre-

5 No. 6 FORM INVARIANCE OF SCMA & EXACT SCMA TOREM 479 sened as he lnear obnaon of e. Furher, a lnear ransforaon an be represened by a arx, so ha P 0 s ulpled on he lef by a arx T : P = TP 0,where de( T ) 0. Wh respe o uaon operaons by generaon, P0 P an be denoed as P = T TP 0,where de( T ) 0,. Leng Tµ = T T,hen P = T µ P 0. The operaon of uaon n SGA s a lnear ransforaon on he exen- Proposon 4. son of ares. 2.4 The order of uaon and rossover Sne uaon and rossover are wo an operaors of SGA, s neresng o sudy wheher he perforng sequenes of rossover and uaon operaons wll affe he evoluon of populaon. In oher words, wha s he relaon beween P + and P +? T T P T P = P T P = T T P = P, () T T P T P = P T P = TT P = P. (2) Whou loss of generaly, assue ha rossover s perfored before uaon. V, V + are woolunveorsofp 0, and he rossover pon s k. The resul of rossover s denoed as V, V + ; he proess an be desrbed by ares operaons as follows: V,,, V,, V +,,, V r,, 2,, T V,, V, V +,, Vr e e en V,2,, V,2, V+,2,, Vr,2 V,, V, V +,, V V = V, V = V, V = V, V = V. where,, +, +, +,2,2 +,2,,,,,2,2,2 r,2 e, e2,, en, Defnon. If he uaon s no perfored on he posons of subsrngs o be rossed, V = V + hen he uaon s alled ndependen. or,2,2, Proposon 5. When he uaon s ndependen, wheher or no he uaon s perfored before or afer rossover, we wll have P+ = P+, and hen he uaon and rossover are alled ouave operaors. 3 Shea spae and exa shea equaon 3. The oneps of shea spae We have defned Ωk, h on he searh spae Ω o represen a shea. Now, we an furher defne a dsree opologal spae (,T ) Ω,whereT s a se of Ω,. Sne he goal of GAs s o fnd he opu soluon n he searh spae Ω, whh eans ha he opu shea s searhed n he shea spae ( T = ΩH ), wha are he relaons of he sheaa n he shea spae T? We an desgn a dvson of he shea spae T, and hen defne he ahng dsane beween sheaa H a and H b by adopng Hang dsane, whh s alled he exended Hang dsane. k h

6 480 SCIENCE IN CHINA (Seres F) Vol. 46 Defnon 2 (he exended Hang dsane). The ahng dsane of H a o H b s denoed as d ( H a, Hb ), whh s opued as follows when he fxed posons of H a are no equal o he fxed posons of H b, or he unfxed posons of H a are orrespondng o he fxed posons of H b, hen s added o d ( H, H ) ; oherwse, d ( H, H ) s kep na. a b ( a b b a I s obvous ha d H, H ) d ( H, H ) holds generally. Defnon 3. The shea H a s sad o ah shea H b f d ( H, H ) = 0. Defnon 4. Two sheaa are sad o ah eah oher f and only f a b b a d ( H, H ) = d ( H, H ) = Exa shea equaon In hs seon, we onnue he prevous work [9 5], and we wll gve a dervaon of he shea equaon based on he onep of shea spae. The shea equaon s obaned under he operaons of he hree evoluonary operaors: proporonal seleon, one-pon rossover and uaon. The nubers of hroosoes n he populaons are kep onsan. I an be proved ha he nubers of sheaa n he populaon are nvaran wh evoluon generaons as R = r2 l. The subsrp of he shea sasfes 0,, l, R n he followng dervaon proess. H 3.2. Muaon (uaon afer rossover). Wh uaon and rossover beng ouave operaors, we an alulae he nuber of he sheaa n he nex generaon ( + populaon) when uaon s perfored afer rossover. For any shea of he shea spae T, k s a poson of he shea, where k l, we have he followng equaon: where P ( H, ) = P ( H H) N ( H, )/ R+ P ( H H) N ( H, )/ R, d ( H, H) = d ( H, H ) K d ( H, H) = 0 a N ( H, ) = H, and he order of shea H s oh ( ) = l; N ( H, ) s he expeed nuber of shea H afer proporonal one-pon rossover. Then N ( H, ) / R s he proporon of shea H n he populaon. Leng P ( H, ) = N ( H, ) / R, and sne P ( H H ) = ( p ( k)),wehave k s b N P ( H H ) = p ( k) ( p ( k)), k s k s 2 where p ( k) s he uaon probably of b k, s s he fxed poson se of shea H, s s a subse of he se s n whh he fxed posons of shea H s no equal o ha of shea H,and s 2 s he opleenary se of s over s. a b Crossover. Frs, a sple funon s defned as δ ( x) =, f x= 0,,andhen δ ( x) =, oherwse x 0

7 No. 6 FORM INVARIANCE OF SCMA & EXACT SCMA TOREM 48 we an gve he defnon of he oeffen as: Lk LR H, H δ l L( k) R( k) l C = (ax( d ( H, H ), d ( H, H ))), where d ( )( H, H ) s he exended Hang dsane beween lef subshea of H and lef Rk l subshea of H, and orrespondngly d ( )( H, H ); p ( k) s he rossover probably on he rossover pon k. Then we ge n N = + H, H + l H, Hl l n H, Hl k= P ( H, ) P( H, ) p ( k)( C C ) P( H, ) P( H, ). (3) If p (k) s a onsan, denoed as p, hen forula (3) an be wren as n p PN ( H, ) = PH (, ) + ( CH,, ) (, ) (, ). H + C l H H P H l P Hl (4) n H, Hl k= Seleon and shea evoluonary equaon. Frs, we nrodue soe rearks. Reark. The average populaon fness a e s denoed as f() : f() = f( H,)/ r, OH ( ) = k where OH ( ) s he shea order of H,andk (0 k n) s a onsan. Reark 2. The fness of shea H a e s denoed as f( H, ): f( H, ) f( H, ) H, = d ( H, H) = 0 d ( H, H) = 0 where OH ( ) = l, and OH ( ) s he shea order of H. Then, by defnng PH (, ) = H r, where OH ( ) = l, OH ( ) d ( H, H) = 0 order of H, r s he populaon sze, we ge: s he shea f( H, ) PH (, + ) = P( H, ). (5) f() Now, we an derve he exa fnal evoluonary equaon of SGA. f( H, ) PH (, + ) = P( H, ) f() f( H, ) = ( P ( H H ) PN ( H, ) f() + P ( H H ) P ( H, )). (6) d ( H, H ) = d ( H, H ) K N

8 482 SCIENCE IN CHINA (Seres F) Vol. 46 Leng P ( H H ) = ( p ( k)) = a( k),wehave k s P ( H H ) = p ( k) ( p ( k)) = b ( k), so ha we ge he followng forula: k s k s2 ( d ( H, H ) = d ( H, H ) k) = ond(), f( H, ) a( k) p PH (, ) (()( akph,) ( C C PH,)( PH,) n + = + H, H+ l H, Hl l f () n H, H k= l p + b( k)(( PH (, ) + ( C + C ) PH (, PH ) (, ))). n H, Hl H, Hl l ond() n H, H k= (7) l Furherore, le f( H, ) a( k) p CkPH (, ) (, ) ( C C ) PH (, PH ) (, ), and n = H, H + l H, Hl l f() n H, Hl k= fh (, ) p Q ( ) = b( k)(( PH (, ) + ( C + C ) PH (, PH ) (, ))), f () n H, Hl H, Hl l ond() n H, Hl k= and we subra boh sdes of eq. (7) wh PH (, ) and dvde he by, so ha we an ake as an ordnary lnear dfferenal equaon of he frs order by akng as a onnuous arguen as 0. The soluon of he equaon an be wren as P (, ) (, ) 0 s H d P 0 s H d PH (, ) e PH (,0) Q ( ) e = + d, (8) 0 where f( H, ) Ps ( H, ) = a( k) + C( k, ) f() s alled evoluonary ably of shea H. 3.3 Dsusson Forula (8) deps he hange o a shea under he evoluonary operaors auraely. We an ake a onluson fro forula (8) ha he shea whose evoluonary ably s posve wll reeve an exponenal nreasng n he rals of GAs. The proess of dedung our shea heore s slar o soe exen o Sephens e al. s work, bu he shea heores are represened a a dfferen degree of oarse granng. Ther work s obaned a he level of he rosop degrees of freedo. One of he ore rearkable feaures of our work s ha we deal wh he oal evoluon proess based on shea spae by adopng a new ool. Our shea heore has a bgger oarse granng han ever. In hs paper, seleon s perfored afer rossover and uaon for a onse resul. On he oher hand, Sephens e al. provded a desrpon of shea evoluonary equaon by usng effeve fness funon, whh ephaszes he seleon operaor. The onep of evoluonary ably of shea s advaned n hs paper o dep he effe of hree evoluon operaors. In seon 2, he nonlnear effes of evoluon operaors are represened hrough lnear opera-

9 No. 6 FORM INVARIANCE OF SCMA & EXACT SCMA TOREM 483 ons on exendng populaons, whh akes he desrpon of ehans of GAs ore expl. As shown n fg., he dfferen shea heores have soe relaons. For exaple, Holland s shea heore an be obaned easly by negleng soe paraeers n forula (8). Holland s shea heore and oher exa odels only pred he quanes of he ndvduals over one generaon, suh as Markov odels and sasal ehans odels [6]. When he populaon evolves fro generaon 0 o, forula (8) shows ha he quanes of a shea an be refleed by PH (, ), whh s able o ake predons auraely over ulple generaons. The shea heore s ndeed an effeve ool for undersandng and analyzng buldng bloks hypohess of GAs. The resuls obaned n hs paper an do beer han onvenonal approahes beause of s rearkable feaures. Forula (3) we have derved deps he hange n he quanes of he sheaa afer rossover, where CH,,,, H C l H H he oeffens of he rossrossover operaor, ply ha he sheaa of l hgher order are obned by he sheaa of lower order. The proess onnues an-lokwse unl he lowes order of sheaa reahes. I s no neessary o drll down o order sheaa beause of onvergene of algorhs. Aordng o forula (8) and forula (3), sheaa wh good evoluonary ably wll obne he sheaa of hgher order Fg.. The poson of he proposed shea heore n he and reeve an exponenal nreasng. Ths s an spae of GAs. exa explanaon of buldng bloks hypohess. 4 Conlusons There are dfferen approahes o odelng he evoluonary proess of GAs. Vose used he sybols of absra algebra and geoery o explan. Fogel, Goldgerg Vose and oher expers adoped Markov han and sasal ools o dep. In hs paper, we apply lnear arx ransforaon o, whh akes he sudy on ehans of GAs spler han wh oher ehods. We also oban he ondon under whh rossover and uaon are ouave for GAs. An exa shea equaon s derved on he bass of he onep of shea spae. The resul s slar o Sephens and Waelbroek s work, bu hey have que dfferen eanngs. Forula (8) an be used as a ool for analyzng he buldng bloks hypohess, and s easy o be exended o he odelng of gene prograng. LR RL Aknowledgeens Nos , ). Referenes Ths work was suppored by he Naonal Naural Sene Foundaon of Chna (Gran. Holland, J. H., Adapaon n Naural and Arfal Syses, Cabrdge, Massahuses: MIT Press, 992.

10 484 SCIENCE IN CHINA (Seres F) Vol Goldberg, D. E., Gene Algorhs n Searh, Opzaon, and Mahne Learnng, MA: Addson-Wesley, Pol, R., Exa Shea Theory for Gene Prograng and Varable-Lengh Gene Algorhs wh One-Pon Crossover, Gene Prograng and Evolvable Mahnes 200, Dordreh: Kluwer Aade Publshers, 200, Pol, R., Langdon, W. B., O Relly, U.-M., Analyss of shea varane and shor er exnon lkelhoods, n Gene Prograng 998: Pro. Thrd Annually Conferene (eds. Koza, J. R., Banzhaf, W., Chellaplla, K.), Unversy of Wsonsn, Madson, Wsonsn, USA, San Franso: Morgan Kaufann, 998, Whley, D., An overvew of evoluonary algorhs: praal ssues and oon pfalls, Inforaon and Sofware Tehnology, 200, 43: Nx, A. E., Vose, M. D., Modelng gene algorhs wh Markov hans, Annals of Maheas and Arfal Inellgene, 992, 5: Davs, T. E. Jose, C., Prnple: A Markov han fraework for he sple gene algorh, Evoluonary Copuaon, 993, (3): Spears, W. M., De Jong, K. A., Analyzng GAs usng Markov odels wh seanally ordered and luped saes, Foundaons of GAs IV, San Franso: Morgan Kaufann, 997, Lee, A., The Shea Theore and Pre s Theore, n Foundaons of Gene Algorhs 3 (eds. Whley, D., Vose, M. D.), San Franso: Morgan Kaufann, 995, Sephens, C., Waelbroek, H., Sheaa evoluon and buldng bloks, Evoluonary Copuaon, 999, 7(2): Vose, M. D., Wrgh, A. H., For nvarane and pl parallels, Evoluonary Copuaon, 999, 9(3): Sephens, C., Waelbroek, H., Sheaa as buldng bloks: Does sze aer? n Foundaons of Gene Algorhs 5 (eds. Banzhaf, W., Reeves, C.), San Franso: Morgan Kaufann, 999, Sephens, C. R., Waelbroek, H., Effeve degree of freedo n gene algorhs and he blok hypohess, n Proeedngs of he Sevenh Inernaonal Conferene on Gene Algorhs (ed. Bak, T. ), San Maeo: Morgan Kaufann,997, Vose, M. D., The Sple Gene Algorhs: Foundaons And Theory, Cabrdge, Massahuses: The MIT Press, Sephens, C., Waelbroek, H., Analyss of he effeve degrees of freedo n gene algorhs, Phys. Rev., 998, E57: Nx, A. E., Vose, M. D., Modelng gene algorhs wh Markov Chans, Ann. Mah. Arf. Inell., 992, 5:

Pendulum Dynamics. = Ft tangential direction (2) radial direction (1)

Pendulum Dynamics. = Ft tangential direction (2) radial direction (1) Pendulum Dynams Consder a smple pendulum wh a massless arm of lengh L and a pon mass, m, a he end of he arm. Assumng ha he fron n he sysem s proporonal o he negave of he angenal veloy, Newon s seond law