Gene Expression Programming: A New Adaptive Algorithm for Solving Problems

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1 Gene Expression Progrmming: A New Adptive Algorithm for Solving Prolems Cândid Ferreir Deprtmento de Ciêncis Agráris Universidde dos Açores Terr-Chã Angr do Heroísmo, Portugl Complex Systems, Vol. 13, issue 2: , 2001 Gene expression progrmming, genotype/phenotype genetic lgorithm (liner nd rmified), is presented here for the first time s new technique for the cretion of computer progrms. Gene expression progrmming uses chrcter liner chromosomes composed of genes structurlly orgnized in hed nd til. The chromosomes function s genome nd re sujected to modifiction y mens of muttion, trnsposition, root trnsposition, gene trnsposition, gene recomintion, nd one- nd two-point recomintion. The chromosomes encode expression trees which re the oject of selection. The cretion of these seprte entities (genome nd expression tree) with distinct functions llows the lgorithm to perform with high efficiency tht gretly surpsses existing dptive techniques. The suite of prolems chosen to illustrte the power nd verstility of gene expression progrmming includes symolic regression, sequence induction with nd without constnt cretion, lock stcking, cellulr utomt rules for the density-clssifiction prolem, nd two prolems of oolen concept lerning: the 11-multiplexer nd the GP rule prolem. 1. Introduction Gene expression progrmming (GEP) is, like genetic lgorithms (GAs) nd genetic progrmming (GP), genetic lgorithm s it uses popultions of individuls, selects them ccording to fitness, nd introduces genetic vrition using one or more genetic opertors [1]. The fundmentl difference etween the three lgorithms resides in the nture of the individuls: in GAs the individuls re liner strings of fixed length (chromosomes); in GP the individuls re nonliner entities of different sizes nd shpes (prse trees); nd in GEP the individuls re encoded s liner strings of fixed length (the genome or chromosomes) which re fterwrds expressed s nonliner entities of different sizes nd shpes (i.e., simple digrm representtions or expression trees). If we hve in mind the history of life on Erth (e.g., [2]), we cn see tht the difference etween GAs nd GP is only superficil: oth systems use only one kind of entity which functions oth s genome nd ody (phenome). These kinds of systems re condemned to hve one of two limittions: if they re esy to mnipulte geneticlly, they lose in functionl complexity (the cse of GAs); if they exhiit certin Electronic mil nd we ddresses: cndidf@gene-expressionprogrmming.com; Present ddress: Gepsoft, 37 The Ridings, Bristol BS13 8NU, UK. mount of functionl complexity, they re extremely difficult to reproduce with modifiction (the cse of GP). In his ook, River Out of Eden [3], R. Dwkins gives list of thresholds of ny life explosion. The first is the replictor threshold which consists of self-copying system in which there is hereditry vrition. Also importnt is tht replictors survive y virtue of their own properties. The second threshold is the phenotype threshold in which replictors survive y virtue of cusl effects on something else - the phenotype. A simple exmple of replictor/phenotype system is the DNA/protein system of life on Erth. For life to move eyond very rudimentry stge, the phenotype threshold should e crossed [2, 3]. Similrly, the entities of oth GAs nd GP (simple replictors) survive y virtue of their own properties. Understndingly, there hs een n effort in recent yers y the scientific community to cross the phenotype threshold in evolutionry computtion. The most prominent effort is developmentl genetic progrmming (DGP) [4] where inry strings re used to encode mthemticl expressions. The expressions re decoded using five-it inry code, clled genetic code. Contrry to its nlogous nturl genetic code, this genetic code, when pplied to inry strings, frequently produces invlid expressions (in nture there is no such thing s n invlid protein). Therefore huge mount of computtionl resources goes towrd editing these illegl structures, which limits this system considerly. Not surprisingly, the gin in performnce of DGP over GP is miniml [4, 5]. 1

2 The interply of chromosomes (replictors) nd expression trees (phenotype) in GEP implies n unequivocl trnsltion system for trnslting the lnguge of chromosomes into the lnguge of expression trees (ETs). The structurl orgniztion of GEP chromosomes presented in this work llows truly functionl genotype/phenotype reltionship, s ny modifiction mde in the genome lwys results in syntcticlly correct ETs or progrms. Indeed, the vried set of genetic opertors developed to introduce genetic diversity in GEP popultions lwys produces vlid ETs. Thus, GEP is n rtificil life system, well estlished eyond the replictor threshold, cple of dpttion nd evolution. The dvntges of system like GEP re cler from nture, ut the most importnt should e emphsized. First, the chromosomes re simple entities: liner, compct, reltively smll, esy to mnipulte geneticlly (replicte, mutte, recomine, trnspose, etc.). Second, the ETs re exclusively the expression of their respective chromosomes; they re the entities upon which selection cts nd, ccording to fitness, they re selected to reproduce with modifiction. During reproduction it is the chromosomes of the individuls, not the ETs, which re reproduced with modifiction nd trnsmitted to the next genertion. On ccount of these chrcteristics, GEP is extremely verstile nd gretly surpsses the existing evolutionry techniques. Indeed, in the most complex prolem presented in this work, the evolution of cellulr utomt rules for the density-clssifiction tsk, GEP surpsses GP y more thn four orders of mgnitude. The present work shows the structurl nd functionl orgniztion of GEP chromosomes; how the lnguge of the chromosomes is trnslted into the lnguge of the ETs; how the chromosomes function s genotype nd the ETs s phenotype; nd how n individul progrm is creted, mtured, nd reproduced, leving offspring with new properties, thus, cple of dpttion. The pper proceeds with detiled description of GEP nd the illustrtion of this technique with six exmples chosen from different fields. 2. An overview of gene expression lgorithms Crete Chromosomes of Initil Popultion Express Chromosomes Execute Ech Progrm Evlute Fitness Terminte Iterte or Terminte? Iterte Keep Best Progrm Select Progrms Repliction Muttion IS trnsposition RIS trnsposition Gene Trnsposition 1-Point Recomintion Reproduction End The flowchrt of gene expression lgorithm (GEA) is shown in Figure 1. The process egins with the rndom genertion of the chromosomes of the initil popultion. Then the chromosomes re expressed nd the fitness of ech individul is evluted. The individuls re then selected ccording to fitness to reproduce with modifiction, leving progeny with new trits. The individuls of this new genertion re, in their turn, sujected to the sme developmentl process: expression of the genomes, confronttion of the selection environment, nd reproduction with modifiction. The process is repeted for certin numer of genertions or until solution hs een found. Note tht reproduction includes not only repliction ut lso the ction of genetic opertors cple of creting genetic diversity. During repliction, the genome is copied nd trnsmitted to the next genertion. Oviously, repliction 2-Point Recomintion Gene Recomintion Prepre New Progrms of Next Genertion Figure 1. The flowchrt of gene expression lgorithm. lone cnnot introduce vrition: only with the ction of the remining opertors is genetic vrition introduced into the popultion. These opertors rndomly select the chromosomes to e modified. Thus, in GEP, chromosome might e modified y one or severl opertors t time or not e 2

3 modified t ll. The detils of the implementtion of GEP opertors re shown in section The genome of gene expression progrmming individuls In GEP, the genome or chromosome consists of liner, symolic string of fixed length composed of one or more genes. It will e shown tht despite their fixed length, GEP chromosomes cn code ETs with different sizes nd shpes Open reding frmes nd genes The structurl orgniztion of GEP genes is etter understood in terms of open reding frmes (ORFs). In iology, n ORF, or coding sequence of gene, egins with the strt codon, continues with the mino cid codons, nd ends t termintion codon. However, gene is more thn the respective ORF, with sequences upstrem from the strt codon nd sequences downstrem from the stop codon. Although in GEP the strt site is lwys the first position of gene, the termintion point does not lwys coincide with the lst position of gene. It is common for GEP genes to hve noncoding regions downstrem from the termintion point. (For now we will not consider these noncoding regions, ecuse they do not interfere with the product of expression.) Consider, for exmple, the lgeric expression: ( + ) ( c d), (3.1) which cn lso e represented s digrm or ET: + where represents the squre root function. This kind of digrm representtion is in fct the phenotype of GEP individuls, eing the genotype esily inferred from the phenotype s follows: cd (3.2) which is the strightforwrd reding of the ET from left to right nd from top to ottom. Expression (3.2) is n ORF, strting t (position 0) nd terminting t d (position 7). These ORFs were nmed K-expressions (from the Krv lnguge, the nme I chose for the lnguge of GEP). Note tht this ordering differs from oth the postfix nd prefix expressions used in different GP implementtions with rrys or stcks [6]. c d The inverse process, tht is, the trnsltion of K-expression into n ET, is lso very simple. Consider the following K-expression: (3.3) The strt position (position 0) in the ORF corresponds to the root of the ET. Then, elow ech function re ttched s mny rnches s there re rguments to tht function. The ssemlge is complete when seline composed only of terminls (the vriles or constnts used in prolem) is formed. In this cse, the following ET is formed: Looking only t the structure of GEP ORFs, it is difficult or even impossile to see the dvntges of such representtion, except perhps for its simplicity nd elegnce. However, when ORFs re nlyzed in the context of gene, the dvntges of such representtion ecome ovious. As stted previously, GEP chromosomes hve fixed length nd re composed of one or more genes of equl length. Therefore the length of gene is lso fixed. Thus, in GEP, wht vries is not the length of genes (which is constnt), ut the length of the ORFs. Indeed, the length of n ORF my e equl to or less thn the length of the gene. In the first cse, the termintion point coincides with the end of the gene, nd in the second cse, the termintion point is somewhere upstrem from the end of the gene. So, wht is the function of these noncoding regions in GEP genes? They re, in fct, the essence of GEP nd evolvility, for they llow modifiction of the genome using ny genetic opertor without restrictions, lwys producing syntcticlly correct progrms without the need for complicted editing process or highly constrined wys of implementing genetic opertors. Indeed, this is the prmount difference etween GEP nd previous GP implementtions, with or without liner genomes (for review on GP with liner genomes see [7]) Gene expression progrmming genes GEP genes re composed of hed nd til. The hed contins symols tht represent oth functions (elements from the function set F) nd terminls (elements from the terminl set T), wheres the til contins only terminls. Therefore two different lphets occur t different regions within 3

4 gene. For ech prolem, the length of the hed h is chosen, wheres the length of the til t is function of h nd the numer of rguments of the function with the most rguments n, nd is evluted y the eqution /+ (3.7) giving the ET: t = h (n-1) + 1. (3.4) Consider gene composed of {,, /, -, +,, }. In this cse n = 2. For instnce, for h = 10 nd t = 11, the length of the gene is 10+11=21. One such gene is shown elow (the til is shown in old): / (3.5) nd it codes for the following ET: In this cse the termintion point shifts severl positions to the right (position 14). Oviously the opposite lso hppens, nd the ORF is shortened. For exmple, consider gene (3.5) nd suppose muttion occurred t position 5, chnging the into : / (3.8) Its expression results in the following ET: In this cse, the ORF ends t position 10, wheres the gene ends t position 20. Suppose now muttion occurred t position 9, chnging the into +. Then the following gene is otined: /+ (3.6) nd its ET gives: In this cse, the termintion point shifts two positions to the right (position 12). Suppose now tht more rdicl modifiction occurred, nd the symols t positions 6 nd 7 in gene (3.5) chnge respectively into + nd, creting the following gene: In this cse, the ORF ends t position 7, shortening the originl ET y 3 nodes. Despite its fixed length, ech gene hs the potentil to code for ETs of different sizes nd shpes, the simplest eing composed of only one node (when the first element of gene is terminl) nd the iggest composed of s mny nodes s the length of the gene (when ll the elements of the hed re functions with the mximum numer of rguments, n). It is evident from the exmples ove, tht ny modifiction mde in the genome, no mtter how profound, lwys results in vlid ET. Oviously the structurl orgniztion of genes must e preserved, lwys mintining the oundries etween hed nd til nd not llowing symols from the function set on the til. Section 5 shows how GEP opertors work nd how they modify the genome of GEP individuls during reproduction. 4

5 3.3. Multigenic chromosomes GEP chromosomes re usully composed of more thn one gene of equl length. For ech prolem or run, the numer of genes, s well s the length of the hed, is chosen. Ech gene codes for su-et nd the su-ets interct with one nother forming more complex multisuunit ET. The detils of such interctions re fully explined in section 3.4. Consider, for exmple, the following chromosome with length 27, composed of three genes (the tils re shown in old): (3.9) It hs three ORFs, nd ech ORF codes for su-et (Figure 2). Position 0 mrks the strt of ech gene; the end of ech ORF, though, is only evident upon construction of the respective su-et. As shown in Figure 2, the first ORF ends t position 4 (su-et 1 ); the second ORF ends t position 5 (su-et 2 ); nd the lst ORF lso ends t position 5 (su-et 3 ). Thus, GEP chromosomes code for one or more ORFs, ech expressing prticulr su-et. Depending on the tsk t hnd, these su-ets my e selected individully ccording to their respective fitness (e.g., in prolems with multiple outputs), or they my form more complex, multi-suunit ET nd e selected ccording to the fitness of the whole, multi-suunit ET. The ptterns of expression nd the detils of selection will e discussed throughout this pper. However, keep in mind tht ech su-et is oth seprte entity nd prt of more complex, hierrchicl structure, nd, s in ll complex systems, the whole is more thn the sum of its prts Expression trees nd the phenotype In nture, the phenotype hs multiple levels of complexity, the most complex eing the orgnism itself. But trnas, proteins, riosomes, cells, nd so forth, re lso products of expression, nd ll of them re ultimtely encoded in the genome. In ll cses, however, the expression of the genetic informtion strts with trnscription (the synthesis of RNA) nd, for protein genes, proceeds with trnsltion (the synthesis of proteins) Informtion decoding: Trnsltion In GEP, from the simplest individul to the most complex, the expression of genetic informtion strts with trnsltion, the trnsfer of informtion from gene into n ET. This process hs lredy een presented in section 3.2 where decoding of GEP genes is shown. In contrst to nture, the expression of the genetic informtion in GEP is very simple. Worth emphsizing is the fct tht in GEP there is no need for trnscription: the messge in the gene is directly trnslted into n ET. GEP chromosomes re composed of one or more ORFs, nd oviously the encoded individuls hve different degrees of complexity. The simplest individuls re encoded in single gene, nd the orgnism is, in this cse, the product of single gene - n ET. In other cses, the orgnism is multisuunit ET, in which the different su-ets re linked together y prticulr function. In other cses, the orgnism emerges from the sptil orgniztion of different su-ets (e.g., in plnning nd prolems with multiple outputs). And, in yet other cses, the orgnism emerges from the interctions of conventionl su-ets with different domins (e.g., neurl networks). However, in ll cses, the whole orgnism is encoded in liner genome. () () 6XE(7 6XE(7 6XE(7 Figure 2. Expression of GEP genes s su-ets. () A three-genic chromosome with the tils shown in old. The rrows show the termintion point of ech gene. () The su-ets codified y ech gene. 5

6 Interctions of su-expression trees We hve seen tht trnsltion results in the formtion of su-ets with different complexity, ut the complete expression of the genetic informtion requires the interction of these su-ets with one nother. One of the simplest interctions is the linking of su-ets y prticulr function. This process is similr to the ssemlge of different protein suunits into multi-suunit protein. When the su-ets re lgeric or oolen expressions, ny mthemticl or oolen function with more thn one rgument cn e used to link the su-ets into finl, multisuunit ET. The functions most chosen re ddition or multipliction for lgeric su-ets, nd OR or IF for oolen su-ets. In the current version of GEP the linking function is priori chosen for ech prolem, ut it cn e esily introduced in the genome; for instnce, in the lst position of chromosomes, nd lso e sujected to dpttion. Indeed, preliminry results suggest tht this system works very well. Figure 3 illustrtes the linking of two su-ets y ddition. Note tht the root of the finl ET (+) is not encoded y the genome. Note lso tht the finl ET could e linerly encoded s the following K-expression: (3.10) However, to evolve solutions for complex prolems, it is more effective touse multigenic chromosomes, for they permit the modulr construction of complex, hierrchicl structures, where ech gene codes for smll uilding lock. () These smll uilding locks re seprted from ech other, nd thus cn evolve independently. For instnce, if we tried to evolve solution for the symolic regression prolem presented in section 6.1 with single-gene chromosomes, the success rte would fll significntly (see section 6.1). In tht cse the discovery of smll uilding locks is more constrined s they re no longer free to evolve independently. This kind of experiment shows tht GEP is in effect powerful, hierrchicl invention system cple of esily evolving simple locks nd using them to form more complex structures [8, 9]. Figure 4 shows nother exmple of su-et interction, where three oolen su-ets re linked y the function IF. The multi-suunit ET could e linerized s the following K- expression: IINAIAINu1c32cAO2 (3.11) Figure 5 shows nother exmple of su-et interction, where the su-ets re of the simplest kind (one-element su- ETs). In this cse, the su-ets re linked 3 y 3 with the IF function, then these clusters re, in their turn, linked lso 3 y 3 with nother IF function, nd the three lst clusters re lso linked y IF, forming lrge multi-suunit ET. This kind of chromosoml rchitecture ws used to evolve solutions for the 11-multiplexer prolem of section nd lso to evolve cellulr utomt rules for the density-clssifiction prolem. The individul of Figure 5 could e converted into the following K-expression: IIIIIIIIIIIII131u32u23c3u31333u3 (3.12) And finlly, the full expression of certin chromosomes requires the sequentil execution of smll plns, where the () 6XE(7 6XE(7 (c) (7 Figure 3. Expression of multigenic chromosomes s ETs. () A two-genic chromosome with the tils shown in old. () The su-ets codified y ech gene. (c) The result of posttrnsltionl linking with ddition. 6

7 () IIAIc32cuNNAO2u3c31cAu12u3112cc () 6XE(7 6XE(7 6XE(7 I N A I A I N u 1 c 3 2 c A O 2 (c) (7 I I N A I A I N u 1 c 3 2 c A Figure 4. Expression of multigenic chromosomes s ETs. () A three-genic chromosome with the tils shown in old ( N is function of one rgument nd represents NOT; A nd O re functions of two rguments nd represent respectively AND nd OR; I is function of three rguments nd represents IF; the remining symols re terminls). () The su-ets codified y ech gene. (c) The result of posttrnsltionl linking with IF. O 2 () 131u32u23c3u31333u3 () (7 I I I I I I I I I I I I I u 3 2 u 2 3 c 3 u u 3 Figure 5. Expression of multigenic chromosomes s ETs. () A 27-genic chromosome composed of one-element genes. () The result of posttrnsltionl linking with IF. 7

8 first su-et does little work, the second continues from tht, nd so on. The finl pln results from the orderly ction of ll suplns (see the lock stcking prolem in section 6.3). The type of linking function, s well s the numer of genes nd the length of ech gene, re priori chosen for ech prolem. So, we cn lwys strt y using singlegene chromosome, grdully incresing the length of the hed; if it ecomes very lrge, we cn increse the numer of genes nd of course choose function to link them. We cn strt with ddition or OR, ut in other cses nother linking function might e more pproprite. The ide, of course, is to find good solution, nd GEP provides the mens of finding one. 4. Fitness functions nd selection In this section, two exmples of fitness functions re descried. Other exmples of fitness functions re given in the prolems studied in section 6. The success of prolem gretly depends on the wy the fitness function is designed: the gol must e clerly nd correctly defined in order to mke the system evolve in tht direction Fitness functions One importnt ppliction of GEP is symolic regression or function finding (e.g., [9]), where the gol is to find n expression tht performs well for ll fitness cses within certin error of the correct vlue. For some mthemticl pplictions it is useful to use smll reltive or solute errors in order to discover very good solution. But if the rnge of selection is excessively nrrowed, popultions evolve very slowly nd re incple of finding correct solution. On the other hnd, if the opposite is done nd the rnge of selection is rodened, numerous solutions will pper with mximum fitness tht re fr from good solutions. To solve this prolem, n evolutionry strtegy ws devised tht permits the discovery of very good solutions without hlting evolution. So, the system is left to find for itself the est possile solution within minimum error. For tht very rod limit for selection to operte is given, for instnce, reltive error of 20%, tht llows the evolutionry process to get strted. Indeed, these founder individuls re usully very unfit ut their modified descendnts re reshped y selection nd popultions dpt wonderfully, finding etter solutions tht progressively pproch perfect solution. Mthemticlly, the fitness f i of n individul progrm i is expressed y eqution (4.1) if the error chosen is the solute error, nd y eqution (4.1) if the error chosen is the reltive error: (4.1) (4.1) where M is the rnge of selection, C (i,j) the vlue returned y the individul chromosome i for fitness cse j (out of C t fitness cses), nd T j is the trget vlue for fitness cse j. Note tht for perfect fit C (i,j) = T j nd f i = f mx = C t. M. Note tht with this kind of fitness function the system cn find the optiml solution for itself. In nother importnt GEP ppliction, oolen concept lerning or logic synthesis (e.g., [9]), the fitness of n individul is function of the numer of fitness cses on which it performs correctly. For most oolen pplictions, though, it is fundmentl to penlize individuls le to solve correctly out 50% of fitness cses, s most proly this only reflects the 50% likelihood of correctly solving inry oolen function. So, it is dvisle to select only individuls cple of solving more thn 50 to 75% of fitness cses. Below tht mrk symolic vlue of fitness cn e ttriuted, for instnce f i = 1. Usully, the process of evolution is put in motion with these unfit individuls, for they re very esily creted in the initil popultion. However, in future genertions, highly fit individuls strt to pper, rpidly spreding in the popultion. For esy prolems, like oolen functions with 2 through 5 rguments, this is not relly importnt, ut for more complex prolems it is convenient to choose ottom line for selection. For these prolems, the following fitness function cn e used: (4.2) where n is the numer of fitness cses correctly evluted, nd C t is the totl numer of fitness cses Selection If n 1 C, then = ; else = 1 2 t f t n ft In ll the prolems presented in this work, individuls were selected ccording to fitness y roulettewheel smpling [10] coupled with the cloning of the est individul (simple elitism). A preliminry study of different selection schemes (roulettewheel selection with nd without elitism, tournment selection with nd without elitism, nd vrious kinds of deterministic selection with nd without elitism) suggests tht there is no pprecile difference etween them s long s the cloning of the est individul is gurnteed (results not shown). Some schemes perform etter in one prolem, others in nother. However, for more complex prolems it seems tht roulettewheel selection with elitism is est. 5. Reproduction with modifiction According to fitness nd the luck of the roulette, individuls re selected to reproduce with modifiction, creting the necessry genetic diversifiction tht llows evolution in the long run. Except for repliction, where the genomes of ll the selected individuls re rigorously copied, ll the remining opertors rndomly pick chromosomes to e sujected to certin modifiction. However, except for muttion, ech 8

9 opertor is not llowed to modify chromosome more thn once. For instnce, for trnsposition rte of 0.7, seven out of 10 different chromosomes re rndomly chosen. Furthermore, in GEP, chromosome might e chosen y none or severl genetic opertors tht introduce vrition in the popultion. This feture lso distinguishes GEP from GP where n entity is never modified y more thn one opertor t time [9]. Thus, in GEP, the modifictions of severl genetic opertors ccumulte during reproduction, producing offspring very different from the prents. We now proceed with the detiled description of GEP opertors, strting oviously with repliction. (Reders less concerned with implementtion detils of genetic opertors my wish to skip this section.) 5.1. Repliction Although vitl, repliction is the most uninteresting opertor: lone it contriutes nothing to genetic diversifiction. (Indeed, repliction, together with selection, is only cple of cusing genetic drift.) According to fitness nd the luck of the roulette, chromosomes re fithfully copied into the next genertion. The fitter the individul the higher the proility of leving more offspring. Thus, during repliction the genomes of the selected individuls re copied s mny times s the outcome of the roulette. The roulette is spun s mny times s there re individuls in the popultion, lwys mintining the sme popultion size Muttion Muttions cn occur nywhere in the chromosome. However, the structurl orgniztion of chromosomes must remin intct. In the heds ny symol cn chnge into nother (function or terminl); in the tils terminls cn only chnge into terminls. This wy, the structurl orgniztion of chromosomes is mintined, nd ll the new individuls produced y muttion re structurlly correct progrms. Typiclly, muttion rte (p m ) equivlent to two point muttions per chromosome is used. Consider the following 3-genic chromosome: //+ Suppose muttion chnged the element in position 0 in gene 1 to ; the element in position 3 in gene 2 to ; nd the element in position 1 in gene 3 to, otining: /+ Note tht if function is mutted into terminl or vice vers, or function of one rgument is mutted into function of two rguments or vice vers, the ET is modified drsticlly. Note lso tht the muttion on gene 2 is n exmple of neutrl muttion, s it occurred in the noncoding region of the gene. It is worth noticing tht in GEP there re no constrints neither in the kind of muttion nor the numer of muttions in chromosome: in ll cses the newly creted individuls re syntcticlly correct progrms. In nture, point muttion in the sequence of gene cn slightly chnge the structure of the protein or not chnge it t ll, s neutrl muttions re firly frequent (e.g., muttions in introns, muttions tht result in the sme mino cid due to the redundncy of the genetic code, etc.). Here, lthough neutrl muttions exist (e.g., muttions in the noncoding regions), muttion in the coding sequence of gene hs much more profound effect: it usully drsticlly reshpes the ET Trnsposition nd insertion sequence elements The trnsposle elements of GEP re frgments of the genome tht cn e ctivted nd jump to nother plce in the chromosome. In GEP there re three kinds of trnsposle elements. (1) Short frgments with function or terminl in the first position tht trnspose to the hed of genes, except to the root (insertion sequence elements or IS elements). (2) Short frgments with function in the first position tht trnspose to the root of genes (root IS elements or RIS elements). (3) Entire genes tht trnspose to the eginning of chromosomes. The existence of IS nd RIS elements is remnnt of the developmentl process of GEP, s the first GEA used only single-gene chromosomes, nd in such systems gene with terminl t the root ws of little use. When multigenic chromosomes were introduced this feture remined s these opertors re importnt to understnd the mechnisms of genetic vrition nd evolvility Trnsposition of insertion sequence elements Any sequence in the genome might ecome n IS element, therefore these elements re rndomly selected throughout the chromosome. A copy of the trnsposon is mde nd inserted t ny position in the hed of gene, except t the strt position. Typiclly, n IS trnsposition rte (p is ) of 0.1 nd set of three IS elements of different length re used. The trnsposition opertor rndomly chooses the chromosome, the strt of the IS element, the trget site, nd the length of the trnsposon. Consider the 2-genic chromosome elow: Suppose tht the sequence in gene 2 (positions 12 through 14) ws chosen to e n IS element, nd the trget site ws ond 6 in gene 1 (etween positions 5 nd 6). Then, cut is mde in ond 6 nd the lock is copied into the site of insertion, otining: 9

10 During trnsposition, the sequence upstrem from the insertion site stys unchnged, wheres the sequence downstrem from the copied IS element loses, t the end of the hed, s mny symols s the length of the IS element (in this cse the sequence ws deleted). Note tht, despite this insertion, the structurl orgniztion of chromosomes is mintined, nd therefore ll newly creted individuls re syntcticlly correct progrms. Note lso tht trnsposition cn drsticlly reshpe the ET, nd the more upstrem the insertion site the more profound the chnge. Thus, this kind of opertor (IS trnsposition nd RIS trnsposition elow) my e seen s hving high hit rte t the lowest levels of ETs [7] Root trnsposition All RIS elements strt with function, nd thus re chosen mong the sequences of the heds. For tht, point is rndomly chosen in the hed nd the gene is scnned downstrem until function is found. This function ecomes the strt position of the RIS element. If no functions re found, it does nothing. Typiclly root trnsposition rte (p ris ) of 0.1 nd set of three RIS elements of different sizes re used. This opertor rndomly chooses the chromosome, the gene to e modified, the strt of the RIS element, nd its length. Consider the following 2-genic chromosome: //+ Suppose tht the sequence + in gene 2 ws chosen to e n RIS element. Then, copy of the trnsposon is mde into the root of the gene, otining: /+/+ During root trnsposition, the whole hed shifts to ccommodte the RIS element, losing, t the sme time, the lst symols of the hed (s mny s the trnsposon length). As with IS elements, the til of the gene sujected to trnsposition nd ll nery genes sty unchnged. Note, gin, tht the newly creted progrms re syntcticlly correct ecuse the structurl orgniztion of the chromosome is mintined. The modifictions cused y root trnsposition re extremely rdicl, ecuse the root itself is modified. In nture, if trnsposle element is inserted t the eginning of the coding sequence of gene, cusing frmeshift muttion, it rdiclly chnges the encoded protein. Like muttion nd IS trnsposition, root insertion hs tremendous trnsforming power nd is excellent for creting genetic vrition Gene trnsposition In gene trnsposition n entire gene functions s trnsposon nd trnsposes itself to the eginning of the chromosome. In contrst to the other forms of trnsposition, in gene trnsposition the trnsposon (the gene) is deleted in the plce of origin. This wy, the length of the chromosome is mintined. The chromosome to undergo gene trnsposition is rndomly chosen, nd one of its genes (except the first, oviously) is rndomly chosen to trnspose. Consider the following chromosome composed of 3 genes: /+ Suppose gene 2 ws chosen to undergo gene trnsposition. Then the following chromosome is otined: /-+ Note tht for numericl pplictions where the function chosen to link the genes is ddition, the expression evluted y the chromosome is not modified. But the sitution differs in other pplictions where the linking function is not commuttive, for instnce, the IF function chosen to link the su-ets in the 11-multiplexer prolem in section However, the trnsforming power of gene trnsposition revels itself when this opertor is conjugted with crossover. For exmple, if two functionlly identicl chromosomes or two chromosomes with n identicl gene in different positions recomine, new individul with duplicted gene might pper. It is known tht the dupliction of genes plys n importnt role in iology nd evolution (e.g., [11]). Interestingly, in GEP, individuls with duplicted genes re commonly found in the process of prolem solving Recomintion In GEP there re three kinds of recomintion: one-point, two-point, nd gene recomintion. In ll cses, two prent chromosomes re rndomly chosen nd pired to exchnge some mteril etween them One-point recomintion During one-point recomintion, the chromosomes cross over rndomly chosen point to form two dughter chromosomes. Consider the following prent chromosomes: / /-/-- Suppose ond 3 in gene 1 (etween positions 2 nd 3) ws 10

11 rndomly chosen s the crossover point. Then, the pired chromosomes re cut t this ond, nd exchnge etween them the mteril downstrem from the crossover point, forming the offspring elow: /-- /-/ With this kind of recomintion, most of the time, the offspring creted exhiit different properties from those of the prents. One-point recomintion, like the ove mentioned opertors, is very importnt source of genetic vrition, eing, fter muttion, one of the opertors most chosen in GEP. The one-point recomintion rte (p 1r ) used depends on the rtes of other opertors. Typiclly glol crossover rte of 0.7 (the sum of the rtes of the three kinds of recomintion) is used Two-point recomintion In two-point recomintion the chromosomes re pired nd the two points of recomintion re rndomly chosen. The mteril etween the recomintion points is fterwrds exchnged etween the two chromosomes, forming two new dughter chromosomes. Consider the following prent chromosomes: ccccc-[1] c+ccccc++c-[2] Suppose ond 7 in gene 1 (etween positions 6 nd 7) nd ond 3 in gene 2 (etween positions 2 nd 3) were chosen s the crossover points. Then, the pired chromosomes re cut t these onds, nd exchnge the mteril etween the crossover points, forming the offspring elow: cccc++c-[3] c+cccccc-[4] Note tht the first gene is, in oth prents, split downstrem from the termintion point. Indeed, the noncoding regions of GEP chromosomes re idel regions where chromosomes cn e split to cross over without interfering with the ORFs. Note lso tht the second gene of chromosome 1 ws lso cut downstrem from the termintion point. However, gene 2 of chromosome 2 ws split upstrem from the termintion point, profoundly chnging the su-et. Note lso tht when these chromosomes recomined, the noncoding region of gene 2 of chromosome 1 ws ctivted nd integrted into chromosome 3. The trnsforming power of two-point recomintion is greter thn one-point recomintion, nd is most useful to evolve solutions for more complex prolems, especilly when multigenic chromosomes composed of severl genes re used Gene recomintion In gene recomintion n entire gene is exchnged during crossover. The exchnged genes re rndomly chosen nd occupy the sme position in the prent chromosomes. Consider the following prent chromosomes: /-//+ /-/+-/ Suppose gene 2 ws chosen to e exchnged. In this cse the following offspring is formed: /-+/+ /-//-/ The newly creted individuls contin genes from oth prents. Note tht with this kind of recomintion, similr genes cn e exchnged ut, most of the time, the exchnged genes re very different nd new mteril is introduced into the popultion. It is worth noting tht this opertor is unle to crete new genes: the individuls creted re different rrngements of existing genes. In fct, when gene recomintion is used s the unique source of genetic vrition, more complex prolems cn only e solved using very lrge initil popultions in order to provide for the necessry diversity of genes (see section 6.1). However, the cretive power of GEP is sed not only in the shuffling of genes or uilding locks, ut lso in the constnt cretion of new genetic mteril. 6. Six exmples of gene expression progrmming in prolem solving The suite of prolems chosen to illustrte the functioning of this new lgorithm is quite vried, including not only prolems from different fields (symolic regression, plnning, Boolen concept lerning, nd cellulr utomt rules) ut lso prolems of gret complexity (cellulr utomt rules for the density-clssifiction tsk) Symolic regression The ojective of this prolem is the discovery of symolic expression tht stisfies set of fitness cses. Consider we re given smpling of the numericl vlues from the function y = (6.1) over 10 chosen points nd we wnt to find function fitting those vlues within 0.01 of the correct vlue. First, the set of functions F nd the set of terminls T must e chosen. In this cse F = {+, -,, /} nd T = {}. Then 11

12 the structurl orgniztion of chromosomes, nmely the length of the hed nd the numer of genes, is chosen. It is dvisle to strt with short, single-gene chromosomes nd then grdully increse h. Figure 6 shows such n nlysis for this prolem. A popultion size P of 30 individuls nd n evolutionry time G of 50 genertions were used. A p m equivlent to two one-point muttions per chromosome nd p 1r = 0.7 were used in ll the experiments in order to simplify the nlysis. The set of fitness cses is shown in Tle 1 nd the fitness ws evluted y eqution (4.1), eing M = 100. If C (i,j) -T j is equl to or less thn 0.01 (the precision), then C (i,j) -T j = 0 nd f (i,j) = 100; thus for C t = 10, f mx = Note tht GEP cn e useful in serching the most prsimonious solution to prolem. For instnce, the chromosome / with h = 6 codes for the ET: 90 Success rte (%) which is equivlent to the trget function. Note lso tht GEP cn efficiently evolve solutions using lrge vlues of h, tht is, it is cple of evolving lrge nd complex su-ets. It is worth noting tht the most compct genomes re not the most efficient. Therefore certin redundncy is fundmentl to efficiently evolve good progrms. In nother nlysis, the reltionship etween success rte nd popultion size P, using n h = 24 ws studied (Figure 7). These results show the supremcy of genotype/pheno Chromosome length Figure 6. Vrition of success rte (P s ) with chromosome length. For this nlysis G = 50, P = 30, nd P s ws evluted over 100 identicl runs Tle 1 Set of fitness cses for the symolic regression prolem. f() Success rte (%) Popultion size Figure 7. Vrition of success rte (P s ) with popultion size. For this nlysis G = 50, nd medium vlue of 49 for chromosome length (h = 24) ws used. P s ws evluted over 100 identicl runs. 12

13 type representtion, s this single-gene system, which is equivlent to GP, gretly surpsses tht technique [9]. However, GEP is much more complex thn single-gene system ecuse GEP chromosomes cn encode more thn one gene (see Figure 8). Suppose we could not find solution fter the nlysis shown in Figure 6. Then we could increse the numer of genes, nd choose function to link them. For instnce, we could choose n h = 6 nd then increse the numer of genes grdully. Figure 8 shows how the success rte for this prolem depends on the numer of genes. In this nlysis, the p m Success rte (%) Success rte (%) Numer of genertions Figure 9. Vrition of success rte (P s ) with the numer of genertions. For this nlysis P = 30, p m = 0.051, p 1r = 0.7 nd chromosome length of 79 ( single-gene chromosome with h = 39) ws used. P s ws evluted over 100 identicl runs Numer of genes Figure 8. Vrition of success rte (P s ) with the numer of genes. For this nlysis G = 50, P = 30, nd h = 6 ( gene length of 13). P s ws evluted over 100 identicl runs. ws equivlent to two one-point muttions per chromosome, p 1r = 0.2, p 2r = 0.5, p gr = 0.1, p is = 0.1, p ris = 0.1, p gt = 0.1, nd three trnsposons (oth IS nd RIS elements) of lengths 1, 2, nd 3 were used. Note tht GEP cn cope very well with n excess of genes: the success rte for the 10-genic system is still very high (47%). In Figure 9 nother importnt reltionship is shown: how the success rte depends on evolutionry time. In contrst to GP where 51 genertions re the norm, for fter tht nothing much cn possily e discovered [7], in GEP, popultions cn dpt nd evolve indefinitely ecuse new mteril is constntly eing introduced into the genetic pool. Finlly, suppose tht the multigenic system with su-ets linked y ddition could not evolve stisfctory solution. Then we could choose nother linking function, for instnce, multipliction. This process is repeted until good solution hs een found. As stted previously, GEP chromosomes cn e esily modified in order to encode the linking function s well. In this cse, for ech prolem the idel linking function would e found in the process of dpttion. Consider, for instnce, multigenic system composed of 3 genes linked y ddition. As shown in Figure 8, the success rte hs in this cse the mximum vlue of 100%. Figure 10 shows the progression of verge fitness of the popultion nd the fitness of the est individul for run 0 of the experiment summrized in Tle 2, column 1. In this run, correct solution ws found in genertion 11. The su-ets re linked y ddition: / nd mthemticlly corresponds to the trget function (the contriution of ech su-et is indicted in rckets): y = ( 4 ) + ( ) + (0) = The detiled nlysis of this progrm shows tht some of the ctions re redundnt for the prolem t hnd, like the ddition of 0 or multipliction y 1. However, the existence of these unnecessry clusters, or even pseudogenes like gene 3, is importnt to the evolution of more fit individuls (compre, in Figures 6 nd 8, the success rte of compct, single-gene system with h = 6 with other less compct systems oth with more genes nd h greter thn 6). The plot for verge fitness in Figure 10 (nd lso Figures 12, 13 nd 17 elow) suggests different evolutionry 13

14 Fitness (mx 1000) Best Ind Avg fitness Genertions Figure 10. Progression of verge fitness of the popultion nd the fitness of the est individul for run 0 of the experiment summrized in Tle 2, column 1 (symolic regression). dynmics for GEP popultions. The oscilltions on verge fitness, even fter the discovery of perfect solution, re unique to GEP. A certin degree of oscilltion is due to the smll popultion sizes used to solve the prolems presented in this work. However, n identicl pttern is otined using lrger popultion sizes. Figure 11 compres six evolutionry dynmics in popultions of 500 individuls for 500 genertions. Plot 1 (ll opertors ctive) shows the progression of verge fitness of n experiment identicl to the one summrized in Tle 2, column 1, tht is, with ll the genetic opertors switched on. The remining dynmics were otined for muttion lone (Plot 2), for gene recomintion comined with gene trnsposition (Plot 3), for one-point recomintion (Plot 4), two-point recomintion (Plot 5), nd gene recomintion (Plot 6). It is worth noticing the homogenizing effect of ll kinds of recomintion. Interestingly, this kind of pttern is similr to the evolutionry dynmics of GAs nd GP popultions [9, 10]. Also worth noticing is the plot for gene recomintion lone (Figure 11, Plot 6): in this cse perfect solution ws not found. This shows tht sometimes it is impossile to find perfect solution only y shuffling existing uilding locks, s is done in ll GP implementtions without muttion. Indeed, GEP gene recomintion is similr in effect to GP recomintion, for it permits exclusively the recomintion of mthemticlly concise locks. Note tht even more generlized shuffling of uilding locks (using gene recomintion comined with gene trnsposition) results in oscilltory dynmics (Figure 11, Plot 3). Tle 2 Prmeters for the symolic regression (SR), sequence induction (SI), sequence induction using ephemerl rndom constnts (SI), lock stcking (BS), nd 11-multiplexer (11-M) prolems. SR SI SI BS 11-M Numer of runs Numer of genertions Popultion size Numer of fitness cses Hed length Numer of genes Chromosome length Muttion rte One-point recomintion rte Two-point recomintion rte Gene recomintion rte IS trnsposition rte IS elements length 1,2,3 1,2, RIS trnsposition rte RIS elements length 1,2,3 1,2, Gene trnsposition rte Rndom constnts muttion rte Dc specific IS trnsposition rte Selection rnge % 20% Error % 0.0% Success rte

15 Figure 11. Possile evolutionry dynmics for GEP popultions. For this nlysis P = 500. The plots show the progression of verge fitness of the popultion. Plot 1: All opertors switched on with rtes s shown in Tle 2, column 1; in this cse perfect solution ws found in genertion 1. Plot 2: Only muttion t p m = 0.051; in this cse perfect solution ws found in genertion 3. Plot 3: Only gene recomintion t p gr = 0.7 plus gene trnsposition t p gt = 0.2 were switched on; in this cse perfect solution ws found in genertion 2. Plot 4: Only one-point recomintion t p 1r = 0.7; in this cse perfect solution ws found in genertion 3. Plot 5: Only two-point recomintion t p 2r = 0.7; in this cse perfect solution ws found in genertion 1. Plot 6: Only gene recomintion t p gr = 0.7; in this cse perfect solution ws not found: the est of run hs fitness 980 nd ws found in genertion Sequence induction nd the cretion of constnts The prolem of sequence induction is specil cse of symolic regression where the domin of the independent vrile consists of the nonnegtive integers. However, the sequence chosen is more complicted thn the expression used in symolic regression, s different coefficients were used. The solution to this kind of prolem involves the discovery of certin constnts. Here two different pproches to the prolem of constnt cretion re shown: one without using ephemerl rndom constnts [9], nd nother using ephemerl rndom constnts. In the sequence 1, 15, 129, 547, 1593, 3711, 7465, 13539, 22737, 35983, 54321,..., the nth (N) term is N = 5n + 4n + 3n + 2n + 1 (6.2) where n consists of the nonnegtive integers 0, 1, 2, 3,... For this prolem F = {+, -,, /} nd T = {}. The set of fitness cses is shown in Tle 3 nd the fitness ws evluted y eqution (4.1), eing M = 20%. Thus, if the 10 fitness cses were computed exctly, f mx = 200. Figure 12 shows the progression of verge fitness of the popultion nd the fitness of the est individul for run 1 of the experiment summrized in Tle 2, column 2. In this Tle 3 Set of fitness cses for the sequence induction prolem. N run, perfect solution ws found in genertion 81 (the su- ETs re linked y ddition): /+-/++++-+/ //+ 15

16 Fitness (mx 200) Best Ind Avg fitness Consider the single-genic chromosome with n h = 7: ??+????? (6.3) where? represents the ephemerl rndom constnts. The expression of this kind of chromosome is done exctly s efore, otining: The? symols in the ET re then replced from left to right nd from top to ottom y the symols in Dc, otining: Genertions Figure 12. Progression of verge fitness of the popultion nd the fitness of the est individul for run 1 of the experiment summrized in Tle 2, column 2 (sequence induction without ephemerl rndom constnts). nd mthemticlly corresponds to the trget sequence (the contriution of ech su-et is indicted in rckets): y = ( 2 +)+( 4-3 )+( )+( )+( 3 )+(-)+( 2 +2). As shown in column 2 of Tle 2, the proility of success for this prolem using the first pproch is Note tht ll the constnts re creted from scrtch y the lgorithm. It seems tht in rel-world prolems this kind of pproch is more dvntgeous ecuse, first, we never know eforehnd wht kind of constnts re needed nd, second, the numer of elements in the terminl set is much smller, reducing the complexity of the prolem. However, ephemerl rndom constnts cn e esily implemented in GEP. For tht n dditionl domin Dc ws creted. Structurlly, the Dc comes fter the til, hs length equl to t, nd consists of the symols used to represent the ephemerl rndom constnts. For ech gene the constnts re creted t the eginning of run, ut their circultion is gurnteed y the genetic opertors. Besides, specil muttion opertor ws creted tht llows the permnent introduction of vrition in the set of rndom constnts. A domin specific IS trnsposition ws lso creted in order to gurntee n effective shuffling of the constnts. Note tht the sic genetic opertors re not ffected y the Dc: it is only necessry to keep the oundries of ech region nd not mix different lphets. The vlues corresponding to these symols re kept in n rry. For simplicity, the numer represented y the symol indictes the order in the rry. For instnce, for the 10 element rry A = {-0.004, 0.839, , 0.05, -0.49, , 0.43, , 0.576, } the chromosome (6.3) ove gives: To solve the prolem t hnd using ephemerl rndom constnts F = {+, -, }, T = {,?}, the set of rndom constnts R = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, nd the ephemerl rndom constnt? rnged over the integers 0, 1, 2, nd 3. The 16

17 prmeters used per run re shown in Tle 2, column 3. In this experiment, the first solution ws found in genertion 91 of run 8 (the su-ets re linked y ddition): Gene 0: -??-?? A 0 = {3, 1, 0, 0, 3, 3, 2, 2, 2, 3} Gene 1: ---???? A 1 = {0, 1, 2, 3, 1, 3, 0, 0, 1, 3} Gene 2: +??-+??? A 2 = {1, 2, 1, 3, 3, 2, 2, 2, 1, 3} Gene 3: +??? A 3 = {3, 0, 1, 3, 0, 2, 2, 2, 2, 0} Gene 4: +?? A 4 = {2, 3, 3, 2, 1, 3, 0, 0, 2, 3} Gene 5: +++???? A 5 = {1, 3, 3, 1, 0, 0, 2, 0, 0, 2} Gene 6: +?-??? A 6 = {3, 0, 0, 2, 1, 1, 3, 1, 3, 2} Gene 7: +-?????? A 7 = {2, 2, 3, 1, 3, 1, 0, 0, 1, 0} nd mthemticlly corresponds to the trget function (the contriution of ech su-et is indicted in rckets): y = (-2)+(-3)+(+3)+( )+(4 4 )+( )+(3). As shown in column 3 of Tle 2, the proility of success for this prolem is 0.31, considerly lower thn the 0.83 of the first pproch. Furthermore, only the prior knowledge of the solution enled us, in this cse, to correctly choose the rndom constnts. Therefore, for rel-world pplictions where the mgnitude nd type of coefficients is unknown, it is more pproprite to let the system find the constnts for itself. However, for some numericl pplictions the discovery of constnts is fundmentl nd they cn e esily creted s indicted here Block stcking In lock stcking, the gol is to find pln tht tkes ny initil configurtion of locks rndomly distriuted etween the stck nd the tle nd plces them in the stck in the correct order. In this cse, the locks re the letters of the word universl. (Although the word universl ws used s illustrtion, in this version the locks eing stcked my hve identicl lels like, for instnce, in the word individul.) The functions nd terminls used for this prolem consisted of set of ctions nd sensors, eing F = {C, R, N, A} (move to stck, remove from stck, not, nd do until true, respectively), where the first three tke one rgument nd A tkes two rguments. In this version, the A loops re processed t the eginning nd re solved in prticulr order (from ottom to top nd from left to right). The ction rgument is executed t lest once despite the stte of the predicte rgument nd ech loop is executed only once, timing out fter 20 itertions. The set of terminls consisted of three sensors {u, t, p} (current stck, top correct lock, nd next needed lock, respectively). In this version, t refers only to the lock on the top of the stck nd whether it is correct or not; if the stck is empty or hs some locks, ll of them correctly stcked, the sensor returns True, otherwise it returns Flse; nd p refers oviously to the next needed lock immeditely fter t. A multigenic system composed of three genes of length 9 ws used in this prolem. The linking of the su-ets consisted of the sequentil execution of ech su-et or supln. For instnce, if the first su-et empties ll the stcks, the next su-et my proceed to fill them, nd so on. The fitness ws determined ginst 10 fitness cses (initil configurtions of locks). For ech genertion, n empty stck plus nine initil configurtions with one to nine letters in the stck were rndomly generted. The empty stck ws used to prevent the untimely termintion of runs, s fitness point ws ttriuted to ech empty stck (see elow). However, GEP is cple of efficiently solving this prolem using 10 rndom initil configurtions (results not shown). The fitness function ws s follows: for ech empty stck one fitness point ws ttriuted; for ech prtilly nd correctly pcked stck (i.e., with 1 to 8 letters in the cse of the word universl ) two fitness points were ttriuted; nd for ech completely nd correctly stcked word 3 fitness points were ttriuted. Thus, the mximum fitness ws 30. The ide ws to mke the popultion of progrms hierrchiclly evolve solutions towrd perfect pln. And, in fct, usully the first useful pln discovered empties ll the stcks, then some progrms lern how to prtilly fill those empty stcks, nd finlly perfect pln is discovered tht fills the stcks completely nd correctly (see Figure 13). Figure 13 shows the progression of verge fitness of the popultion nd the fitness of the est individul for run 2 of the experiment summrized in Tle 2, column 4. In this run, perfect pln ws found in genertion 50: ARCuptppuApNCptuutNtpRppptp Note tht the first su-pln removes ll the locks nd stcks correct letter; the second su-pln correctly stcks ll the remining letters; nd the lst su-pln does nothing. It should e emphsized tht the plns with mximum fitness evolved re in fct perfect, universl plns: ech genertion they re tested ginst nine rndomly generted initil configurtions, more thn sufficient to llow the lgorithm to 17

18 Fitness (mx 30) Best Ind Avg fitness (IC) hs higher density of 1s, or into stte of ll 0s if the IC hs higher density of 0s. The ility of GAs to evolve CA rules for the densityclssifiction prolem ws intensively investigted [12-15], ut the rules discovered y the GA performed poorly nd were fr from pproching the ccurcy of the GKL rule, humn-written rule. GP ws lso used to evolve rules for the density-clssifiction tsk [16], nd rule ws discovered tht surpssed the GKL rule nd other humn-written rules. This section shows how GEP is successfully pplied to this difficult prolem. The rules evolved y GEP hve ccurcy levels of % nd 82.55%, thus exceeding ll humn-written rules nd the rule evolved y GP The density-clssifiction tsk Genertions Figure 13. Progression of verge fitness of the popultion nd the fitness of the est individul for run 2 of the experiment summrized in Tle 2, column 4 (lock stcking). generlize the prolem (s shown in Figure 13, once reched, the mximum fitness is mintined). Indeed, with the fitness function nd the kind of fitness cses used, ll plns with mximum fitness re universl plns. As shown in the fourth column of Tle 2, the proility of success for this prolem is very high (0.70) despite using nine (out of 10) rndom initil configurtions. It is worth noting tht GP uses 167 fitness cses, cleverly constructed to cover the vrious clsses of possile initil configurtions [9]. Indeed, in rel-life pplictions it is not lwys possile to predict the kind of cses tht would mke the system discover solution. So, lgorithms cple of generlizing well in fce of rndom fitness cses re more dvntgeous Evolving cellulr utomt rules for the density-clssifiction prolem Cellulr utomt (CA) hve een studied widely s they re idelized versions of mssively prllel, decentrlized computing systems cple of emergent ehviors. These complex ehviors result from the simultneous execution of simple rules t multiple locl sites. In the density-clssifiction tsk, simple rule involving smll neighorhood nd operting simultneously in ll the cells of one-dimensionl cellulr utomton, should e cple of mking the CA converge into stte of ll 1s if the initil configurtion The simplest CA is wrp-round rry of N inry-stte cells, where ech cell is connected to r neighors from oth sides. The stte of ech cell is updted y defined rule. The rule is pplied simultneously in ll the cells, nd the process is iterted for t time steps. In the most frequently studied version of this prolem, N=149 nd the neighorhood is 7 (the centrl cell is represented y u ; the r = 3 cells to the left re represented y c,, nd ; the r = 3 cells to the right re represented y 1, 2, nd 3 ). Thus the size of the rule spce to serch for this prolem is the huge numer of Figure 14 shows CA with N = 11 nd the updted stte for the cellulr utomton u upon ppliction of certin trnsition rule. t= t=1 c u Figure 14. A one-dimensionl, inry-stte, r = 3 cellulr utomton with N = 11. The rrows represent the periodic oundry conditions. The updted stte is shown only for the centrl cell. The symols used to represent the neighorhood re lso shown. The tsk of density-clssifiction consists of correctly determining whether ICs contin mjority of 1s or mjority of 0s, y mking the system converge, respectively, to n ll 1s stte (lck or on cells in spce-time digrm), nd to stte of ll 0s (white or off cells). Becuse the density of n IC is function of N rguments, the ctions of locl cells with limited informtion nd communiction must e coordinted with one nother to correctly clssify the ICs. Indeed, to find rules tht perform well is chllenge, nd severl lgorithms were used to evolve etter rules [14-17]. The est rules with performnces of 86.0% (coevolution 2) nd 85.1% (coevolution 1) were discovered using coevolutionry pproch etween GA-evolved rules nd ICs [17]. However, the im of this section is to compre the performnce of GEP with GAs nd GP when pplied to difficult 18

19 prolem. And, in fct, GEP does evolve etter rules thn the GP rule, using computtionl resources tht re more thn four orders of mgnitude smller thn those used y GP Two gene expression progrmming discovered rules In one experiment F = {A, O, N, I} ( A represents the oolen function AND, O represents OR, N represents NOT, nd I stnds for IF) nd T = {c,,, u, 1, 2, 3}. The prmeters used per run re shown in Tle 4, column 1. Tle 4 Prmeters for the density-clssifiction tsk. GEP 1 GEP 2 Numer of genertions Popultion size Numer of ICs Hed length 17 4 Numer of genes 1 3 Chromosome length Muttion rte Point recomintion rte IS trnsposition rte IS elements length 1,2,3 -- RIS trnsposition rte RIS elements length 1,2,3 -- The fitness ws evluted ginst set of 25 unised ICs (i.e., ICs with equl proility of hving 1 or 0 t ech cell). In this cse, the fitness is function of the numer of ICs i for which the system stilizes correctly to configurtion of ll 0s or 1s fter 2xN time steps, nd it ws designed in order to privilege individuls cple of correctly clssifying ICs oth with mjority of 1s nd 0s. Thus, if the system converged, in ll cses, indiscrimintely to configurtion of 1s or 0s, only one fitness point ws ttriuted. If, in some cses, the system correctly converged either to configurtion of 0s or 1s, f = 2. In ddition, rules converging to n lternted pttern of ll 1s nd ll 0s were eliminted, s they re esily discovered nd invde the popultions impeding the discovery of good rules. And fi- nlly, when n individul progrm could correctly clssify ICs oth with mjorities of 1s nd 0s, onus equl to the numer of ICs C ws dded to the numer of correctly clssified ICs, eing in this cse f = i + C. For instnce, if progrm correctly clssified two ICs, one with mjority of 1s nd nother with mjority of 0s, it receives 2+25=27 fitness points. In this experiment totl of 7 runs were mde. In genertion 27 of run 5, n individul evolved with fitness 44: OAIIAucONOAIANI1u23u312c3c212c3ccuc13 Note tht the ORF ends t position 28. This progrm hs n ccurcy of tested over 100,000 unised ICs in 149x298 lttice, thus etter thn the of the GP rule tested in 149x320 lttice [16, 17]. The rule tle of this rule (GEP 1 ) is shown in Tle 5. Figure 15 shows three spce-time digrms for this new rule. As comprison, GP used popultions of 51,200 individuls nd 1000 ICs for 51 genertions [16], thus totl of 51,200. 1, = 2,611,200,000 fitness evlutions were mde, wheres GEP only mde = 37,500 fitness evlutions. Therefore, in this prolem, GEP outperforms GP y more thn four orders of mgnitude (69,632 times). In nother experiment rule slightly etter thn GEP 1, with n ccurcy of , ws otined. Agin, its performnce ws determined over 100,000 unised ICs in 149x298 lttice. In this cse F = {I, M} ( I stnds for IF, nd M represents the mjority function with three rguments), nd T ws oviously the sme. In this cse, totl of 100 unised ICs nd three-genic chromosomes with su-ets linked y IF were used. The prmeters used per run re shown in the second column of Tle 4. The fitness function ws slightly modified y introducing rnking system, where individuls cple of correctly clssifying etween 2 nd 3/4 of the ICs receive one onus equl to C; if etween 3/4 nd 17/20 of the ICs re correctly clssified two onus C; nd if more thn 17/20 of the ICs re correctly clssified three onus C. Also, in this experiment, individuls cple of correctly clssifying only one kind of sitution, lthough not indiscrimintely, were differentited nd hd fitness equl to i. Tle 5 Description of the two new rules (GEP 1 nd GEP 2 ) discovered using GEP for the density-clssifiction prolem. The GP rule is lso shown. The output its re given in lexicogrphic order strting with nd finishing with GEP 1 GEP 2 GP rule

20 Figure 15. Three spce-time digrms descriing the evolution of CA sttes for the GEP 1 rule. The numer of 1s in the IC ( 0 ) is shown ove ech digrm. In () nd () the CA correctly converged to uniform pttern; in (c) it converged wrongly to uniform pttern. By genertion 43 of run 10, n individul evolved with fitness 393: MIuu111321cMIM3u32233M1MIcc1c1 Its rule tle is shown in Tle 5. Figure 16 shows three spce-time digrms for this new rule (GEP 2 ). Agin, in this cse the comprison with GP shows tht GEP outperforms GP y fctor of 10, Boolen concept lerning The GP rule nd the 11-multiplexer re, respectively, oolen functions of seven nd 11 ctivities. Wheres the solution for the 11-multiplexer is well-known oolen function, the solution of the GP rule is prcticlly unknown, s the progrm evolved y GP [16] is so complicted tht it is impossile to know wht the progrm relly does. This section shows how GEP cn e efficiently pplied to evolve oolen expressions of severl rguments. Furthermore, the structurl orgniztion of the chromosomes used to evolve solutions for the 11-multiplexer is n exmple Figure 16. Three spce-time digrms descriing the evolution of CA sttes for the GEP 2 rule. The numer of 1s in the IC ( 0 ) is shown ove ech digrm. In () nd () the CA converges, respectively, to the correct configurtion of ll 0s nd ll 1s; in (c) the CA could not converge to uniform pttern. 20