Exact GP Schema Theory for Headless Chicken Crossover and Subtree Mutation

Exact GP Schema Theory for Healess Chcken Crossover an Subtree Mutaton Rccaro Pol School of Computer Scence The Unversty of Brmngham Brmngham, B5 TT, UK R.Pol@cs.bham.ac.uk Ncholas F. McPhee Dvson of Scence an Mathematcs Unversty of Mnnesota, Morrs Morrs, MN, USA mcphee@mrs.umn.eu Abstract- Here a new general GP schema theory for healess chcken crossover an subtree mutaton s presente. The theory gves an exact formulaton for the expecte number of nstances of a schema at the next generaton ether n terms of mcroscopc quanttes or n terms of macroscopc ones. The paper gves examples whch show how the theory can be specalse to specfc operators. Introucton The theory of schemata n genetc programmng has ha a ffcult chlhoo. After some excellent early efforts leang to fferent worst-case-scenaro schema theorems [,, 3, 4, 5, 6, 7], exact schema theores have become avalable only very recently [8, 9,, ]. These new theores gve exact formulatons (rather than lower bouns) for the expecte number of nstances of a schema at the next generaton, an are applcable to GP wth varous types of subtree crossover. No exact schema theory for subtree mutaton (or any other type of GP mutaton) has ever been propose. Ths paper flls ths theoretcal gap an presents a new general GP schema theory for subtree mutaton an healess chcken crossover. Healess chcken crossover s a varant of crossover, ntrouce for GAs n [] an for GP n [3], n whch one of the parents s ranomly generate whle the other s selecte from the populaton. Our theory gves an exact formulaton for the expecte number of nstances of a schema at the next generaton for these operators. The paper s organse as follows. Frstly, we prove a revew of earler relevant work on schemata n Secton. Most of the concepts ntrouce n that secton are escrbe extensvely, snce they are necessary to unerstan the rest of the paper. Then, we erve general schema theorems for GP wth healess chcken crossover an subtree mutaton n Sectons 3 an 4, respectvely. In Secton 5 we gve examples that show how the theory can be specalse to obtan schema theorems for specfc operators an prmtve sets. Some conclusons are rawn n Secton 6. Backgroun Schemata are sets of ponts of the search space sharng some syntactc features. For example, n the context of GAs operatng on bnary strngs, syntactcally a schema s a strng of symbols from the alphabet,,*, where the character * s nterprete as a on t care symbol. Typcally schema theorems are escrptons of how the number of members of the populaton belongng to a schema vary over tme. If «À Øµ s the probablty that a newly create nvual samples the schema À, whch we term the total transmsson probablty of À, an exact schema theorem s smply [4] Ñ À Ø ½µ Å«À Øµ () where Å s the populaton sze, Ñ À Ø ½µs the number of nvuals n À at generaton Ø ½an s the expectaton operator. Hollan s [5] an other worst-case-scenaro schema theores normally prove a lower boun for «À Øµ or, equvalently, for Ñ À Ø ½µ. One of the ffcultes n obtanng theoretcal results on GP usng the ea of schema s that ts efnton s much less straghtforwar than for GAs. Varous efntons have been propose n the lterature [,, 3, 4, 5, 7], but for brevty here we wll escrbe only the efnton of fxe-sze-an-shape schema ntrouce n [5, 6] whch s what s use n ths paper an n other recent work [8, 9,,, 6].. GP Schemata Syntactcally a GP fxe-sze-an-shape schema (or just schema for smplcty) s a tree compose of functons from the set an termnals from the set Ì, where an Ì are the functon an termnal sets use n a GP run [5, 6]. The prmtve s a on t care symbol whch stans for a sngle termnal or functon. A schema À represents programs havng the same shape as À an the same labels for the non- noes. For example, f =+, * an Ì =x, y the schema (+ x (= y =)) represents the four programs (+ x (+ y x)), (+ x (+ y y)), (+ x (* y x)) an (+ x (* y y)). Usng ths efnton, n [5, 6] a worst-case-scenaro schema theorem was erve for GP wth pont mutaton an one-pont crossover. Ths result was mprove n [8, 9] where an exact schema theory for GP wth one-pont crossover (but no mutaton) was erve.. Cartesan Noe Reference Systems In [] a general schema theory for GP wth subtreeswappng crossover was presente whch was base on the noton of varable arty hyperschema an on the concepts of

3 A 4 5... s the efnng noe functon, Àµ, whch returns f the noe at coornates µ s a efnng noe, f t s a = symbol, ½ f t s not n À. 3 B C E D F G Fgure : Tree-nepenent Cartesan noe reference system. Noes an lnks of the maxmal tree are rawn wth ashe lnes. Only four layers are shown. Cartesan noe reference systems an probablty strbutons over them. These are also the bass for the new theory presente n ths paper. They are escrbe n ths an the followng sectons. A Cartesan noe reference system can be efne by frst conserng the largest possble tree that can be create wth noes of arty ÑÜ. Ths maxmal tree woul nclue noe of arty ÑÜ at epth, ÑÜ noes of arty ÑÜ at epth, ¾ ÑÜ noes of arty ÑÜ at epth, etc.. Then one can organse the noes n the tree nto layers of ncreasng epth an assgn an nex to each noe n a layer. We can then efne a coornate system base on the layer number an the nex. Ths reference system can also be use to locate the noes of non-maxmal trees by usng a subset of the noes an lnks n the maxmal tree. So, for example, f ÑÜ, the noes n the expresson (A (B C D) (E F (G H))) woul be place n a noe reference system as ncate n Fgure where, for example, F s nexe by (,3). It shoul be note that n the ths kn of reference system t s possble to transform pars of coornates nto ntegers by countng the noes n breath-frst orer (an vce versa). So, noes A, B, C, D, E, F an G woul have nces,, 4, 5,, 7, 8 an 5, respectvely. We wll use ths property to smplfy the notaton n some of the followng sectons..3 Functons over Noe Reference Systems Gven a noe reference system t s possble to efne functons over t. An example of such functons s the name functon Æ µ whch returns the noe at poston µ n a partcular tree ; f oes not have a noe at poston µ, a efault value of s returne. For example, for the tree n Fgure, Æ ¼ ¼µ an Æ ¾ ¼µ, whle Æ ¾ ¾µ. Another example of a noe functon s the arty functon µ whch returns the arty of the noe at coornates µ n. The functon returns ½ f µ s not n. For example, for the tree n Fgure, ¼ ¼µ¾, ½ ¼µ ¾, ¾ ½µ¼an ¾ µ½. Fnally, t shoul be note that these functons can be apple to schemata too. A useful functon n hanlng schemata...... H.4 Moellng the Selecton of Crossover an Mutaton Ponts Most genetc operators use n GP requre the selecton of a noe where to perform a transformaton (e.g. the nserton of a ranom subtree, or of a subtree taken from another parent). In most cases the selecton of the noe s performe wth a stochastc process of some sort. It s possble to moel ths process by assumng that a probablty strbuton s efne over the noes of each nvual. If we use the noereference system ntrouce n the prevous secton, ths can be expresse as the functon: Ô µ ÈÖÒ A noe at coornates µ s selecte n program Ó () where we assume that Ô µ s zero for all the unefne coornates µ n. For example, f we select noes wth unform probablty from the tree n Fgure, then Ô µ ½ f µ exsts n, an Ô µ ¼otherwse. There are many possble uses for probablty strbutons over noe reference systems. In the followng secton we wll concentrate on ther use n moellng crossover operators. Later t wll become clear how these can be use to moel healess chcken crossover an subtree mutaton..5 Moellng Subtree-swappng Crossover In general n orer to moel crossover operators we nee to use the followng contonal probablty strbuton: Ô ½ ½ ¾ ¾ ½ ¾ µ ÈÖÒ A noe at coornates ½ ½ µ s selecte n parent ½ an a noe at coornates ¾ ¾ µ s selecte n parent ¾ wth the conventon Ô ½ ½ ¾ ¾ ½ ¾ µ ¼ f Æ ½ ½ ½ µ or Æ ¾ ¾ ¾ µ, where Æ µ s the name functon efne n Secton.3. If the selecton of the crossover ponts s performe nepenently n the two parents, then Ô ½ ½ ¾ ¾ ½ ¾ µô ½ ½ ½ µ Ô ¾ ¾ ¾ µ where Ô µ s efne n Equaton. We wll call crossover operators for whch ths relaton s true separable. Stanar crossover wth unform selecton of the crossover ponts s a separable operator wth Æ Æ µ µ Ô µ Ë µ where Ë µ s the number of noes n an Æ Üµ s a functon whch returns f Ü s true, otherwse. For ths probablty strbuton we use the notaton Ô µ rather than Ô µ snce ths can be seen as the contonal probablty of selectng noe µ f (or gven that) the program beng consere s. Ó

Also stanar crossover wth a 9%-functon/%-anynoe selecton polcy s separable. However, t shoul be note that some crossover operators, lke for example onepont crossover an strongly type GP crossover, are not separable. Moels for these an other crossover operators are escrbe n []. Thanks to these probablstc moels of crossover, t s possble to evelop a general schema theory for GP as escrbe n the followng sectons. Ths theory s the bass for the schema theory for healess chcken crossover an subtree mutaton presente later n ths paper..6 Exact GP Schema Theorems for Subtree-swappng Crossovers For smplcty n ths an the followng sectons we wll use a sngle nex to entfy noes unless otherwse state. We can o ths because, as ncate prevously, there s a one-to-one mappng between pars of coornates an natural numbers. In orer to state a schema theorem val for subtreeswappng crossovers, we nee to ntrouce new form of schema: the Varable Arty Hyperschema, orva hyperschema for brevty. A VA hyperschema s a roote tree compose of nternal noes from the set an leaves from Ì []. The operator = s a on t care symbols whch stans for exactly one noe, the termnal # stans for any val subtree, whle the functon # stans for exactly one functon of arty not smaller than the number of subtrees connecte to t. For example, the VA hyperschema (# x (+ = #)) represents all the programs wth the followng characterstcs: a) the root noe s any functon n the functon set wth arty or hgher, b) the frst argument of the root noe s the varable x, c) the secon argument of the root noe s +, ) the frst argument of the + s any termnal, e) the secon argument of the + s any val subtree. If the root noe s matche by a functon of arty greater than, the thr, fourth, etc. arguments of such a functon are left unspecfe,.e. they can be any val subtree. We can use VA hyperschemata an the noton of probablty strbutons over noe reference systems to obtan the followng general result []: Theorem. The total transmsson probablty for a fxesze-an-shape GP schema À uner a subtree-swappng crossover operator an no mutaton s «À Øµ ½ Ô ÜÓ µô À Øµ Ô ÜÓ Ô ½ ØµÔ ¾ Øµ (3) ¾À ½¾È ¾¾È Ô ½ ¾ µæ ½ ¾ Í À µµæ ¾ ¾ Ä À µµ where: Ô ÜÓ s the crossover probablty; Ô À Øµ s the selecton probablty of the schema À; È s the set of unque Ñ ÀØµ ÀØµ In ftness proportonate selecton Ô À Øµ Å, where Øµ Ñ À Øµ s the number of programs matchng the schema À at generaton nvuals n the populaton; Ô ½ Øµ an Ô ¾ Øµ are the selecton probabltes of parents ½ an ¾, respectvely; the thr summaton s over all the crossover ponts (noes) n the schema À; the fourth summaton s over all the crossover ponts n the noe reference system; Ô ½ ¾ µ s the probablty of selectng crossover pont n parent ½ an crossover pont n parent ¾ ; Ä À µ s the VA hyperschema obtane by rootng at coornate n an empty reference system the subschema of À below crossover pont, then by labellng all the noes on the path between noe an the root noe wth # functon noes, an labellng the arguments of those noes whch are to the left of such a path wth # termnal noes; Í À µ s the hyperschema obtane by replacng the subtree below crossover pont wth a # noe. The functons Ä À µ an Í À µ are esgne to return exactly the hyperschemata neee to create À usng crossover. Í À µ s the hyperschema representng all the trees that match the upper porton of À (.e., the parts of À not below crossover pont ). Ä À µ s the hyperschema representng all the trees that match the lower porton of À, but where the matchng porton s at some arbtrary poston. The combne effect of these efntons s that f one crosses over any nvual matchng Í À µ at pont wth any nvual matchng Ä À µ at pont, the resultng offsprng s always an nstance of À. Further, ths s the only way to construct an nstance of À. 3 To better unerstan how Í À µ an Ä À µ are constructe, let us conser an example; throughout ths example we wll use the D coornate system, so postons an wll n fact be orere pars. Let us take our schema to be À (* = (+ x =)), an our coornates to be ½ ¼µ an ½ ½µ. Then Fgure llustrates how we construct Í À µ (the top two coornate grs) an Ä À µ (the lower three coornate grs). The top coornate gr shows the ntal schema À, wth the crossover pont marke, an the lower part of the schema shae. The next gr then shows Í À µ, whch s obtane by smply replacng the shae subtree (n ths case just the termnal = ) wth a #. The upper of the three coornate grs for Ä À µ agan llustrates the ntal schema À wth the crossover pont marke. Now, however, the shae area (the part of À below ) nees to be translate to poston as shown n the secon coornate gr. The thr coornate gr then shows the nserton of # symbols (a) along the path from the root to (n ths case just ¼ ¼µ) an (b) n all argument postons to the left of # symbols (n ths case just ½ ¼µ). Ths placement of # symbols, combne wth the fact that we allow # s to represent functons of varyng arty, ensures that Ä À µ (# # =) represents all the possble trees whose subtrees at poston match the lower part of À (.e., the part below poston ). Let us enote wth Àµ the schema obtane by replac- Ø, À Øµ s the mean ftness of such programs, an Øµ s the mean ftness of the programs n the populaton. 3 Ä À µ an Í À µ are scusse n more etal n [].

Crossover Pont U(H,(,)) 3 * = + x = 3 * # + Empty Noe ng all the efnng noes n the schema À wth = noes. We wll refer to Àµ as the shape of À. If the choce of the crossover ponts n any two parents, ½ an ¾, epens only on ther shapes, ½ µ an ¾ µ,.e. f Ô ½ ¾ µô ½ µ ¾ µµ, we term the operators noe-nvarant. For noe-nvarant subtree-swappng crossovers Equaton 3 can be transforme nto the followng exact macroscopc escrpton of schema propagaton: Theorem. The total transmsson probablty for a fxesze-an-shape GP schema À uner a noe-nvarant subtree-swappng crossover operator an no mutaton s «À Øµ ½ Ô ÜÓ µô À Øµ Ô ÜÓ Ô Ð µ (4) Ð ¾À Ô Í À µ ØµÔ Ä À µ Ð Øµ Crossover Pont x = L(H,(,),(,)) 3 * = + Empty Noe where the schemata ½, ¾, are all the possble program shapes (.e. all the fxe-sze-an-shape schemata nclung only = symbols) an the other symbols have the same meanng as n Theorem. The sets Í À µ an Ä À µ Ð ether are (or can be represente by) fxe-sze-an-shape schemata or are the empty set. So, the theorem ncates whch pars of schemata can contrbute to the creaton of nstances of a schema an wth whch relatve probablty. Such schemata can be consere the bulng blocks for the schema. x = 3 = 3 # # Path to Root = Fgure : Phases n the constructons of the VA hyperschema bulng blocks Í À ½ ¼µµ an Ä À ¼ ½µ ½ ½µµ of the schema À (* = (+ x =)) wthn a noe coornate system wth ÑÜ ¾..7 Prevous Schema Theores for Mutaton We are aware of only two schema-theory results for mutaton applcable to the stanar GP representaton. We brefly summarse them below. In [7] Rosca erve a worst-case-scenaro schema theorem for roote-tree schemata, whch can be efne as hyperschemata wthout = symbols an # functon noes. In the case n whch only subtree mutaton an ftness proportonate selecton are present the theorem s equvalent to: È¾ÀÈ Ñ À Ø ½µ ÅÔ À Øµ ½ ÔÑ ¾ È ¾ÀÈ Ñ Øµ µ Ë µ Ñ Øµ µ Ç Àµ (5) where Ô Ñ s the mutaton probablty (per nvual), Ë µ s the sze of a program matchng the schema À, µ s ts ftness, an Ç Àµ s the orer of a schema efne as the number of efnng symbols t contans. A secon result for mutaton can be obtane from the worst-case-scenaro GP schema theorem for fxe-sze-anshape schemata uner pont mutaton an one-pont crossover erve n [5, 6]. In the absence of crossover, ths leas to: Ñ À Ø ½µ ÅÔ À Øµ ½ Ô Ñ µ Ç Àµ (6) where Ô Ñ s the mutaton probablty (per noe) an Ç Àµ (the orer of À) s the number of non- symbols n À.

3 Schema Theory for Subtree-swappng Healess Chcken Crossover Dfferent forms of subtree-swappng healess chcken crossover can be efne epenng on whether one returns one or two offsprng an whether such offsprng nhert ther root noes from the parent whch has been ranomly generate or the one selecte from the populaton [3]. In ths paper we wll concentrate on the case n whch we generate a sngle offsprng, an the offsprng nherts the root from the parent selecte from the populaton. The schema theory for subtree-swappng healess chcken crossover s a natural extenson of the theory for subtreeswappng crossover snce the only fference between the two operators s the source of the non-root-onatng parent: the populaton through ftness proportonate selecton n the latter case, a stochastc tree generaton algorthm n the former case. Therefore, the theorems (an the proofs) prove n ths secton are also very smlar to the corresponng results for subtree-swappng crossover. Inee, for the class of operators healess chcken crossover operators efne above we have: Theorem 3. The total transmsson probablty for a fxesze-an-shape GP schema À uner a subtree-swappng healess chcken crossover s «À Øµ ½ Ô ÜÓ µô À Øµ Ô ÜÓ Ô ½ Øµ ¾ Øµ (7) ¾À ½¾È ¾¾Ë Ô ½ ¾ µæ ½ ¾ Í À µµæ ¾ ¾ Ä À µµ where: Ë s the space of all possble programs that can be bult wth the gven termnal an functon sets, ¾ Øµ s the probablty that the ranom tree generaton algorthm use wll prouce program ¾ at generaton Ø, an the other symbols have the same meanng as n Theorem. Proof. Let Ô ½ ¾ Øµ be the probablty that, at generaton Ø, the selecton/crossover/ranomsaton process wll choose parent ½ taken from the populaton, parent ¾ ranomly generate an crossover ponts an n ½ an ¾, respectvely. Then, let us conser the functon ½ ¾ ÀµÆ ½ ¾ Í À µµæ ¾ ¾ Ä À µµ Gven two parent programs, ½ an ¾, an a schema of nterest À, ths functon returns the value f crossng over ½ at poston an ¾ at poston yels an offsprng n À. It returns otherwse. Ths functon can be consere as a measurement functon (see [7]) that we want to apply to the probablty strbuton of parents an crossover ponts at tme Ø, Ô ½ ¾ Øµ. If ½, ¾,, an are stochastc varables wth jont probablty strbuton Ô ½ ¾ Øµ, the functon ½ ¾ Àµ can be use to efne a stochastc varable ½ ¾ Àµ. The expecte value of s: ½ We can wrte ¾ ½ ¾ ÀµÔ ½ ¾ Øµ (8) Ô ½ ¾ ØµÔ ½ ¾ µô ½ Øµ ¾ Øµ (9) where Ô ½ ¾ µ s the contonal probablty that crossover ponts an wll be selecte when the parents are ½ an ¾, Ô ½ Øµ s the selecton probablty for the root-onatng parent an ¾ Øµ s the probablty that the ranom tree generaton algorthm wll prouce program ¾ at generaton Ø. Substtutng Equaton 9 nto Equaton 8 an notng that f crossover pont s outse the schema À, then Ä À µ an Í À µ are empty sets, lea to () ½ ¾È ¾ ¾Ë Ô ½ Øµ ¾ Øµ ¾ À ½ ¾ ÀµÔ ½ ¾ µ Snce s a bnary stochastc varable, ts expecte value also represents the probablty that the offsprng prouce by healess chcken crossover s n À. So, the contrbuton to «À Øµ ue to selecton followe by healess chcken crossover s. By multplyng ths by Ô ÜÓ an ang the term ½ Ô ÜÓ µô À Øµ ue to selecton followe by clonng one obtans the r.h.s. of Equaton 7. ¾ Ths result allows one to calculate the expecte proporton of nvuals belongng to a schema n the next generaton. Ths s a mcroscopc moel snce t requres to conser the propertes of each member of the search space, whch makes t har to use t for computatonal stues. However, ths moel can be transforme nto a macroscopc moel for a very general class of healess chcken crossovers. If we efne as noe nvarant a healess chcken crossover n whch Ô ½ ¾ µô ½ µ ¾ µµ, then we can obtan a macroscopc verson of the prevous theorem by followng a strategy smlar to the one use n the proof of Theorem, obtanng Theorem 4. The total transmsson probablty for a fxesze-an-shape GP schema À uner a noe-nvarant subtree-swappng healess chcken crossover s «À Øµ ½ Ô ÜÓ µô À Øµ () Ô ÜÓ Ô Ð µ Ð ¾À Ô Í À µ Øµ Ä À µ Ð Øµ where Ä À µ Ð Øµ s the probablty of ranomly generatng programs n Ä À µ Ð an the other symbols have the same meanng as n Theorem. Proof. We prove the theorem by transformng Equaton 7 nto Equaton. The schemata ½, ¾, represent sjont sets of programs. Ther unon represents the whole search space.

So, È Æ ½ ¾ µ ½. Lkewse, È Ð Æ ¾ ¾ Ð µ ½. If we multply the terms wthn the quaruple summaton n Equaton 7 by the l.h.s. of these equatons an reorer the terms, we obtan: Ð ½ ¾È ¾ ¾Ë Ô ½ Øµ ¾ Øµ ¾ À Ô ½ ¾ µ Æ ½ ¾ Í À µµæ ½ ¾ µæ ¾ ¾ Ä À µµæ ¾ ¾ Ð µ Ð ½ ¾È ¾ ¾Ë Ð Ô ½ Øµ ¾ Øµ Æ ½ ¾ Í À µµæ ¾ ¾ Ä À µµ ¾ À Ô ½ ¾ µ For noe-nvarant healess chcken crossover operators Ô ½ ¾ µô ½ µ ¾ µµ, whch substtute nto the prevous equaton gves: Ð ½ ¾È ¾ ¾Ë Ð Ô ½ Øµ ¾ Øµ ¾ À Æ ½ ¾ Í À µµæ ¾ ¾ Ä À µµ Ð ½ ¾È ¾ ¾Ë Ð Ô ½ Øµ ¾ Øµ Æ ½ ¾ Í À µµæ ¾ ¾ Ä À µµ Ð ¾À Ô Ð µ Ô ½ ØµÆ ½ ¾ Í À µµ ½¾È ¾ ØµÆ ¾ ¾ Ä À µµ ¾¾Ë Ð Ô ½ µ ¾ µµ ¾ À Ô Ð µ È Snce ½¾È È Ô ½ ØµÆ ½ ¾ Í À µµ Ô Í À µ Øµ an ¾¾Ë Ð ¾ ØµÆ ¾ ¾ Ä À µµ Ä À µ Ð Øµ, ths completes the proof. ¾ Ths an the prevous theorems are qute smlar to the corresponng theorems for crossover. However, there s one mportant fference. Once the stochastc tree generaton algorthms s known, the quanttes À Øµ are numerc constants. So, the schema theorems for healess chcken crossover are lnear n the schema selecton probabltes, whle those for crossover are quaratc. Theorem 4 ncates whch schemata can contrbute to the creaton of nstances of a schema an wth whch relatve probablty. 4 Schema Theory for Subtree Mutaton Once the theory for healess chcken crossover s avalable t s very easy to mofy t to become a theory for subtree mutaton. It s suffcent to constran the choce of the crossover pont n the ranom parent to always be the root noe. Ths can be moelle by settng: Ô ½ ¾ µô ½ µæ ¼µ () where Ô ½ µ s the probablty of selectng mutaton pont n the root onatng parent ½. As a consequence, the result n Theorem 3 smplfes conserably, leang rectly to the followng Corollary 5. The total transmsson probablty for a fxesze-an-shape GP schema À uner subtree mutaton s «À Øµ ½ Ô Ñ µô À Øµ Ô Ñ Ô ½ Øµ ¾ Øµ (3) ¾À ½¾È ¾¾Ë Ô ½ µæ ½ ¾ Í À µµæ ¾ ¾ Ä À ¼µµ where Ô Ñ s the probablty of mutaton (per nvual) an all the other symbols have the same meanng as n Theorem 3. If the choce of the mutaton pont n the parent program,, epens only on ts shape, µ,.e. Ô µ Ô µµ, we term the mutaton operator noe-nvarant. For noenvarant mutaton operators t s possble to specalse the results n Theorem 4 obtanng Corollary 6. The total transmsson probablty for a fxesze-an-shape GP schema À uner noe-nvarant subtree mutaton s «À Øµ ½ Ô Ñ µô À Øµ (4) Ô Ñ Ô µô Í À µ Øµ Ä À ¼µØµ ¾À where all the symbols have the same meanng as n Theorem 4. Proof. For a noe nvarant mutaton operator, the quantty Ô Ð µ n Equaton becomes Ô µæ ¼µ. So, only terms where ¼reman. The VA hyperschema Ä À ¼µ has no # symbols snce t s smply a subtree of À. So, there exsts only one shape Ð such that Ä À ¼µ Ð µ ¼. Let us call t Ð. So, only the terms n Equaton where Ð Ð reman. The proof s complete by notng that Ä À ¼µ Ð Ä À ¼µ. ¾ So, also mutaton s a lnear operator. 5 Specalsatons an Example In orer to use the theory presente n the prevous sectons t s necessary to efne the quanttes Øµ an Ä À µ Ð Øµ. All other quanttes are efne once one chooses a partcular crossover-/mutaton-pont selecton algorthm an a partcular selecton algorthm. It shoul be note that Ä À µ Ð s always ether the empty set or a set whch can be represente by fxe-sze-an-shape schema, so we wll nee to be able to express À Øµ for a generc schema À. In the followng subsectons we wll prove expressons for Øµ an À Øµ for two very wely use ranomtree generaton algorthms: the full metho an the grow metho [, 8]. Startng from the root noe, both methos

use the strategy of creatng trees by selectng ranom noes recursvely along each branch untl ether a termnal s chosen, or a maxmum epth s reache; only termnals are then chosen at epth. The two methos ffer n that the full metho only chooses from untl the epth lmt s reache, guaranteeng that each branch s full out to epth, whereas grow chooses from Ì, whch makes t possble for some branches to have length less than. 5. Probablty Dstrbutons for the Full Metho Let us start by recursvely efnng a functon µ over a noe reference system whch returns the probablty that the subtree roote at poston µ n s create when usng the full metho. Ths s gven by: µ (5) Æ µ ¼µÆ µ Ì Æ µ ¼µÆ µ µ ½ Ò¼ ½ ÑÜ Ò µ By mofyng approprately the expresson for µ we can generalse t so as to return the probablty that the subtree roote at poston µ create when usng the full metho belongs to the subschema of À roote at the same poston, obtanng: Àµ (6) Àµ Æ Àµ ¼µÆ µ ½ Àµ Ì Æ Àµ ¼µÆ µ Àµ ½ Àµ ½ Ò¼ Àµµ Àµ ½ ÑÜ Ò Àµ where s the subset of nclung the functons/termnals of arty. So, ¼, ½ an ¼ Ì. Then, clearly for the full metho we can efne À Øµ ¼ ¼Àµ an Øµ ¼ ¼µ (whch, ncentally, are nepenent from Ø). 5. Probablty Dstrbutons for the Grow Metho We procee n a smlar way for the grow metho. We efne a functon µ over a noe reference system whch returns the probablty that the subtree of roote at poston µ be create when usng the grow metho. Then, we generalse the expresson for µ so as to return the probablty that the subtree roote at poston µ create when usng the grow metho belongs to the subschema of À roote at the same poston, obtanng: Àµ Æ Àµ ¼µÆ µ (7) Àµ ½ Àµ Ì Æ Àµ ½µÆ µ Àµ ½ Àµ ½ Ò¼ Àµµ Àµ ½ ÑÜ Ò Àµ É ½ wth the conventon that Ò¼ ½ ÑÜ Ò Àµ ½. Then, for the grow metho we efne À Øµ ¼ ¼Àµ an Øµ ¼ ¼µ. 5.3 Example Let us wrte a macroscopc, exact schema theorem equaton for the schema µµ assumng that we are usng mutaton base on the grow metho wth a maxmum allowe epth, Ô Ñ ½an unform selecton of the crossover ponts (.e. n Equaton 4 Ô µ½ë µ). Let us conser the prmtve set ÁÆ ÁÆÇÊ ¼ whch can be ecompose nto ½ ÁÆ ÁÆÇÊ an ¼ ¼. The semantcs of these prmtves (see [9, 6]) s unmportant for our example. We also assume that at generaton Ø the populaton oes not contan nvuals wth more than 3 noes. In these contons, by applyng Corollary 6 an smplfyng we obtan: «µµµ (8) Ô µ µµµ ½ Ô µµ µµ µµµ ¾ ½ Ô µµµ µ µµ µµµ By usng Equaton 7 we then can calculate À Øµ for the schemata, µ, an µµ. For À µµ we obtan À Øµ ¼ ¼Àµ Æ ¼ ¼Àµ¼µÆ ¼ µ Æ ¼ ¼Àµ ½µÆ ¼ µ ¾ ½ ¼Àµ ¾ ½ ¼Àµ ¾ Æ ½ ¼Àµ¼µÆ ½ µ ¾ Æ ½ ¼Àµ ½µÆ ½ µ ¾ ¾ ¼Àµ ¾ ¼Àµ Æ ¾ ¼Àµ¼µÆ ¾ µ Æ ¾ ¼Àµ ½µÆ ¾ µ ½ ¾ Lkewse, we obtan Øµ ½ an µøµ ¾. By substtutng these values n Equaton 8, we obtan «µµøµ ¾ Ô Øµ ¾ Ô ½ µøµ Ô ½ µµøµ

Ths equaton shows how mutaton an selecton nteract n the creaton of nstances of (= (= =)). It s partcularly nterestng to stuy two cases. Frstly, let us conser a neele n a haystack stuaton n whch only programs of length 3 are ft at all, whle all other programs have zero ftness. Then, clearly Ô µµøµ½an Ô ØµÔ µøµ ¼. In ths crcumstances one woul expect the algorthm to keep samplng programs of length 3. However, the expecte proporton of programs n (= (= =)) s only about 4%. Ths means that the bases of mutaton work aganst the ntene bases of selecton. In fact, even on a flat lanscape the mutaton bases mpose a ynamcs on the populaton. 6 Conclusons Here we have presente the frst ever exact schema theory for GP wth healess chcken crossover an subtree mutaton, thus fllng an mportant theoretcal gap. The theory s not only an exact formulaton for the expecte number of nstances of a schema at the next generaton but t s also very general. So, t s applcable to most subtree-swappng healess chcken crossovers an mutaton operators use n practce. In the paper we have also prove examples whch show how the theory can be specalse to specfc operators. As shown by some recent exploratons reporte n [, 6], exact schema theores have many purposes. They can be use, for example, to stuy the exact schema evoluton n nfnte populatons over multple generatons, to make comparsons between fferent operators an entfy ther bases, to stuy the evoluton of sze, an nvestgate bloat. The exact theory presente here also offers these possbltes as shown n [], where we have use t to characterse the behavours an bases of fferent mutaton operators. Acknowlegements The authors woul lke to thank the members of the EE- BIC (Evolutonary an Emergent Behavour Intellgence an Computaton) group at Brmngham, n partcular Jon Rowe, for useful scussons an comments. 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