Proeedngs of the World Congress on Engneerng and Computer Sene 2009 Vol II WCECS 2009, Otober 20-22, 2009, San Franso, USA Parallel Grammatal Evoluton for Crut Optmzaton 2OGLFK Kratohvíl, Pavel Ošmera, 2QGHM Popelka Abstrat Ths paper desrbes a Parallel Grammatal Evoluton (PGE) that an evolve omplete ruts usng a varable length of lnear genome to govern the mappng of a Bakus Naur Form grammar defnton. In order to nrease the effeny of Grammatal Evoluton (GE) the nfluene of bakward proessng and an nfluene of several ftness funtons were tested. PGE wth bakward proessng an also take advantage of progressve rossover and mutaton operators. The algorthm s nternally parallel and onssts of three dfferent nteronneted populatons. A new non-tree struture of GE was tested. Index Terms grammatal evoluton, rut optmzaton, parallel evoluton. I. INTRODUCTION Grammatal Evoluton (GE) [1] an be onsdered as a form of grammar-based genet programmng (GP). In partular, Koza s genet programmng has enjoyed onsderable popularty and wdespread use. Unlke a Koza-style approah, there s no dstnton made at ths stage between what he desrbes as funton (operator n ths ase) and termnals (varables). Koza orgnally employed Lsp as hs target language. Ths dstnton s more of an mplementaton detal than a desgn ssue. Grammatal evoluton an be used to generate programmes n any language, usng Bakus Naur Form (BNF). BNF grammars onsst of termnals, whh are tems that an appear n the language,.e. +, -, sn, log et. and nontermnal, whh an be expanded nto one or more termnals and non-termnals. A non-termnal symbol s any symbol that an be rewrtten to another strng, and onversely a termnal symbol s one that annot be rewrtten. The major strength of GE wth respet to GP s ts ablty to generate mult-lne funtons n any language. Rather than representng the programs as parse tree, as n GP, a lnear genome s used [1]-[3]. A genotype-phenotype mappng s employed suh that eah ndvdual s varable length byte strngs, ontans the nformaton to selet produton rules from a BNF grammar. Manusrpt reeved July 14, 2009. Ths work was supported n part by the Czeh Mnstry of Eduaton under Grant No: MSM 00216305529,QWHOOLJHQW 6\VWHPV LQ $XWRPDWLRQ DQG E\ WKH JUDQW DJHQF\ *$ ý5 1R 102/09/1668: Control Algorthm Desgn by Means of Evolutonary Approah. O. Kratohvíl s wth the European Polytehnal Insttute Kunove, (emal: kratohvl@edukomplex.z). P. Ošmera s wth the Brno Unversty of Tehnology, Faulty of Mehanal Engneerng, Brno, Czeh Republ (e-mal: osmera @fme.vutbr.z). O. Popelka s wth the Insttute Department of nformats, Mendel Unversty, of Agrulture and Forestry, FBE, Brno, Czeh Republ, (xpopelka@node.mendelu.z). The grammar allows the generaton of programs, n an arbtrary language that are guaranteed to be syntatally orret. The user an talor the grammar to produe solutons that are purely syntatally onstraned, or they may norporate doman knowledge by basng the grammar to produe very spef form of sentenes. Beause, GE mappng tehnque employs a BNF defnton, the system s language ndependent, and theoretally an generate arbtrarly omplex funtons. There s qute an unusual approah n GEs, as t s possble for ertan genes to be used two or more tmes f the wrappng operator s used. BNF s a notaton that represents a language n the form of produton rules. It s possble to generate programs usng the Grammatal Swarm Optmzaton (GSO) tehnque [2] wth a performane, whh s smlar to the GE. The relatve smplty, the small populaton szes, and the omplete absene of a rossover operator synonymous wth program evoluton n GP or GE are man advantages of GSO. In the grammatal evoluton GE the dfferent approah to the genotype and phenotype s used. GE evolves a sequene of rule numbers that are translated, usng a predetermned grammar set, nto a phenotyp tree. Our approah uses a parallel struture of GE (PGE). A populaton s dvded nto several sub-populatons that are arranged n the herarhal struture [4]. Every subpopulaton has two separate parts: a male group and a female group. Every group uses qute a dfferent type of seleton. In the frst group a lassal type of GA seleton s used. Only dfferent ndvduals an be added to the seond group. Ths strategy was nspred by harem system n Nature that solves problem of an adaptaton of a omplex organsms to mroorgansms [5]. The bologally nspred strategy nreases an nner adaptaton of PGE. Ths analogy would lead us one step further, namely, to the belef that the ombnaton of GE wth 2 dfferent seletons that are smultaneously used an mprove an adaptve behavour of GE [5], [6], [7]. On the prnple of two seletons we an reate a parallel GE wth a herarhal struture. II. PARALLEL GRAMMATICAL EVOLUTION The PGE s based on the grammatal evoluton GE [1], where BNF grammars onsst of termnals and non-termnals. Termnals are tems, whh an appear n the language. Non-termnals an be expanded nto one or more termnals and non-termnals. Grammar s represented by the tuple {N,T,P,S}, where N s the set of nontermnals, T the set of termnals, P a set of produton rules whh map the elements of N to T, and S s a start symbol whh s a member of N. For example, the BNF s used for our problem below: N = {expr, fn} T = {sn, os, +, -, /, *, X, 1, 2, 3, 4, 5, 6, 7, 8, 9}
Proeedngs of the World Congress on Engneerng and Computer Sene 2009 Vol II WCECS 2009, Otober 20-22, 2009, San Franso, USA S = <expr> and P an be represented as 4 produton rules: 1. <expr> := <fn><expr> <fn><expr><expr> <fn><num><expr> <var> 2. <fn> := sn os + * - U- 3. <var> := X 4. <num> := 0,1,2,3,4,5,6,7,8,9 The produton rules and the number of hoes assoated wth them are n Table 1. The symbol U- denotes an unary mnus operaton. Table 1: The number of avalable hoes for every produton rule rule no hoes 1 4 2 6 3 1 4 10 There are notable dfferenes when ompared wth [1]. We don t use two elements <pre_op> and <op>, but only one element <fn> for all funtons wth n arguments. There are not rules for parentheses; they are substtuted by a tree representaton of the funton. The element <num> and the rule <fn><num><expr> were added to over generatng numbers. The rule <fn><num><expr> s derved from the rule <fn><expr><expr>. Usng ths approah we an generate the expressons more easly. For example when one argument s a number, then +(4,x) an be produed, whh s equvalent to (4 + x) n an nfx notaton. The same result an be reeved f one of <expr> n the rule <fn><expr><expr> s substtuted wth <var> and then wth a number, but t would need more genes. There are not any rules wth parentheses beause all nformaton s nluded n the tree representaton of an ndvdual. Parentheses are automatally added durng the reaton of the text output. If n the GE soluton s not restrted anyhow, the searh spae s too large and an have nfnte number of solutons. For example the funton os(2x), an be expressed as os(x+x); os(x+x+1-1); os(x+x+x-x); os(x+x+0+0+0...) et. It s desred to lmt the number of elements n the expresson and the number of repettons of the same termnals and non-termnals. III. BACKWARD PROCESSING OF THE GE The hromosome s represented by a set of ntegers flled wth random values n the ntal populaton. Gene values are used durng hromosome translaton to dede whh termnal or nontermnal to pk from the set. When seletng a produton rule there are four possbltes, we use gene_value mod 4 to selet a rule. However the lst of varables has only one member (varable X) and gene_value mod 1 always returns 0. A gene s always read; no matter f a deson s to be made, ths approah makes some genes n the hromosome somehow redundant. Values of suh genes an be randomly reated, but genes must be present. The Fg. 1 shows the genotype-phenotype translaton sheme. The ndvdual body s shown as a lnear struture, but n fat t s stored as a one-way tree (hld objets have no lnks to parent objets). In the dagram we use abbrevated notatons for nontermnal symbols: f - <fn>, e - <expr>, n - <num>, v - <var>. IV. PROCESSING THE GRAMMAR Fg. 1 Relatons between genotype and phenotype n the GE wth bakward proessng [6] The proessng of the produton rules s done bakwards from the end to the begnnng of the rule (Fg. 2). Then produton rule <fn><expr1><expr2> s proessed as <expr2><expr1><fn>. We use <expr1> and <expr2> at ths pont to denote whh expresson wll be the frst argument of <fn>.
Proeedngs of the World Congress on Engneerng and Computer Sene 2009 Vol II WCECS 2009, Otober 20-22, 2009, San Franso, USA Fg. 2 Proposed bakward notaton of a funton tree struture Fg. 3. Crossover n GE wth bakward proessng [6] The man dfferene between <fn> and <expr> nontermnals s n the number of real objets they produe n the ndvdual s body. Nontermnal <fn> always generates one and only one termnal; on the ontrary <expr> generates an unknown number of nontermnal and termnal symbols. If the phenotype s represented as a tree struture then a produt of the <fn> nontermnal s the parent objet for handlng all objets generated by <expr> nontermnals ontaned n the same rule. Therefore the rule <fn><expr1><expr2> an be represented as a tree (Fg. 4). Fg. 4 Produton rule shown as a tree To selet a produton rule (seleton of a tree struture) only one gene s needed. To proess the seleted rule a number of n genes are needed and fnally to selet a spef nontermnal symbol agan one gene s needed. If the proessng s done bakwards the frst proessed termnals are leafs of the tree and the last proessed termnal n a rule s the root of a subtree. The very last termnal s the root of the whole tree. Note that n a forward proessng (<fn><expr1><expr2>) the frst proessed gene odes the rule, the seond gene odes the root of the subtree and the last are leafs. When usng the forward proessng and odng of the rules desrbed n [1] t s not possble to easly reover the tree struture from genotype. Ths s aused wth <expr> nontermnals usng an unknown number of suessve genes. The last proessed termnal beng just a leaf of the tree. The proposed bakward proessng s shown n Fg. 1. 4.1 Phenotype to genotype projeton Usng the proposed bakward proessng system the translaton to a phenotype subtree has a ertan sheme. It begns wth a produton rule (seletng the type of the subtree) and ends wth the root of the subtree (n our ase wth a funton). In the genotype ths means that one gene used to selet a produton rule s followed by n genes wth dfferent ontexts whh are followed by one gene used to translate <fn>. Therefore a gene odng a produton rule forms a par wth a gene odng termnal symbol for <fn> (root of the rule). Those genes an be marked when proessng the ndvdual. Ths s an example of a smple markng system: BB Begn blok (a gene odng a produton rule) IB Insde blok EB End blok (a gene odng a root of a subtree) The EB and BB marks are par marks and n the hromosome they defne a blok (Fg. 1G). Suh bloks an be nested but they don t overlap (the same way as parentheses). The IB mark s not a par mark, but t s always ontaned n a blok (IB marks are presently generated by <num> nontermnals). Gven a BB gene a orrespondng EB gene an be found usng a smple LIFO method. A blok of hromosome enlosed n a BB-EB gene par then odes a subtree of the phenotype. Suh blok s fully autonomous and an be exhanged wth any other blok or t an serve as ompletely new ndvdual. Only BB genes ode the tree of ndvdual s body, whle EB and IB genes ode the termnal symbols n the resultng phenotype. The BB genes ode the struture of the ndvdual, hangng ther values an ause hange of the appled produton rule. Therefore hange (e.g. by mutaton) n the value of a strutural gene may trgger hange of ontext of many, or all followng genes. Ths smple markng system ntrodues a phenotype feedbak to phenotype; however t doesn t affet the unversalty of the algorthm. It s not dependent on the used termnal or nontermnal symbols; t only requres the result
Proeedngs of the World Congress on Engneerng and Computer Sene 2009 Vol II WCECS 2009, Otober 20-22, 2009, San Franso, USA to be a tree struture. Usng ths system t s possble to ntrodue a progressve rossover and mutaton. 4.2 Crossover When usng grammatal evoluton the resultng phenotype oded by one gene depends on the value of the gene and on ts ontext. If a hromosome s rossed at random pont, t s very probable that the ontext of the genes n seond part wll hange. Ths way rossover auses destruton of the phenotype, beause the newly added parts ode dfferent phenotype than n the orgnal ndvdual. Ths behavour an be elmnated usng a blok markng system. Crossover s then performed as an exhange of bloks. The rossover s made always n an even number of genes, where the odd gene must be BB gene and even must be EB gene. Startng BB gene s presently hosen randomly; the frst gene s exluded beause t enapsulates (together wth the last used gene) the whole ndvdual. The operaton takes two parent hromosomes and the result s always two hld hromosomes. It s also possble to ombne the same ndvduals, whle the resultng hld hromosomes an be entrely dfferent. Gven the parents: 1) os( x + 2 ) + sn( x * 3 ) 2) os( x + 2 ) + sn( x * 3 ) The operaton an produe hldren: 3) os( sn( x * 3 ) + 2 ) + sn( x * 3 ) 4) os( x + 2 ) + x Ths rossover method works smlar to dret ombnng of phenotype trees, however ths method works purely on the hromosome. Therefore phenotype and genotype are stll separated. The result s a hromosome, whh wll generate an ndvdual wth a struture ombned from ts parents. Ths way we reeve the enodng of an ndvdual wthout bakward analyss of hs phenotype. To perform a rossover the phenotype has to be evaluated (to mark the genes), but t s nether used nor know n the rossover operaton (also t doesn t have to exst). 4.3 Mutaton Mutaton an be dvded nto mutaton of strutural (BB) genes and mutaton of other genes. Mutaton of one strutural gene an affet other genes by hangng ther ontext therefore strutural mutaton amount should be very low. On the other hand the amount of mutaton of other genes an be set very hgh and t an speed up searhng an approxmate soluton. Gven an ndvdual: sn( 2 + x ) + os( 3 * x ) and usng only mutaton of non-strutural genes, t s possble to get: os( 5 x ) * sn( 1 * x ) Therefore the struture doesn t hange, but we an get a lot of new ombnatons of termnal symbols. The dvded mutaton allows usng the benefts of hgh mutaton whle elmnatng the rsk of damagng the struture of an ndvdual. 4.4 Populaton model The system uses three populatons formng a smple tree struture (Fg. 5). There s a Master populaton and two slave populatons, whh smulate dfferent genders. The lnks among the populatons lead only one way - from bottom to top. Fg. 5 The populaton model 4.5 Female populaton When a new ndvdual s to be nserted n a populaton a hek s preformed whether t should be nserted. If a same or smlar ndvdual already exsts n the populaton then the new ndvdual s not nserted. In a female populaton every genotype and phenotype ours only one. The populaton mantans a very hgh dversty; therefore the mutaton operaton s not appled to ths populaton. Removng the ndvduals s based on two rterons. The frst rteron s the age of an ndvdual - length of stay n the populaton. The seond rteron s the ftness of an ndvdual. Usng the seond rteron a maxmum populaton sze s mantaned. Parents are hosen usng the tournament system seleton. 4.5 Ftness funton Around the searhed funton there s defned an equdstant area of a gven sze. Ftness of an ndvdual s phenotype s omputed as the number of ponts nsde ths area dvded by the number of all heked ponts (a value n <0,1>). Ths ftness funton forms a strong seleton pressure; therefore the system fnds an approxmate soluton very qukly. 4.6 Logal funton XOR as test funton Input values are two nteger numbers a and b; a, b 2< 0, 1 >. Output number s the value of logal funton XOR. Tranng data s a set of trples (a, b, ): P = {(0, 0, 0); (0, 1, 1); (1, 0, 1); (1, 1,0)}. Thus the tranng set represents the truth table of the XOR funton. The funton an be expressed usng _, ^, funtons: a + b = (a ^ b) _ ( a ^ b) = (a _ b) ^ ( a _ b) = (a _ b) ^ (a ^ b) The grammar was smplfed so that t does not ontan ondtonal statement and numer onstants, on the other hand three new termnals were added to generate funtons _, ^,. Thus the grammar generates representatons of the XOR funtons usng other logal funtons. funton xxor($a,$b) { $result = "no_value"; $result = ($result) (((~$b & ($a & ($a & ~$b))) & $a) (~$a & $b)); return $result; Number of generatons: 53
Proeedngs of the World Congress on Engneerng and Computer Sene 2009 Vol II WCECS 2009, Otober 20-22, 2009, San Franso, USA 9 &,5&8,7 237,0,=$7,21 The objetve s to generate the struture of a ombnatoral log rut performng as full bnary adder. Bnary adder an be represented wth the followng equatons: s y x = 1 x y x = + 1 + 1 y GRHV QRW VROYH WKH FRUH RI WKH SUREOHP RI JHQHUDWLQJ SDUDOOHO VWUXFWXUH $V PHQWLRQHG DERYH D UHZULWLQJ JUDPPDU XVHV D VHW RI QRQWHUPLQDO V\PEROV ZKLFK DUH EHLQJ UHZULWWHQ LQWR WHUPLQDO V\PEROV 7HUPLQDO V\PEROV UHSUHVHQW WKH DFWXDO EORFNV RI D ORJLF FLUFXLW $OO QRQ WHUPLQDOV QHHG WR EH WUDQVODWHG LQWR WHUPLQDOV EHIRUH WKH LQGLYLGXDO LV XVDEOH KHQFH RQFH D QRQWHUPLQDO LV WUDQVODWHG LW KDV WR EH UHPRYHG IURP WKH LQGLYLGXDOV ERG\ 7HUPLQDOV LQ WKH ERG\ DUH QR IXUWKHU SURFHVVHG The rut has three nputs x, y, -1 and two output varables s,, where s s the atual sum result, s arry bt, x, y are the atual bnary nputs and -1 s arry bt from prevous addton. The truth table of bnary adder has 16 output values, where equatons (1) and (2) eah defne eght of them. 0HWKRGV The parallel grammatal evoluton wth bakward proessng was used to solve the problem. The ore of the method s a genet algorthm extended wth several supportng algorthms. The man extenson added to the genet algorthm s a translaton layer nserted between the hromosome and the atual soluton whh s formed by a proessor of ontext-free grammar. The man advantage of suh extenson s the ablty to reate gener tree strutures and retreve them n reusable format. Grammatal evoluton wth bakward proessng an also take advantage of progressve rossover and mutaton operators. The system s nternally parallel as t onssts of three dfferent nteronneted populatons. Tab. 1 Produton rules The produton rules that are shown n Tab.1 are the most gener ones, allowng any syntatally orret soluton to be generated. Ths an however be adjusted n ase we would lke to use spef sets of gates. For example we an defne a rule so that frst nput of a gate s onneted to OR gate and the other s onneted to AND gate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g. 6 Example of a found soluton These prnples mean that t s only possble to generate tree strutures. Log ruts annot however be represented as a smple tree. A sgnal from s output an use the same gates as the output sgnal of. At ths pont we have hosen to aept ths lmtaton and test overall performane of grammatal evoluton appled to ths problem. Ths means that the generated solutons an never reah the optmal number of buldng bloks sne t s not possble to reuse exstng bloks. For the bnary adder rut t should be possble to reah the optmal tme-delay, although wth a more omplated rut. Also t s neessary to note that many of possble optmal solutons are ruled out smply beause reusng of gates s prohbted; ths makes fndng the optmal tme-delay soluton more dffult. However we are onfdent that ths lmtaton of the algorthm an be overome and t would be possble to generate truly parallel strutures. )LWQHVV IXQFWLRQ There are several optons how to ompute ftness and ompare dfferent hypotheses produed by genet algorthm. As the man rteron we hoose the number of mathes aganst the nput truth-table of ombnatons. However ths rteron alone s nsuffent, sne there are only 16 output values, there are also only 16 values of ftness. The number of possble solutons ether orret or norret s only lmted by arbtrary sze of the searh spae, whh an be adjusted by the length of the hromosome (n the experments set so that the effetve maxmum of elements n the struture s approxmately 70). A ftness value for n-th ndvdual s then defned as: n 8 n 8 n n n ( MS j, MC j, CS CC ) F =, j= 1 j= 1
Proeedngs of the World Congress on Engneerng and Computer Sene 2009 Vol II WCECS 2009, Otober 20-22, 2009, San Franso, USA where MS = 1 f the j-th output value of varable s mathes the desred value n truth table and smlarly MC = 1 f the value of varable s mathed. C s the ount of nodes n the generated struture. CS s the ount of nodes n the tree branh responsble for omputng output of varable s, and CS s the omplexty of the branh )LJ 7UHH UHSUHVHQWDWLRQ RI WKH VROXWLRQ RQ ILJXUH It s mportant to note that the arbtrary hromosome sze does not lmt the soluton sze relably, usng the rossover operator the algorthm an bypass the lmtaton and generate solutons wth up to approxmately 1000 elements. Therefore the ftness onsstng of only 16 values s napproprate sne t does not value lower omplexty of the soluton. The smplest soluton to ompute ftness value as a weghted sum of mathes and omplexty of an ndvdual ddn t ft our needs and led to premature onvergene. Ths problem was solved replang salar ftness value by a vetor of ftness. It allows evaluatng ndvduals wth a fner granularty then number of mathed values alone. To ompare the vetor ftness values a herarhal set of rules was used. Com plexty 200 180 160 140 120 100 80 60 40 20 Optmzaton Generatons 0 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 Fg. 8 Two step optmzaton proess One the algorthm s adapted to vetor ftness a vast number of defntons of ftness arse. As the qualtatve haratersts the number of mathes and soluton omplexty were hosen. Another possble hoe would be the maxmum tme-delay of the rut. In our ase where the output varable paths are not nteronneted the tmedelay s orrelated wth omplexty of eah path and thus makes no dfferene to onvergene of the algorthm. The hoe of omplexty above tme-delay was therefore drven only by mplementaton. Complexty of a soluton s defned as number of termnals n the strng representaton of the struture, ths s slghtly hgher than the atual number of gates. Fgure 7 shows the tree representaton of a soluton shown on Fg.6. The omplexty s defned as number of nodes n the tree. The frst approah was to smply use both the sum of mathes and omplexty of both tree root branhes. VI. CONCLUSION PGE has proved suessful for rut optmzaton. Parallel GEs wth herarhal struture an nrease the effeny and robustness of systems, and thus they an trak better optmal parameters n a hangng envronment. From the expermental sesson t an be onluded that modfed standard GEs wth only two sub-populatons an reate PGE muh better than lassal versons of GEs. The parallel grammatal evoluton an be used for the automat generaton of rut strutures. We are far from supposng that all dffultes are removed but frst results wth PGEs are very promsng. Although we are at early stages of experments t seems that t s possble to use parallel grammatal evoluton wth bakward proessng to generate ombnatoral log ruts. The grammatal algorthm an be outperformed wth algorthms, whh are desgned spefally for ths purpose. ACKNOWLEDGMENTS Ths work has been supported by Czeh Mnstry of Eduaton No: MSM 00216305529 Intellgent Systems n Automaton and *$ ý5 1R &RQWURO $OJRULWKP 'HVLJQ E\ 0HDQV RI Evolutonary Approah REFERENCES [1] O Nell, M., Ryan, C.: Grammatal Evoluton: Evolutonary Automat Programmng n an Arbtrary Language Kluwer Aadem Publshers 2003. [2] O Nell, M., Brabazon, A., Adley C.: The Automat Generaton of Programs for Classfaton Problems wth Grammatal Swarm, Proeedngs of CEC 2004, Portland, Oregon (2004) 104 110 [3] Pasezny, W., Suzuk. H., Sawa, H.: Chemal Genet Programmng Evoluton of Amno Ad Rewrtng Rules Used for Genotype-Phenotype Translaton, Proeedngs of CEC 2004, Portland, Oregon (2004) 1639-1646. [4] Ošmera, P., Šmoník, I, Roupe, J.: Multlevel dstrbuted genet algorthms. In Proeedngs of the Internatonal Conferene IEE/IEEE on Genet Algorthms, Sheffeld (1995) 505 510. [5] L Z., Halang W. A., Chen G.: Integraton of Fuzzy Log and Chaos Theory; paragraph: Osmera P.: Evoluton of Complexty, Sprnger, 2006 (ISBN: 3-540-26899-5) 527 578. [6] Ošmera, 33DQþHN 7 3LYRND 3 3DUDOOHO *UDPPDWLFDO Evoluton, Proeedngs of WCECS 2007, San Franso,24-26 Otober 2007, USA, 897-902 [7] RUKOVANSKÝ, I. Evoluton of Complex Systems. 8th Jont Conferene on Informaton Senes. Salt Lake Cty, Utah, USA. July 21-25, 2005.