Investigating Cellular Automata

Transcription

1 Researcher: Taylor Dupuy Advsor: Aaro Wootto Semester: Fall 4 Ivestgatg Cellular Automata A Overvew of Cellular Automata: Cellular Automata are smple computer programs that geerate rows of black ad whte cells based o rules whch update each cell based o ts eghborg cells. A example of ths would be a cellular automata havg a row cotag oe black cell surrouded by two whte cells where ths specfc automata t has a rule that says for each cell of ths type, that s, for each black cell surrouded by two whte cells t wll be updated to a black cell the ext row. Rules lke ths for all possble codtos whch are defed exhaustvely. A example of a cellular automata rule 3 s show below. The box drectly below ths text depcts the rules by whch the cellular automata s updated. The mage below ad to the left depcts the cells beg updated step by step for the frst fve steps. The mage to the rght depcts the evoluto of cellular automata rule3 after steps. A terestg fact about rule 3 s the that the ceter colum of rule 3 has passed every radomess test t has goe through ad that t s fact s used the radom umber geerator for Stephe Wolfram s Mathematca program ever sce the program s cepto. Uversalty s the property of beg able to perform dfferet tasks wth the same uderlyg costructo by beg programmed a dfferet way. 3 I pla Eglsh ths s to say that a computer program, upo gvg t dfferet tal codtos, wll perform tasks that mmc aother computer program the same way the program whch t s Namg scheme: readg the bottom row from rght to left we have bary whch s equal to 3. Also ote that from rght to left the tal codtos for whch the gve rule are appled also go from to 7 bary. These ad more mages ca be foud at 3 Weste, E

2 mmckg would costruct the sad costructo. I beleve t would be helpful to see a pcture of a uversal cellular automata whch wll llustrate the oto of uversalty better tha ay cofusg setece :-). 4 The above pcture shows a 9-color cellular automato wth sgfcatly more rules tha the elemetary cellular automata dscussed prevously. Nevertheless I thk the llustrato of the oto of uversalty s clear from these mages. I must ote that ths property of uversalty s ot lmted to cellular automata. The property of uversalty ca be foud Turg maches, tag-cyclc systems, ad regster maches, whch are other dealzed computatoal systems. Backgroud: I Stepha Wolfram of Wolfram Research publshed A New Kd of Scece a hstory ad vestgato of cellular automata ad other computatoal systems. The results preseted A New Kd of Scece are atypcal of a maor work because t does ot cota a formal mathematcal structure by whch the deas preseted the book are dscussed. After havg read A New Kd of Scece I developed a terest the mathematcal structure that les behd these dealzed computatoal systems. I wsh to expla why computato behaves a way t does. I wsh to expla formally why certa automata 4 Images take from

3 have the property of uversalty or complexty, what propertes (rules) determe the behavor of the automata (complexty, ad uversalty). I order to do so, I beleve I eed to uderstad ad/or develop a mathematcal structure whch I may exame automata as a operato o a set. Because I m most famlar wth elemetary cellular automata, I wsh frst uderstad a partcular uversal elemetary automata, elemetary cellular automata. Rule s the smallest uversal system whch has bee show to be uversal. It was show to be uversal by both Cook 4 ad Wolfram. 5 At the ed of the semester I wsh to expla why rule s uversal ad be able to prove t formally. I ve commeted o the mathematcal structure below ad I must ote that my research wll clude fdg/uderstadg a rgorous represetato these operatos performed by dealzed computatoal systems. Structure: The structure whch oe ca represet a automata seems ot to be uque. I m cosderg represetg cellular automata as a set wth the operato o the set of the updatg rule for the partcular cellular automata. I ve show below a way of represetg a elemetary cellular automata a fte feld. Ths method below s ot ecessarly the oe I wll choose represetg the automata mdterm of fal reports. 5 Cook, M. "Uversalty Elemetary Cellular Automata." Complex Systems 5, -4, 4., Wolfram, S. A New Kd of Scece. Champag, IL: Wolfram Meda, pp ,.

4 . Defg Elemetary Automata over a Fte Feld: We shall call the black ad whte states elemets of a set B {, } whch we shall defe coucto wth operatos whch wll allows us to classfy ths set as a fte feld. If we have a cellular automata row wth compoets, or otrval rows we shall defe ths row as follows. r (,,, ) r b b b B where b, b,, b B. Ths allows us to use a vector space structure o the cellular automata ad we make wrte r as a lear combato of bass elemets Ad we wrte, (,,,), (,,,),, (,,,) e e e r be... Defg Automata Fucto: Our goal s to defe the geeralzed automata fucto A : B B + for whch we may take ay rules for a gve cellular automata ad use them to operate o the row as to defe the recurrece relato A r r ad. It s coveet at ths r for r B r B + pot to defe the fuctos for updatg each cell based o the adacet cells... Updatg Cells 3 We desre the fucto : B B. s defed exhaustvely, gvg all the cases for 3 each possble elemet B.,, (,, ) (,,) for,,, 7 B. Now order to get these sgle bt values to a form whch we k ca use we eed to do perform a trasformato φ k : B B whch maps the sgle bt k to a subspace of B wth the same dmeso (whch ths specfc case s, but whch could possble be geeralzed for other computatoal systems). Now all that s left s do t decostructo of some elemet of B to a form where we may apply the fucto ad the recostruct the compoets uder the mage to a B + vector space. 7

5 .. Applcato of alpha-fucto Allow us to decostruct 3 Let Ψ : B B defed by for β r to compoets whch wll be more useful our veture. Ψ ( r) β ( b, b+, b + ) ad let us create some β β β ( ( b,,) whch all ca be used to apply to part of b the mage of for p,,,, b,, b, b, b, b,), ad we get that ( ) β β β +. 6 B. Takg all Now applyg : + φ ( + ) B B to each,, + where φ ( + ) s defed by φ ( + ) (,,,,,,) thspot + terms β p uder we get compoets of Ar ( ) whch whe we take the sum of all of them we get Ar ( ). So we have Ar r where s gve by + φ ( + ) ( ) (,,, ) ( ( r )) Ψ The sloppess of ths argumet s to be corrected 7 ote that ca also be as coeffcets for a lear combato of the bass elemets for a bass of B +.