LIP. Laboratoire de l Informatique du Parallélisme. Ecole Normale Supérieure de Lyon

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1 LIP Lortoire de l Informtique du Prllélisme Eole Normle Supérieure de Lyon Institut IMAG Unité de reherhe ssoiée u CNRS n 1398 One-wy Cellulr Automt on Cyley Grphs Zsuzsnn Rok Mrs 1993 Reserh Report N o Eole Normle Supérieure de Lyon 46, Allée d Itlie, Lyon Cedex 07, Frne, Téléphone : ; Téléopieur : ; Adresses életroniques : lip@frensl61.itnet; lip@lip.ens lyon.fr (uup).

2 One-wy Cellulr Automt on Cyley Grphs Zsuzsnn Rok Mrs 1993 Astrt The notion of one-dimensionl one-wy ellulr utomt hs een introdued to model ellulr utomt with only onewy ommunition etween two neighor ells. In this pper, we generlize this notion to ellulr utomt working on dierent ommunition grphs. We present some neessry nd/or suient onditions for ellulr utomton to e simulted y one-wy ellulr utomton hving the sme underlying grph, nd we give some ounds on the simultion-time of this mimi. Keywords: ellulr utomt, one-wy ellulr utomt, Cyley grphs Resume L notion d'utomte ellulire ve diretions privilegiees ete introduite omme modele pour les utomtes ellulires ve ommunition unidiretionelle entre ellules voisines. Dns et rtile, nous generlisons ette notion des utomtes ellulires denis sur divers grphes de ommunition. Nous presentons des onditions, neessires et/ou susntes, pour qu'un utomte ellulire puisse ^etre simule pr un utomte ellulire ve diretions privilegiees gissnt sur le m^eme grphe. Nous donnons des ornes sur l perte de temps osionnee pr e type de mime. Mots-les: Cyley utomtes ellulires, utomtes unidiretionels, grphes de

3 One-wy Cellulr Automt on Cyley Grphs Zsuzsnn Rok Lortoire de l'informtique du Prllelisme ENS-Lyon, 46 Allee d'itlie Lyon Cedex 07 Frne Mrh 18, Introdution The notion of one-wy ellulr utomton (OCA for short) hs een introdued mny yers go in order to model ellulr utomt (CA) with direted ommunitions. A mhine with direted ommunitions is esier to onstrut nd llows fster ommunitions. Using trellis utomt, K. Culik nd C. Chorut [6] hve shown tht n- time one-dimensionl ellulr utomt re equivlent to n-time one-wy ellulr utomt (i.e. if lnguge is reognized with CA in time n, then there exists n OCA whih reognizes this lnguge in time n, nd onversely). We rst rell the denition of OCA given y K. Culik, nd in the rest of the pper, we intend to study CA nd OCA dened with more omplex ommunitions etween ells. It is well-known ft tht CA n e dened in higher dimensionl spes nd with ritrry neighorhoods. In order to study these CA s well s CA dened on more omplex ut regulr rhitetures, we introdue in Setion 3 the notion of CA nd This work ws prtilly supported y the Esprit Bsi Reserh Ation \Algeri nd Semntil Methods In Computer Siene", y the PRC \Mthemtique et Informtique" nd y Grnt of the Frenh Gouvernement. 1

4 OCA dened on Cyley grphs (previously introdued y severl uthors in severl wys). The following setion (Setion 4) is devoted to the study of two lssil neighorhoods in terms of Cyley grphs: von Neumnn nd Moore neighorhoods. We prove tht the simultion-time of the mimi of von Neumnn CA y von Neumnn OCA depends on the dimension n of the spe, while it is not the se for the mimi of Moore CA y Moore OCA. The Moore OCA whih simultes the Moore CA works in two times slower, ut the von Neumnn OCA whih simultes the von Neumnn CA works in n + 1 times slower. In one-dimensionl spe, the two ftors re the sme, euse the two notions orrespond extly with the sme CA. In the 5th setion we study hexgonl nd tringulr CA nd OCA. We point out tht CA do not depend only on the physil rhiteture (hexgonl in this se), ut lso on the lol spe orienttion etween the neighors of two djent ells. It explins why our denition of CA sed on Cyley grphs is so well dpted. Conerning tringulr CA, we prove tht whtever the underlying Cyley grph is, there does not exist one-wy tringulr ellulr utomton. In our lst setion, we study ellulr utomt dened on ritrry Cyley grphs: we give some suient nd neessry onditions for CA to e simulted y n OCA sed on the sme rhiteture nd we present some ounds of the simultion time of this mimi. Some rells on OCA One of the simplest models of prllel omputtion is the one-wy ellulr utomton (OCA) whih hs lredy een studied y severl uthors [6, 8, 10]. They dene this notion s n rry of n identil nite-stte mhines (ells) tht operte synhronously t disrete time steps. The input 1 : : : n where i is in the nite lphet, is pplied to the rry in prllel t time 0 y setting the sttes of the ells to 1 ; : : :; n. Then the stte of ell t time t + 1 is deterministi funtion of its stte nd the stte of its right neighor t time t. In ellulr utomton (CA) the stte of ell t time t + 1 is funtion of its stte nd the sttes of oth its right nd left neighors t time t. More formlly, we present elow Culik's denition [6] for one-dimensionl one-wy ounded ellulr utomt (BOCA). A is xed nite nonempty lphet not ontining the symol ] nd

5 A 0 A distinguished suset. The free monoid (resp. semigroup) generted y A is denoted y A (resp. A + ) nd the empty word of A is denoted y ": A + = A n f"g. A one-dimensionl one-wy ounded ellulr utomton is triple O = (A; A 0 ; R) where R: (A[f]g)! (A[f]g) is trnsition funtion stisfying R(; ) = ] i = = ]. For ny word 1 : : : n A + dene: ( 1 ) = R( 1 ; ]) if n = 1 ( 1 : : : n ) = R( 1 ; )R( ; 3 ) : : :R( n?1 ; n )R( n ; ]) otherwise: Then is length-preserving mpping of A + into itself. Given funtion f : N! N nd suset A. We sy tht the lnguge L + is reognized y O in time f(n), if the words w L re extly those words of + for whih f(jwj) (w) strts with letter in A 0 : L = fw + j f(jwj) (w) A 0 A g: We lso sy tht L is f(n)-time OCA lnguge. In the speil se when f(n) = n? 1, L is lled rel-time OCA lnguge. It is often onvenient to represent the funtioning of ellulr utomton on time-spe digrm (see Figure 1). Eh row orresponds to the ongurtion t ertin time t 0 of n-ell rry. At time t = 0, the i-th ell ontins the i-th letter of the input word w. As times goes y, the dierent vlues re omputed ording to the trnsition funtion R, the new vlue of ell t time t depends on its own vlue t time t? 1 nd the vlue of its right neighor t time t? 1. The vlue of the ell i t time t is denoted y t i. In our pper, we generlize this notion for innite (non-ounded) onewy ellulr utomt working on more omplex rhitetures. 3 Cellulr utomt nd one-wy ellulr utomt on Cyley grphs 3.1 Cyley grphs In order to generlize Culik's notion, we dene ellulr utomt on Cyley grphs. Cyley grphs hve een introdued y Cyley [3, 4, 5] in 1878 nd hve een studied for themselves y severl uthors [, 11, 9, 1]. 3

6 t= n t= n 4 t= n Figure 1: The time-spe digrm of OCA In order to give our denition for ellulr nd one-wy ellulr utomt, we use denition of Cyley grph sed upon White's denition [11]. Our denition is very similr to his one, so we present only our denition, nd we indite the point where we hve mde some moditions. Let G e group, with G = fg 1 ; g ; : : :g (possily innite set) suset of its element set X. We denote y G?1 the set of the inverse genertors: G?1 = fg?1 1 ; g?1 ; : : :g. A word w in G is nite produt f 1 f : : :f n, where eh f i elongs to the set G[G?1. If every element of G n e expressed s word in G, then g 1 ; g ; : : : re sid to e genertors for G. We dene reltion s n equlity etween two words in G. If G is generted y fg 1 ; g ; : : :g nd if every reltion in G n e dedued from reltions fp = p 0, q = q 0, r = r 0, : : : g, then we write G = hg 1 ; g ; : : : j p = p 0 ; q = q 0 ; r = r 0 ; : : :i (= hg j Ri) nd hg 1 ; g ; : : : j p = p 0 ; q = q 0 ; r = r 0 ; : : :i is sid to e presenttion of G. A presenttion is sid to e nitely generted (nitely relted) if the numer of genertors (dening reltions) is nite. A nite presenttion is oth nitely generted nd nitely relted. In our pper, we re only interested in nite presenttions, nd we present elow our denition for Cyley grphs. Denition 1 For every group presenttion there is n ssoited Cyley 4

7 grph? = (V; A): the verties orrespond to the elements of the group, nd the rs re olored with genertors nd inverse genertors in the following wy. If verties v 1 nd v orrespond to group elements h 1 nd h respetively, then there is n r olored with genertor g from v 1 to v, nd there is n r olored with the inverse genertor g?1 from v to v 1 if nd only if h 1 g = h. In White's denition, Cyley grph ontins no rs olored with inverse genertors. We hve mde this ddition, euse our denition ts etter to the ommunitions etween the ells of ellulr utomt. We summerize the min properties of Cyley grphs? whih help to etter understnd the denition of ellulr utomt. 1.? is the Cyley grph of group G whose presenttion is nite.. In? the rs hve regulr oloring with the genertors: for eh vertex v nd genertor g, there exists extly one r olored with g strting t v, nd extly one r olored with g ending t v. This property lso holds for the inverse genertors. It implies lso tht there exists pth etween every pir of verties. 3. If there exists n r olored with g from the vertex i to the vertex j, then there exists n r olored with g?1 from j to i. 4. If there exists n r olored with g from the vertex i to the vertex j, then it is the only one from i to j (tht is, there does not exist nother r from i to j). The properties, 3, 4 hold for ny Cyley grph. The Cyley grph of the group G = h; j = i is shown in Figure. We present some more notions used y White, whih we shll use in the following. An element g of generting set for group G is sid to e redundnt if it n e written s word w of the non-redundnt genertors. The order of g is the length of w. The order of non-redundnt genertors is 1. A generting set is sid to e miniml if it ontins no redundnt genertors: the group G = h; j = i is miniml, while the group G 0 = h; ; j = ; = ; = i is not miniml. The Cyley grph of G 0 is shown in Figure. 5

8 Figure : The Cyley grph of the groups G = h; j = i nd G 0 = h; ; j = ; = ; = i. Then we oserve tht the following orrespondenes our: GROUP CAYLEY GRAPH n element vertex genertor g the set of ll rs olored y g the inverse of genertor g the set of ll rs olored y g?1 word wlk the multiplition of elements the suession of wlks n identity word losed wlk A hrteriztion of grphs? whih n e oriented nd olored so s to form Cyley grphs is given in [7]. 3. Cellulr utomt Let? = (V; A) e Cyley grph. Imgine, tht we lote nite stte utomt in the verties of?. Then we simply ll utomton (or ell) v the utomton loted in the vertex v. A ell v n ommunite with some ells lled \neighor" ells. Formlly, we dene ellulr utomton on? in the following wy. Denition A ellulr utomton on Cyley grph? = (V; A) is 4- tuple C = (S;?; N; ) where 6

9 S is nite set, lled the set of sttes. S n lso e onsidered s the lphet of ommunition etween the ells;? is the Cyley grph of nitely generted innite group G = hg j Ri; The neighorhood of ell v? is vetor of ells dened y words of G: N(v) = (vw 1 ; vw ; : : :; vw k ; v) where 8i; w i is word in G. The neighorhood of ell v onsists of those ells, whih v n ommunite with (the sttes of whih re needed to ompute the new stte of v); : S k! S is the lol trnsition funtion. The stte of eh ell evolves in disrete time steps omputed y the trnsition funtion. The new stte of the ell v is (vw 1 ; vw ; : : :; vw k ; v). In this denition, the neighorhood is not symmetril, tht is, if ell i is the neighor of ell j, the ell j is not neessrily the neighor of the ell i. If it is the se for every neighor of ell, it mens tht this CA is ertin kind of OCA, euse the informtion n move in only one diretion etween the neighor ells. As we wnt to prtiulrize CA with one-wy ommunition, we rther onsider CA where informtion moves in oth diretions etween the neighor ells, tht is, where the neighorhood of ell is symmetril. We loose it of the generlity of our denition for ellulr utomt, ut it is not diult to see, tht for given ellulr utomton C the ellulr utomton C 0 dened on the sme Cyley grph nd with symmetrized neighorhood n simulte C in the sme time. It hs een shown for ellulr utomt dened in Z n, tht ellulr utomt n e simulted y ellulr utomt dened with rst neighors neighorhood. We show here, tht it is the se for ellulr utomt dened on Cyley grphs. Theorem 1 Let C = (S;?; N; ) e ellulr utomton dened on the Cyley grph? of some group G = hg 1 ; g ; : : :; g m j r 1 ; r ; : : :; r p i. The neighorhood of ell v is dened y N(v) = (vw 1 ; vw ; : : :; vw k ) where 8i, w i is word in G. Then we n onstrut ellulr utomton C 0 = (S 0 ;? 0 ; N 0 ; 0 ) with \rst neighors" neighorhood whih n simulte C. Proof. First, we dene the Cyley grph? 0 in the following wy:? 0 is the Cyley grph of the group G 0 onstruted from G s follows. 1. Let G 0 = G nd R 0 = R. 7

10 . Let us onsider rst the neighor vw 1 of the ell v. Let us denote this group element with h 1. If h 1 6 G then we dd h 1 in the set of genertors: G 0 := G 0 [ fh 1 g nd the reltion h 1 = vw 1 in the set of reltions: R 0 := R 0 [ fh 1 = vw 1 g (we dd n r from vertex v to vertex vw 1 olored with h 1 in the Cyley grph). 3. We repet the proedure desried in for eh neighor vw i of the ell v. Then we otin group G 0 with the sme group elements, where jg 0 j = e( k + m) nd R 0 is redundnt generting set. Then we dene the ellulr utomton C 0 = (S 0 ;? 0 ; N 0 ; 0 ) in the following wy: S 0 = S;? 0 is the Cyley grph of the group G 0 ; N 0 (v) = (vg 1 ; vg ; : : :; vg e ) where 8i, g i G 0 ; 0 : (S 0 ) e! S 0 nd 0 (s 0 1 ; s0 ; : : :; s0 e) = (s 1 ; s ; : : :; s k ). It is esy to see tht C 0 n simulte C without ny loss of time. In the following, we study only ellulr utomt with symmetril, rstneighors neighorhood. It is onvenient to do, euse we n hrterize the ommunitions etween the ells with the Cyley grph itself, without ny loss of generlity. We hve shown ove tht ellulr utomt dened with more generl neighorhood n e simulted y suh dened ellulr utomt. Hene we modify our denition (Denition ) in the following wy: Denition 3 A ellulr utomton on Cyley grph? = (V; A) is 4- tuple C = (S;?; N; ) where S is nite set lled the set of sttes. S is the lphet of ommunition etween the ells;? is the Cyley grph of nitely generted innite group G = hg j Ri, where jgj = k; The neighorhood of ell v? is the vetor of those ells whih re onneted to v y n r olored with genertor or n inverse genertor: where 8i; g i G; N(v) = (vg 1 ; vg ; : : :; vg k ; vg?1 ; 1 vg?1 ; : : :; vg?1 k ; v) 8

11 : S jgj+1! S is the lol trnsition funtion. The stte of eh ell evolves in disrete time steps omputed y the trnsition funtion. In the following we use this denition for ellulr utomt. For etter hrterizing the glol ehviour of ellulr utomt, we introdue the notion of ongurtion. Denition 4 Let C = (S;?; N; ) e ellulr utomton on the Cyley grph? = (V; A). A ongurtion is ny pplition from V to S. Denition 5 A stte ] is quiesent if (]; ]; : : :; ]) = ]. Denition 6 A nite ongurtion is ongurtion with quiesent stte lmost everywhere. 3.3 One-wy ellulr utomt In the notion of OCA, informtion n move in only one diretion: for onedimensionl OCA, the stte of ell depends on its stte nd the stte of its right neighor, while in CA the stte of ell depends on its stte nd the sttes of its right nd left neighors. We hve supposed tht the neighorhood of ell is symmetril in ellulr utomt. It mens tht the informtion n move in oth senses etween two neighor ells. In generl, this is not the se. Among the neighors of ell x there n e neighor y suh tht etween x nd y informtion n move in oth diretions, nd there n lso e neighor z suh tht etween x nd z informtion n move in only one diretion. In Culik's denition for OCA, for ll ell, etween ell nd its neighor ells informtion n move in only one diretion. We wnt to generlize this notion, thus we prtiulrize ellulr utomt with one-wy ommunition etween every pir of neighor ells. In order to dene one-wy ellulr utomt, we rst dene one-wy Cyley grphs. Denition 7 Let? e the Cyley grph of the group G. We dene the one-wy Cyley grph of the group G s the grph? o otined from? y deleting ll rs olored with the genertors g in G. In Figure 3 we show the one-wy Cyley grph of the group G = h; j = i. Remrk 1 For group G = hg j Ri where there exists genertor g suh tht g = 1, in the one-wy Cyley of G, g does not pper. 9

12 We dene now one-wy ellulr utomt on Cyley grphs in the following wy. Denition 8 A one-wy ellulr utomton is 4-tuple O = (S o ;? o ; N o ; o ) where S o is nite set lled the set of sttes. S o is the lphet of ommunition etween the ells;? o is the one-wy Cyley grph of nitely generted innite group G = hg j Ri; The neighorhood of ell v? o is the vetor of those ells whih re onneted to v y n r olored with n inverse genertor: where 8i; g i G; N o (v) = (vg?1 ; 1 vg?1 ; : : :; vg?1 k ; v) o : So jgj+1! S o is the lol trnsition funtion. The stte of eh ell evolves in disrete time steps omputed y the trnsition funtion Figure 3: The one-wy Cyley grph of the group G = h; j = i. 4 The von Neumnn nd Moore neighorhoods Cellulr utomt with these two lssil neighorhoods hve extensively een studied in Z n. We introdue elow dierent denitions for these ellulr utomt, whih re sed on Cyley grphs. In our denitions, the 10

13 dimension of the spe orresponds to the numer of the non-redundnt genertors. In the following, insted of giving omplete forml denitions y dening the set of sttes, the Cyley grph, the neighorhood, nd the trnsition funtion for ll kinds of ellulr utomt, we dene only the group on the Cyley grph of whih the CA is dened. It implies tht the neighorhood of ell is dened y the genertors nd y the inverse genertors for CA (y the inverse genertors for OCA), nd tht the trnsition funtion is dened ording to this neighorhood nd the set of sttes. 4.1 The von Neumnn neighorhood In the n-dimensionl spe Z n, the von Neumnn neighorhood of the ell 0 is dened with n-dimensionl unit-vetors nd their inverses. We dene von Neumnn CA nd OCA in the following wy. Denition 9 An n-dimensionl von Neumnn ellulr utomton is ellulr utomton on the Cyley grph of the group G = hg 1 ; g ; : : :; g n j g i g j = g j g i ; 81 i < j ni: The Cyley grph of this group for the -dimensionl se is shown in Figure. Denition 10 An n-dimensionl one-wy von Neumnn ellulr utomton is ellulr utomton on the one-wy Cyley grph of the group G = hg 1 ; g ; : : :; g n j g i g j = g j g i ; 81 i < j ni: The one-wy Cyley grph of this group for the -dimensionl se is shown in Figure 3. K. Culik nd C.Chorut [6] hve shown tht m-time CA n e simulted y m-time OCA. We prove elow, tht n-dimensionl von Neumnn CA n e simulted y n-dimensionl von Neumnn OCA. Theorem For eh ellulr utomton C with von Neumnn neighorhood in the n-dimensionl spe there exists one-wy ellulr utomton O with von Neumnn neighorhood suh tht if ongurtion ppers in the evolution of C t time t, then this ongurtion ppers in the evolution of O t time (n + 1)t. We shll sy tht the OCA n simulte the CA with simultion-time of ftor n

14 Proof. First, we prove this ssertion in the -dimensionl se. Let C = (S;?; N; ) e von Neumnn ellulr utomton. Then we onstrut the one-wy ellulr utomton O = (S o ;? o ; N o ; o ) with S o = S [ S 3 [ S 4? o is the one-wy Cyley grph of the group G = h; j = i N o (v) = (v?1 ; v?1 ; v) o : S 3 o! S o o (x; y; z) = (x; y; z) o ((x 1 ; x ; x 3 ); (y 1 ; y ; y 3 ); (z 1 ; z ; z 3 )) = (x 1 ; x ; z 1 ; z 3 ) o ((x 1 ; x ; x 3 ; x 4 ); (y 1 ; y ; y 3 ; y 4 ); (z 1 ; z ; z 3 ; z 4 )) = (y 4 ; z 3 ; y 1 ; y ; y 3 ): The trnsition funtion is not totlly dened ut the trnsitions desried ove re the only trnsitions used during the simultion. In order to etter understnd the forml denition of the trnsition, we detil here wht o (x; y; z) = (x; y; z) mens: let v e ell, then its neighors in the OCA re ells v?1, v?1 nd v itself. In o (x; y; z), z denotes v's stte t ertin time t, x denotes the stte of its neighor v?1 nd y denotes the stte of its neighor v?1 t this time t. Then the new stte of v t time t + 1 is triple (x; y; z). We illustrte the dierent steps of the rst itertion of this simultion in Figure 4,5,6. In the gures, v denotes the stte of ell v t time 0, nd [v] t ertin time t. Let e ongurtion of the CA t time 0, nd 0 e the ongurtion fter the rst itertion omputed for eh ell i y (i; i; i?1 ; i?1 ; i). We wnt to simulte these trnsitions y the OCA. At the eginning (step 0), we injet in the ells of the OCA: eh ell knows its own stte (Figure 4). In the rst step of the simultion shown in Figure 5, the new stte of eh ell is triple (s 1 ; s ; s 3 ) ording to the trnsition funtion o (x; y; z) = (x; y; z): The stte of eh ell is omputed synhronously in this wy, so in the end of this step, the stte of eh ell is triple ontining the sttes of the ell nd its neighors. We detil elow the sttes of ells v, v, v nd v in the rst step of the simultion. 1

15 [ v -1-1 ]=v -1-1 [ v -1 ]=v -1 [ v -1 ]=v [ v -1 ]=v -1 [ v ]=v [ v ]=v [ v -1 ]=v -1 [ v]=v [ v ]=v step 0 Figure 4: The initil stte. CELL STEP 0 STEP 1 v v (v?1, v?1, v) v v ((v)?1 = v, v?1, v) v v (v?1, (v)?1 = v, v) v v ((v)?1 = v, (v)?1 = v, v) In the seond step of the simultion shown in Figure 6 nd detiled elow, the trnsition funtion is omputed s funtion from (S 3 ) 3 to S 4 y o ((x 1 ; x ; x 3 ); (y 1 ; y ; y 3 ); (z 1 ; z ; z 3 )) = (x 1 ; x ; z 1 ; z 3 ): CELL STEP STEP 3 v (v?1, v?1, v, v) ( : : :) v ( : : :, v?1, v?1, v) ( : : :) v (v?1, v, v, v) (v; v; v?1 ; v?1 ; v) If we look t the trnsition funtion o ((x 1 ; x ; x 3 ; x 4 ); (y 1 ; y ; y 3 ; y 4 ); (z 1 ; z ; z 3 ; z 4 )) = (y 4 ; z 3 ; y 1 ; y ; y 3 ) for ell v nd the sttes of ells v, v, v fter the seond step of the itertion, we nd tht y 4 is the fourth omponent of the old stte of its neighor dened y genertor?1 tht is ell v, whih is the stte of ell v t time 0. 13

16 [ v -1-1 ]=( ) [ v -1 ]=( ) [ v -1 ]=( ) [ v -1 ]=( ) -1-1 [ v ] -1 [ v ]=( v, v, v ) [ v -1 ]=( ) [ v ]=(v, v -1, v ) [ v ]=( v -1, v -1, v ) [ v ]=( v, v, v ) step 1 Figure 5: The 1st step of the simultion. [ v -1-1 ]=( ) [ v -1 ]=( ) [ v -1 ]=( ) -1-1 [ v ]=( ) [ v -1 ]=( ) [ v ]=( v -1, v -1, v, v ) [ v -1 ]=( ) [ v ]=(, v -1, v -1, v ) -1 [ v ]=( v, v, v, v ) step Figure 6: The nd step of the simultion. z 3 is the third omponent of its own old stte whih is the stte of ell v t time 0. y 1 is the rst omponent of the old stte of its neighor dened y genertor?1 tht is ell v, whih is the stte of ell v?1 t time 0. y is the seond omponent of the old stte of its neighor dened y genertor?1 tht is ell v, whih is the stte of ell v?1 t time 0. y 3 is the third omponent of the old stte of its neighor dened y genertor?1 tht is ell v, whih is the stte of ell v t time 0. 14

17 It mens, tht in the following (third) step of the simultion, the trnsition of ell v of the CA n e omputed in ell v of the OCA. We hve seen, tht fter the simultion of the rst itertion of the CA we hve the sme ongurtion in the OCA, ut the stte of eh ell i of the CA is moved in the ell i of the OCA. It is ler, tht fter the simultion of the seond itertion, it moves into the ell i() = i() of the OCA. Then, fter the t-th itertion, the stte of the ell i of the CA moves into the ell i() t of the OCA. It is ler, tht if the ongurtion 0 ppers in the OCA 3 times lter, then the ongurtion 00 ppering t time t in the CA ppers in the OCA t time 3t. It mens tht the OCA n simulte the CA with simultiontime of ftor 3 in the -dimensionl se. n-dimensionl se: n-dimensionl von Neumnn ellulr (one-wy ellulr) utomt re dened on the Cyley grph (one-wy Cyley grph) of the group G = hg 1 ; g ; : : :; g n j g i g j = g j g i ; 81 i < j ni. The neighorhood of ell v is N(v) = (vg 1 ; : : :; vg n ; vg?1 1 ; : : :; vg n?1 ; v) nd N o(v) = (vg?1 1 ; : : :; vg n?1 ; v). We study the rst itertion. Let us onsider rst, in whih ell of the OCA n e omputed the trnsition of the ell v of the CA. Aording to the neighorhood, in the rst step of the simultion, the sttes of the ells vg?1 1, : : :, vg n?1, v n rrive in the ell v of the OCA. Hene, its trnsition n e omputed in ell of the OCA, where the informtions of ll ells vg 1, : : :, vg n, v n rrive. As the generting set is miniml, this ell is the ell vg 1 g : : : g n. It is ler, tht the simultion-time is n + 1. As in the -dimensionl se, the stte of eh ell i of the CA moves into the ell ig 1 g : : :g n of the OCA. After the seond itertion, it moves into the ell i(g 1 g : : : g n ), et: : :. If ongurtion ppers in the CA t time t, it ppers in the OCA t time (n + 1)t, nd the stte of eh ell v of the CA moves into the ell v(g 1 g : : : g n ) t of the OCA. Formlly, we only dene the set of sttes. The trnsition funtion n e dened in similr wy s in the -dimensionl se. Consider P i representing the numer of ells whih n trnsmit their stte to ell v in time shorter thn i: P i is the numer of solutions of Rell tht x 1 + x + : : : + x n i, x k N. P i = ix k=1 k + n? 1 n? 1 = n + i i 15

18 whih is proved y onsidering the 1-ry representtion of the x k 's. Then it is suient to onsider the set of sttes of the OCA s eing S o = S P0 [ S P1 [ : : : [ S Pn+1 : Notie tht P 0 = 1 nd P 1 = n The Moore neighorhood The Moore neighorhood is lso lled \full" neighorhood, euse the neighors of ell 0 in in the n-dimensionl spe Z n re f(i 1 ; i ; : : :; i n ) j 8j f1; : : :; ng; i j f?1; 0; 1gg nd the numer of the neighors is 3 n. As in the se of the von Neumnn neighorhood, we dene only the group on the Cyley grph of whih we dene ellulr utomt, nd we dene it y indution on the dimension of the spe. We do not egin with the denition of one-dimensionl Moore CA, euse they hve the sme rhiteture s one-dimensionl von Neumnn CA, nd thus the speil properties of their underlying grph re not pereptile. So our method is: 1. We dene the Cyley grph of -dimensionl Moore ellulr utomt.. We dene the Cyley grph of (n + 1)-dimensionl Moore ellulr utomt sed on the denition in dimension n. Before giving our forml denition, we present some nottions whih we use in the following. NOTATIONS: g n m denotes the genertor g m in the n-dimensionl spe. R n m denotes the reltion R m in the n-dimensionl spe. num n r = r?1 n denotes the numer of the r-ordered genertors in r the n-dimensionl spe. (A redundnt genertor is lled r-ordered genertor, if the lst letter of the word of length r with whih it is de- ned is genertor. All the other letters of this word n e genertors or inverse genertors.) 16

19 P sum n = r r k=1 k?1 n denotes the numer of ll genertors with k order t most r in the n-dimensionl spe. In the denition of -dimensionl Moore CA we give some exmples for these nottions. In the following, we use the notions num n i nd sum n i for numering the genertors nd the reltions in the (n + 1)-dimensionl denition. Denition 11 A -dimensionl Moore ellulr utomton is ellulr utomton on the Cyley grph of the group G = hg 1 ; g ; g 3 ; g 4 j R 1 : g 1 g = g g 1 ; R : g 3 = g 1 g ; R 3 : g 4 = (g 1 )?1 g i: The Cyley grph of G is shown in Figure 7 where we denote y the genertor g 1, y the genertor g, y the genertor g 3, y d the genertor g d d Figure 7: The Cyley grph of G. In this denition, 1. g 1 nd g re non-redundnt genertors;. g 3 nd g 4 re redundnt genertors; 17

20 3. R 1 is ommuttivity reltion etween the non-redundnt genertors; 4. R nd R 3 re reltions dening the redundnt genertors; We give some exmples for the nottions num n i nd sum n i : 1. num denotes the numer of the -ordered genertors: = : num =?1 These re the genertors g 3 = g 1 g nd g 4 = (g 1 )?1 g.. sum 1 denotes the numer of ll 1-ordered genertors, i.e. the numer of the non-redundnt genertors: sum 1 = 1X k=1 k?1 k These re the genertors g 1 nd g. = 0 1 = : 3. sum denotes the numer of ll 1- nd -ordered genertors, i.e. the numer of ll genertors: sum = X k=1 k?1 k = 0 1 These re ll genertors g 1, g, g 3 nd g = + = 4: In generl, for eh dimension, we n sy tht the generting set onsists of non-redundnt nd redundnt genertors, nd the relting set onsists of ommuttivity reltions nd reltions dening the redundnt genertors. More preisely, in dimension n, the following properties hold: PROPERTIES: 1. The numer n of the non-redundnt genertors orresponds to the dimension of the spe. ( in the denition of -dimensionl Moore CA). All possile redundnt genertors re dened for whih the following properties hold: Let G 1 = fg 1 ; : : :; g n g e the set of the non-redundnt genertors. Then for ll redundnt genertors g 0 = h 1 : : : h k (m > n) 18

21 () k f; : : :; ng () 8i f1; : : :; k? 1g; h i G 1 [ G?1 1 nd h k G 1 () 8i; j f1; : : :; kg; h i 6= h j (d) If genertor is letter in the word dening g 0, then its inverse is not. (e) For ll i nd j > i, the genertor g i preedes the genertor g j in the word dening g 0 (it mens, tht in the set of ll words dened in the sme genertors there is only one word whih denes redundnt genertor, nd this is the word in whih the indies of the genertors re in inresing suession). Remrk tht g 0 is k-ordered genertor y the notion introdued in Setion The numer of ll genertors is (3 n?1)= (the numer of ll genertors nd inverse genertors is 3 n? 1 not onerning the genertor dened y the empty word), so the numer of ll redundnt genertors is (3 n? 1)=? n. ((3? 1)=? = in the denition of -dimensionl Moore CA) n 4. The numer of the ommuttivity reltions is reltion for every pir of non-redundnt genertors. ( = 1 in the denition of -dimensionl Moore CA), euse we dene 5. The numer of the reltions dening redundnt genertors is equl to the numer of the redundnt genertors, tht is (3 n? 1)=? n. ( in the denition of -dimensionl Moore CA) 6. The numer of ll reltions is (the numer of the ommuttivity reltions)+(the numer of reltions dening redundnt genertors)= + (3 n? 1)=? n. n (1 + = 3 in the denition of -dimensionl Moore CA) Then we dene the Cyley grph of (n + 1)-dimensionl Moore ellulr utomt sed on the n-dimensionl denition. Our method is the following: 19

22 METHOD: 1. We keep ll non-redundnt genertors dened in the n-dimensionl spe;. By dding new non-redundnt genertor to the generting set, we enter in the n + 1-dimensionl spe; 3. We keep the ommuttivity reltions etween the non-redundnt genertors dened in the n-dimensionl spe; 4. We dene new ommuttivity reltions with the n-dimensionl nonredundnt genertors nd the new non-redundnt genertor. Then we hve ll the needed ommuttivity reltions; 5. We give reltions dening i-ordered redundnt genertors for ll i from to n + 1: () We keep ll reltions dening redundnt genertors in the n- dimensionl spe; () We dene some new reltions: reltions dening new redundnt genertors y ontention of the n-dimensionl genertors nd the new non-redundnt genertor, nd reltions dening new redundnt genertors y ontention of the inverses of the n- dimensionl genertors nd the new non-redundnt genertor. We dd these new redundnt genertors to the generting set. Before dening the Cyley grph of (n + 1)-dimensionl Moore ellulr utomt, we give n exmple for this method: we dene the Cyley grph of 3-dimensionl Moore ellulr utomt sed on the -dimensionl denition. EXAMPLE: A -dimensionl Moore ellulr utomton is dened y Denition 11. Then we dene the group G 0 on the Cyley grph of whih we dene 3-dimensionl Moore ellulr utomt in the following wy: where G 0 = hg 3 1 ; g3 ; : : :; g3 13 j R 3 1 ; R3 ; : : :; R3 13i 0

23 1. g 3 ; 1 g3 re non-redundnt genertors dened y g 3 1 = g 1 g 3 = g. g 3 3 is new non-redundnt genertor 3. R 3 1 is ommuttivity reltions dened y R 3 = 1 R : 1 g3 1 g3 = g3 g3 1 (g 1 g = g g) 1 4. R 3; R3 3 re new ommuttivity reltions dened y R 3 : g 3 1 g3 3 = g 3 3 g3 1 R 3 3 : g 3 g3 3 = g 3 3 g3 5. R 3 ; : : :; 4 R3 13 re reltions dening redundnt genertors R 3 ; : : :; 4 R3 9 re reltions dening ll -ordered genertors () R 3 ; 4 R3 5 re reltions dening -ordered genertors in the - dimensionl spe: R 3 : 4 g3 = 4 g = 3 g 1 g = g3 1 g3 R 3 5 : g 3 5 = g 4 = (g 1)?1 g = (g 3 1)?1 g 3 () R 3 6 ; : : :; R3 9 re reltions dening new -ordered genertors: R 3 6 : g 3 6 = g 3 1 g3 3 R 3 7 : g 3 7 = (g1) 3?1 g 3 3 R 3 : 8 g3 = 8 g3 g3 3 R 3 9 : g 3 9 = (g) 3?1 g 3 3 R 3 ; : : :; 10 R3 13 re reltions dening ll 3-ordered genertors R 3 10 : g3 10 = g 3 g3 3 = g3 1 g3 g3 3 R 3 11 : g 3 11 = (g 3)?1 g 3 3 = (g 3 1)?1 (g 3 )?1 g 3 3 R 3 1 : g 3 1 = g 4 g3 3 = (g 3 1)?1 g 3 g3 3 R 3 13 : g 3 13 = (g 4)?1 g 3 3 = g 3 1(g 3 )?1 g 3 3 1

24 Let us now dene formlly (n + 1)-dimensionl Moore ellulr nd onewy ellulr utomt sed on the denition of n-dimensionl Moore ellulr utomt whih is dened on the Cyley grph of the group i: +(3 n?1)=?n G = hg n 1 ; gn ; : : :gn (3 n?1)= j R n 1 ; Rn ; : : :; Rn n where g n i s nd R n i s denote genertors nd reltions s desried in properties 1? 6, tht is 1. g n 1 ; : : :; gn n re non-redundnt genertors;. g n ; : : n+1 :gn (3 n?1)= re redundnt genertors; 3. R n; : : :; 1 Rn n re ommuttivity reltions; 4. R n n genertors. +1; : : :; R n n re reltions dening the redundnt +(3 n?1)=?n Denition 1 A (n+1)-dimensionl Moore ellulr utomton is ellulr utomton dened on the Cyley grph of the group n + 1 G = hg n+1 1 ; g n+1 ; : : :g n+1 j (3 n+1?1)= Rn+1 1 ; R n+1 ; : : :; R n+1 where i +(3 n+1?1)=?n?1 1. g n+1 1 ; g n+1 ; : : :; gn n+1 re non-redundnt genertors dened y 81 i n; g n+1 i = g n i (METHOD STEP1: we keep ll non-redundnt genertors dened in the n-dimensionl spe). g n+1 n+1 is new non-redundnt genertor. (METHOD STEP: y dding new non-redundnt genertor to the generting set, we enter in the n + 1-dimensionl spe) 3. R n+1 1 ; : : :; R n+1 re ommuttivity reltions dened y n 81 i n ; R n+1 i = R n i (METHOD STEP3: we keep the ommuttivity reltions etween the non-redundnt genertors dened in the n-dimensionl spe)

25 4. We dene new ommuttivity reltions with the n-dimensionl nonredundnt genertors nd the new non-redundnt genertor: For let l = k? n. Then (METHOD STEP4) n R n+1 n +1 n < k n l n; R n+1 k : g n+1 l g n+1 n+1 = g n+1 n+1 gn+1 l n + 1 ; : : :; R n+1 re reltions dening i-or- +(3 n+1?1)=?n?1 dered genertors for every i from to n + 1 in the following wy: () We keep the reltions dening the i-ordered genertors in the n- dimensionl denition. For + 1 +sum n+1 i?1?sum n+1 n < k +sum n+1 i?1?sum n+1 1 +num n i n let l = k?? sum n+1 i?1 + sum n+1 1. Then l num n i ; Rn+1 k : g n+1 = sum n+1 i?1 +l gn sum n i?1 +l n + 1 (METHOD STEP5) Remrk, tht in the n-dimensionl spe there is no n+1-ordered genertor, this se does not our when i = n + 1. () We dene new i-ordered genertors: for every i? 1 -ordered genertor dened in the n-dimensionl se, we dene pir of new i-ordered genertors. We dene rst n i-ordered genertor y ontention of the i? 1-ordered genertor nd the new nonredundnt genertor g n+1 n+1 (se \l is odd", desried elow), then we dene nother i-ordered genertor y ontention of the inverse of this i? 1-ordered genertor nd the new non-redundnt genertor g n+1 n+1 (se \l is even"). For i?1? sum n num n i < k n + sum n+1 i? sum n sum n+1 let l = k? n ? sum n+1 i?1 + sum n+1 1? num n i. 3

26 Then 81 l num n i?1, if l is odd: R n+1 k : g n+1 = sum n+1 i?1 +numn +l gn sum n i i? +(l+1)=gn+1 if l is even: R n+1 k : g n+1 = sum n+1 i?1 +numn +l (gn sum g n+1 n i i? +(l+1)=)?1 n+1 (METHOD STEP5) The numering of the redundnt genertors nd the reltions dening them seem to e it omplited. So we give here more detiled explntion. The rst n + 1 genertors re non-redundnt genertors, fter we hve ll -ordered genertors, ll 3-ordered genertors, : : :, ll n + 1-ordered genertors. In generl, the indies of the i-ordered genertors egin with sum n+1 i?1 +1, tht is fter ll genertors with order t most i?1, nd the lst i- ordered genertor hs index sum n+1 i. In the denition of i-ordered genertors we rst keep ll i-ordered genertors lredy dened in the n-dimensionl spe (the numer of these genertors is num n i ): the indies egin with sum n+1 i?1 +1 nd the lst index is sum n+1 i?1 +num n i (STEP 5). Then we dene new i-ordered genertors y ontention of the i? 1-ordered genertors dened in the n-dimensionl spe nd the new redundnt genertor, nd y the ontention of the inverses of these i?1-ordered genertors nd the new redundnt genertor. As the numer of ll i? 1-ordered genertors in the n-dimensionl spe is num n i?1, the indies egin with sum n+1 i?1 +num n i +1 nd the lst index is sum n+1 i?1 + num n + i numn = i?1 sumn+1 i (STEP 5). The indies of the reltions dening these genertors re otined from n + 1 the previous indies y dding to them (the rst reltions re ommuttivity reltions) nd sutrting sum n+1 1 (the rst-ordered genertors re not redundnt, we do not dene them with reltions). We dene lso Moore OCA s follows. Denition 13 A n-dimensionl Moore one-wy ellulr utomton is ellulr utomton on the one-wy Cyley grph of the group G = hg n 1 ; gn ; : : :gn (3 n?1)= j R n 1 ; Rn ; : : :; Rn n i +(3 n?1)=?n where g n i s nd R n i s re dened s in the denition of n-dimensionl Moore ellulr utomt. We hve seen tht von Neumnn CA n e simulted y von Neumnn OCA, nd the simultion-time of this mimi hs ftor n + 1 in the n- dimensionl spe. In the denition of von Neumnn CA ll genertors 4

27 re non-redundnt. In the se of the Moore neighorhood there re lso redundnt genertors, so we ould think, tht if Moore CA n e simulted y Moore OCA, then the simultion-time is less thn for von Neumnn CA. Indeed, it is the se. Theorem 3 In the n-dimensionl spe, Moore CA n e simulted y Moore OCA with simultion-time of ftor. Proof. First, we prove this ssertion in the -dimensionl se. In order to simplify the nottion, we denote y the genertor g1, y the genertor g, y the genertor g3, y d the genertor g4. Let C = (S;?; N; ) e -dimensionl Moore ellulr utomton with N(v) = (v; v; v; vd; v?1 ; v?1 ; v?1 ; vd?1 ; v) nd : S 9! S. Then we onstrut the one-wy ellulr utomton O = (S o ;? o ; N o ; o ) with S o = S [ S 4? o is the one-wy Cyley grph of the group N o (v) = (v?1 ; v?1 ; v?1 ; vd?1 ; v) o : S 5 o! S o o (x; y; z; v; w) = (x; y; z; w) G = h; j = ; = ; d =?1 i o ((x 1 ; x ; x 3 ; x 4 ); (y 1 ; y ; y 3 ; y 4 ); (z 1 ; z ; z 3 ; z 4 ); (v 1 ; v ; v 3 ; v 4 ); (w 1 ; w ; w 3 ; w 4 )) = (w ; w 1 ; w 4 ; x 1 ; z 1 ; z ; z 3 ; y ; w 3 ) The trnsition funtion is not totlly dened, ut the trnsitions desried ove re the only trnsitions used during the simultion. We illustrte the dierent steps of the rst itertion in Figure 8,9. Let e ongurtion of the CA t time 0, nd let 0 e the ongurtion fter the rst itertion, t time 1. We ompute the trnsition of ell v y (v; v; v; d; v?1 ; v?1 ; v?1 ; vd?1 ; v) in the CA. We wnt to ompute this trnsition in the OCA. At the eginning, we injet in the ells of the OCA (Figure 8): eh ell knows its own stte. 5

28 -1-1 [ v ]= v -1-1 [ v ]= v -1 [ v -1-1 ]= v -1 d [ vd ]= vd -1 [ v ]= v [ vd ]= vd [ v ]= v [ v ]= v [ v ]= v step 0 Figure 8: The initil stte. CELL STEP 0 v v v v v v v v In the rst step of the simultion, eh ell i reeives the stte of its neighors, nd ording to the trnsition funtion, its stte will e 4- tuple onsisting of the sttes of its neighors dened y genertors,, nd its own stte. The new sttes of ells is shown in Figure [ v ]= ( ) -1 [ v ]= ( ) -1-1 [ v ]= ( ) -1 d [ vd ]= vd [ v ]=(v, vd, v, v ) [ v ]=( v, v, v, v ) [ vd ]= ( ) [ v ]=(vd, v, v -1, v ) [ v ]=( v, v, v, v ) step 1 Figure 9: The rst step of the simultion. 6

29 We present elow the sttes of ells v, v, v, v in eh step of the simultion. CELL STEP 1 v (v?1 ; v?1 ; v?1 ; v) v ((v)?1 = v; (v)?1 = vd?1 ; (v)?1 = (v)()?1 = v?1 ; v) v ((v)?1 = vd; (v)?1 = v; (v)?1 = (v)()?1 = v?1 ; v) v ((v)?1 = (v)?1 = v; (v)?1 =?1 = v; (v)?1 = v; v) In the seond step of the simultion, eh ell i reeives the stte ( 4-tuple) of its neighors. For eh ell i, the informtion rriving in the ell i is lredy suient to ompute the trnsition of ell i of the CA. See Figure 9: onsider the ell v nd its neighors. Aording to the trnsition funtion nd looking t the gure we nd tht w is the seond omponent of its own stte = v, w 1 is the rst omponent of its own stte = v, w 4 is the 4-th omponent of its own stte = v x 1 is the rst omponent of the stte of its neighor dened y genertor whii is vd, et: : : This informtion is extly the needed informtion to ompute the trnsition of the ell v of the CA. CELL STEP v (: : :) v (: : :) v (: : :) v (v; v; v; vd; v?1 ; v?1 ; v?1 ; vd?1 ; v) The ongurtion 0 of the CA ppering t time 1 ppers in the OCA t time, nd the stte of eh ell i of the OCA moves into the ell i of the OCA s in the se of the von Neumnn neighorhood (i = i). It is ler, tht fter the seond itertion, the stte of eh ell i of the CA moves into ell i = i of the OCA. Then the ongurtion ppering in 7

30 the CA t time t ppers in the OCA t time t, nd the stte of eh ell i of the CA moves into ell i t. n-dimensionl se: The neighorhood of ell v in the n-dimensionl Moore CA is N(v) = (vg 1 ; : : :; vg (3?1)=; vg?1 n 1 ; : : :; vg?1 (3 n?1)=; v) nd in the n-dimensionl Moore OCA is N o (v) = (vg?1 1 ; : : :; vg?1 ; v). We study (3 n?1)= the rst itertion. Let us onsider rst, tht the trnsition of the ell v of the CA in whih ell of the OCA n e omputed. Aording to the neighorhood, in the rst step of the simultion, the sttes of ells vg?1 1, : : :, vg?1 (3 n?1)=, v n rrive in the ell v of the OCA. Hene, its trnsition n e omputed in ell w of the OCA, where the informtion of ll ells vg 1, : : :, vg (3?1)=, v n rrive. As every redundnt genertor is dened n s word in the non-redundnt genertors, this ell is w = vg 1 g : : : g n = vg (3n?1)=? n?1 +1 (s there re n?1 n-ordered genertor, vg (3n?1)=? n?1 +1 is the rst one nd it is dened y vg 1 g : : :g n ). It is the sme ell s in the se of the von Neumnn neighorhood. All ells vg 1, : : :, vg (3n?1)=, v re in the neighorhood of w, so it is ler, tht the simultion-time is lwys. As in the -dimensionl se, the stte of eh ell i of the CA moves into nother ell: in the ell ig (3 n?1)=? n?1 +1 of the OCA. After the seond itertion, it moves into the ell ig (3 n?1)=? n?1 +1, et: : :If ongurtion ppers in the CA t time t, it ppers in the OCA t time t, nd the stte of eh ell v of the CA moves into ell vg t (3 n?1)=? n?1 +1 of the OCA. Formlly, we uild the OCA with set of sttes S o = S [ S (3n?1)=+1 nd trnsition funtion o : So 5! S o dened in similr wy s in the -dimensionl se. Remrk In the se of the von Neumnn neighorhood, the simultiontime of the mimi of CA y OCA depends on the dimension n of the spe (n + 1), ut it is independent in the se of the Moore neighorhood (lwys ). In the se of the Moore neighorhood, the distne etween ny two ells in the neighorhood of ell i of the OCA is. The trnsition funtion of ell v of the CA n e omputed in one of its neighor ells w in the OCA, so every needed informtion n rrive in this ell in two units of time. In the von Neumnn neighorhood it is not the se: informtion must follow edges of the n-dimensionl hyperue for rriving in ell w, thus n + 1 units of time re needed. For n = 3 these pths re shown in Figure 10. 8

31 i i i i Figure 10: The pths of informtion in the von Neumnn nd Moore neighorhoods in the 3-dimensionl spe. 5 The hexgonl nd tringulr neighorhoods In this setion we study hexgonl nd tringulr neighorhoods. While von Neumnn nd Moore neighorhoods n e dened in Z n, it is not the se for the two other ones. We present our denitions with the help of Cyley grphs. 5.1 The hexgonl neighorhood A hexgonl ellulr utomton is usully dened s ellulr utomton in the plne R, where the ells re t the enters of hexgons whih tile the plne, nd the neighors of ell re the ells loted t the enter of the djents hexgons (see Figure 11). We present elow forml denition with the help of Cyley grphs. Denition 14 A hexgonl ellulr utomton is ellulr utomton de- ned on the Cyley grph of the group G = h; ; j = ; = ; = ; = 1i The Cyley grph of G is shown in Figure 1. This denition nnot e \nturlly" extended to the n-dimensionl spe: we should study the tilings of R n y regulr polyhedr hving more thn n summits ( n is the numer of the summits of the n-dimensionl hyperue whih orresponds to the von Neumnn neighorhood). But in R n, very few is known on suh tilings. 9

32 Figure 11: The hexgonl neighorhood Figure 1: The Cyley grph of G. Denition 15 A hexgonl one-wy ellulr utomton is ellulr utomton dened on the one-wy Cyley grph of the group G = h; ; j = ; = ; = ; = 1i In the generting set of the group on the Cyley grph of whih we hve dened von Neumnn ellulr utomt, there is no redundnt genertor. The simultion of CA y OCA is slow, in the n-dimensionl spe it is n+1. In the se of the Moore neighorhood, in the generting set there re \ll possile" redundnt genertors. Then the simultion of CA y OCA is muh more fster: it is lwys, independently of the dimension of the spe. In the -dimensionl spe, in ll the three ses (von Neumnn, Moore, hexgonl neighorhoods), there re two non-redundnt genertors in the generting set. In the se of the von Neumnn neighorhood there is no redundnt genertor, in the Moore neighorhood there re two ones (ll 30

33 possile), nd in the hexgonl neighorhood there is only one (not ll possile). For Moore CA, the presene of ll redundnt genertors llows its fster simultion y Moore OCA. We study elow if the simultion of hexgonl CA y hexgonl OCA is fster thn in the se of the von Neumnn neighorhood. Theorem 4 Hexgonl CA n e simulted y hexgonl OCA with simultion-time of ftor. Proof. Let C = (S;?; N; ) e hexgonl ellulr utomton with N(v) = (v; v; v; v?1 ; v?1 ; v?1 ; v) nd : S 7! S. Then we onstrut the onewy ellulr utomton O = (S o ;? o ; N o ; o ) with S o = S [ S 4? o is the one-wy Cyley grph of the group G = h; ; j = ; = ; = ; = 1i N o (v) = (v?1 ; v?1 ; v?1 ; v) o : S 4 o! S o o (x; y; z; v) = (x; y; z; v) o ((x 1 ; x ; x 3 ; x 4 ); (y 1 ; y ; y 3 ; y 4 ); (z 1 ; z ; z 3 ; z 4 ); (v 1 ; v ; v 3 ; v 4 ) = (z ; z 1 ; x ; v 1 ; v ; v 3 ; v 4 ) The trnsition funtion is not totlly dened, ut the trnsitions desried ove re the only trnsitions used during the simultion. We illustrte the dierent steps of the rst itertion in Figures 13, 14. In these gures the stte of ell v t ertin time t is denoted y [v], nd its ste t time 0 is denoted y v. Let e the initil ongurtion of the CA, nd let 0 e the ongurtion fter the rst itertion, t time 1. In the CA the imge of ell v is given y (v; v; v; v?1 ; v?1 ; v?1 ; v). We wnt to ompute this trnsition in the OCA: in the rst step, ording to the neighorhood, the stte of ells v?1, v?1, v?1, v n rrive in ell v, so the trnsition of ell v of the CA n e omputed in ell w, where the informtions of v, v, v, v n rrive. This ell is w = v = v ( = 1). It mens, tht the trnsition of ell of the CA is omputed in the sme ell y the OCA, so the stte of 31

34 ell in the CA does not move into nother ell in the OCA (ontrsted with the se of von Neumnn nd Moore neighorhoods). Formlly, t time 0, we injet in the ells of the OCA: eh ell knows its own stte (see Figure 13). We detil elow the sttes of ells v, v?1, v?1, v?1 in eh step of the simultion. [ v ]= v [ v -1 ]= v -1 [ v -1 ]= v [ v ]= v [ v ]= v [ v ]= v [ v -1 ]= v -1 Figure 13: The initil stte. CELL STEP 0 v v v?1 v?1 v?1 v?1 v?1 v?1 In the rst step of the simultion, eh ell i reeives the stte of its neighors, nd ording to the trnsition funtion, its stte will e 4- tuple onsisting of the sttes of its neighors dened y genertors?1,?1,?1 nd its own stte. The new sttes of ells is shown in Figure 14: onsider ell v nd its neighors. Looking t the gure nd the trnsition funtion we nd tht o ((x 1 ; : : :; x 4 ); (y 1 ; : : :; y 4 ); (z 1 ; : : :; z 4 ); (v 1 ; : : :; v 4 ) = (z ; z 1 ; x ; v 1 ; v ; v 3 ; v 4 ) z is the seond omponent of the stte of its neighor dened y genertor?1, whih is v. 3

35 z 1 is the rst omponent of the stte of its neighor dened y genertor?1, whih is v. x is the seond omponent of the stte of its neighor dened y genertor?1, whih is v. v 1 is the rst omponent of its own stte whih is v?1. et: : : This informtion is extly the needed informtion to ompute the trnsition of ell v of the CA. [ v ]= ( ) [ v -1 ]= (v, v,, v -1 ) -1-1 [ v -1 ]= (, v, v, v -1 ) -1 [ v ]= ( ) [ v ]= (v -1, v -1, v -1, v ) [ v ]= ( ) [ v -1 ]= (v,, v, v -1 ) Figure 14: The rst step of the simultion. CELL STEP 1 v (v?1 ; v?1 ; v?1 ; v) v?1 (v(?1 ) ; (v?1 )?1 = v; (v?1 )?1 = (v?1 )() = v; v?1 ) v?1 ((v?1 )?1 = v; v(?1 ) ; (v?1 )?1 = (v?1 )() = v; v?1 ) v?1 ((v)?1 = v; (v)?1 = v; v(?1 ) ; v?1 ) So in the seond step of the simultion, eh ell i reeives the stte ( 4-tuple) of its neighors, nd this informtion is suient to ompute the trnsition of ell i of the CA. 33

36 CELL STEP v (v; v; v; v?1 ; v?1 ; v?1 ; v) v?1 (: : :) v?1 (: : :) v?1 (: : :) The ongurtion 0 of the CA ppering t time 1 ppers in the OCA t time. Then the ongurtion ppering in the CA t time t ppers in the OCA t time t. Let us onsider now the Cyley grph of the group G 0 = h; ; j 3 = 1; 3 = 1; 3 = 1; = 1i shown in Figure 15. We hve only mrked the genertors,,, the inverse genertors re the olors of the orresponding opposite rs. Figure 15: The Cyley grph of G 0. We nd, tht the Cyley grph of G 0 forms hexgonl rhiteture, ut the lol spe orienttion of the neighorhood of ell \turns" in eh ell, while in the Cyley grph on whih we hve dened hexgonl ellulr utomt, it is homogeneous. In Figure 16 we show the dierent lol spe orienttions. Remrk 3 To given physil rhiteture orrespond severl Cyley grphs dened y dierent groups: Cyley grph denes oth the physil rhiteture of the ellulr utomton, the lol orienttion of the neighorhood of eh ell. 34

37 G G' Figure 16: The dierent lol spe orienttions of the neighorhoods. If two Cyley grphs forming the sme physil rhitetures re not isomorphi, they my led to dierent notions of ellulr utomt: some ellulr utomt dened y denition 14 nnot e simulted y ellulr utomton dened on the Cyley grph of the group G The tringulr neighorhood A tringulr ellulr utomton is usully dened s ellulr utomton in the plne R, where the ells re t the enter of equilterl tringles, nd the neighors of ell re the ells loted t the enter of the tringles whih re djent side y side (see Figure 17). Figure 17: Tringulr neighorhood. Let us now ome k to the hexgonl neighorhood shown in Figure 11. Remrk tht while in hexgonl ellulr utomt the ells re loted t the 35

38 enters of the hexgons, in tringulr ellulr utomt they re loted t the verties of hexgons. On the other hnd, in tringulr ellulr utomt the ells re loted t the enters of the tringles, nd in hexgonl ellulr utomt they re loted t the verties of tringles. So there is dulity etween these two rhitetures. We dene formlly tringulr ellulr nd one-wy ellulr utomt s we hve done in the se of the other neighorhoods. Denition 16 A tringulr ellulr utomton is ellulr utomton de- ned on the Cyley grph of the group G = h; ; j = 1; = 1; = 1; () = 1i In Cyley grphs, etween two neighor ells there is n r olored with genertor nd there is nother r, olored with the inverse of this genertor. Figure 18 shows the Cyley grph of G. As in this group the inverse of genertor is the genertor itself, we denote this pir of rs y single edge. This denition is it speil reltively to ll the denitions we hve given efore: the neighorhood of ell formlly onsists of 7 neighors (N(v) = (v; v; v; v?1 ; v?1 ; v?1 ; v)). In relity, it onsists of only 4 neighors, euse eh neighor dened y genertor g is the sme ell s the neighor dened y the inverse genertor g?1. Figure 18: The Cyley grph of G. 36