Lortoire de l Informtique du Prllélisme Ecole Normle Supérieure de Lyon Unité de recherche ssociée u CNRS n 1398 Simultions Between Cellulr Automt on Cyley Grphs Zsuzsnn Rok Decemre 1994 Reserch Report N o 94-40 Ecole Normle Supérieure de Lyon 46 Allée d Itlie, 69364 Lyon Cedex 07, Frnce Téléphone : (+33) 72.72.80.00 Télécopieur : (+33) 72.72.80.80 Adresse électronique : lip@lip.ens lyon.fr
Simultions Between Cellulr Automt on Cyley Grphs Zsuzsnn Rok Decemre 1994 Astrct We consider cellulr utomt on Cyley grphs nd compre their computtionl powers ccording to the rchitecture on which they work. We show tht, if there exists homomorphism with nite kernel from group into nother one such tht the imge of the rst group hs nite index in the second one, then every cellulr utomton on the Cyley grph of one of these groups cn e uniformlly simulted y cellulr utomton on the Cyley grph of the other one. This simultion cn e constructed in liner time. With the help of this result we lso show tht cellulr utomt working on ny Archimeden tiling cn e simulted y cellulr utomton on the grid of Z 2 nd conversely. Keywords: cellulr utomt, Cyley grphs, simultions Resume Nous comprons l puissnce de clcul des utomtes cellulires gissnt sur dierents grphes de Cyley. Nous montrons que, s'il existe un morphisme noyu ni d'un groupe dns un utre tel que l'indice de l'imge du premier groupe est ni dns le deuxieme, lors tout utomte cellulire sur le grphe de Cyley d'un de ces groupes peut-^etre simule pr un utomte cellulire sur le grphe de Cyley de l'utre groupe vec un fcteur de perte de temps lineire. Nous montrons ussi, que les utomtes cellulires gissnt sur les pvges Archimediens peuvent ^etre simules pr un utomte cellulire sur l grille de Z 2 et reciproquement. Mots-cles: utomtes cellulires, grphes de Cyley, simultions
Simultions Between Cellulr Automt on Cyley Grphs Zsuzsnn Rok Lortoire de l'informtique du Prllelisme ENS-Lyon, 46 Allee d'itlie 69364 Lyon Cedex 07 Frnce zrok@lip.ens-lyon.fr 1 Introduction A cellulr utomton (CA) is network of identicl nite utomt which work in prllel nd synchronously. It is lso required tht the network e regulr, thus, it cn e considered s Cyley grph of nitely presented group. Automt re plced on the vertices of the grph, nd they communicte with ech other through the edges. Some recent ppers [5, 8,6]hve lredy considered this generlized notion of cellulr utomt, s we do in this pper. Our gol is to compre the computtionl power of dierent models, more precisely, the power of cellulr utomt working on dierent Cyley grphs. In order to do tht, we study simultions etween them. The notion of simultion is very intuitive ut hs never een studied for itself. In [7], we hve shown with the help of vrious exmples tht this notion is very complicted, nd we hve given some possile denitions for it. Here we shll use simplied denition which is not the most generl one, ut ts every simultion presented in our pper. First, we study some exmples: we construct simultions etween cellulr utomt with von Neumnn, hexgonl, tringulr neighorhoods nd lso cellulr utomt on trees. These exmples llow us to give sucient condition for every cellulr utomton on Cyley grph to e simulted y This work ws prtilly supported y the Esprit Bsic Reserch Action \Algeric nd Syntcticl Methods In Computer Science", y the PRC \Mthemtique et Informtique" nd y the grnt of the French Government. 1
cellulr utomton on nother Cyley grph. We show tht this condition is not necessry. When it holds, simultions re rther simple. We remrk tht the sme underlying grph cn e colored in severl wys corresponding to siclly dierent groups. We present more complicted simultion etween hexgonl cellulr utomt on two dierent Cyley grphs. Then, we give sucient condition for simultion to e possile in oth directions etween cellulr utomt. In the lst section, we showthtevery plnr, modulr structure is equivlent to the grid Z 2 with respect to liner time simultions. 2 Denitions In this section, we recll some lgeric notions in order to dene cellulr utomt on Cyley grphs. 2.1 Presenttion of group Let G e group nd X its element set, nd let G = fg1 g2 :::g (possily innite set) e suset of X. We denote y G ;1 the set of the inverse elements of G: G ;1 = fg 1 ;1 g;1 2 :::g. Ifwe consider the free monoid on G[G;1, tht is, the set of words on G [ G ;1,we cn ssocite to word w n element [w] ofg. More thn one words cn correspond to one element ofg. Ifevery element of G cn e expressed s word on G [ G ;1,wesy tht G is generting set for G. We dene reltion s n equlity etween two words in G. A genertor g is sid to e idempotent if g 2 =1. IfG is generted y G = fg1 g2 :::g nd if every reltion in G cn e deduced from reltions R = fp = p 0, q = q 0, r = r 0, :::g, then we write G = hg1 g2 :::j p = p 0 q= q 0 r= r 0 :::i (G = hg j Ri) nd hg1 g2 :::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) ifthenumer of genertors (dening reltions) is nite. A nite presenttion is oth nitely generted nd nitely relted. In this pper, we shll only study nitely presented innite groups. 2.2 Cyley grphs For every group presenttion G = hg j Ri there is n ssocited Cyley grph ;=(V A): the vertices (V ) correspond to the elements of the group, nd the rcs (A) re colored with genertors in the following wy. There exists n rc colored with genertor g fromvertex x to vertex y, if nd only if y = xg in G. Remrk tht the Cyley grph depends on the group presenttion nd not on the group itself. 2
Remrk 1 From now on, if we refer to group G = hg j Ri, we refer to its presenttion nd not to the group itself. Thus, we shll tlk out cellulr utomton dened on the Cyley grph of group G = hg j Ri nd not cellulr utomton dened on the Cyley grph of the presenttion hg j Ri of group G. The following properties hold in Cyley grphs ;: Property 1(Cyley grphs 1): In ; the rcs hve regulr coloring with the genertors: for ech vertex v nd genertor g, there exists exctly one rc colored with g strting t v, nd exctly one rc colored with g ending t v. If, in group, the reltion g = g 0 holds for two genertors g nd g 0,we delete one of them from the generting set nd replce ll of its occurrences y the other one in ll reltions. Thus, we hve lso the following property: Property 2(Cyley grphs 2): If there exists n rc colored with g from the vertex i to the vertex j, then it is the only one from i to j. Remrk tht the sme undirected grph cn sometimes e colored in severl wys. See n exmple for the grph which gives the tringulr tiling of the plne. There re seven possile colorings, here we present onlytwo of them: the Cyley grphs of groups G = h c j c =1 = i nd G 0 = h c j 3 =1 3 =1 c 3 =1 c=1i re shown in Figure 1, 1 (the other colorings re presented in Section 7). c c.. Figure 1: Two colorings for tringulr tiling. 2.3 Cellulr utomt on Cyley grphs As Cyley grphs re grphicl representtions of groups, they hve very regulr structures. Hence, y putting utomt in the vertices, we cn otin generl notion of cellulr utomt in n nlog wy s A. Mch nd F. Mignosi in [5]. 3
Denition 1 A cellulr utomton on Cyley grph ; = (V A) is 4-tuple A =(S ; N ) where S is nite set, clled theset of sttes, ; is the Cyley grph of nitely presented group G = hg j Ri, The neighorhood N is vector dened y words of G: N =(w1 w2 ::: w m ) where 8i w i is in G [ G ;1 [f1g, : S m! S is the locl trnsition rule In n nlog wy s for cellulr utomt in Z n,we cn chrcterize the glol ehvior of the cellulr utomton. A congurtion is n ppliction c from G to S, so the set C of ll congurtions is S G. A cellulr utomton trnsforms congurtion into nother one: 8c 2 C 8i 2 G A(c)(i) =(c(iw1) c(iw2) ::: c(iw m )): Remrk tht more generl denition of cellulr utomton cn lso e done y dening the neighorhood with words on G [ G ;1, nd not only with genertors nd their inverses. However, we hve shown in [7] tht we do not loose ny of the generlity of the denition for cellulr utomt if we mkethisrestriction. Moreover, this denition llows us to consider tht cells communicte through the rcs of the grph. For short, we shll cll utomton (or cell) v the utomton put in vertex v of ;. 2.4 The notion of simultion etween cellulr utomt. Mny ppers hve lredy studied vrious simultions etween cellulr utomt, ut the forml denition of simultion hs not een clerly given. In[7, 8, 6] we study this notion in detils, nd we show through some exmples, why this notion is not esy to formlly dene nd to work with. In this pper, we shll compre the computtionl power of cellulr utomt whose neighorhoods re complete, tht is, they contin ll genertors, ll inverse genertors nd lso the neutrl element. We design some simultions in the sense of the following denition: Denition 2 Let A =(S ; N ) e cellulr utomton nd C A the set of its congurtions. Let B =(S 0 ; 0 N 0 0 ) e cellulr utomton nd C B the set of its congurtions. We sy tht B simultes A, if there exist n injective ppliction f : C A! C B nd constnt T in N such tht for ll c in C A f(a(c)) = B T (f(c)): 4
T is the simultion time fctor, tht is, the time which is necessry for B to simulte one itertion of A. It depends on f ut not on c. This denition cn e illustrted y the following digrm: c A w A(c) f ;! f(c) w BT f ;! B T (f(c)) This denition does not cover ll simultions. In [7] wehve studied the prolem of cellulr utomt with only one-wy communictions etween cells: \given Cyley grph, whether ll idirectionl cellulr utomton cn e simulted y one-wy cellulr utomton on this grph?" Sometimes, such simultion is possile, ut fter the simultion of ech itertion of the idirectionl cellulr utomton, this congurtion is \shifted" in the one-wy cellulr utomton. In order to understnd this phenomenon, let us consider the exmple of the line, tht is, the Cyley grph of the group G = h j i. The neighorhood of idirectionl cellulr utomt is given y N =( ;1 1) nd the neighorhood of one-wy cellulr utomt is (for exmple) y N = ( 1). Let A e idirectionl cellulr utomton nd O one-wy cellulr utomton simulting A. Let the initil congurtion of O e the sme s A's. It is cler tht the trnsition of cell v of A cnnot e computed in the cell itself: it cnnot know the stte of its neighor dened y ;1.However, ll needed sttes cn rrive in cell v ;1. It mens then, tht the congurtion of A fter the rst itertion cn e found in O, ut with \shift" ;1. Simultions without shifts re not possile. The Denition 2 does not llow suchsimultions. However, ll long this pper, we restrict ourselves to this denition, ecuse we do not need shift when simulting: shifts re needed only in the cse of cellulr utomt with one-wy communictions. This denition, even if it is not the most generl one, covers lso very complicted simultions. In [7] wehve shown with two exmples tht f is not necessrily recursive function, stte of simulted cellulr utomton cn e \splittered" in the simulting cellulr utomton, more sttes of simulted cellulr utomton cn e grouped in the simulting one. In this pper we do not consider simultions where stte cn e splittered: we consider tht every stte of simulted cellulr utomton is n \tomic" informtion. If every stte in the initil congurtion of the simulting cellulr utomton depends on only one stte of the simulted cellulr utomton, we shll sy tht the simultion is elementry (ie. the sttes re not grouped). 5
First, we shll present some simultion results etween cellulr utomt with some clssicl neighorhoods s von Neumnn, hexgonl nd tringulr ones. 3 Some exmples with clssicl neighorhoods 3.1 Von Neumnn neighorhood In the n-dimensionl spce Z n, the von Neumnn neighorhood is dened with n-dimensionl unit-vectors nd their inverses. These unit-vectors form n independent vector system, so we cn dene them in similr wy with the help of Cyley grphs: the grid of Z n cn e colored s the Cyley grph of the Aelin group with miniml generting set of n genertors. Denition 3 An n-dimensionl von Neumnn cellulr (resp. one-wy cellulr) utomton is cellulr (resp. one-wy cellulr) utomton on the Cyley grph of the group G n VN = hg1 g2 ::: g n j g i g j = g j g i 1 i<j ni: The Cyley grph of this group for the 2-dimensionl cse is shown in Figure 2. Figure 2: The Cyley grph of group G = h j = i. Here, we study, whether n-dimensionl von Neumnn CA cn e simulted y m-dimensionl von Neumnn CA. Denition 4 Let ; e the Cyley grph of group G n VN.Wecll ll of rdius k nd denote y Bk n the set of ll vertices eing t distnce t most k from 1. We denote y #(Bk n) the numer of vertices in Bn k. Theorem 1 For n>m, n-dimensionl von Neumnn CA cnnot e simulted y m-dimensionl von Neumnn CA. 6
Proof. We show this result y the simple fct tht the growth of two Cyley grphs is dierent. We suppose tht every n VNCA cn e simulted y m VNCA. Study the rst itertion of the n VNCA. In order to compute the trnsition of cell 1 for the n VNCA, the sttes of ll B1 n re needed. Let T e the simultion time fctor. Then, these sttes must e found in the initil congurtion of m VNCA in the ll BT m. In order to simulte the second itertion of n VNCA, the sttes of cells of ll B2 n re necessry, hence they must e in the ll B2T m. The trnsition of cell of n VNCA t time t is computed in function of the sttes of cells eing t distnce t most t, hence the sttes of cells of ll Bt n must e in the ll BtT m of m VNCA, nd so on. It mens tht if the simultion is elementry, then#(bt n) #(Bm tt ), for ll t. If it is not elementry, then let k is the mximl numer of sttes elonging to cell of m VNCA then, we hve #(Bt n ) #(BtT m k ). As #(Bn t )=O(t n )nd #(BtT m k )=O(tm ), if t is ig enough, then #(Bt n ) > #(BtT m k ), which leds to contrdiction. As G m VN is sugroup of G n VN,ifn m, simultion in the other direction is lwys possile without ny loss of time (y \ignoring" the other dimensions). With n nlog proof, we cn show the following theorem. Theorem 2 Let B k nd Bk 0 ethells of rdius k in the Cyley grphs ; nd ; 0,respectively. If #(B k ) is polynomil of degree p nd #(Bk 0 ) is polynomil of degree q, ifpq, then not every CA on Cyley grph ; cn e simulted y CAonCyleygrph ; 0. This result is not very stonishing. S. Cole hs een shown in [2], tht the lnguge recognition power of CA increses with the dimension of the spce, which is similr sttement. We shll now study simultions etween CA with dierent kinds of neighorhoods. 3.2 Von Neumnn nd hexgonl neighorhoods First of ll, we present nintuitive denition of hexgonl cellulr utomton, then we dene it formlly, on Cyley grphs. A hexgonl cellulr utomton is usully dened s cellulr utomton in the plne R 2, where the cells re t the centers of hexgons which tile the plne, nd the neighors of cell re the cells locted t the center of the djcents hexgons (see Figure 3). We present elow forml denition with the help of Cyley grphs. Denition 5 A hexgonl cellulr (one-wy cellulr) utomton is cellulr (one-wy cellulr) utomton dened on the Cyley grph of the group G h = h c j = c =1i 7
Figure 3: The hexgonl neighorhood. c c.. Figure 4: Cyley grphs of G h nd G h2. The Cyley grph of G h is shown in Figure 4. Physicists hve shown in [3] tht, in certin cses, 3 VN nets cn e simulted y hexgonl nets. The ide comes from the identicl numer of neighorhoods of cell: 7 in oth cses. Unfortuntely, in generl the simultion is not possile. Here, we give ll possile simultions etween CA with these two kinds of neighorhoods, nd show when simultion is not possile. For short, we denote hexgonl CA y HCA. In the plne, ll given simultions re very simple, ecuse these cellulr utomt re dened on the Cyley grphs of isomorphic groups. Lter we give some more complicted exmples. Proposition 1 For n 2, every HCA cn e simulted ynn VNCA. Proof. First we prove this proposition for n =2: leta =(S h ; h N h h )e hexgonl CA dened y Denition 5 with n initil congurtion c 0 A. We construct 2 VNCA B =(S2 VN ;2 VN N2 VN 2 VN) which simultes A. First, let us consider the elements of G h s words in f g. We cn do this, ecuse c = ;1 ;1. Let : G h! G2 VN e dened y (u) = u for ll u 2 G h.lets2 VN e sup-set of S h.we dene the initil congurtion c 0 B of B y c 0 B ((u)) = c0 A (u) for ll u in G h, see Figure 5. 8
1 2 3 4 1 2 3 5 6 7 8 c 5 6 7 9 10 11 12.. 9 10 Figure 5: Initil congurtions of 2D VNCA nd HCA. For computing the trnsition of cell v of the HCA, the sttes of ll of its neighors dened y,, c, ;1, ;1, c ;1 re needed. In c 0 B,theycne found in cells dened y v, v, v ;1 ;1, v ;1, v ;1 nd v, respectively. As not ll of these cells re neighors of v y the von Neumnn neighorhood, the simultion cnnot e given with T = 1. They re in ll of rdius 2, so T =2 is possile. This simultion cn e done s in ll previous simultions: the rst step of the simultion of n itertion is memorizing step, nd in the second step cells cn compute the trnsitions of A. Formlly, we dene B y S2 VN = S h [ S 5 h 2 VN : S 5 2 VN! S 2 VN 2 VN(x y z r s)=(x y z r s) 2 VN((x1 x2 ::: x5) (y1 y2 ::: y5) (z1 z2 ::: z5) (r1 r2 ::: r5) (s1 s2 ::: s5)) = h (x1 x2 x3 x4 r4 y2 x5): For n =3: sg2 VN G3 VN, we cn dene the simultion s efore (y \ignoring" the third dimension): we dene the initil congurtion of B with c 0 (u) =!, B! 62 S h, for ll u not eing n imge y ;, nd with little modiction of the trnsition function. We give nother simultion: we cn use the third dimension in order to decrese the simultion time fctor down to T =1.We construct 3 VNCA B simulting A s follows. Let : G h!p(g3 VN) e dened y (1) = f1 c (c) ;1 (c) 2 (c) ;2 :::g (u) =fu uc u(c) ;1 u(c) 2 u(c) ;2 :::g 8u 2 G h Remrk tht ech element ofg h hs n innite numer of imges y ;, which ws not the cse in the previous simultions. WedeneS3 VN s sup-set of S h the initil congurtion c 0 B of B is dened y c 0 ((u)) = B c0 A (u) nd is shown in Figure 6. With this construction, if two 9
5 c 4 1 v 2 3 5 6 7 1 2 3 c 4 1 4 6 v 5 8 9 8 9 10 6 9 5.. Figure 6: Initil congurtions of HCA nd 3 VNCA. informtion re neighors in c 0 A, they re lso neighors in c0 B.SoBcn simply dened y S3 VN = S h 3 VN : S 7 3 VN! S 3 VN 3 VN(x y z r s t u)= h (x y z r s t u) For n>3, s ;3 VN is sugrph of ; n VN nd every CA on ;3 VN cn e simulted y CA on ; n VN with T = 1, hence every HCA cn e simulted lso y n VNCA n>2, with simultion time fctor T =1. Proposition 2 Every 2 VNCA cn e simulted y HCA. Proof. The simultion cn e given s in the converse direction the von Neumnn neighorhood is included in the hexgonl neighorhood, so for computing the trnsition of cell of the 2 VNCA in the HCA, it is sucient tochoose the needed informtion. Formlly, let A =(S2 VN ;2 VN N2 VN 2 VN) e2vnca. We construct hexgonl CA B =(S h ; h N h h ) in \nturl" wy which simultes it with T =1. Let : G2 VN! G h e n ppliction dened y (u) =u for ll u 2 G2 VN. Let S h = S2 VN. We dene the initil congurtion c 0 B of B y c0 B ((u)) = c0 A (u). We cn see, tht neighor informtion in c 0 A re lso neighors in c0 B, hence we cn dene A y S h = S2 VN h (x y z r s t u)=2 VN(x y r s u) 10
Theorem 3 For n>2, n VNCA cnnot e simulted y HCA. Proof. Hexgonl CA cn e simulted y 2 VNCA, nd conversely. Ifn VNCA could e simulted y HCA, then it would imply tht n VNCA cn e simulted y 2VNCA which isincontrdiction with Theorem 1. In Section 2.2 we hve remrked tht the sme non-oriented grph cn e colored in dierent wys. We hve lsoshown in Figure 1 the Cyley grph of group G h2 = h c j 3 =1 3 =1 c 3 =1 c=1i. Its underlying grph gives the sme tringulr tiling of the plne, so we could hve use it to dene hexgonl. Similr simultion results cn e done ut these simultions re it more complicted, ecuse G2 VN nd G h2 re not isomorphic, while G2 VN nd G h re. This mens tht not only the physicl rchitecture of cellulr utomt is importnt for simultions, ut lso the locl communictions, tht is, Cyley grphs on which we dene them. We shll study this prolem in Section 5.1. 3.3 Simultions etween CA on the Cyley grph of freegroups Here we study it more complicted simultions. First of ll, we study the Cyley grphs ; nd ; 0 of the groups FR3 = h c j i nd FR2 = h j i, respectively (see Figure 7). 1 c 1 Figure 7: The Cyley grphs of FR3 nd FR2. Let A =(S ; N ) e cellulr utomton. Wewnt to construct cellulr utomton B =(S 0 ; 0 N 0 0 ) which simultes A. Let : FR3! FR2 e 11
homomorphism dened y (1)=1 () = () = (c) = 2 (uv) =(u)(v) u v 2 FR3: (see Figure 8) We dene S 0 s sup-set of S. Let the initil congurtion of B given y c 0 B((u)) = c 0 A(u) u2 FR3: γ (c) 1 γ () γ () Then, B is dened y Figure 8: The mpping : FR3! FR2 S 0 = S [ S 5 0 : S 05! S 0 0 (x y z r s)=(x y z r s) 0 ((x1 ::: x5) (y1 ::: y5) (z1 ::: z5) (r1 ::: r5) (s1 ::: s5)) = (x2 y1 x1 r3 z4 z3 s5): With this construction, neighor sttes in A re not neighors in B: while the needed sttes to compute the new stte of cell rrive y single rcs in A, they rrive y pirs of rcs in B. So, in the rst step of the simultion of n itertion, ll cells store the sttes of ll of its neighors, nd in the second step they compute the trnsition of A (the simultion time fctor is T = 2). Let us now study free groups generted y ny numer of genertors. 12
Theorem 4 Every CA A dened on the Cyley grph of free-group with n genertors FR n cn e simulted ycab dened on the Cyley grph of nother free-group with m(> 1) genertors FR m with simultion time fctor dlog m ne. Proof. The ssertion is true if m n, ecuse we cn dene the initil congurtion of simulting CA y \ignoring" some rnches of the tree. If m<n, then we dene mpping : FR n! FR m with (g1) =w1 (g2) =w2. (g n )=w n (uv) =(u)(v) u v 2 FR n where w1 ::: w n re dierent words of the sme length k. Ifk dlog m ne, then these w i 's cn e given. We cn construct simultion with simultion time fctor k in similr wy s in the cse of CA on the Cyley grph of FR3 nd FR2. 3.4 An exmple for simultion y grouping sttes Sometimes, only non-elementry simultions re possile etween two cellulr utomt. Here we give n exmple for such simultion. Exmple 1(Cylinder utomton on the line): Let G = h j = 2 =1i nd G 0 = h j i nd ;, ; 0 their Cyley grphs, respectively. LetA =(S ; N )eca.let : G! G 0 e homomorphism dened y (1) = 1, () =1,() = nd for ll u, for ll v in G, (uv) = (u)(v) (see Figure 9). We uild CA B =(S 0 ; 0 N 0 0 )simulting A y S 0 = S [ S 2 0 :(S 0 ) 3! S 0 0 ((x1 x2) (y1 y2) (z1 z2)) = ((z2 x2 z1) (x1 z1 x2)) strting from the initil congurtion given y c 0 B ((u)) = c0 A (u) for ll u in G. 4 A sucient condition In the simultion etween CA dened on the Cyley grph of free groups, we cn remrk tht is n injective homomorphism. We cn then remrk lso tht in Exmple 1, is homomorphism with nite kernel f1 g. In generl we cn stte tht: 13
1 2 3 γ: 1 2 3 2 3 Figure 9: The homomorphism : h j = 2 =1i!h j i. Theorem 5 If there exists homomorphism with nite kernel from group G into nother group G 0, then every cellulr utomton dened on the Cyley grph of G cn e simulted y cellulr utomton dened on the Cyley grph of G 0. Proof. Let A =(S ; N ) e cellulr utomton nd : G! G 0 homomorphism with nite kernel: for ech genertor g i in G, (g i )=w i where for ll i, w i is word in G 0 nd mxfjw i jg = m. Wewnt to construct cellulr utomton B =(S 0 ; 0 N 0 0 ) which simultes A. First, let e injective. Let n e the numer of neighors of cell in B. Let! e stte not elonging to S. We dene the set of sttes of B y S 0 =(S [f!g) [ (S [f!g) n [ (S [f!g) n2 [ :::[ (S [f!g) nm nd the trnsition function y 0 : S 0n! S 0 in the following wy. Cells with no preimge re in stte!. At time 1, they store ll sttes of ll of their neighors t time 0: 0 (x1 x2 ::: x n )=(x1 x2 ::: x n ): At time 2, they store ll sttes of ll of their neighors t time 1, tht is, the sttes of those cells t time 0, which re t distnce t most 2 from the cell: 0 ((x11 x12 ::: x1n) ::: (x n1 x n2 ::: x nn )) = (x11 x12 ::: x nn ): At timem, cell v stores the sttes of ll cells which re dened y genertors, inverse genertors, words of length two, words of length 3, :::, words of length m in G 0.Asevery w i is word of length t most m, its stte is known y the cell v. As the locl function is the sme for ech cellnd is homomorphism, for every cell v in B, for ll i, the stte of the cell vw i rrives y the sme pth 14
nd s the sme component ofvector, hence, t time m, the trnsitions of A cn e computed in B. We now study the cse where : G! G 0 is not injective utitskernel is nite. As the kernel of homomorphism forms group, this kind of simultion is possile only if G hs nitely presented non-trivil nite sugroup. The simultion cn e dened s efore, the only dierence is tht, while in the previous cse, in ech cellofb, the trnsition of single cell of A is computed, here the trnsitions of ll cells which hve the sme imge y re computed. Remrk tht if is injective, the simultion constructed in this wy is elementry. Corollry 1 If two groups re isomorphic, then cellulr utomt dened on their Cyley grphs cn e simulted y ech other in n elementry wy. We cnnot sy nything out the simultion time fctor, it depends on the Cyley grph. 5 Elementry simultions In this section, we study only elementry simultions. We hve seen, tht the existence of n injective homomorphism llows elementry simultions. On the other hnd, if there did not exist homomorphism with nite kernel from group into nother one, then we could not dene ny simultion. In the following exmple we show, tht in some cses, there does not exist homomorphism with nite kernel, ut the simultion (even elementry) is possile: the condition given in Theorem 5 is not necessry. In Section 6, we study other, non-elementry simultions. 5.1 Hexgonl nd tringulr neighorhoods First, we present the intuitive denition of tringulr cellulr utomton, then we dene it formlly,on Cyley grphs. A tringulr cellulr utomton is usully dened s cellulr utomton in the plne R 2, where the cells re t the center of equilterl tringles, nd the neighors of cell re the cells locted t the center of the tringles which re djcent side y side (see Figure 10). Denition 6 A tringulr cellulr utomton is cellulr utomton dened on the Cyley grph of the group G t = h c j 2 =1 2 =1 c 2 =1 (c) 2 =1i: 15
Figure 10: Tringulr neighorhood. In the Cyley grph of G t, s for ll genertor g in G t, g 2 =1,etween every pir of neighor vertices there re two rcs colored with g: we replce them y single, non-oriented edges colored with g. This denition is it specil reltively to ll the denitions we hve given efore: the neighorhood of cell formlly consists of 7 neighors (N =( c ;1 ;1 c ;1 1)). In relity, it consists of only 4 neighors, ecuse ech neighor dened y genertor g is the sme cell s the neighor dened y the inverse genertor g ;1. c Figure 11: The Cyley grph of G t. The denition of hexgonl cellulr utomt hd lredy een given in Section 3.2 (Denition 5). We recll tht they re dened on the Cyley grph of the group G h = h j = c =1i. We shll show tht n elementry simultion cn lso e dened sometimes without the existence of ny homomorphism with nite kernel. We shll denote tringulr cellulr utomt y TCA. 16
Lemm 1 There does not exist homomorphism with nite kernel from G t to G h. Proof. We suppose tht : G t! G h is homomorphism with nite kernel, () =w1, () =w2 nd (c) =w3. If w1 = w2 = w3 =1,thenthekernel of is innite. Hence, t most one of the w i 's must e dierent from1. We suppose tht it is w1. As w1 is n element ofg h,ndg h is commuttive, w1 cn e expressed s w1 = n m 6=1. Then,1=(1) = ( 2 )=()() = w1 2 = 2n 2m 6= 1, which leds to contrdiction. Lemm 2 There exists n injective homomorphism from G h to G t. Proof. Let : G h! G t dened y (1)=1,() =, () =c, (c) =c nd for ll u nd for ll v in G h, (uv) =(u)(v). See Figure 12. c Figure 12: The injective homomorphism : G h! G t. In order to show tht is homomorphism, it is sucient toshowtht () = () nd (c) = 1: () = ()() = :c = c () =()() =c: = c::c:c = c (c) =:c:c =1 It is cler tht is n injective homomorphism. Proposition 3 Every TCA cn e simulted y HCA in n elementry wy, nd conversely. 17
Proof. Lemm 2 nd Theorem 5 imply tht every hexgonl CA cn e simulted y tringulr CA. Let us study now the converse simultion, when we wnt to simulte every TCA T with HCA H. We dene the set of sttes of H y S h = S t [f!g (! 62 S t ). The initil congurtions of T nd H re shown in Figure 13, cells without pre-imge re in stte!. The trnsition function of H is given y h (x y z r s t u)= t (x y z u) ifs = t = u =! h (x y z r s t u)= t (r s t u)ifx = y = z =! h (x y z r s t u)=! if u =!: ω ω ω c 1 c 3 5 7 2 4 6 8 10 12 14 9 11 13 15 17 19 21 16 18 20 1 3 5 7 2 4 ω ω ω 8 10 12 9 11 13 ω ω ω 22 23 24 25 26 27 28 21 15 16 17 18 19. ω ω ω 22 24 26. Figure 13: Initil congurtions of TCA nd HCA. This construction is it dierent from the others. For exmple, consider the cell hving informtion 12 in the initil congurtion of T. The sttes of its neighor dened y the genertor (resp., c) (numer 4 (resp. 11,13)) is in neighor cell dened y genertor ;1 (resp. ;1, c ;1 ) in the initil congurtion of H. Let us study now the cell numerted 11 in the initil congurtion of T. The stte of its neighor dened y the genertor (resp., c) is in neighor cell dened y (resp., c) in the initil congurtion of H. So there re two 18
types of cells, ut we cn dene the trnsition function without contrdiction, ecuse if the needed sttes re in neighors dened y,, c (resp. ;1, ;1, c ;1 ), then the others re in stte! (only one choice is ville). As we hve lredy noticed, the sme grph cn e colored in dierent wys. Here, we do not give ll simultions etween cellulr utomt on the other Cyley grphs, we study this prolem in more generl wy in Section 7. We tke only one exmple. In Figure 4, we show the Cyley grph of group G h2 = h c j 3 =1 3 =1 c 3 =1 c =1i: As this grph gives lso the tringulr tiling of the plne, it cn e used in order to dene hexgonl cellulr utomt. For short, we denote cellulr utomt on the Cyley grph of G h2 y HCA2. In order to show tht every HCA cn e simulted y HCA2, it is sucient to give n injective homomorphism from G h to G h2. Lemm 3 There exists n injective homomorphism from G h to G h2. Proof. Let : G h! G h2 dened y () =c () =c (c) =c nd for ll u, for ll v in G h, (uv) =(u)(v), see Figure 14. In order to prove tht is n injective homomorphism, it is sucient to show tht (c) = 1 nd () = (): (c) =c:c:c = :(cc)::():c = :(c ;1 )::( ;1 ):c = :(::)c:c = :():(cc): = :( ;1 :c ;1 ): = =1: () =c:c = :(c ;1 ): = (:)(:) = ( ;1 )( ;1 ) =(:)(c:c)(:)= ;1 c ;1 ;1 () = c:c = c()c = c(c ;1 )c = c(:c)(c:)c = c( ;1 )( ;1 )c = (c:)(:)(:c) = ;1 c ;1 ;1 : It is cler tht is n injective homomorphism. In similr wy sinlemm1,we cn show tht there does not exist ny homomorphism with nite kernel in the converse direction. Thus, we do not know whether every HCA2 cn e simulted y HCA in n elementry wy or not. In the following section, we give nother, non-elementry simultion. 6 Other simultions Here, we study other, more complicted simultions. We hve seen tht every HCA cn e simulted y HCA2. Here we construct the converse simultion. 19
c Figure 14: The injective homomorphism : G h! G h2. Theorem 6 Every HCA2 cn e simulted yhca. Proof. Let A =(S ; N ) e HCA2 nd let c 0 A e its initil congurtion. Let : G h! G h2 ehomomorphism dened y (1)=1 () =c () =c (c) =c: We construct CA B =(S 0 ; 0 N 0 0 ) which simultes A: lets = S 9,ndwe dene the initil congurtion c 0 B of B y c 0 B (u) =(c0 A ((u)) c0 A ((u)) c0 A ((u)c) c0 A ((u);1 ) c 0 A ((u);1 ;1 ) c 0 A ((u)) c0 A ((u)) c0 A ((u)c;1 ) c 0 A ((u)c;1 )): The denition of this initil congurtion is shown in Figure 15. The grey tringle is the imge of tringle c of HCA, in the lrge shpe re cells whose informtion re grouped in c 0 B. The tuple (1 2 ::: 9) denotes the order of informtion in stte-vector of HCA. With this construction, for ll informtion of c 0, the neighor informtion re locted in uniform wy A inc0 B nd t distnce 1: simultion cn e given without ny loss of time. 20
c A B 1 C 1 6 7 D 2 4 9 3 8 5 E F G h 2 (A,) (B,) c (C,) (1,2,3,4,5,6,7,8,9) 1 (D,) (E,) (F,) G h 1 γ : G G h 1 h 2 γ (1)=1 γ ( )= c γ ( )= c γ ( c )= c Figure 15: Initil congurtions. 21
We dene the trnsition function 0 : S 7! S y 0 ((x1 x2 ::: x9) (y1 y2 ::: y9) (z1 z2 ::: z9) (r1 r2 ::: r9) (s1 s2 ::: s9) (t1 t2 ::: t9) (w1 w2 ::: w9)) = ((w6 w2 y5 t3 w4 r7 w1) (r7 w4 w3 r9 w1 w8 w2) (z1 x9 w8 z4 r4 w2 w3) (w8 w1 w9 w5 w2 w6 w4) (w4 x3 z7 w8 w9 s1 w5) (t3 w7 w4 w1 t8 w9 w6) (w9 t8 x1 x2 w6 t5 w7) (t3 w7 w4 w1 t8 w9 w8) (x2 w5 w6 w7 x3 w4 w9)): Remrk tht in the proof of Theorem 6, the grouped cells re fx x xc x ;1 x ;1 ;1 x x xc ;1 xc ;1 g where x is n imge element. Notice tht x is in (G h ), x is in the left-coset (G h ) of (G h ), nd so on, xc ;1 is in (G h )c ;1. In generl, we cn stte the following theorem: Theorem 7 Let e homomorphism from G to G 0. If the index of the imge of G is nite, then every cellulr utomton on the Cyley grph of G 0 cn e simulted y cellulr utomton on the Cyley grph of G. Proof. We rst study the cse when is injective. Let H e the sugroup of G 0 such tht H = (G). Let H = fh Hu1 Hu2 ::: Hu m g e the set of ll distinct left-cosets of H. Let A =(S ; N ) e CA on the Cyley grph of G 0 nd c 0 its initil congurtion. We dene CA A B =(S0 ; 0 N 0 0 ) on the Cyley grph of G which simultes A. WedeneS 0 s sup-set of S m+1. In order to dene the initil congurtion c 0 B,we group the sttes of c0 A in the following wy: for ll v in G, let c 0 B(v) =(c 0 A((v)) c 0 A((v)u1) c 0 A((v)u2) ::: c 0 A((v)u m )): We hve to show, tht neighor informtion of c 0 A re uniformlly plced in c 0 for every components of every vectors: for ll i, if for some B u in G0, c 0 A (u) is the i-th component in the stte-vector c 0 B (v) for some v in G, nd neighor informtion c 0 (ug) isthej-th component inthestte-vector A c0 B (w) for some g in G 0 nd w in G such thtw = vx, then for ll U in G 0 eing the i-th component in stte-vector c 0 B (V ) for some V in G, the neighor informtion c 0 (Ug)must e the j-th component in the stte-vector A c0 B (W ) where W is in G nd W = Vx. Let X 2 G nd (X) = x. Consider cell xu i (2 (G)u i )ina. By the denition of c 0 B, its stte cn e found s the i-th component inthestteofcell X in B: c 0 B(X) =(c 0 A(x) c 0 A(xu1) ::: c 0 A(xu i ) ::: c 0 A(xu m )): 22
In order to compute the new stte of xu i in B, for ll g in G 0 [ G 0;1, the sttes of its neighor cells xu i g in A re needed. Let xu i g e in (G)u j for some j. It mens tht there exists ~ X in G such tht ( ~ X)=~x nd xui g =~xu j,hence c 0 B( ~ X)=(c 0 A (~x) c 0 A(~xu1) ::: c 0 A(~xu j )=c 0 A(xu i g) ::: c 0 A(~xu m )) nd ~ X = XU for some U in G. Let Y 2 G nd (Y )=y. Consider cell yu i in A. Its stte cn e found s the i-th component in the stte of cell Y in B: c 0 B(Y )=(c 0 A(y) c 0 A(yu1) ::: c 0 A(yu i ) ::: c 0 A(yu m )): We wnt toknow, in which cellofb the stte of the cell yu i g of A cn e found. Becuse of the properties of groups, we know tht it will e the j-th component in the stte-vector of some cell ~ Y in B: let ( ~ Y )=~y, then yu i g =~yu j nd c 0 B ( ~ Y )=(c 0 A (~y) c0 A (~yu 1) ::: c 0 A (~yu j)=c 0 A (yu ig) ::: c 0 A (~yu m)): We must show tht ~ Y = YU. As is homomorphism, weknowtht nd hence ( ~ X)=(XU)=(X)(U) =x(u) =~x (U) =x ;1 ~x: On the other hnd, from xu i g =~xu j,wehve From yu i g =~yu j,wehve x ;1 ~x = u i gu ;1 j : ~Y = ;1 (~y) = ;1 (y) ;1 (U) =YU: Let us consider the cse when is not injective. Then there exists n injective homomorphism 0 : G= ker()! G 0, nd we cn construct simultion in similr wy s efore. Remrk tht the simultion time fctor is not lwys 1, it depends on the generting sets. Then, from Theorem 5, the following ssertion holds. Theorem 8 Let e homomorphism from group G in nother group G 0.If hs nite kernel, if the index of the imge of G is nite, then every cellulr utomton on the Cyley grph of G cn e simulted y cellulr utomton on the Cyley grph of G 0 nd conversely. 23
Denition 7 We shll sy tht G1 B G2 if nd only if there exists homomorphism : G1! G2 with nite kernel such tht the index of the imge of G1 is nite. If G1 B G2, then there exist simultions in oth directions etween cellulr utomt dened on G1 nd G2. We re interested y the symmetrized reltion of B. As simultions re trnsitives, we study the symmetric nd trnsitive closure of B. Denition 8 We dene on nitely presented groups the reltion G1 G2 s the symmetric nd trnsitive closure of the reltion B. From Theorem 8, the following proposition holds. Proposition 4 G1 G2 if nd only if there exists suite of nitely presented groups G 0 0 G 0 1 ::: G 0 n such tht G 0 0 = G1 nd G 0 n = G2 nd for ll i 0, G 0 i B G 0 i+1 or G 0 i+1 B G 0 i. Conjecture: If there exist simultion etween cellulr utomt on G1 nd cellulr utomt on G2 in oth directions, then G1 G2. If this ssertion is true, it would imply tht the existence of simultions in oth directions etween cellulr utomt dened on two groups is n indecidle prolem. 7 Cellulr utomt on Archimeden tilings In the previous section we hve studied hexgonl nd tringulr CA. Let us denote y ; H nd ; T, respectively, the underlying grphs on which these CA work. T. Choud ([1]) hs shown tht Figures 16 nd 17 show ll possile colorings for these grphs. The Archimeden tilings re presented in Figure 18 they re exctly the tilings using nite numer of regulr nd convex polygons such tht the degree of every vertex nd the order of polygons round every vertex is the sme. In [1] it is shown tht ll Archimeden tilings cn e colored s Cyley grphs he hs lso given ll possile colorings. We show now tht cellulr utomt on ll these grphs re equivlent from computtionl point of view: they cn e simulted y ech other in liner time. In order to show the existence of injective homomorphisms from G2 VN to groups corresponding to these tilings, we introduce the following notion. Denition 9 Let ; e n Archimeden tiling colored s Cyley grph. Consider vertex 1, genertor g. Let us denote y ~x the rc strting t 1 colored y g nd considered svector in R 2.Let A e vertex nd h genertor or n inverse genertor. We denote y \ (~x h) A the ngle etween ~x nd the rc colored 24
c c c d c c c d c c Figure 16: All Cyley grphs for hexgonl CA. 25
c c c.. c. d. e. Figure 17: All Cyley grphs for tringulr CA. c. y h strting t A. We sy tht two vertices A nd B hve the sme sitution if for ll genertor nd inverse genertor h, \ (~x h) A = \ (~x h) B. See n exmple for sme nd dierent situtions in Figure 19. Vertices A nd B hve the sme sitution, ut not A nd C. The following lemm is consequence of the fcts tht in n Archimeden tiling, the type of vertices is the sme ([4]) nd tht the geometricl order of genertors in every vertex is the sme ([1]). Lemm 4 Let x nd y e two vertices hving the sme sitution. Then for ll genertor (or inverse genertor) g, xg nd yg hve the sme sitution. Now we cn show the following proposition: Proposition 5 There is n injective homomorphism from G2 VN to ll groups whose Cyley grphs hve n underlying grph corresponding to n Archimeden tiling. Proof. Let ; e the Cyley grph of group G such tht its underlying grph corresponds to n Archimeden tiling. Let x, y e two vertices hving the sme sitution nd let p1 e pth from x to y: y = xp1. Let z e third vertex hving the sme sitution s x nd y nd let p2 e pth from x to z: z = xp2. 26
Figure 18: Archimeden tilings. 27
A C B Figure 19: Dierent vertex-situtions. We suppose tht x, y nd z re chosen in such wy thtp n 1 6= pn0 2 for ll n>0 nd n 0 > 0in N. It is possile, ecuse these tilings re periodic in the plne in two independnt directions. Then, from Lemm 4, for ll n>0in N, p n 1 6= 1 nd p n 2 6= 1. On the other hnd, lso from Lemm 4, the pth p1 strting t z is \prllel" to the pth etween x nd y, nd the pth p2 strting t y is \prllel" to the pth etween x nd z, hence, zp1 = yp2, tht is, p1p2 = p2p1. Recll tht two-dimensionl von Neumnn CA re dened on the Cyley grph of the group G2 VN = h j = i. Let : G2 VN! G e mpping dened y (1) = 1 () =p1 () =p2 nd (uv) =(u)(v) u v 2 G2 VN: It is cler tht is n injective homomorphism. Then, from Theorems 5 nd 7 nd Proposition 5, the following ssertion holds. Theorem 9 Every two-dimensionl von Neumnn cellulr utomton cn e simulted y cellulr utomton on ny Archimeden tiling nd conversely. This result cn lso e interpreted in the following wy. If we consider Cyley grphs s possile rchitectures for prllel mchines, we cnchoose ny of Archimeden tilings for such nrchitecture in the plne, they hve thesme computtionl power. However, s simultions etween cellulr utomt on these grphs require mny sttes, it is necessry tht mchines hve sucient mount of locl memory. 8 Open prolems In this pper, we hve only studied simultions, where the sttes of the simulted cellulr utomton re considered s tomic informtions. However, mny simultions exist with splitting sttes, they should lso e studied. 28
We hve given sucient condition for converse simultions etween cellulr utomt on Cyley grph. Cn ny sucient nd necessry condition e given? This is the question tht we hve sked ourselves when we hve dened the reltion etween groups. If the nswer is \no" in the generl cse, whether does suclss of groups exist for which such condition cn e given? References [1] T. Choud. Plnr Cyley grphs. preprint, 1994. [2] S. Cole. Rel time computtions y n-dimensionl itertive rys of nitestte mchines. IEEE Trnsctins on Computers, 4:349{365, 1969. [3] U. Frisch, D. d'humieres, B. Hsslcher, P. Lllemnd, Y. Pomeu, nd J.- P. Rivet. Lttice gs hydrodynmics in two nd three dimensions. Complex Systems, 1(4):649{708, 1987. [4] B. Grunum nd G. C. Shephrd. Tilings nd Ptterns. Freemn nd Co, 1987. [5] A. Mch nd F. Mignosi. Grden of Eden congurtions for cellulr utomt on Cyley grphs of groups. SIAM Journl on Discrete Mthemtics, 6(1):44{56, 1993. [6] Zs. Rok. One-wy cellulr utomt on Cyley grphs (short version). In FCT'93, volume 710of LNCS, pges 406{417. Springer Verlg, August 1993. [7] Zs. Rok. Cellulr utomt on Cyley grphs. PhD thesis, Ecole Normle Superieure de Lyon, Frnce, July 1994. [8] Zs. Rok. One-wy cellulr utomt on Cyley grphs. Theoreticl Computer Science, 132:259{290, 1994. 29