SANTA FE TRAIL FOR ARTIFICIAL ANT BY MEANS OF ANALYTIC PROGRAMMING AND EVOLUTIONARY COMPUTATION

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1 SANTA FE TRAIL FOR ARTIFICIAL ANT BY MEANS OF ANALYTIC PROGRAMMING AND EVOLUTIONARY COMPUTATION ZUZANA OPLATKOVA, IVAN ZELINKA, ROMAN SENKERIK Faculty of Applied Infomatics Tomas Bata Univesity in Zlin Nad Stanemi 4511, Zlin Czech Republic {oplatkova, zelinka, Abstact: This pape deals with the usage of an altenative tool fo symbolic egession analytic pogamming which is able to solve vaious poblems fom the symbolic domain as well as genetic pogamming and gamatical evolution. The main tasks of analytic pogamming in this pape, is setting an optimal tajectoy fo a obot (atificial ant on Santa Fe tail). In this contibution main pinciples of analytic pogamming ae descibed and explained. In the second pat of the aticle is in detail descibed how analytic pogamming was used fo setting an optimal tajectoy fo an atificial ant accoding the use equiements. An ability to ceate so called pogams, as well as genetic pogamming o gammatical evolution do, is shown in that pat. In this contibution thee evolutionay algoithms wee used Self Oganizing Migating Algoithm, Diffeential Evolution and Simulated Annealing. The total numbe of simulations was 150 and esults show that the fist two used algoithms wee moe successful than not so obust Simulated Annealing. Keywods: Analytic Pogamming, symbolic egession, optimization, evolutionay algoithms 1. INTRODUCTION The tem symbolic egession epesents a pocess duing which measued data is fitted and a suitable mathematical fomula is obtained in an analytical way. This pocess is well known fo mathematicians. It is used when a mathematical model of unknown data is needed. Fo long time symbolic egession was a domain of humans but in few last decades computes have gone to foegound of inteest in this field. Fistly, the idea of symbolic egession done by means of compute has been poposed in Genetic Pogamming (GP) by John Koza [Koza 1998], [Koza 1999], [ The othe two appoaches ae Gammatical Evolution (GE) developed by Cono Ryan [O Neill, Ryan 2003], [O Sullivan, Ryan 2002], [ and hee descibed Analytic Pogamming (AP) designed in [Zelinka 2002], [Zelinka, Oplatkova 2003], [Zelinka, Oplatkova, Nolle 2004], [Zelinka, Oplatkova, Nolle 2005]. Genetic pogamming was the fist tool fo symbolic synthesis of so called pogams done by means of compute instead of humans. The main idea comes fom genetic algoithms (GA) [Davis 1996] which John Koza uses in his GP. The ability to solve vey difficult poblems was poved many times, and hence, GP today can be applied, e.g. to synthesize highly sophisticated electonic cicuits, obot tajectoy, biochemisty poblems, etc. [Koza 1999]. The othe tool is GE which was developed in last decade of 20th centuy by Cono Ryan. Gamatical evolution has one advantage compaed to GP and this is ability to use abitay pogamming language not only LISP as is in the case of the cannonical vesion of GP. In contast to othe evolutionay algoithms, GE was used with a few seach stategies with a binay epesentation of the populations [O Sullivan, Ryan 2002], as well as with othe algoithms like [O'Neill M., Babazon A. 2006], [O'Neill, Leahy, Babazon, 2006]. Othe inteesting investigations using symbolic egession wee caied out by Johnson [Johnson 2003] woking on Atificial Immune Systems and Pobabilistic Incemental Pogam Evolution (PIPE) [Salustowicz, Schmidhube, 1997] geneates functional pogams fom an adaptive pobability distibution ove all possible pogams. To Gammatical Evolution foeuns GADS which solves the appoach to gamma [Pateson, 2003], [Pateson, Livesey, 1996]. Othe example is atificial immune system pogamming which was evolved fo symbolic egession fom an evolutionay algoithm called atificial immune systems [Johnson, 2003]. Appoaches which diffe in epesentation and gamma ae descibed in gene IJSSST, Vol. 9, No. 3, Septembe ISSN: x online, pint

2 expession pogamming [Feeia, 2006], multiexpession pogamming [Oltean, Gosan, 2003], meta-modelling by symbolic egession and paeto Simulated Annealing [Stinsta, Rennen, Teeuwen, 2006]. To the goup of hybid appoaches, belongs mainly numeical methods connected with evolutionay systems, e.g. [Davidson, Savic, Waltes, 2003]. This contibution demonstates use of a method which is independent on compute platfom, pogamming language and can use any evolutionay algoithm (as demonstated by [Zelinka 2002], [Zelinka, Oplatkova 2003], [Zelinka, Oplatkova, Nolle 2004], [Zelinka, Oplatkova, Nolle 2005]) to find an optimal solution of equied task. 2. ANALYTIC PROGRAMMING 2.1. Desciption Basic pinciples of the AP wee developed in the Until that time mainly GP and GE existed. GP uses genetic algoithms while AP can be used with any evolutionay algoithm, independently on individual epesentation. To avoid any confusion based on the use of names accoding to the used algoithm, name - Analytic Pogamming was chosen, because AP stands fo synthesis of analytical solution by means of evolutionay algoithms [Zelinka 2002], [Zelinka, Oplatkova 2003], [Zelinka, Oplatkova, Nolle 2004], [Zelinka, Oplatkova, Nolle 2005]. AP was inspied, in geneal, by numeical methods in Hilbet spaces and by GP. Pinciples of AP [Zelinka, Oplatkova, Nolle 2005] ae somewhee between these two philosophies. Fom GP an idea of evolutionay ceation of symbolic solutions is taken into AP while fom Hilbet spaces an idea of synthesis of moe complicated functions fom elementay functions is adopted into AP. Analytic pogamming is based as well as GP on the set of functions, opeatos and so-called teminals, which ae usually constants o independent vaiables like fo example: vaiable. The stuctue of GFS is fucated i.e. it is ceated by subsets of functions accoding to the numbe of thei aguments. The content of GFS is dependent only on a use. Vaious functions and teminals can be mixed togethe. Fo example GFS all is a set of all functions, opeatos and teminals, GFS 3ag is a subset containing functions with only thee aguments, GFS 0ag epesents only teminals, etc. GFS all = {LechPhi, BetaRegulaized, +, -, /, *, cos, tan, sin, logγ, x, t, π} GFS 2ag = {+, -, /, *, cos, tan, sin, logγ, x, t, π} GFS 1ag = {cos, tan, sin, logγ, x, t, π} GFS 0ag = {x, t, π} Fig. 1: Geneal Function Set GFS This fucated stuctue is necessay the main pinciple of AP to be able to wok without any difficulties. The coe of AP is based on discete set handling, poposed in [Zelinka 2004a], [Lampinen, Zelinka 1999], see Fig. 2. Discete set handling (DSH) shows itself as univesal inteface between EA and symbolically solved poblem. That is why AP can be used almost by any evolutionay algoithm. Biefly said, DSH woks with intege indexes which epesent numeical o non-numeical expessions (opeatos, functions, ) in a discete set. This index then seves like a pointe into a discete set. Based on that, appopiate objects ae chosen fo cost function evaluation [Lampinen, Zelinka 1999]. Duing an evolutionay pocess only indexes ae used fo all evolutionay opeations. Objects fom the discete set ae used (by means of intege index) only in cost function, whee is accoding to intege index a symbolic stuctue is synthesized and consequently evaluated. functions: sin, tan, And, O opeatos: +, -, *, /, dt, teminals: 2.73, 3.14, t, All these mathematical objects ceate a set which AP ties to synthesize the appopiate solution fom. The set of mathematical objects ae functions, opeatos and so-called teminals (usually constants o independent vaiables). All these objects ae mixed togethe as is shown in Fig. 1 and consists of functions with diffeent numbe of aguments. The set is called fo aticle puposes geneal functional set GFS because the content of this set is vey Fig. 2: Discete set handling 2.2. Mapping method in AP The nested stuctue pesence in GFS is vitally impotant fo AP. It is used to avoid synthesis of IJSSST, Vol. 9, No. 3, Septembe ISSN: x online, pint

3 pathological pogams, i.e. pogams containing functions without aguments, etc. Pefomance of AP is, of couse, impoved if functions of GFS ae expetly chosen based on expeiencies with the poblem. The impotant pat of the AP is a sequence of mathematical opeations which ae used fo pogam synthesis. These opeations ae used to tansfom an individual of a population into a suitable pogam. Mathematically said, it is mapping fom an individual domain into a pogam domain. This mapping consists of two main pats. The fist pat is called discete set handling (DSH) and the second one ae secuity pocedues which do not allow to synthesize pathological pogams. Discete set handling poposed in [Zelinka 2004a], [Lampinen, Zelinka 1999] is used fo ceating intege indeces which ae used as values fo the evolutionay pocess. The method of DSH, when used, allows to handle abitay objects including nonnumeical objects like linguistic tems {hot, cold, dak,...}, logic tems (Tue, False) o othe use defined functions. In the AP, DSH is used to map an individual into GFS and togethe with secuity pocedues (SP) ceates above mentioned mapping which tansfoms abitay individual into a pogam. Individuals in the population consist of intege paametes, i.e. an individual is an intege index pointing into GFS. Analytic pogamming is basically a seies of function mapping. In Fig. 3 an atificial example is demonstated how a final function is ceated fom an intege individual. and because thee is one unused pointe 11 tan should be mapped on Mod (modulo) which is on 11 th position in GFS all. But Modulo needs othe aguments and individual has no othe indexes. Theefoe suboutines ae used which ae descibed bette in next paagaph and instead of modulo is used vaiable t fom GFS 0ag. To avoid synthesis of pathological functions a few secuity ticks is used in AP. The fist one is that GFS consists of subsets containing functions with the same numbe of aguments. Existence of this fucated stuctue is used in the special secuity suboutine which is measuing how fa the end of individual is and accoding to these objects fom diffeent subsets ae selected to avoid pathological function synthesis. Peciselly if moe aguments ae desied than possible (the end of the individual is nea) function will be eplaced by anothe function with the same index pointe fom subset with lowe numbe of aguments. Fo example, it may happen that the last agument fo one agument function will not be a teminal (zeo-agument function). If pointe is bigge than length of subset, i.e. pointe is 5 and is used GFS 0, then element is selected accoding to element = pointe_value mod numbe_of_elements_in_gfs 0. In this example case selected element would be vaiable t (see GFS 0 in Fig. 1). GFS need not to be constucted only fom clea mathematical functions as is demonstated but also othe use-defined functions such as logical functions, functions which epesent elements of electical cicuits o obot movement commands can be used Cossove, Mutations and Othe Evolutionay Opeations Fig. 3: Main pinciples of AP Numbe 1 in the individual fist position means that the opeato + fom GFS all is used. Because the opeato + must have at least two aguments next two index pointes 6 (sin fom GFS all ) and 7 (cos fom GFS all ) ae dedicated to this opeato as its aguments. Both functions, sin and cos, ae oneagument functions so the next unused pointes 8 (Tan fom GFS all ) and 9 (t fom GFS all ) ae dedicated to sin and cos function. Because vaiable t is used as an agument of cos, this pat of esulting function is closed (t is zeo-agument) in its AP development. One-agument function tan emains Duing evolution of a population a diffeent opeatos ae used, such as cossove and mutation. In compaison with GP o GE evolutionay opeatos like mutation, cossove, tounament, selection ae fully in the competence of used evolutionay algoithm. Analytic pogamming does not contain them in any point of view of its intenal stuctue. Analytic pogamming is ceated like a supestuctue of EAs fo symbolic egession independent on thei algoithmic stuctue. Opeations used in evolutionay algoithms ae not influenced by AP and vice vesa. Fo example if Diffeential Evolution (DE) is used fo symbolic egession in AP then all evolutionay opeations ae done accoding to the DE ules. IJSSST, Vol. 9, No. 3, Septembe ISSN: x online, pint

4 2.4. Vesions of AP Today, AP exists in thee vesions. In all thee vesions the same sets of functions, teminals, etc. as Koza use in GP Koza [Koza 1998], [Koza 1999], [ ae necessay fo the pogam synthesis. The second vesion (AP meta, lets call the fist vesion AP basic ) is modified in the sense of constant estimation. Fo example, when Koza uses in so called sextic poblem [Koza 1999] andomly geneated constants, AP uses only one, called K, which is inseted into the fomula at vaious places by evolutionay pocessing. The function can look like as in (1). (1) x " K # K When the pogam is synthesized, then all K ae indexed so that K 1, K 2,, K n, ae obtained in fomula (2), and then all K n ae estimated by the second evolutionay algoithm and esult is in (3). (2) (3) Because EA woks unde EA (i.e. EA maste pogam K indexing EA slave estimation of K n ), this vesion is called AP with metaevolution - AP meta. As this vesion was quite time consuming, anothe modification of AP meta was done extending the second vesion by estimation of K. It is done by suitable methods of nonlinea fitting (AP nf ). This method has shown the most pomising pefomance when unknown constants ae pesent Secuity pocedues x " K 1 # K 2 x "1.289 # "112 Secuity pocedues (SP) ae used in the AP to avoid vaious citical situations as well as in GP. In the case of AP secuity pocedues wee not developed fo AP puposes afte all, but they ae mostly integated pat of AP. Howeve sometimes they have to be defined as a pat of cost function, based on kind of situation (fo example situation 2, 3 and 4, see below). Citical situations ae like: 1. pathological function (without aguments, selflooped...) 2. functions with imaginay o eal pat (if not expected)) 3. infinity in functions (dividing by 0,...) 4. fozen functions (an extemely long time to get a cost value - hs...) 5. etc... Simply as a SP can be egaded hee mapping fom an intege individual to the pogam whee is checked how fa the end of the individual is and based on this infomation a sequence of mapping is ediected into a subset with lowe numbe of aguments. This satisfies that no pathological function will be geneated. Othe activities of SP ae integated pat of the cost function to satisfy items 2-4, etc Similaities and Diffeences Because AP was patly inspied by GP, then between AP, GP and GE ae some diffeences as well as some logical similaities. A few of the most impotant ones ae: Similaity Synthesized pogams: AP as well as GP and GE is able to do symbolic egession in a geneal point of view. It means that output of AP is accoding to all impotant simulations [Zelinka 2002], [Zelinka, Oplatkova 2003], [Zelinka, Oplatkova, Nolle 2004], [Zelinka, Oplatkova, Nolle 2005] simila to pogams fom GP and GE (see Functional set: AP basic opeates in pinciple on the same set of teminals and functions as GP o GE. Diffeences AP meta o AP nf use univesal constant K (diffeence) which is indexed afte pogam synthesis. Individual coding: coding of an individual is diffeent. Analytic pogamming uses an intege index instead of diect epesentation as in canonical GP. Gammatical evolution uses binay epesentation of an individual, which is consequently conveted into integes fo mapping into pogams by means of BNF [O Neill, Ryan 2003]. Individual mapping: AP uses discete set handling, [Zelinka, Oplatkova, Nolle 2005] while GP in its fundamental fom uses diect epesentation in Lisp [Koza 1998] and GE uses gamma - Backus-Nau fom (BNF) [O Neill, Ryan 2003]. Constant handling: GP uses a andomly geneated subset of numbes constants, GE utilizes use detemined constants and AP uses only one constant K fo AP meta and AP nf, which is estimated by othe EA o by nonlinea fitting. Secuity pocedues: to guaantee synthesis of non-pathological functions, pocedues IJSSST, Vol. 9, No. 3, Septembe ISSN: x online, pint

5 ae used in AP which ediect the flow of mapping into subsets of a whole set of functions and teminals accoding to the distance of the end of the individual. If a pathological function is synthesized in GP, then synthesis is epeated. In the case of GE, when the end of an individual is eached, then mapping continues fom the individual beginning, which is not the case of AP. It is designed so that a nonpathological pogam is syntesized befoe the end of the individual is eached (maximally when the end is eached) Selected Solved Poblems Duing AP development and eseach simulations, a lot of vaious kind of pogams has been synthesized. In (3) a mathematical fomula is shown to demonstate complexity of synthesized fomulas, which wee andomly geneated amongst 1000 fomulas to check if the final stuctue is fee of pathologies i.e. if all functions has the ight numbe of aguments, etc. In this case no attention was paid on mathematical easonability of the following test pogams based on clea mathematical functions. In the pevious text, a diffeent appoach to the symbolic egession called analytic pogamming was descibed. Based on its esults and stuctue, it can be stated that AP seems to be a univesal candidate fo symbolic egession by means of diffeent seach stategies. Poblems on which AP was utilized wee selected fom test and theoy poblems domain as well as fom eal life poblems. The selected issues ae following: Random synthesis of function fom GFS, 1000 times epeated. The aim of this simulation was to check if pathological function can be geneated by AP. In this simulation andomly geneated individuals wee ceated and consequently tansfomed into pogams and checked fo thei intenal stuctue. No pathological pogam was identified [Zelinka 2002]. sin(t) appoximation was epeated 100 times. Hee AP was used to synthesize the pogam function sin(x) fitting [Zelinka 2002]. cos(t) + sin(t) appoximation was epeated 100 times, the same like in the pevious example. Main aim was again fitting of dataset geneated by a given fomula [Zelinka 2002]. Solving of odinay diffeential equations (ODE): u (t) = cos(t), u(0) = 1, u(π) = -1, u (0) = 0, u (π) = 0, was epeated 100 times, in that case AP was looking fo suitable function, which would solve this case of ODE [Zelinka 2002]. Solving of ODE: ((4 + x)u (x)) + 600u(x) = 5000(x-x2), u(0)=0, u(1)=0, u (0)=0, u (1)=0, was epeated 5 times (due to longe time of simulation in the Mathematica envionment). Again as in the pevious case, AP was used to synthesize a suitable function solution of this kind of ODE. This ODE was used fom and epesents a civil engineeing poblem in eality [Zelinka 2002]. Boolean even and symmety poblems accoding to the [Koza 1999] fo compaative easons [Zelinka, Oplatkova 2003], [Zelinka, Oplatkova, Nolle 2004]. Sextic and Quintic poblems [Zelinka, Oplatkova 2003]. Simple neual netwok synthesis by means of AP a simple few layeed NN synthesis was tested by AP [Zelinka, Vaacha, Oplatkova, 2006] Optimal obot tajectoy estimation [Oplatkova 2005], [Oplatkova, Zelinka 2006]. Such elementay objects ae usually simple mathematical opeatos (+, -, *, ), simple functions (sin, cos, And, No, ), use-defined functions (simple commands fo obots MoveLeft, TunRight, ), etc. Output of symbolic egession is a moe complex object (fomula, function, command, ), solving a given poblem like data fitting of so called Sextic and Quintic poblem descibed by Eq. (4), [Koza 1999], [Zelinka, Oplatkova 2003], andomly synthesized function Eq. (5) [Zelinka, Oplatkova 2003], Boolean poblems of paity and symmety solution (basically logical cicuits synthesis) Eq. (6) [Koza 1999], [Zelinka, Oplatkova, Nolle 2005] o synthesis of quite complex obot contol command Eq. (7) [Koza 1998], [Oplatkova 2005]. Equations (4) - (7) mentioned hee ae just only a few samples numeous epeated expeiments done by AP and ae used to demonstate how complex stuctues can be poduced by symbolic egession in geneal sense fo diffeent poblems. ( ) " x 2 K % x 3 K 1 + $ # K 4 ( K 5 + K 6 )' ( )1+ K 2 + 2x )x ) K 7 & " 1 % t$ ' # log( t) & sec (1 ( 1.28) log sec(1 ( 1.28) ( ( )) ( sinh( sec( cos( 1) ))) (4) (5) IJSSST, Vol. 9, No. 3, Septembe ISSN: x online, pint

6 No[(Nand[Nand[B B, B && A], B]) && C && A && B, No[(!C && B && A!A && C && B!C &&!B &&!A) && (!C && B && A!A && C && B!C &&!B &&!A) A && (!C && B && A!A && C && B!C &&!B &&!A), (C!C && B && A!A && C && B!C &&!B &&!A) && A]] Pog2[ Pog3[ Move, Right, IfFoodAhead[ Left, Right]], IfFoodAhead[ IfFoodAhead[ Left, Right], Pog2[ IfFoodAhead[ IfFoodAhead[ IfFoodAhead[ Left, Right], Right], Right], IfFoodAhead[ Pog2[ Move, Move], Right]]]] (6) (7) colou. The gey fields epesent obstacles (fields without food on the oad) fo the ant. If thee would not be these holes the ant could go diectly though the way. It would be enough to go and see befoe ant if thee is food. If yes ant would go staight and eat the bait. If not it would tun aound and see whee is food and the cycle would epeat till the ant would eat the last bait. In the eal wold obots have obstacles in thei moving. Theefoe also in this case such appoach was chosen. The fist poblem which ant has to ovecome is the simple hole (position {8, 27} in [Koza 1999]). Second one is two holes in the line (positions {13,16} and {13,17}), o thee holes ({17,15}, {17,16}, {17,17}). Next poblem is holes in the cones one (position {13,8}, two ({1,8},{2,8}) and thee holes ({17,15}, {17,16}, {17,17}). Remaining text in this aticle is investigation on evolutionay synthesis of obot commands, which is well known in genetic pogamming like Santa Fe tail of atificial ant. 3. PROBLEM DESIGN 3.1. Santa Fe Desciption The Santa Fe tail, demonstated in Fig. 4, was chosen fom [Koza 1999] to make a compaative study with the same poblem which was solved by Koza in Genetic Pogamming Set of functions The set of functions used fo movements of the ant is following. As a set of vaiables GFS 0, i. e. in the case of this aticle functions, which povide moving of an ant, without any agument which could be add duing the pocess of evolution. The set consist of GFS 0 = {Left, Right, Move}, whee GFS 0 a set of vaiables and teminals, zeo agumented functions GFS 0ag Left function fo tuning aound in the anticlockwise diection Right function fo tuning aound in the clockwise diection Move function fo moving staight and if a bait is in the field whee the ant is moved, it is eaten. This set of functions is not enough to make successfully a desied task. Moe functions ae necessay. Then a GFS 2 and GFS 3 wee set up. GFS 2 = {IfFoodAhead, Pog2} GFS 3 = {Pog3} Fig. 4: Santa Fe tail The aim of the task is that an atificial ant should go though defined tail and eat all food what is thee. The SantaFe tail is defined as a 32 x 31 fields whee food is set out. In Fig. 4 a black field is food fo the ant. The gey one is basically the same as a white field but fo cleaness was used the gey Whee the numbe in GFS means the aity of the functions inside, i.e. numbe of aguments which ae needed to be evaluated coectly. Aguments ae added to those functions duing evolution pocess see above Desciption of AP. IfFoodAhead is a decision function the ant contols the field in font of it and if thee is food, the function in the field fo tuth agument is executed; othewise function in false position is pefomed. Pog2 and Pog3 ae the same function in the pinciple. They do 2 o 3 functions in the same time. These two functions wee oiginally defined also in IJSSST, Vol. 9, No. 3, Septembe ISSN: x online, pint

7 Koza s appoach but in AP it is necessay because of stuctue of geneating the pogam Fitness function The aim of the ant is to eat all food on the way. Thee ae 89 baits. This is so called aw fitness. And the value of cost function (8) is calculated as a diffeence between aw fitness and a numbe of baits eaten by an ant [Koza 1999] which went though the gid accoding to just geneated way. CV = 89" Numbe _ of _ Food (8) Numbe_of_Food - numbe of eaten baits by an ant accoding to synthesized way The aim is to find such fomula whose cost value is equal zeo. To obtain an appopiate solution two constaints should be set up into a cost function. One is a limitation concened to numbe of steps. It is not desied ant to go field by field in the gid. A equiement to the fastest way and the most effective is desied. Then a limit of steps was equal to 600. Accoding to oiginal assignment 400 steps should be sufficient. But as [Maik et al. 2004] says Koza s optimal solution was as in (9). But as simple solution showed 545 steps ae necessay fo an ant to eat all food in the Santa Fe tail. IfFoodAhead[Move, Pog3[Left, Pog2[IfFoodAhead[Move, Right], Pog2[Right, Pog2[Left, Right]]], Pog2[IfFoodAhead[Move, Left], Move]]] (9) Functionality of the equation (9) can be descibed by following. If bait is in font of the ant it moves on the field and eats the food. If thee is nothing it does following 3 commands. If food is in font of the ant it moves and eats the food, if not it tuns twice ight. Next Pog2(Left, Right) is not necessay thee, this is the eason why all pogam takes 545 steps instead of 404 in the case of no Pog2(Left,Right). Then next contol of food in font of ant is again, if yes ant moves and eats the food. If not it tuns left to the oiginal diection as it was at the beginning of the pogam. If the cycle is somewhee inteupt, e.g. in the case of tuth in the fist function IfFoodAhead, the cycle is epeated still fom the beginning until all food is not eaten o constained steps ae not eached. The second constaint could be concened to the length of the list of commands fo an ant. The moe longe list could cause the moe steps to each all food is eaten. In this peliminay study this constaint was not set up. But in futhe studies a penalization concened to this constaint will be suely used. 4. USED EVOLUTIONARY ALGORITHMS In this pape SelfOganizing Migating Algoithm (SOMA), Diffeential Evolution (DE) and Simulated Annealing (SA) wee used as an evolutionay algoithm. Fo detailed infomation see [Zelinka 2004a], [Pice, Ston, Lampinen 2005], [Kikpatick, Gelatt, Vecchi 1983] Diffeential Evolution (DE) Diffeential Evolution is a population-based optimization method that woks on eal-numbe coded individuals [Pice, Ston, Lampinen 2005]. Fo each individual x i,g in the cuent geneation G, " x i,g by adding DE geneates a new tial individual the weighted diffeence between two andomly x selected individuals 1,G and x 2,Gto a thid x 3,G. The esulting andomly selected individual individual x " i,gis cossed-ove with the oiginal x individual i,g. The fitness of the esulting u i,g +1, individual, efeed to as petubated vecto x is then compaed with the fitness of i,g. If the u fitness of i,g +1 is geate than the fitness of x i,g is eplaced with u i,g +1, othewise x i,g +1. in the population as x i,g, x i,g emains Diffeential Evolution is obust, fast, and effective with global optimization ability. It does not equie that the objective function is diffeentiable, and it woks with noisy, epistatic and time-dependent objective functions Self-Oganizing Migating Algoithm (SOMA) SOMA is a stochastic optimization algoithm that is modelled on the social behaviou of coopeating individuals [Zelinka 2004a] It was chosen because it has been poven that the algoithm has the ability to convege towads the global optimum [Zelinka 2004a]. SOMA woks on a population of candidate solutions in loops called migation loops. The population is initialized andomly distibuted ove the seach space at the beginning of the seach. In each loop, the population is evaluated and the solution with the highest fitness becomes the leade L. Apat fom the leade, in one migation loop, all individuals will tavese the input space in the diection of the leade. Mutation, the andom petubation of individuals, is an impotant opeation fo evolutionay stategies (ES). It ensues the divesity amongst the individuals and it also IJSSST, Vol. 9, No. 3, Septembe ISSN: x online, pint

8 povides the means to estoe lost infomation in a population. Mutation is diffeent in SOMA compaed with othe ES stategies. SOMA uses a paamete called PRT to achieve petubation. This paamete has the same effect fo SOMA as mutation has fo GA. The novelty of this appoach is that the PRT Vecto is ceated befoe an individual stats its jouney ove the seach space. The PRT Vecto defines the final movement of an active individual in seach space. The andomly geneated binay petubation vecto contols the allowed dimensions fo an individual. If an element of the petubation vecto is set to zeo, then the individual is not allowed to change its position in the coesponding dimension. An individual will tavel a cetain distance (called the PathLength) towads the leade in n steps of defined length. If the PathLength is chosen to be geate than one, then the individual will oveshoot the leade. This path is petubed andomly Simulated Annealing (SA) Simulated Annealing is one of olde algoithms compaed to SOMA and DE. It was intoduced by Kikpatick et al. fo the fist time [Kikpatick, Gelatt, Vecchi 1983]. An inspiation fo developing this algoithm was annealing of metal. In the pocess metal is heated up to tempeatue nea the melting point and then it is cooled vey slowly. The pupose is to eliminate unstable paticles. In othe wods, paticles ae moved towads an optimum enegy state. Metal is then in moe unifom cystalline stuctue. This appoach was used in the case of simulated annealing including tems. It stat off fom a andomly selected point. Then a cetain (depends on use) numbe of points is geneated in the neighbouhood. The point with the best cost value is selected to be the middle of new neighbouhood (stat point fo a new loop). But it is possible to accept also wose value of cost function. The acceptance is based on a pobability which deceases with numbe of iteations. In the case that the best cost value is in the stat point this one is chosen fo the next loop. 5. EXPERIMENTAL RESULTS The main idea is to show that SOMA, DE and SA ae able to solve such poblems of symbolic egession setting a tajectoy - unde Analytic Pogamming. 50 simulations wee caied out fo each algoithm, i.e. 150 simulations in total. SOMA and DE have almost all simulations with positive esults; only one case in both algoithms did not each the exteme. SA was not so successful, only 14 positive esults. To show that AP is able to wok with abitay evolutionay algoithms we suppose to cay simulations out with Genetic Algoithms (GA) and othe algoithms and also paallel computing is intended in this field. Data fom all simulations wee pocessed and vizualized in [Oplatkova 2005], [Oplatkova, Zelinka 2006]. In simulations done fo puposes of this aticle following setting was used to un SOMA, DE and SA accoding to Tab. 1 - Tab. 3. Tab. 1: Setting of SOMA Paamete PathLength 3 Value Step 0.22 PRT 0.21 PopSize 200 Migations 50 MinDiv -0.1 Individual Length 50 Tab. 2: Setting of DE Paamete Value NP 200 F 0.8 CR 0.2 Geneations 700 Individual Length 50 Tab. 3: Setting of SA Paamete Value T T min α MaxIte MaxIteTemp 93 Individual Length 50 Fistly, the esults show values of cost function evaluations. This paamete shows good pefomance of Analytic Pogamming. As you can see in Tab. 4 the lowest numbe of cost function evaluations equals to fo SA and 3396 fo SOMA. DE was also not so fa with its 4030 cost function evaluations. IJSSST, Vol. 9, No. 3, Septembe ISSN: x online, pint

9 The Fig. 5 uses data fom Tab. 4 and shows it in a gaphical way whee the diamond means the aveage value. As can be seen, SA had the lowest aveage value. But this might be caused by only 14 cases which wee included in the chat while SOMA and DE had 49 positive cases. Tab. 4: Cost function evaluation fo SOMA, DE and SA Cost Function Evaluation SOMA DE SA Minimum Maximum Aveage Fig. 7: Histogam of DE algoithm Fig. 8: Histogam of SA algoithm Fig. 5: Gaphical epesentation of minimal, maximal and aveage values of cost function evaluation fo SOMA, DE and SA Anothe ceation of histogams can be made fom the point of view of numbe of cases (axe y) which appeaed in some inteval of cost function values (axe x). This appoach can be seen in Fig. 9 - Fig. 11. Second indicato depicts histogam of successful hits and the numbe of cost function evaluations fo each hit Fig. 6 - Fig. 8. Negative esults ae not included. Fig. 9: Histogam of SOMA algoithm numbe of cases in specific intevals of cost function values Fig. 6: Histogam of SOMA algoithm IJSSST, Vol. 9, No. 3, Septembe ISSN: x online, pint

10 The following pictue depicts that the ant went though all the fields (Fig. 12); the white X shows fields which wee attended by the ant. The notation (10) contains set of ules fo the ant how to go successfully though the tail. In the notation (11) the whole desciption of the oute can be seen whee Ea, So, We, No means east, south, west and noth (which cadinal point the ant is tuned into). The numbes in backets ae positions on the gid. Fig. 10: Histogam of DE algoithm numbe of cases in specific intevals of cost function values Fig. 11: Histogam of SA algoithm numbe of cases in specific intevals of cost function values Next point, which we wee inteested in, was a numbe of commands fo the ant and numbe of steps equied to eat all baits (Tab. 5, Tab. 6). In the Tab. 6 you can see that DE found a oute which is ovecome in the least numbe of steps. Soted lists of pais commands and steps ae seen fo all 3 algoithms in Tab. 5. As it is shown, it can be stated that the smallest numbe of commands does not have to cause the smallest numbe of steps. And vice vesa, that small numbe of steps does not mean the small set of commands. Tab. 5: Numbe of commands Numbe of leaves (commands) SOMA DE SA Minimum Maximum Aveage Tab. 6: Numbe of steps Numbe of steps SOMA DE SA Minimum Maximum Aveage Fig. 12: Santa Fe Tail ovecome by ant found by DE IfFoodAhead[ Move, IfFoodAhead[ Move, Pog2[ Pog2[ Right, IfFoodAhead[ Pog2[ IfFoodAhead[ IfFoodAhead[ Move, Move], Move], Move], Pog3[ IfFoodAhead[ Move, IfFoodAhead[ Pog3[ Right, Right, Pog2[ Left, Pog2[ IfFoodAhead[ Pog2[ Pog2[ Left, Move], Right], IfFoodAhead[ Move, Left]], Pog2[ IfFoodAhead[ Move, Move], Pog2[ IfFoodAhead[ Move, Right], Right]]]]], Left]], Left, IfFoodAhead[ Move, Right]]]], Move]]] {{32, 1}, {32, 2}, {32, 3}, {32, 4}, {So}, {31, 4}, {30, 4}, {29, 4}, {28, 4}, {27, 4}, {We}, {So}, {Ea}, {27, 5}, {27, 6}, {27, 7}, {So}, { Ea}, {No}, {Ea}, {27, 8}, {27, 9}, {27, 10}, {27, 11}, {27, 12}, {27, 13}, {So}, {26, 13}, {25, 13}, {24, 13}, {23, 13}, {We}, {So}, {Ea}, {So}, {22, 13}, {21, 13}, {20, (10) IJSSST, Vol. 9, No. 3, Septembe ISSN: x online, pint

11 13}, {19, 13}, {18, 13}, {We}, {So}, {Ea}, {So}, {17, 13}, { We}, {So}, {Ea}, {So}, {16, 13}, {15, 13}, {14, 13}, {13, 13}, {12, 13}, {11, 13}, {10, 13}, {9, 13}, {We}, {So}, {Ea}, { So}, {8, 13}, {We}, {8, 12}, {8, 11}, {8,10}, {8, 9}, {8, 8}, {No}, {We}, {So}, {We}, {8, 7}, {No}, {We}, { So}, {We}, {8, 6}, {8, 5}, {8, 4}, {No}, {We}, {So}, {We}, {8, 3}, {No}, {We}, {So}, {We}, {8, 2}, {No}, {We}, {So}, {7, 2}, {6, 2}, {5, 2}, {4, 2}, {We}, {So}, {Ea}, {So}, {3, 2}, {We}, {So}, { Ea}, {So}, {2, 2}, {We}, {So}, {Ea}, {2, 3}, {2, 4}, {2, 5}, { 2, 6}, {So}, {Ea}, {No}, {Ea}, {2, 7}, {So}, {Ea}, {No}, {Ea}, {2, 8}, {So}, {Ea}, {No}, {3, 8}, {4, 8}, {Ea}, {No}, {We}, {No}, { 5, 8}, {Ea}, {5, 9}, {5, 10}, {5,11}, {5, 12}, {5, 13}, {5, 14}, {5, 15}, {So}, {Ea}, {No}, {Ea}, { 5, 16}, {So}, {Ea}, {No}, {Ea}, {5, 17}, {So}, {Ea}, {No}, {6, 17}, {7, 17}, {8, 17}, {Ea}, {No}, {We}, {No}, {9, 17}, {Ea}, {No}, {We}, {No}, {10, 17}, {11, 17}, {12, 17}, {13, 17}, {14, 17}, {Ea}, {No}, {We}, {No}, {15,17}, {Ea}, {No}, {We}, {No}, {16, 17}, {Ea}, {No}, {We}, {No}, {17, 17}, {Ea}, {17, 18}, {17, 19}, {17, 20}, {So}, {Ea}, {No}, {Ea}, { 17, 21}, {So}, {Ea}, {No}, {18, 21}, {19, 21}, {Ea}, {No}, {We}, {No}, {20, 21}, {Ea}, {No}, {We}, {No}, {21, 21}, {22, 21}, {23, 21}, {24, 21}, {25, 21}, {Ea}, {No}, {We}, {No}, {26, 21}, {Ea}, {No}, {We}, {No}, {27, 21}, {Ea}, {27, 22}, {27, 23}, {27, 24}, {So}, {Ea}, {No}, {Ea}, {27, 25}, {So}, {Ea}, {No}, {28, 25}, {29, 25}, {Ea}, {No}, {We}, {No}, { 30, 25}, {Ea}, {30, 26}, {30, 27}, {30, 28}, {So}, {Ea}, {No}, {Ea}, {30, 29}, {So}, {Ea}, {No}, {Ea}, {30, 30}, {So}, {29, 30}, {28, 30}, {27, 30}, {26, 30}, {We}, {So}, {Ea}, {So}, {25, 30}, {We}, {So}, {Ea}, {So}, {24, 30}, {23, 30}, {We}, {So}, {Ea}, {So}, {22, 30}, {We}, { So}, {Ea}, {So}, {21, 30}, {20, 30}, {We}, {So}, {Ea}, {So}, {19, 30},{We}, {So}, {Ea}, {So}, {18, 30}, {We}, {18, 29}, {18, 28}, {18, 27}, {No}, {We}, {So}, {We}, {18, 26}, {No}, {We}, {So}, {We}, {18, 25}, {No}, {We}, {So}, {We}, {18, 24}, {No}, {We}, {So}, {17, 24}, {16, 24}, {We}, {So}, {Ea}, {So}, {15, 24}, {We}, {So}, {Ea}, {So}, {14, 24}, {We}, {So}, {Ea}, {14, 25}, {14, 26}, {So}, {Ea}, {No}, {Ea}, {14, 27}, {So}, {Ea}, { No}, {Ea}, {14, 28}, (11) {So}, {13, 28}, {12, 28}, {11, 28}, {We}, {So}, {Ea}, {So}, {10, 28}, {We}, {10, 27}, {10, 26}, {10, 25}, {No}, {We}, {So}, {We}, {10, 24}, {No}, {We}, {So}, {9, 24}, {8, 24}} 6. CONCLUSIONS This contibution deals with an altenative algoithm fo symbolic egession. This study shows that this algoithm is suitable not only fo mathematical egession but also fo setting of optimal tajectoy fo atificial ant which can be eplaced by obots in eal wold, in industy. To compae with standad GP it can be stated on the basic esults above that AP can solve this kind of poblems in shote times as cost function evaluations ae counted. The aim of this study was not to show that AP is bette o wose than GP (o GE when compaed) but that AP is also a poweful tool fo symbolic egession with suppot of diffeent evolutionay algoithms. The main object of the pape was to show that symbolic egession done by AP is able to solve also cases whee linguistic tems as fo example commands fo movement of atificial ant o obots in eal wold ae. Hee simulations fo 3 algoithms SOMA, DE and SA wee caied out. As figues showed SOMA and DE wee moe successful in positive esults than SA was. This poved that a good pefomance of AP depends on a choice of suitable obust and poweful evolutionay algoithms. Duing simulations caied in this poblem wee eached following esults: 50 simulations fo each algoithm means 150 in total fo all 3 algoithms. Positive esults - 49 fom 50 simulations fo SOMA - 49 fom 50 fo DE - and 14 fom 50 fo SA which accomplished the equied tasks thus Analytic Pogamming is able to solve such kind of poblems in symbolic egession. This esult also says that Simulated Annealing is not so poweful tool as othe two evolutionay algoithms ae. It is supposed that the cost function is vey complicated with quite a lot of local optima and theefoe the Simulated Annealing was not so successful as SOMA o DE wee. IJSSST, Vol. 9, No. 3, Septembe ISSN: x online, pint

12 Solutions which fulfil conditions which wee laid down by John Koza [Koza 1998] concened to numbe of steps, wee found (2 by SOMA and 3 by DE). It means 5 solutions wee successful unde the 400 steps. Moeove, 17 (SOMA) + 20 (DE) + 6 (SA), in total 43 fom 150 wee successful unde the 545 steps which was intoduced by John Koza [Koza 1998], [Maik et al., 2004] as an optimal one. Futue eseach is key activity in this field. The following steps ae to finished simulations with GA and othe evolutionay algoithms and to ty some othe class of poblems to show that Analytic Pogamming is poweful tool as Genetic Pogamming o Gammatical Evolution ae. 7. ACKNOWLEDGEMENTS This wok was suppoted by the gant NO. MSM of the Ministy of Education of the Czech Republic and by gants of Gant Agency of Czech Republic GACR 102/06/ REFERENCES Davidson J.W., Savic D.A., Waltes G.A., 2003, Symbolic and numeical egession: expeiments and applications, Infomatics and Compute Science An intenational Jounal, Vol. 150, Elsevie, USA, 2003, pages , ISSN: Davis L. 1996, Handbook of Genetic Algoithms Intenational Thomson Compute Pess, ISBN , 1996 Feeia C., 2006, Gene Expession Pogamming: Mathematical Modeling by an Atificial Intelligence, Spinge, 2006, ISBN: Johnson C. G., 2003, Atificial Immune System Pogamming fo Symbolic Regession, Lectue notes in Compute Sciences seies, Spinge, Volume 2610/2003, 2003, ISSN Johnson Colin G. 2003, Atificial immune systems pogamming fo symbolic egession, In C. Ryan, T. Soule, M. Keijze, E. Tsang, R. Poli, and E. Costa, editos, Genetic Pogamming: 6th Euopean Confeence, LNCS 2610, p , 2003 Kikpatick S., Gelatt C. D., Vecchi M. P. 1983, Optimization by Simulated Annealing, Science, 13 May 1983, 220 (4598), p Koza J.R. 1998: Genetic Pogamming MIT Pess, ISBN , 1998 Koza J.R., Bennet F. H., Ande D., Keane M. 1999: Genetic Pogamming III Mogan Kaufnamm pub., ISBN , 1999 Lampinen J., Zelinka I. 1999, New Ideas in Optimization Mechanical Engineeing Design Optimization by Diffeential Evolution, Vol. 1. London: McGaw-Hill, 1999, ISBN Mařík V. a kol. 2004, Atificial Intelligence IV. 2004, Academia, Paha, Czech edition O Neill M., Ryan C. 2003: Gammatical Evolution. Evolutionay Automatic Pogamming in an Abitay Language Kluwe Academic Publishes, 2003, ISBN Oltean M., Gosan C., 2003, Evolving Evolutionay Algoithms using Multi Expession Pogamming, The 7th Euopean Confeence on Atificial Life, Septembe 14-17, 2003, Dotmund, Edited by W. Banzhaf (et al), LNAI 2801, pp , Spinge-Velag, Belin, 2003 O'Neill M., Babazon A Gammatical Diffeential Evolution. In Poc. Intenational Confeence on Atificial Intelligence (ICAI'06), pp , CSEA Pess. O'Neill M., Leahy F., Babazon A Gammatical Swam: A vaiable-length Paticle Swam Algoithm. Swam Intelligent Systems, pp.59-74, Spinge. Oplatkova Z. 2005, Optimal Tajectoy of Robots Using Symbolic Regession, In: CD-ROM of Poc. 56 th Intenational Astonautical Congess 2005, Fukuoka, Japan, 2005, pape n. IAC-05- C Oplatkova Z., Zelinka I., Investigation on Atificial Ant using Analytic Pogamming, GECCO 2006, Seattle, Washington, USA, 8 12 July 2006, ISBN O'Sullivan J., Ryan C., 2002, An Investigation into the Use of Diffeent Seach Stategies with Gammatical Evolution, Poceedings of the 5th Euopean Confeence on Genetic Pogamming, p , 2002, Spinge-Velag London, UK, ISBN: Pateson N., 2003 Genetic Pogamming with context sensitive gammas, doctoal thesis, Univesity of St. Andews, 2003 Pateson N., Livesey M.,1996, Distinguishing genotype and phenotype in genetic pogamming, In Koza, Goldbeg, Fogel & Riolo, eds. Late Beaking Papes at GP 1996, MIT Pess, 1996, ISBN Pice K., Ston R. M., Lampinen J. A. 2005: Diffeential Evolution : A Pactical Appoach to Global Optimization (Natual Computing Seies) Spinge; 1 edition, 2005, ISBN: Salustowicz R. P., Schmidhube J. 1997: Pobabilistic Incemental Pogam Evolution, Evolutionay Computation, vol. 5, n. 2, 1997, pages , MIT Pess, ISSN Stinsta E., Rennen G., Teeuwen G., 2006, Metamodelling by symbolic egession and Paeto Simulated Annealing, Tilbug Univesity, Nethelands, n , ISSN IJSSST, Vol. 9, No. 3, Septembe ISSN: x online, pint

13 Zelinka I. 2002: Analytic pogamming by Means of Soma Algoithm. Mendel 02, In: Poc. 8th Intenational Confeence on Soft Computing Mendel 02, Bno, Czech Republic, 2002, , ISBN Zelinka I. 2004a, SOMA Self Oganizing Migating Algoithm, In: B.V. Babu, G. Onwubolu (eds), ), New Optimization Techniques in Engineeing Spinge-Velag, 2004, ISBN X Zelinka I., Oplatkova Z. 2003: Analytic pogamming Compaative Study. CIRAS 03, The second Intenational Confeence on Computational Intelligence, Robotics, and Autonomous Systems, Singapoe, 2003, ISSN Zelinka I., Vaacha P., Oplatkova Z., Evolutionay Synthesis of Neual Netwok, Mendel th Intenational Confeence on Softcomputing, Bno, Czech Republic, 31 May 2 June 2006, pages 25 31, ISBN Zelinka I.,Oplatkova Z, Nolle L. 2004b: Boolean Symmety Function Synthesis by Means of Abitay Evolutionay Algoithms-Compaative Study, ESM '2004, In: Poc. 18 th Euopean Simulation Multiconfeence, Magdebug, Gemany 2004 Zelinka I.,Oplatkova Z, Nolle L. 2005: Boolean Symmety Function Synthesis by Means of Abitay Evolutionay Algoithms-Compaative Study, Intenational Jounal of Simulation Systems, Science and Technology, Volume 6, Numbe 9, August 2005, pages 44-56, ISSN: , online 6/No.9/cove.htm, ISSN: x 9. AUTHORBIOGRAPHIES ZUZANA OPLATKOVA was bon in Czech Republic, and went to the Tomas Bata Univesity in Zlin, whee she studied technical cybenetics and obtained he MSc. degee in Ph.D. she defended in Mach She is a senio lectue (atificial intelligence) at the same univesity. He addess is: oplatkova@fai.utb.cz IVAN ZELINKA was bon in Czech Republic, and went to the Technical Univesity of Bno, whee he studied technical cybenetics and obtained his degee in He obtained Ph.D. degee in technical cybenetics in 2001 at Tomas Bata Univesity in Zlin. Now he is Associate Pofesso (atificial intelligence, theoy of infomation), head of depatment and Vice dean fo Science and Intenational Relations. His addess is: zelinka@fai.utb.cz and his Web-page can be found at Tab. 7: Soted numbes of steps and commands fo all algoithms SOMA DE SA soted by steps soted by commands soted by steps soted by steps soted by commands soted by steps IJSSST, Vol. 9, No. 3, Septembe ISSN: x online, pint

14 IJSSST, Vol. 9, No. 3, Septembe ISSN: x online, pint