Optal Mutaton Rates and Selecton Pressure n Genetc Algorths Gabrela Ochoa, Inan Harvey and Hlary Buxton Centre for Coputatonal Neuroscence and Robotcs COGS { Unversty of Sussex, Brghton BN 9QH, UK E-al: fgabro, nanh, hlarybg@sussex.ac.uk Abstract It has been argued that optal per-locus utaton rates n GAs are proportonal to selecton pressure and the recprocal of genotype length. In ths paper we suggest that the noton of error threshold, borrowed fro olecular evoluton, sheds new lght onths arguent. We show eprcally the exstence of error thresholds n GAs runnng on a sple abstract landscape and then nvestgate a real-world ndustral proble, deonstratng coparable phenoena n a practcal applcaton. We study the correspondence between error thresholds and optal utaton rates on these two probles, and explore the eect of derent selecton pressures. Results suggest that error thresholds and optal utaton rates are ndeed correlated. Moreover, as the selecton pressure ncreases, both error thresholds and optal utaton rates ncrease. These ndngs ayhave practcal consequences, as heurstcs for easurng error thresholds n real-world applcatons wll provde useful gudelnes for settng optal utaton rates. INTRODUCTION The perforance of a GA heavly depends on the choce of ts an control paraeters: populaton sze, utaton rate, and recobnaton rate. Despte research so far, there are no general heurstcs on how to set the. It has been suggested that the utaton rate s the ost senstve GAcontrol paraeter (Schaer et al., 989 Back, 996). Several studes n the lterature look for \optal" utaton rates (Hesser & Manner, 99 Muhlenben, 99 Back, 99, 99), and optal schees for varyng the utaton rate over a sngle run (Fogarty,989 Back &Schutz, 996). The studes by Hesser and Manner (99), Muhlenben (99), and Back (99, 99, 996) concde n that optal per-locus utaton rates depend anly on = (the recprocal of the genotype length). Moreover, Back (996) suggests that as the selecton pressure ncreases the optal utaton rate also ncreases. In ths paper we suggest new foundatons for the dependency of optal utaton rates on both the selecton pressure and the recprocal of the genotype length. Also, the senstvty oftheutaton rate s explaned by ths new vewpont. Ths knowledge coes fro the eld of olecular evoluton, n partcular fro the noton of error thresholds (Secton ). The error threshold s the crtcal utaton rate beyond whch structures created by anevolutonary process are destroyed ore frequently than selecton can reproduce the. The exstence of ths phenoenon n GAs and ts relatonshp wth the ore falar noton of optal utaton rates, has been ntroduced n Ochoa et al. (999) for sple landscapes. Here, we explore eprcally optal utaton rates and error thresholds on a real-world engneerng proble the Wng-Box desgn optzaton proble. In partcular, we study the dependence of both optal utaton rates and error thresholds on the selecton pressure. Also, we are nterested n the relatonshp between error thresholds and optal utaton rates. These ssues are ntally explored on abstract toy probles (e.g. the Royal Starcase functons, descrbed n secton.), thus the followng step would be to explore whether slar phenoena occurs n a real-world proble. The Wng- Box proble was forulated orgnally n the fraework of the GAME (Genetc Algorths n Manufacturng Engneerng) project at COGS, Unversty of Sussex. Brtsh Aerospace provded ndustral realstc data for the denton of ths proble (Secton.).
The reander of ths docuent s organzed as follows. Secton dscusses the noton of selecton pressure n GAs, descrbes rankng and how ths selecton schee allows control over the selecton pressure. Secton ntroduces the notons of quasspeces and error thresholds fro the eld of olecular evoluton, t also dscusses the hypotheszed relatonshp between error thresholds and optal utaton rates. Secton descrbes the test probles used n ths paper: the Royal Starcase, and the Wng-Box proble. Sectons and 6 descrbe our ethods and results respectvely, and, nally, Secton 7 suarzes our ndngs. SEECTION PRESSURE In GAs, selecton allocates reproductve opportuntes for each organs n the populaton. The tter the organs, the ore tes t s lkely to be selected for reproducton. Selecton has to be balanced wth varaton fro utaton and recobnaton the explotaton-exploraton balance. Several selecton echanss have been suggested n the lterature, although there are no general gudelnes on whch to use on a gven crcustance. There s, however, the noton of derent selecton pressures assocated wth each selecton echans. Selecton pressure s an nforal ter that ndcates the strength of a selecton echans. oosely, the selecton pressure easures the rato of axu to average tness n the populaton. For the experents n ths paper we used rank selecton. Ths selecton schee s paraeterzed n such a way that allows control over the selecton pressure. It also elnates the need for tness scalng echanss.. RANK SEECTION In rank selecton, ndvduals n the populaton are ranked accordng to tness. The expected value of each ndvdual depends on ts rank rather than on ts absolute tness. The lnear rankng ethod proposed by Baker (98) works as follows: Organss n the populaton are ranked n ncreasng order of tness, fro to M (the populaton sze). The user chooses the expected value Max (Max ) of the ndvdual wth rank M. The expected value of each ndvdual n the populaton at te t s gven by: rank( t) ; ExpV al( t) =Mn+(Max ; Mn) M ; () Where Mn s the expected value of the ndvdual wth rank. Gven the constrants Max and ExpV al( t) =M, t s requred that Max P and Mn =;Max.Ateach generaton the ndvduals n the populaton are ranked and assgned expected values accordng to equaton, Baker recoended Max = : and showed that ths schee copared favorably to proportonal selecton on soe selected test probles. The selectve pressure of lnear rankng can be vared by tunng the axu expected value Max (see Equaton ), whch controls the slope of the lnear functon. The value recoended by Baker (98) of Max = : eans that, on average, the best ndvdual s expected to be sapled. tes, ths s a rather oderate selectve pressure, close to the extree case of a rando walk (Max =:). The axu possble expected value for lnear rankng s Max = :. QUASISPECIES AND ERROR THRESHODS Quasspeces theory was derved n the 7s by Egen and Schuster (979) to descrbe the dynacs of replcatng nuclec acd olecules under the nuence of utaton and selecton. Ths theory was orgnally developed n the context of pre-botc evoluton, but n a wder sense t descrbes any populaton of reproducng organss. An portant concept n quasspeces theory s the noton of error threshold of replcaton. If replcaton were error free, no utants would arse and evoluton would stop. On the other hand, evoluton would also be possble f the error rate of replcaton were too hgh (only a few utatons produce an proveent, but ost wll lead to deteroraton). The noton of error threshold allows us to quantfy the resultng nal replcaton accuracy (.e. axal utaton rate) that stll antans adaptaton. Ths can be seen at ts clearest n an extree for of a tness landscape whch contans a sngle peak of tness >, all other sequences havng a tness of. Wth an nnte populaton there s a phase transton at a partcular error rate p, theutaton rate at each ofthe loc n a sequence. In Egen and Schuster (979), ths crtcal error rate s deterned analytcally (Equaton ), and t s dened as the rate above whch the proporton of the nnte populaton on the peak drops to chance levels. p = ln() () In Equaton, represents the selectve advantage of the aster sequence over the rest of the populaton (.e. the selecton pressure), and the chroosoe length. In the splest case, s the rato of the aster sequence reproducton rate (tness) to the average reproducton rate of the rest.
As stated orgnally, the quasspeces odel consdered nnte and asexual populatons. ater on extensons were developed that consder nte populatons and recobnaton. Most quasspeces studes consdered sple landscapes, ncludng sngle peak landscapes, double peak landscapes and at tness landscapes. The work of Bonhoeer and Stadler (99), descrbed below, studed error thresholds on ore coplex landscapes.. ERROR THRESHODS ON COMPEX ANDSCAPES Bonhoeer and Stadler (99) studed the evoluton of olecular quasspeces on two derent coplex tness landscapes, the Sherrngton Krkpatrck spn glass and the Graph Bparttonng landscape. They descrbed an eprcal approach for locatng error thresholds on these hghly correlated landscapes. In order to locate the error threshold eprcally, theysulated the evoluton of a populaton at a constant error rate for, cycles, whch proved long enough to reach equlbru on several paraeters of the populaton (the axal tness, the average tness aong others). The error threshold ay be approached fro below and above, wth both ethods producng slar results. To approach t fro below, the sulaton starts wth a hoogeneous populaton at the global optu. Then the populaton s allowed to reach equlbru at a constant utaton rate of.. Afterwards, the utaton rate s ncreased by a xed, sall step and the coputaton contnues wth the current populaton. Ths process contnues untl a predened axu for the utaton rate, p ax, s reached. To approach the error threshold fro above, the sulaton starts wth a rando populaton. Then the populaton s allowed to reach equlbru at a constant utaton rate of p ax. Afterwards, the utaton rate s decreased by a xed step and the coputaton contnues wth the current populaton. Ths process s repeated untl the utaton rate s. For both approaches, the consensus sequence n the populaton s calculated at the end of each sulaton cycle for each utaton step. The consensus sequence s dened as the sequence of predonant sybols (bts) n each poston t s plotted as follows: f the ajorty of ndvduals has a `' or `' n a poston the eld s plotted whte or black, respectvely. The eld s plotted grey f the poston s undecded (see Fgures -, for slar plots). The error threshold s characterzed by the loss of the consensus sequence,.e. the genetc nforaton of the populaton. Beyond the error threshold the consensus sequence s no longer constant n te. The eprcal ethod descrbed above was developed usng the quasspeces equatons as the underlyng odel of evoluton. In ths paper, we borrow ths approach but use a GA nstead of the quasspeces odel for sulatng evoluton. The resultng ethod can be appled to locate error thresholds n GAs runnng on general coplex landscapes.. ERROR THRESHODS AND OPTIMA MUTATION RATES The noton of error threshold sees to be ntutvely related to the dea of an optal balance between explotaton and exploraton n genetc search. Too low autaton rate ples too lttle exploraton n the lt of zero utaton, successve generatons of selecton reove all varety fro the populaton, and once the populaton converges to a sngle pont n genotype space all further exploraton ceases. On the other hand, utaton rates can be too excessve n the lt, where utaton places a randoly chosen allele at every locus on a genotype, the evolutonary process degenerates nto rando search wth no explotaton of the nforaton acqured n precedng generatons. Any optal utaton rate ust le between these two extrees, but ts precse poston wll depend on several factors ncludng, n partcular, structure of the tness landscape. It can, however, be hypotheszed that where evoluton proceeds through a successve accuulaton of nforaton, then a utaton rate close to the error threshold s an optal utaton rate for the landscape under study. Ths utaton rate should axse the search done through utaton subject to the constrant of not losng nforaton already ganed. TEST PROBEMS. THE ROYA STAIRCASE FAMIY OF FUNCTIONS The Royal Starcase faly of functons was proposed by van Nwegen and Crutcheld (998) for analyzng epochal evolutonary search. These functons are related to the Royal Road functons (Mtchell et al., 99). Although sple, Royal Starcase functons capture soe essental eleents found on coplex probles, naely, the exstence of hghly degenerate genotype-to-phenotype aps (.e. the appng fro genetc speccaton to tness s a any-to-one functon). Next, we present a descrpton of the Royal Starcase class of tness functons:
. Genotypes are speced by bnary strngs s = s s :::s s f g, of length = NK.. Startng fro the rst poston, the nuber I(s) of consecutve s n a strng s counted.. The tness f(s) of strng s wth I(s) consecutve ones, followed by a zero, s f(s) = + bi(s)=kc. The tness s thus an nteger between and N +, correspondng to plus the nuber of consecutve fully-set blocks startng fro the left.. The sngle global optu s s = naely,the strng of all s. Fuselage Cavty Top panel Rbs Rb ptch Fxng N (nuber of blocks) and K (bts per block) deternes a partcular proble or tness landscape. For the experents n ths paper we selected N = and K =, that s a strng length of.. THE WING-BOX PROBEM The Wng-Box proble was forulated as part of the Genetc Algorths n Manufacturng Engneerng (GAME) project at COGS, Unversty of Sussex. An ndustral partner, Brtsh Aerospace, provded data fro a real Arbus wng box. A coon proble faced n the desgn of arcraft structures, s to dene structures of nu weght that can wthstand a gven load. Fg. sketches the eleents of a wng relevant to ths proble. The wng s supported at regular ntervals by sld rbs whch run parallel to the arcraft's fuselage. On the upper part of the wng, thn etal panels cover the gap separatng adjacent rbs. The objectve stondthenuber of panels and the thckness of each of these panels whle nzng the ass of the wng and ensurng that none of the panels buckle under axu operatonal stresses. More detals, and the equatons for calculatng the tness functon, can be found n McIlhagga et al. (996)... GENETIC REPRESENTATION - THE DETA ENCODING A full descrpton of a potental soluton to the Wng- Box proble requres the denton of the nuber of rbs N and the thckness of the N ; panels. There s a constrant on the thckness of these panels whch s that adjacent panels should not der n thckness by ore than.. The splest way to accoplsh ths, s to encode the derences n thckness between adjacent panels rather than the absolute thck- http://www.nforatcs.sussex.ac.uk/projects/gae/ Fgure : Relevant eleents of a wng. Wng densons are xed. The varable eleents are the nuber of rbs and the thckness of the top panels. ness of the panels. If we know the derence n thckness th() between panels and + for ( N;), the absolute thckness of the rst panel s enough to dene everythng else. N th() th()= th()-th() N: Nuber of rbs th th(): Thckness of panel... th()= th(+)-th()... th(n-)= th(n-)-th(n-) Fgure : Genetc representaton of the wng paraeters. Orgnally, the Wng-Box paraeters were encoded followng the order descrbed by Fg..For the experents n ths paper we xed the nuber of panels n (.e N = rbs, snce the nuber of rbs s + the nuber of panels), thus our genetc encodng s the sae, but excludng the rst gene. The thckness of the rst panel was allowed to vary between and by steps of ;. Ths requres values whch can be represented wth a nu of bts. For all subsequent N ; panels the derence n thckness wth the prevous panel s encoded. Accordng to anufacturng tolerance consderatons, only ve values were allowed for these derences n thckness: f;: ;: : : :g. Three bts are needed to encode these ve values. Notce that a change n th() leads to changes n the thckness of panel +, and of all subsequent panels up to the tp of the wng. Notce also that n both the encodng of the rst secton, and the reander N ; sectons, theresanaount of redundancy n the genotype to
phenotype appng. To su up, the nuber of bts needed for encodng an ndvdual s for the rst panel, and for each of the others 9 panels, that s + 9 = 6. METHODS All experents were run usng a generatonal GA wth lnear rankng and stochastc unversal saplng. Three derent selecton pressures { strong, edu, and weak, were tested. These qualtatve agntudes correspond to settng the Max paraeter n Equaton to.,., and., respectvely. The genetc operators were the standard bt utaton, and two-pont recobnaton wth a rate of.6. The utaton rate was expressed as utatons per genotype. Several utaton rates were explored. The populaton sze was always. The strng lengths were for the Royal Starcase functon, and 6 for the Wng-Box proble (see Secton..) Two types of experents were run. Frst, for calculatng and producng the error thresholds plots (Fgs. -8), we used the eprcal descrbed n secton.. We approached the error threshold fro \above", that s, the sulaton started fro a rando populaton. Then the populaton was allowed to reach equlbru at a constant hghutaton rate (:= for the Starcase proble and 6:= for the Wng-Box Proble). Afterwards, the utaton rate was decreased by a sall step (: for the Starcase proble and : for the Wng-Box Proble) and the coputaton was contnued wth the current populaton. Ths process was repeated untl the utaton rate was.. Second, for estatng optal utaton rates, we calculated the nuber of evaluatons before ndng the optu strng on the Royal Starcase proble (averaged over runs). For the ore coplex Wng- Box proble, where the optu strng s not known before-hand, the approach was to calculate the bestso-far tness reached after a xed nuber of functon evaluatons (averaged over runs)., generatons proved to be long enough to reach equlbru on populaton best and average tness on our test probles. evaluatons proved to be enough to reach equlbru on best-so-far average tness. 6 RESUTS 6. ERROR THRESHODS 6.. Royal Starcase Proble Fgure shows results for strong and edu selecton pressures on a Royal Starcase functon wth N =, K =. The pctures llustrate the exstence of a stable consensus sequence for utaton rates below the error threshold. The error threshold s characterzed by the loss of the consensus sequence. Beyond the error threshold the consensus sequence s no longer constant n te. In ths case the consensus sequence below the error threshold s the sngle optu strng n the landscape (the strng of all ones, dsplayed whte below). The error thresholds for each tness level or step can be observed. Error thresholds were shown to be lower for edu selecton pressure. Strong Selecton Pressure Medu Selecton Pressure Fgure : The consensus sequence on the Royal Starcase functon (N = K = ), for strong and edu selecton pressures. The X-axs shows the consensus bt(=black, = whte) for each poston, the Y-axs shows the utaton rate. 6.. Wng-Box Proble Fgure shows results for a strong selecton pressure on the Wng-Box proble. Fgure (eft) shows the exstence of a stable consensus sequence for utaton rates below the error threshold. The error threshold s agan vsualzed as the transton fro a stable consensus sequence to a rando sequence of bts. Notce that there s not a clear and sngle transton, fro approxately bt 7 to bt the error threshold looks hgher than for the rest of the bts. Otherwse the transton sees to occur around. to. utatons per genotype. Fgure (Rght) was produced wth the a of hghlghtng the results, t plots the sae data as gure (eft) but appng the consensus sequence onto the strng of all ones. Thus the error threshold s dstngushed as the transton fro a whte to a rando pattern of bts.
6 Strong Selecton Pressure 7 6 Strong Selecton Pressure 7 Fgure : The consensus sequence on the Wng-Box proble for a strong selecton pressure. eft gure shows the standard consensus sequence plot, whereas Rght gure hghlghts the error thresholds by appng the consensus sequence onto the strng of all ones. standard devatons (not shown for the sake of clarty) were of the sae order of agntude as the average. Thus, there were large run-to-run varatons n the te to reach the optal strng. Optal utaton rates are those whch nd the peak wth the least nuber of evaluatons. Notce that there s not a sngle crtcally precse optal utaton rate, but nstead a range of utaton values producng near-optal results. Optal utaton rates were shown to be lower for the edu selecton pressure. In the plots we ndcate, wth an arrow, the eprcally estated error thresholds. Error thresholds were found to be wthn the range of optal utaton rates for both edu and strong selecton pressures. Thus, these results support the hypotheszed relatonshp between these two easures. Fgures shows error thresholds for edu and weak selecton pressures. The consensus sequence was agan apped onto the strng of all ones to hghlght the transton. Hence, the error threshold s dstngushed as the transton fro a whte to a rando pattern of bts. It can be notced that the error threshold decreases as the selecton pressure decreases. For the edu selecton pressure the transton for ost bts occurs around. utatons per genotype, whereas for the weak selecton pressure t occurs around. utatons per genotype. Agan there s no sngle transton for all bts n the genotype. 6 Medu Selecton Pressure 7 6 Weak Selecton Pressure 7 Fgure : Error thresholds on the Wng-Box proble for edu (Max =:) and weak (Max =:) selecton pressures. 6. OPTIMA MUTATION RATES 6.. Royal Starcase Proble Curves n Fgure 6 show the nuber of evaluatons to reach the global axu as a functon of the utaton rate, for edu and strong selecton pressures. Each data pont gves the average of runs. The Evaluatons 6 Royal Starcase (N =, K = ) Medu......6.7.8...6.8 Mutaton Rate (/) Strong Fgure 6: Nuber of evaluatons for ndng the optu strng as a functon of the utaton rate on a Royal Star case functon. Arrows ndcate the approxate agntude of the error threshold n each case. 6.. Wng-Box Proble Curves n gures 7-9 show the average best-so-far tness attaned after evaluatons as a functon of the utaton rate for strong, edu and weak selecton pressures. Because we are dealng wth a nzaton proble, optal utaton rates are those that produce the lower tness value (the lower wng structure weght). Each data pont gves the average of runs. Notce that derent utaton rate ranges and step szes were used for each selecton pressure. In each case a sngle utaton value can be dstngushed as the one producng the nal average weght. These optal utaton rates are shown to decrease n agntude as the selecton pressure decreases. Agan we ndcate n the plots, wth arrows, the eprcally estated error thresholds. For the three selecton pressures explored, error thresholds were found to be close to the eprcally estated optal uta-
ton rates. Thus, results for ths real-world applcaton also support the hypotheszed relatonshp between error thresholds and optal utaton rates. Weak Selecton Pressure Strong Selecton Pressure Ftness 9 Ftness 9 8......6.7.8 Mutaton Rate (/) 9 8........ Mutaton Rate (/) Fgure 7: Average best-so-far tness after evaluatons on the Wng-Box proble for a strong selecton pressure (Max = :). Error bars show the standard devaton. The arrow ndcates the approxate error threshold. Ftness 9 9 8 Medu Selecton Pressure...7....7. Mutaton Rate (/) Fgure 8: Average best-so-far tness after evaluatons on the Wng-Box proble for a edu selecton pressure (Max = :). The arrow ndcates the approxate error threshold. Error bars show the standard devaton 7 DISCUSSION Our results suggest that error thresholds and optal utaton rates are ndeed correlated. Moreover, ths relatonshp carred over fro sple toy probles such as the Royal Starcase to a real-world applcaton, the wng-box proble. The an plcatons of these ndng are two-fold. Frst, theoretcally, n helpng Fgure 9: Average best-so-far tness after on the Wng-Box proble for a weak selecton pressure (Max = :). Error bars show the standard devaton. The arrow ndcates the approxate error threshold to understand GAs' behavor, as nsghts about error thresholds wll shed lght on our understandng of optal utaton rates. Second, practcally, as heurstcs for ndng error thresholds wll provde useful gudelnes for settng optal utaton rates, thus provng the perforance of GAs. The consensus sequence plots (Fgures - ), borrowed and adapted fro theoretcal bology (Bonhoeer & Stadler, 99), are new to the GA county. They represent anovel way to vsualze the structure of tness landscapes, snce features such as the \step-ness" of the Royal Starcase functon can be clearly notced (Fgure ). They ay serve as a tool to derentate crtcal (and less crtcal) areas n the genotype, whch ayhave practcal plcatons when tacklng real-world probles. Frst, t ay be possble to nfer portant knowledge about an appled proble. Second, t ay be possble to rene the genotype representatons and optal schedules for utaton rates as dscussed below. The strength of selecton was shown to have a sgncant eect on the agntude of both error thresholds and optal utaton rates. The stronger the selecton pressure, the hgher these agntudes. Ths suggests that the selecton schee used has to be taken nto consderaton when settng the utaton rate. Ths s partcularly true when eprcally coparng the perforance of derent selecton echanss on a gven proble. To be far, coparsons should be ade choosng the optal utaton rate for each selecton schee.
As dscussed above, there are derences n the error threshold agntude across the genotype, whch are ore clearly observed for the Royal Starcase proble (Fgure ) than the wng-box proble. Ths supports the dea that a te-varyng schee for the utaton rate would be optal. Ths dea was orgnally proposed by Fogarty (989), who found that varyng the utaton rate over te and across the bt representaton of ndvduals (or both), sgncantly proved the perforance of the GA. More recently, slar ndngs were reported by Back (99) and Muhlenben (99). A clear plcaton of the ndngs here are that not only can useful estates of optal utaton rates be nferred fro error thresholds but also that a systeatc ethod of settng a non-xed schedule of such rates can be devsed for fales of real world applcaton probles. Ths, then, deserves future nvestgaton. Acknowledgeents Many thanks to A. Meer who kndly produced the Wng-Box gure. Specal thanks to M. Sordo for careful correctons and crtcal readng. The Wng-Box tness functon source code was orgnally wrtten by M. McIlhagga. The rst author s funded by CONICIT, Venezuela. Reference Back, T., & Schutz, M. (996). Intellgent utaton rate control n canoncal genetc algorths. In Zbgnew, R., & Mchalewcz, M. (Eds.), Proceedngs of the Nnth Internatonal Syposu on Foundatons of Intellgent Systes, Vol. 79 of NAI. Sprnger. Back, T. (99). The nteracton of utaton rate, selecton, and self-adapton wthn a genetc algorth. In und R. Manderck, B. M. (Ed.), Parallel Proble Solvng fro Nature. North- Holland. Back, T. (99). Optal utaton rates n genetc search. In Forrest, S. (Ed.), Proceedngs of the th Internatonal Conference on Genetc Algorths. Morgan Kaufann. Back, T. (996). Evolutonary algorths n theory and practce. The Clarendon Press Oxford Unversty Press. Evoluton strateges, evolutonary prograng, genetc algorths. Baker, J. E. (98). Adaptve selecton ethods for genetc algorths. In Grefenstette, J. (Ed.), Proceedngs of the st Internatonal Conference on Genetc Algorths. awrence Erlbau Assocates. Bonhoeer, S., & Stadler, P. (99). Error thresholds on correlated tness landscapes. J. Theor. Bol., 6, 9{7. Egen, M., & Schuster, P. (979). The Hypercycle: A Prncple of Natural Self-Organzaton. Sprnger- Verlag. Fogarty, T.(989). Varyng the probablty ofu- taton n the genetc algorth. In Schaer, J. (Ed.), Proceedngs of the rd Internatonal Conference on Genetc Algorths. Morgan Kaufann. Hesser, J., & Manner, R. (99). Investgatons of the M-heurstc for optal utaton probabltes. In und R. Manderck, R. M. (Ed.), Parallel Proble Solvng fro Nature. North-Holland. McIlhagga, M., Husbands, P., & Ives, R. (996). A coparson of search technques on a wng-box optsaton proble. ecture Notes n Coputer Scence,. Mtchell, M., Forrest, S., & Holland, J. H. (99). The Royal Road for genetc algorths: tness landscapes and GA perforance. In Varela, F. J., & Bourgne, P. (Eds.), Proceedngs of the Frst European Conference on Artcal fe. MIT Press, Cabrdge, MA. Muhlenben, H. (99). How genetc algorths really work: I. utaton and hllclbng. In Manner, R., & Manderck, R. (Eds.), Parallel Proble Solvng fro Nature. North-Holland. Ochoa, G., Harvey, I., & Buxton, H. (999). Error thresholds and ther relaton to optal utaton rates. In Floreano, J., Ncoud, D., & Mondada, F. (Eds.), Proceedngs of the Ffth European Conference on Artcal fe. Sprnger- Verlag. Schaer, J., Caruana, R., Eshelan,., & Das, R. (989). A study of control paraeters aectng onlne perforance of genetc algorths for functon optzaton. In Schaer, J. (Ed.), Proceedngs of the rd Internatonal Conference on Genetc Algorths San Mateo CA. Morgan Kaufann. van Nwegen, E., & Crutcheld, J. P. (998). Optzng epochal evolutonary search: Populatonsze ndependent theory. Tech. rep. Preprnt 98-6-6, Santa Fe Insttute.