Investigations
Jim Smith
IntelligentComputerSystemsCentre,
UniversityoftheWestofEngland,
BristolBS1612QY,U.K.
Abstract. Thispaperpresentsandexaminesthebehaviourofasystem
whereby the rules governinglocal searchwithin a MemeticAlgorithm
are co-evolved alongside the problem representation. We describe the
rationaleforsuchasystem,andtheimplementationofasimpleversion
inwhich the evolving rules are encoded as (condition:action) patterns
appliedtothe problemrepresentation,and are eectively self-adapted.
Weinvestigatethebehaviourofthealgorithmonatestsuiteofproblems,
and show signicant performance improvements overa simpleGenetic
Algorithm,aMemeticAlgorithmusingaxedneighbourhoodfunction,
andasimilarMemeticAlgorithmwhichusesrandomrules,i.e.withthe
learningmechanismdisabled.
Analysisoftheseresultsenablesustodrawsomeconclusionsaboutthe
waythateventhesimpliedsystemisabletodiscoverandexploit
dier-entformsofstructureandregularitieswithintheproblems.Wesuggest
thatthis\meta-learning"ofproblemfeaturesprovidesameansbothof
creating highly scaleable algorithms, and of capturing features of the
solutionspaceinanunderstandableform.
1 Introduction
TheperformancebenetswhichcanbeachievedbyhybridisingEvolutionary
Al-gorithms(EAs)withLocalSearch(LS)operators,so-calledMemeticAlgorithms
(MAs),havenowbeenwelldocumentedacrossawiderangeofproblemdomains
suchascombinatorialoptimisation[1],optimisationofnon-stationaryfunctions
[2], andmulti-objectiveoptimisation[3].See[4] foracomprehensive
bibliogra-phy.Commonlyin thesealgorithms,theLocalSearchimprovementstepis
per-formedoneachoftheproductsofthegenerating(recombinationandmutation)
operators,priortoselectionforthenextpopulation.
There are three principal components which aect the workings of the LS
algorithm. Therst is thechoice ofpivot rule, which is usually either Steepest
Ascent orGreedy Ascent.Thesecond componentisthe\depth"oflocalsearch,
whichcanvaryfromoneiteration,tothesearchcontinuingtolocal optimality.
thought of as dening a set of points that can be reached by the application
ofsomemoveoperatortoapoint.Wecanconsiderthegraphsdenedby
dier-entmoveoperatorsas \tness landscapes" [6].Merz andFreisleben[7] discuss
anumberofstatisticalmeasureswhichcanbeusedtocharacterisetness
land-scapes,andhavebeenproposedaspotentialmeasuresofproblemdiÆculty.They
showthatthechoiceofmoveoperatorcanhaveadramaticeectontheeÆciency
andeectiveness oftheLocalSearch,andhenceoftheresultantMA.
Insomecases,domainspecicinformationmaybeusedtoguidethischoice.
However, it hasrecently been shown that the optimal choice of operators can
benotonlyinstancespecic withinaclassofproblems[7,pp254{258],butalso
dependantonthestateoftheevolutionarysearch[8].Theideathatperiodically
changingthemoveoperatormayprovideameansofescapefromlocaloptimaby
redeningtheneighbourhoodstructure, hasbeendemonstratedintheVariable
Neighbourhood Search algorithm[9].
The aim of this work is to provide a means whereby the denition of the
local search operator used within a MA can be varied over time, and then to
examinewhether evolutionaryprocesses canbeused to control that variation,
sothatabenecial adaptationtakesplace.Inorderto accomplishthis aim,we
must address four major issues. The rst of these is the representation of LS
operators in aform that can be processed by an evolutionary algorithm, and
the choice of ofinitialisation andvariation operators. Thesecond is thecredit
assignmentmechanismforassigningtnesstotheLSpopulationmembers.The
third is thechoiceof populationstructures andsizes, along withselection and
replacementmethodsformanagingtheLSpopulation.Finally,werequireaset
ofexperiments,problemsandmeasurementsdesignedtopermitevaluationand
analysisofthebehaviourofthesystem.Thispaperrepresentstherststageof
thisanalysis.
2 Rule-Based Adaptation of Move Operators
TherepresentationchosenfortheLSoperatorsisatuple<Search Depth,PivotRule,
Pairing,Move>.Thersttwoelementsareself-explanatory.ValuesforPairing
aretakenfromthesetfLinked,Random,Fitness Basedg.WhentheLocalSearch
phaseisapplied toamemberofthesolutionpopulation,thevalueofthe
Pair-ing inthecorrespondingmemberoftheLSpopulationisexamined.Ifthevalue
is Linked then that LS member is used to act on the solution. If the Pairing
takesoneofthevaluesRandom orFitness Based thenaselectionismadefrom
theavailable(i.e.unlinked)membersoftheLSpopulationaccording.Bymaking
thisvariablepartoftheruleandsubjecttoevolution,thesystemcanbevaried
betweenthe extremesof afully unlinked system,(in which although still
inter-actingthetwopopulationsevolveseparately),andafullylinkedsystem.Inthe
lattertheLSoperatorscan beconsidered tobeextrageneticmaterialwhich is
anal-symbiosis,i.e.wherethetwopopulationsshareacommonupdatingmechanism.
The representation chosen for the move operators was as condition:action
pairs, which specify apattern to belooked for in the problem representation,
and adierent pattern it should be changed to. Although this representation
at rst appears very simple, it has the potential to represent highly complex
moves viathe use of symbolsto denote not only wildcardcharacters but also
the specications of repetitions and iterations. Further, permitting the use of
dierent length patterns in the two parts of the rule givesscope for cut and
splice operatorsworkingonvariablelengthsolutions.
Whilethe framework that we havedescribeabove is intended to permit a
full explorationof severalresearch issues,weshallinitially restrictourselvesto
consideringasimplesystem,andexaminingitsbehaviouronawellunderstood
setofbinaryencodedtestproblems.Fortheseinitialinvestigationswetherefore
restricted theLS operatorsto asingleimprovementstep,a greedyacceptance
mechanism,andfulllinkage.Thisrestrictiontowhatareeectivelyself-adaptive
systemsprovidesameansofdealingwiththecreditassignmentandpopulation
managementisssuesnotedabove.
Wealsoinitiallyrestrictourselvestoconsideringonlyruleswherethe
condi-tion and action patternsareof equallengthandare composed ofvaluestaken
from the set f0,1,#g.Thelast of these is a\don't care" symbol which is only
allowedto appearinthecondition partoftherule.
The neighbourhood of a point i is dened by nding the (unordered) set
of positions where the substring denoted by condition is matched in the
rep-resentation of i. Foreach of these anew string is then madeby replacing the
matching substring with the action pattern. To givean example, ifwehavea
solutionrepresentedbythebinarystring1100111000andarule1#0:111,then
this matchesthe rst,second, sixthand seventhpositions, and the
neighbour-hood isthesetf1110111000,11111111000,1100111100,1100111110g.Note that
in thisinitialwork wedonotconsideredthestringastoroidal.
3 Related Work
TheCOMAsystemcanberelatedtoanumberofdierentbranchesofresearch,
allof which oerdierentwaysofperspectivesandmeansof analysingit's
be-haviour.Space constraintsprecludeafulldiscussionofeachofthese,so wewill
brieyoutlinesomeoftheseperspectives.
Although we are not aware of other algorithms in which the LS operators
used by an MA are adapted in this fashion, there are other examples of the
useofmultipleLSoperatorswithin evolutionarysystems.KrasnogorandSmith
[8]describewhat theycall a\MultiMemeticAlgorithm",in theyused asimple
Self-Adaptivemechanismto evolvethechoice of which axed set of staticLS
operators(\memes")shouldbeappliedtoindividualsolutions.Theyreportthat
tobelinked,thenthisinstantiationoftheCOMAframeworkcanbeconsidered
as a type of Self Adaptation. The use of the intrinsic evolutionary processes
to adapt search strategies, and the conditions necessary, is well documented
e.g. for mutation step sizes [10,11], mutation probabilities [12],recombination
operators[13,14]andgeneralvariationoperators[15],amongstmanyothers.
If the two populations are not linked, then we have a co-operative
coevo-lutionary system, where the tness of the members of the LS population is
assigned as some function of the relative improvement they cause in the
\so-lution"population.Co-operativeco-evolutionary(or\symbiotic")systemshave
beenusedwithgoodreportedresultsforfunction optimisation[16{18]andBull
conducted aseries of more generalstudies on the conditions necessaryfor
co-operativeco-evolutionto occur[19{21].These issueswill beexploredin future
work.
IfweweretosimplyapplytheruleselectedfromintheLSpopulation
(possi-blyiteratively)totransformanindividualsolution,withoutconsideringapivot
rule,then wecouldalsoviewthesystemasatypeof\developmentallearning"
akintothestudiesintheevolutionofGeneticCode[22]
COMA diersfrom the last two paradigmsbecause theLS populationcan
potentiallymodifythegenotypeswithinthesolutionpopulation.Thisphaseof
improvementbyLScanbeviewedasakindoflifetimelearning,whichleads
nat-urallytothequestionofwhetheraBaldwinianapproachmightbepreferableto
theLamarkianLearningcurrentlyimplemented. However,evenifaBaldwinian
approachwas used, the principal dierencebetweenthe COMA approach and
theco-evolutionarysystemsaboveliesin theuseofapivotrulewithin theLS,
suchthatdetrimentalchangesarerejected.
Finally, andperhapsmostimportantly,weshould consider thatifthe same
rulehasanimprovingeectondierentparts ofasolutionchromosomeoveras
numberof generations, then the evolutionof rules canbeseen asaprocess of
learninggeneralisationsaboutpatternswithin theproblem representation,and
hencethesolutionspace.ThispointofviewisakintothatofLearningClassier
Systems.Forthecaseofunlinkedtness-basedselectionofLSoperators,insight
fromthis eldcanbeusedtoguidethecreditassignmentprocess.
4 The Test Suite and Experimental set-up
Inordertoexaminethebehaviourofthesystemitwasusedwithasetofvariants
of atest function whose properties are well known. This was a sixty four bit
problem composed of 16 subproblems of Deb's 4-bit fully deceptive function
givenin[23].Thetnessofeachsubproblemiisgivenbyitsunitationu(i)(i.e.
thenumberofbits setto1):
f(i)=
0:6 0:2u(i) : u(i)<4
1 : u(i)=4
(1)
Thisseparationensuresthateventhelongestrulesallowedintheseexperiments
wouldbeunabletoaltermorethanoneelementin anyofthesubfunctions.
Athirdvariantofthis problem(Shifted-Trap)wasdesignedto bemore
dif-cult than the rst for the COMA algorithm to learn a single generalisation,
bymakingpatterns which were optimalin onesub-problem, sub-optimalin all
others. Thiswas achievedbynotingthat eachsub-problemasdened aboveis
afunction ofunitation, and thereforecanbearbitrarily translatedbydening
a4-bitstringand usingthe Hammingdistance fromthis stringin place ofthe
unitation.Since wehave16sub-problems,wesimply usedthebinarycodingof
thesub-problem'sindexasbasisforitstnesscalculation.
Weusedagenerationalgeneticalgorithm,withdeterministicbinary
tourna-mentselectionforparentsandnoelitism.OnePointCrossover(withprobability
0.7)andbit-ippingmutation (withabitwiseprobabilityof0.01)wereusedon
theproblemrepresentation.Thesevaluesweretakenas\standard"fromthe
lit-erature,bearinginmindthenatureofthe4-Trapfunction.Mutationwasapplied
to therules withaprobabilityof 0.0625of selectinganewallele valuein each
locus(theinverseofthemaximumrulelengthallowedtotheadaptiveversion).
Foreachproblem,20runsweremadeforeachpopulationsizef100,250,500g.
Eachrunwascontinueduntiltheglobaloptimumwasreached,subjecttoa
max-imumof 1millionevaluations.Note thatsince oneiterationof LSmay involve
several evaluations, this allows more generations to the GA, i.e. we compare
algorithmsstrictlyonthebasisofthenumberofcallstotheevaluationfunction.
The algorithms used are: a \vanilla" GA i.e. with no use of Local Search
(GA),asimpleMAusingoneiterationofgreedyascentovertheneighbourhood
at Hamming distance 1 (MA), a version of COMA using a randomly created
ruleineachapplication,(i.e.withthelearningdisabled)(RandComa),variants
of COMA using rules of xed lengths in the ranges f1;:::;9g (1-Coma,::
:,9-Coma), and nally anadaptive versionof COMA (A-Coma). ForA-Comathe
rule lengths are randomly initialised in the range [1,16]. During mutation, a
valueof+= 1israndomlychosenandaddedwithprobability0.0625,subject
to stayinginrange.
5 Comparison Results
Figure1showstheresultsofthese experimentsasaplotofmean timeto
opti-mumfor4-Trapwiththreedierentpopulationsizes.Whenanalgorithmfailed
to reach theoptimumin alltwentyruns,the meanistakenoverthesuccessful
runs,andthisnumberisshown.Theerrorbarsrepresentonestandarddeviation.
It should benotedthat thescale onthey-axisislogarithmic. Wecansee that
theGAandMA,and1-Comaalgorithmsfailtondtheoptimumasfrequently,
orwhentheydoasfast,forthesmallerpopulationsizes.Forallpopulationsizes
testtothesolutiontimesforthesuccessfulrunstodetectstatisticallysignicant
dierencesinperformance.TheGA,MAand1-comaalgorithmsaresignicantly
slowerthantherestatthe5%levelforapopulationof100.Forapopulationsize
of250theGAandMAalgorithmsaresignicantlyslower.Whenthepopulation
sizeisincreasedto500,theworseperformanceseenwiththeGAandMAisno
longersignicant,accordingtothisconservativetest.However,theRand-Coma
is now signicantly slowerthan all but the GA, MA and 2-Coma. 1-Coma is
signicantlyslowerthanallbuttheGA,and2-ComaisslowerthanallbutGA,
MAandRand-Coma.
Inshort,whatwecanobserveisthatforxedrulelengthsofbetween3and
9,andfortheadaptiveversion,theCOMAsystemderivesperformancebenets
from evolvingLSrules. Signicantly, and unlike theGA and MA, the COMA
algorithmdoesnotdependonacertainsizepopulationbeforeitisabletosolve
theproblemreliably.Thisisindicativeofafarmorescaleablealgorithm.
Figure 2 shows the results of the experiments on the variants of the trap
functions.Forthe\Shifted"trapfunction,theperformancesoftheGAandMA
arenotsignicantlydierentfromthoseontheunshifted version.thisrefelects
the fact that these algorithms solve the sub-problems independently and are
\blind" to whether the optimal stringfor each is dierent. Whenweexamine
theresultsfortheCOMAalgorithms, weseeslowersolutiontimesthanonthe
previousproblem,resultingfromthefactthatnoonerulecanbefoundwhichwill
givengoodperformancein everysubproblem.Howeverweseeasimilarpattern
ofreliableproblemsolvingforallbut1-Comaand2-Coma.Analysisrevealsthat
even these last twoare statistically signicantlybetterthan GAorMA forall
butthelargestpopulationsize.Interestingly,theRandComaalgorithmperforms
wellhere,probablyasanaturalconsequenceofusingarandomruleeverytime,
sopromotingdiversityintherulebase.
Considering Dist-Trap, we rst note that the GA, MA and Rand-COMA
failed to solve the function to optimality in any run, regardless of population
size.ThepoorresultsoftheGAcanbeattributedtothemismatchbetweenthe
distributed nature of the representationand the high positional bias of the
1-pointCrossoverused.WhenweconsidertheactionofthebitippingLSoperator
onasubproblem,this willleadtowardsthesub-optimal solution,wheneverthe
unitationis0,1 or2,andthegreedysearchof theneighbourhoodwillalsolead
towardsthe deceptiveoptimum75%of thetime when theunitation is3. This
observationhelpsus to understandthepoorresultsofthe simpleMA, andthe
1-Comaalgorithm.
When we examine the other COMA results, noting that the success rate
is less than for the other problems, we again see the same pattern of better
performance for the adaptive version and xed rulelengths in the range 3-5,
tailing o at the extreme lengths. Note that although themean solution time
dropsfor longrulelength,sotoodoesthenumberof successfulruns which we
Theresultsgivenabovearepromisingfromthepointofviewofimproved
opti-misationperformance,butrequiresomeanalysisandexplanation.Thedeceptive
functionsusedwerespecicallychosenbecauseGAtheorysuggeststhattheyare
solvedbyndingandmixingoptimalsolutionstosub-problems.Whenwe
con-sidertheresults,wecanseethat theperformanceisbeston4-Trap,witharule
length of4,which wouldsupport the hypothesis that the systemis \learning"
thestructureofthesub-problems.Althoughnotimmediatelyapparentfromthe
logarithmicscales,thesolutionstimesherearelessthanhalfthoseontheother
problems.
Howeverweshouldnotethatthealgorithmsarenotawareofthesub-problem
boundaries.On4-TrapandShifted-Trap,forlengthsof4orlessoccasionally,and
always for lengths greater than 4, the changes made will overlap several
sub-problems. This must always happen for Dist-Trap. It is clear from theresults
with dierent rule lengths, and from the distributed problem, that there is a
second form of improvement working on a longer timescale., which does not
arisesimplyfromtheuseofrandomrules.
Inordertoexaminethebehaviourofthealgorithmweplottedthepopulation
meansof theeectiverulelength(onlyrelevantfor A-Coma),the \specicity"
(i.e.theproportionofvaluesintheconditionnotset to#)andthe\unitation"
(theproportionofbitsintheactionsetto1),andalsothehighesttnessinthe
population(with 100as theoptimum) asafunction ofthe number of elapsed
generations. Figure3showstheA-Comaresultsaveragedover20 runsoneach
ofthethreeproblems,withapopulationof250.Wealsomanuallyinspectedthe
evolvingrulebasesonalargenumberofrunsforeachproblem.
Forthe4-Trapfunction(lefthandgraph),thesystemrapidlyevolvesmedium
length (3 4), general (specicity < 50%) rules whose action is to set all the
bitsto 1(meanunitationapproaches100%).Notethat in theabsenceof
selec-tivepressure(i.e.thepivotrulesmeantthatthesolutionswereleftunchanged),
allthree of thesevalueswouldbeexpected to remainat theirinitialvalues, so
these changes resultfrombenecial adaptation.Closerinspection ofthe
evolv-ing rulebase conrmsthat theoptimalsubproblem stringis being learnedand
applied.
For the Shifted-Trap function, where the optimal sub-blocks are all
dier-ent(middle)therulelengthdecreasesmoreslowly.Thespecicityalsoremains
higher, and the unitation remains at 50%, indicating that dierent rules are
beingmaintained.Thisisborneoutbycloserexaminationoftherulesets.
ThebehaviouronDist-Trapissimilartothaton4-Trap,albeitoveralonger
timescale.Ratherthanlearningspecicrulesaboutsub-problems,whichcannot
possibly be happening (since no rule is able to aect more than one locus of
anysubproblem), thesystem isapparentlylearningthe generalrule of setting
allbitsto1.
The rules are generally shorter than for 4-Trap, (although this is slightly
length of the rules used denes a maximum radius in Hamming space for the
neighbourhood, rather than a xed distance from the original solution. Both
of theseobservations,whentakenin tandem withthelongertimes tosolution,
suggest that when thesystem is unable to nd a singlerule that matches the
problems'structure,amorediversesearchusingamorecomplexneighbourhood
is used, which slowly adapts itself to the state of the current population of
solutions.
7 Conclusions
We havepresented a framework in which rules governing LS operators to be
usedin memeticalgorithms canbeco-evolvedwiththepopulationofsolutions.
A simpleversionwas implementedwhich used Self-Adaptation ofthe patterns
dening themoveoperators,and thiswasshowntogiveimprovedperformance
over both GAs and simple MAs on the test set. We showed that the system
was able to learn generalisations about the problem when these were useful.
WealsonotedthattheCOMAalgorithmswerefarlessdependantonacritical
populationsize tolocatetheglobaloptima,andsuggestedthat thisindicatesa
farmorescaleabletypeofalgorithm.
Thetestset usedherewasdesignedtomaximisedierenttypesofdiÆculty
forCOMA,namelydeception,inappropriaterepresentationordering,and
multi-pledierentoptima.Althoughspaceprecludestheirinclusion,wehaverepeated
theseexperimentswithawiderangeofdierentproblemtypesandfound
simi-larperformancebenets.Clearlyfurtherexperimentationandanalysisisneeded,
and there are many issues to be explored,however we believethat this paper
representstherststagein apromisingline ofresearch.
References
1. Merz,P.,Freisleben,B.: AcomparionofMemeticAlgorithms,TabuSearch,and
ant coloniesfor the quadratic assignment problem. In:Proceedings of the 1999
CongressonEvolutionaryComputation,IEEEServiceCenter(1999)2063{2070
2. Vavak, F., Fogarty, T., Jukes, K.: A genetic algorithm with variable range of
local search for tracking changing environments. In Voigt, H.M., Ebeling, W.,
I.Rechenberg,Schwefel,H.P.,eds.:ProceedingsoftheFourthConferenceon
Par-allel ProblemSolvingfromNature,SpringerVerlag(1996)376{385
3. Knowles, J.,Corne,D.: Acomparativeassessment ofmemetic,evolutionaryand
constructive algorithms for the multi-objective d-msatproblem. In: Gecco-2001
WorkshopProgram.(2001)162{167
4. Moscato, P.: Memetic algorithms' home page. Technical report,
http://www.densis.fee.unicamp.br/~moscato/memetichome.html(2002)
5. Hart,W.E.: AdaptiveGlobalOptimization withLocal Search. PhDthesis,
Uni-versityofCalifornia,SanDiego(1994)
Corne,D.,Dorigo,M.,Glover, F.,eds.:NewIdeasinOptimization,McGrawHill
(1999)245{260
8. Krasnogor, N., Smith, J.: Emergenceof protiable searchstrategies based ona
simpleinheritance mechanism. InSpector,L., Goodman, E., Wu,A., Langdon,
W.B.,Voigt,H.M.,Gen,M.,Sen,S.,Dorigo,M.,Pezeshk,S.,Garzon,M.,Burke,
E., eds.:Proceedings of the Geneticand EvolutionaryComputation Conference,
GECCO-2001,MorganKaufmann(2001)432{439
9. Hansen,P.,Mladenovic,N.: Anintroductiontovariableneighborhoodsearch. In
Vo,S.,Martello,S.,Osman,I.H.,Roucairol,C., eds.:Meta-Heuristics:Advances
and trends in local search paradigms for optimization. Proceedings of MIC 97
Conference. KluwerAcademicPublishers,Dordrecht,TheNetherlands(1998)
10. Schwefel, H.P.: Numerical Optimisation of Computer Models. John Wiley and
Sons, NewYork(1981)
11. Fogel, D.: Evolving Articial Intelligence. PhDthesis, University of California
(1992)
12. Back,T.: Selfadaptationingeneticalgorithms. InVarela,F.,Bourgine,P.,eds.:
Towards aPractice onAutonomousSystems:Proceedingsof theFirst European
ConferenceonArticialLife,MITPress(1992)263{271
13. Schaer,J.,Morishima,A.: Anadaptivecrossoverdistributionmechanismfor
ge-neticalgorithms.InJ.J.Grefenstette,ed.:ProceedingsoftheSecondInternational
ConferenceonGeneticAlgorithms,LawrenceErlbaum(1987)36{40
14. Smith, J.,Fogarty,T.C.: An adaptive poly-parental recombinationstrategy. In
Fogarty,T.C.,ed.:EvolutionaryComputing2,SpringerVerlag(1995)48{61
15. Smith,J.,Fogarty,T.: Adaptivelyparameterisedevolutionarysystems:Self
adap-tiverecombinationandmutationinageneticalgorithm.InVoigt,Ebeling,
Rechen-berg, Schwefel, eds.: Proceedings of the Fourth Conference on Parallel Problem
SolvingfromNature,SpringerVerlag(1996)441{450
16. Husbands, P.: Distributed co-evolutionary genetic algorithms for multi-criteria
andmulti-constraintoptimisiation. InFogarty,T.,ed.:EvolutionaryComputing:
ProceedingsoftheAISBworkshop.LNCS865,SpringerVerlag(1994)150{165
17. Paredis, J.: The symbiotic evolution of solutions and their representations. In
Eshelman,L.J.,ed.:ProceedingsoftheSixthInternationalConferenceonGenetic
Algorithms,MorganKaufmann(1995)359{365
18. Potter, M.A., DeJong, K.: A cooperative coevolutionary approach to function
optimisation. InDavidor,Y.,ed.:ProceedingsoftheThirdConferenceonParallel
ProblemSolvingfromNature,SpringerVerlag(1994)248{257
19. Bull, L.: Articial Symbiology. PhD thesis, University of the West of England
(1995)
20. Bull,L.: Evolutionarycomputinginmultiagentenvironments:Partners. InBack,
T., ed.: Proceedings of the Seventh International Conference on Genetic
Algo-rithms,MorganKaufmann(1997)370{377
21. Bull,L.,Holland,O.,Blackmore,S.: Onmeme-genecoevolution. ArticialLife6
(2000)227{235
22. Keller,R.E.,Banzhaf,W.:Theevolutionofgeneticcodeingeneticprogramming.
InBanzhaf,W.,Daida,J.,Eiben,A.E.,Garzon,M.H.,Honavar,V.,Jakiela,M.,
Smith,R.E.,eds.:ProceedingsoftheGeneticandEvolutionaryComputation
Con-ference.Volume2.,Orlando,Florida,USA,MorganKaufmann(1999)1077{1082
23. Back, T., Fogel, D., Michalwicz, Z.: Handbook of Evolutionary Computation.
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